arX

iv1

006

2739

v2 [

hep-

th]

23

Jun

2010

Oscillating Universe in Horava-Lifshitz Gravity

Kei-ichi Maeda1 2 lowast Yosuke Misonoh1 dagger and Tsutomu Kobayashi3 Dagger

1Department of Physics Waseda University Okubo 3-4-1 Shinjuku Tokyo 169-8555 Japan2RISE Waseda University Okubo 3-4-1 Shinjuku Tokyo 169-8555 Japan

3Research Center for the Early Universe (RESCEU) Graduate School of ScienceThe University of Tokyo Tokyo 113-0033 Japan

(Dated November 12 2018)

We study the dynamics of isotropic and homogeneous universes in the generalized Horava-Lifshitzgravity and classify all possible evolutions of vacuum spacetime In the case without the detailedbalance condition we find a variety of phase structures of vacuum spacetimes depending on thecoupling constants as well as the spatial curvature K and a cosmological constant Λ A bounceuniverse solution is obtained for Λ gt 0K = plusmn1 or Λ = 0 K = minus1 while an oscillation spacetime isfound for Λ ge 0 K = 1 or Λ lt 0 K = plusmn1 We also propose a quantum tunneling scenario from anoscillating spacetime to an inflationary universe resulting in a macroscopic cyclic universe

PACS numbers 0460-m 9880Cq 9880-k

I INTRODUCTION

Recently Horava proposed a power-counting renor-malizable theory of gravity [1] which has attractedmuch attention over the past year In Horavarsquos theoryLorentz symmetry is broken and it exhibits a Lifshitz-likeanisotropic scaling in the ultraviolet (UV) t rarr ℓzt ~x rarrℓ~x with the dynamical critical exponent z = 3 (For thisreason the theory is called Horava-Lifshitz (HL) gravity)It is then natural to expect that the UV behavior of thetheory would give rise to new scenarios of cosmology [2ndash4] Earlier works have indeed revealed some interestingaspects of HL cosmology such as dark matter as integra-tion constant [5] the generation of chiral gravitationalwaves from inflation [6] scale-invariant fluctuations with-out inflation [7] and possible dark energy scenario [8ndash10]There are also some discussion closely related to obser-vational cosmology and astrophysics such as cosmologi-cal perturbation [7 11ndash22] observational constraints [23ndash25] primordial magnetic field without inflation [26] anda relativistic star [27 28]

Though the viability of HL gravity is still under in-tense debate [5 29ndash41] we give the theory the benefit ofthe doubt and will furthermore pursue consequences ofHoravarsquos intriguing idea

Here we focus on the dynamics of Friedmann-Lemaitre-Robertson-Walker (FLRW) universe in HL gravity whichmay provide us new aspect of the early universe Westudy the FLRW spacetime with non-zero spatial curva-ture in the context of HL gravity The non-trivial cosmo-logical evolution is brought by the various terms in thepotential which are constructed from the spatial curva-ture R ij In particular we find non-singular behavior inhigh curvature region which may lead to avoidance of a

lowastElectronic address maedawasedajpdaggerElectronic address yrdquounderscorerdquomisonoumoegiwasedajpDaggerElectronic address tsutomuresceusu-tokyoacjp

big bang initial singularity [4] Consequently many stud-ies on the dynamics of the FLRW universe in HL gravitycome out within the last few years [2ndash4 42ndash60]

In the original HL gravity the so-called detailed bal-ance condition is assumed However this condition canbe loosened to have arbitrary coupling constants [61]Hence both models have been so far discussed in the anal-ysis of the FLRW universe As for matter componentsa perfect fluid with the equation of state P = wρ and ascalar field have been discussed As a result we can clas-sify the analyzed models into four types (1) The modelwith the detailed balance condition and with a perfectfluid [42ndash51] (2) The model with the detailed balancecondition and with a scalar field [2 4 58] (3) The modelwithout the detailed balance condition and with a per-fect fluid [52ndash57 60] and (4) The model without thedetailed balance condition and with a scalar field [3 59]

So far many works have been done assuming the de-tailed balance condition and they reveal the possibilityof singularity avoidance such as a bounce universe [2ndash4 8 21 43 44 58] and an oscillating spacetime [43ndash45 48] The initial singularity is avoided because ofldquodarkrdquo radiation which is a negative aminus4 term It comesfrom higher curvature terms The ldquodarkrdquo radiation wasfirst introduced in the context of a brane world [62] Al-though such an effect is very interesting and importantthe ldquodarkrdquo radiation term may fail to avoid a singularityif one include radiation or massless field The conven-tional radiation behaves as aminus4 with positive coefficientIf we have a sufficient amount of real radiation the uni-verse will inevitably collapse to a big-crunch singularityFurthermore if we assume that radiation field has alsothe same scaling law as gravity in the UV limit the en-ergy density of radiation field changes as aminus6[23] whichis the same scaling law of stiff matter in the conventionaltheory Inclusion of the positive aminus6 term will kill thepossibility of singularity avoidance by ldquodarkrdquo radiationIn order to save the present mechanism for singularityavoidance one needs a negative aminus6 term which may beobtained in the generalized HL gravity model [61]

2

Recently some papers have discussed the case withoutthe detailed balance and studied a singularity avoidance(a bounce universe or an oscillating behavior) One isby use of a phase space analysis [53 59] and the other isthe case with perfect fluid with time-evolving equation ofstate [56 60] The former analysis was not properly per-formed because they introduce the dynamical variablesmore than the degrees of freedom In the latter casealthough they discuss some interesting transitions theassumption of the equation of state is not so clearIn the present paper since there has so far not been a

systematic and substantial analysis in cosmology basedon this most general potential without the detailed bal-ance condition we provide a complete classification ofthe cosmological dynamics We do not include any mat-ter fields not only for simplicity but also to avoid unclearassumption It is just straightforward to include perfectfluid with the equation of state P = wρ (w=constant) Inparticular our analysis includes matter fluid with radia-tion and stiff matter as it is We clarify which conditionsshould be satisfied for singularity avoidance We alsopropose some possible scenario for a cyclic universe iethe oscillating spacetime will transit by quantum tunnel-ing to an inflationary phase resulting in a cyclic universeafter reheatingThe paper is organized as follows After giving the

generalized model of Horava-Lifshitz gravity in sectII westudy the isotropic and homogeneous vacuum spacetimein sectIII We find a variety of phase structures including abounce universe and an oscillating universe We then in-voke a more realistic cosmological model which may leadto a macroscopic cyclic universe via quantum tunnelingfrom an oscillating universe

II HORAVA-LIFSHITZ GRAVITY AND THE

COUPLING CONSTANTS

The basic variables in HL gravity are the lapse func-tion N the shift vector Ni and the spatial metric gij These variables are subject to the action [1 61]

S =1

2κ2

int

dtd3xradicgN (LK minus VHL[gij ]) (21)

where κ2 = 1M2PL and the kinetic term is given by

LK = K ijKij minus λK

2 (22)

with

K ij =1

2N(gij minusnablaiNj minusnablajNi) (23)

being the extrinsic curvature The potential term VHL

will be defined shortly In general relativity we haveλ = 1 only for which the kinetic term is invariant undergeneral coordinate transformations In HL gravity how-ever Lorentz symmetry is broken in exchange for renor-malizability and the symmetry of the theory is invariance

under the foliation-preserving diffeomorphism transfor-mations

t rarr t(t) xi rarr xi(t xj) (24)

As implied by the symmetry (24) it is most natural toconsider the projectable version of HL gravity for whichthe lapse function is dependent only on t N = N(t) [1]Since the Hamiltonian constraint is derived from the vari-ation with respect to the lapse function in the pro-jectable version of the theory the resultant constraintequation is not imposed locally at each point in spacebut rather is an integration over the whole space In thecosmological setting the projectability condition resultsin an additional dust-like component in the Friedmannequation [see Eq (32) below] [5]The most general form of the potential VHL is given

by [61]

VHL = 2Λ+ g1R

+κ2(

g2R 2 + g3R ijR

ji

)

+ κ3g4ǫijkR iℓnablajR ℓ

k

+κ4(

g5R3 + g6R R

ijR

ji + g7R

ijR

jkR

ki

+g8R ∆R + g9nablaiR jknablaiR jk)

(25)

where Λ is a cosmological constant R ij and R are the

Ricci and scalar curvatures of the 3-metric gij respec-tively and girsquos (i = 1 9) are the dimensionless cou-pling constants (See Appendix A for some conditionson these coupling constants)In the original proposal [1] Horava assumed the

detailed balance condition by which the potentialterm (25) is simplified to some extent The potentialunder the detailed balance condition is given by

VDB = minus3κ2micro2Λ2W

2(3λminus 1)+

κ2micro2ΛW

2(3λminus 1)R

minus (4λminus 1)κ2micro2

8(3λminus 1)R 2 +

κ2micro2

2R

jiR

ij

minus2κ2micro

ω2C ijR ij +

2κ2

ω4C ijC ij (26)

where

C ij = ǫikℓnablak

(

Rjℓ minus

1

4R δjℓ

)

(27)

is the Cotton tensor and ΛW micro and ω are constants Thepotential (26) is therefore reproduced by identifying

Λ = minus3(3λminus 1)

2micro2κ2 (28)

g1 = minus1 (29)

g2 = minus (4λminus 1)

4(3λminus 1)micro2κ2 g3 = micro2κ2 (210)

g4 = minus4microκ2

ω2 g5 =

2κ2

ω4 g6 = minus10κ2

ω4

g7 =12κ2

ω4 g8 =

3κ2

2ω4 g9 =

4κ2

ω4 (211)

3

and ΛW = minus(3λ minus 1)(micro2κ2) In the detailed balancecase micro and ω are two free parametersIn what follows we adopt the unit of κ2 = 1(M2

PL = 1)for brevity

III FLRW UNIVERSE IN HORAVA-LIFSHITZ

GRAVITY

We discuss an isotropic and homogeneous vacuum uni-verse in Horava-Lifshitz gravity Note that such a vac-

uum spacetime is not realized in general relativity Wewill extend our analysis to anisotropic spacetime (Bianchicosmology) in the separate paper

Assuming a FLRW spacetime which metric is givenby

ds2 = minusdt2 + a2(

dr2

1minusKr2+ r2dΩ2

)

(31)

with K = 0 or plusmn1 We find the Friedmann equation as

H2 +2

(3λminus 1)

K

a2=

2

3(3λminus 1)

[

Λ +gda3

+gra4

+gsa6

]

(32)

where H = aa

gd = 8C

gr = 6(g3 + 3g2)K2

gs = 12(9g5 + 3g6 + g7)K3 (33)

A constant C may appear from the projectability condi-tion and could be ldquodark matterrdquo[5] For a flat universe(K = 0) the higher curvature terms do not give anycontribution and then the dynamics is almost trivialHence in this paper we discuss only non-flat universe(K = plusmn1)If λ = 1 we find a usual Friedmann equation for an

isotropic and homogeneous universe in GR with a cos-mological constant dust radiation and stiff matter Ifgd gr and gs are non-negative such a spacetime givesa conventional FLRW universe model However sincethose coefficients come from higher curvature terms theirpositivity is not guaranteed Rather some of them couldbe negative As a result we find an unconventional cos-mological scenario which we shall discuss here In whatfollows we assume that λ gt 13 but do not fix it to beunityIn this paper we assume C = 0 just for simplicity The

Friedmann equation is written as

1

2a2 + U(a) = 0 (34)

where

U(a) =1

3λminus 1

[

K minus Λ

3a2 minus gr

3a2minus gs

3a4

]

(35)

Since the scale factor a changes as a particle with zeroenergy in this ldquopotentialrdquo U the condition U(a) le 0gives the possible range of a when the universe evolvesSo we can classify the ldquomotionrdquo of the universe by thesigns of K and Λ and by the values of gr and gs Note

that in the case with the detailed balance condition wehave

gr = 6(g3 + 3g2) = minus 3micro2

2(3λminus 1)lt 0 for λ gt 13

gs = 12(9g5 + 3g6 + g7)K = 0 (36)

It is some special case of our analysis although its dy-namics will be completely different from generic casesbecause gs vanishesWe find mainly the following four types of the FLRW

universe

(1) [BB rArrBC ] Suppose U(a) le 0 for a isin (0 aT ] andthe equality is true only when a = aT A spacetimestarts from a big bang (BB) and expands but it even-tually turns around at a = aT to contract finding a bigcrunch (BC ) aT is a scale factor when the universe turnsaround from expansion to contraction

(2)[BB rArr infin or infin rArr BC ] If U(a) lt 0 for any posi-tive values of a a spacetime starts from a big bang andexpands forever or its time reversal (A spacetime con-tracts to a big crunch) As for the asymptotic spacetimewe find a prop t [MilneM ] (K = minus1) for Λ = 0 while

a prop exp(radic

Λ3 t) [de SitterdS ] for Λ gt 0 We denotethem as BB rArr M and BB rArr dS respectively For thecontracting cases we describe them as M rArr BC anddS rArr BC respectively

(3) [Bounce] If U(a) le 0 for a isin [aT infin) and the equalityholds only when a = aT a spacetime initially contractsfrom an infinite scale and it eventually turns around ata finite scale aT and expands forever The asymptoticspacetimes are the same as the case (2) M and dS

(4) [Oscillation] If U(a) le 0 for a isin [amin amax] and theequality holds only when a = amin and a = amax aspacetime oscillates between two finite scale factorsFor some specific values (or specific relations) of gr and

gs which divides two different phases of spacetimes we

4

find a static universe (S)(5) [S ] A spacetime is static with a constant scale factoraS if U(aS) = 0 and Uprime(aS) = 0There are two types of static universes one is stable

(Ss) and the other is unstable (Su) When we have an un-stable static universe we also find the following types ofdynamical universes with a static spacetime as an asymp-totic state as well(6) [Su rArr infin or infin rArrSu] If U(a) le 0 for a isin [aS infin) andthe equality holds only at aS a spacetime starts from astatic state in the infinite past and expands forever or itinitially contracts from an infinite scale and eventuallyreach a static state in the infinite future We then haveSu rArr dS M or dS M rArrSu

(7) [BBrArr Su or Su rArr BC ] If U(a) le 0 for a isin (0 aS ] andthe equality holds only at aS A spacetime starts from abig bang and expands to a static state with a finite scaleaS or its time reversal (A spacetime contracts from astatic state to a big crunch)

(8) [Su rArr Bounce rArr Su] If U(a) le 0 for a isin [aS aT ] (ora isin [aT aS]) and the equality holds only at aS and aT aspacetime starts from a static state in the infinite pastand expands (or contracts) It eventually bounces at afinite scale aT and then reach a static state again in theinfinite futureFor the case of Λ 6= 0 introducing the curvature scale

ℓ which is defined by

Λ

3=

ǫ

ℓ2 (37)

where ǫ = plusmn1 we can rescale the variables and rewritethe ldquopotentialrdquo U by the rescaled variables as

U(a) =1

3λminus 1

[

K minus ǫa2 minus gr3a2

minus gs3a4

]

(38)

where a = aℓ gr = grℓ2 and gs = gsℓ

4 Using thispotential and variables we can discuss the fate of theuniverse without specifying the value of ΛA static universe will appear if we find a solution a =

aS(gt 0) which satisfies U(aS) = 0 and Uprime(aS) = 0 IfΛ 6= 0 (ǫ = plusmn1) it happens if there is a relation betweengr and gs which is defined by

gs = g [ǫK](plusmn)s (gr)

=1

9ǫ2

[

2K minus 3ǫKgr plusmn 2(1minus ǫgr)32]

(39)

This gives the curve ΓǫK(plusmn) on the gr-gs plane whichgives the boundary between two different phases of space-time The radius of a static universe is given by

aS = a[ǫK](plusmn)S =

radic

1

3ǫ

[

K plusmnradic

1minus ǫgr

]

(310)

if it is real and positive Here plusmn correspond to the curvesΓǫ(plusmn) Since U(a) = 0 is the cubic equation with respect

to a2 and a2 = a2S is the double root we have the thirdroot which is given by

aT = a[ǫK](plusmn)T =

radic

1

3ǫ

[

K ∓ 2radic

1minus ǫgr

]

(311)

where the universe turns around (or bounces) To existsuch a point it must be real and positiveIf Λ = 0 we find

gs = minusK

12g2r (312)

which is found from (39) in the limit of ǫ = 0 The cor-responding curve on the gr-gs plane is denoted by Γ0K The radius is given by

aS = a[0K]S =

radic

gr6K

(313)

assuming Kgr gt 0Note that our classification depends just on gr and gs

(or gr and gs) apart from K and Λ Since gr and gs aregiven by g2 g3 g5 g6 and g7 but do not include g4 g8and g9 the fate of the universe is classified only by theconditions on the coupling constants of higher curvatureterms but not on those of their derivatives such as nablajR ℓ

kIn the case with the detailed balance conditions we

find Λ lt 0 from Eq (28) and then obtain from Eq(36)

gr = minus94 gs = 0 (314)

Now we shall discuss what kind of spacetimes are real-ized under which conditions in the following three casesseparately [A Λ = 0 B Λ gt 0 and C Λ lt 0]

A Λ = 0

If a cosmological constant is absent the ldquopotentialrdquo iswritten as

U(a) =1

(3λminus 1)a4

[

Ka4 minus gr3a2 minus gs

3

]

(315)

In Fig 1 we show the fate of the universe which dependson the values of gr and gs For the case ofK = 1 there aretwo types of spacetime phases One is BB rArr BC and theother is an oscillating universe In fact if gr gt 0 gs lt 0and g2r + 12gs gt 0 we find the scale factor a is boundedin a finite range as (0 lt) amin le a le amax (lt infin) where

a2min equiv 1

6

[

gr minusradic

g2r + 12gs

]

a2max equiv 1

6

[

gr +radic

g2r + 12gs

]

(316)

which gives an oscillating universe The condition for anoscillating universe is written as

gr gt 0 minus g2r12

le gs lt 0 (317)

5

(a) K = 1

(b) K = minus1

FIG 1 Phase diagram of spacetimes for Λ = 0 The os-cillating universe is found only for the case of K = 1 Thestable and unstable static universes (Ss and Su) exist on theboundary Γ01 and Γ0minus1 respectively On Γ0minus1 we also finddynamical universes with an asymptotically static spacetimeSu rArr BC Su rArr M BBrArr Su or MrArr Su

which is shown in Fig 1(a) by ldquoOscillationrdquo (the light-orange colored region) in the gs-gr plane The equality inEq (317) which is the curve Γ01 gives a static universe

with the scale factor a = aS =radic

gr6

For the case of K = minus1 we find three types of space-time phases BB rArr M (or M rArr BC) BB rArr BC andBounce (see Fig1(b)) On the boundary curve Γ0minus1which is defined by Eq (312) ie gs = g2r 12 (gr lt 0)we find an unstable static universe Su and the dynamicaluniverses with an asymptotically static spacetime Su rArrBC Su rArr M BBrArr Su or MrArr Su

The bounce universe is found if gs lt 0 or gs = 0 withgr lt 0 which is shown by rdquoBouncerdquo (the light-green re-gion) The radius at a turning point aT is given by

aT =

radic

1

6

(

minusgr +radic

g2r minus 12gs

)

(318)

Next we shall evaluate the period of an oscillating uni-verse in the case of K = 1 The solution for Eq (34) isgiven by

tminus tmax = minusint a

amax

daradic

minus2U(a)

= amax

radic

3λminus 1

2E (φ[a] k) (319)

where E(φ k) is the elliptic integral of the second kind

which is defined by

E(φ k) =

int φ

0

dθradic

1minus k2 sin2 θ (320)

k and φ[a] are given by

k =

radic

a2max minus a2min

amax (321)

φ[a] = sinminus1

(

a2max minus a2

a2max minus a2min

)

(322)

The period T is given by

T = 2(tmin minus tmax) = 2amax

radic

3λminus 1

2E (k) (323)

where E(k) is the complete elliptic integral of the secondkind defined by E(k) = E(π2 k)In order to evaluate the period we consider some lim-

iting cases which are the boundaries of the region ofOscillation In Fig 2 we show the potential U(a) by theblue curve for one boundary curve Γ01 which is givenby gs = minusg2r 12 It gives a stable static universe withthe scale factor aS We also show the potential near theother boundary of Oscillation (the positive gr-axis) by thedashed orange curve Choosing for example gr = 1 andgs = minus0001 we find an oscillating univese with the scalefactor a isin [00316705 0576481] Since these two poten-

FIG 2 The potential U(a) for a stable static universe andan oscillating universe near the gr-axis The ldquocouplingrdquo con-stants are chosen as gr = 1 and gs = minus112 on Γ01 for astatic universe which radius is shown by aS = 1

radic6 We

also show the case with gr = 1 and gs = minus0001 for an oscil-lating universe which maximum and minimum radii are givenby amax = 0576481 and amin = 00316705 respectively

tials give the limiting cases we find that 0 lt amin le aSand aS le amax lt

radic2 aS for an oscillating universe

In the limit of a static universe (near Γ01) we find theperiod TS as

TS = π

radic

(

3λminus 1

2

)

gr6 (324)

while in the other boundary limit (gs rarr 0) we obtain

T0 =

radic

(

3λminus 1

2

)

4gr3

(325)

6

From these evaluations giving the value of gr we findthe period T of any oscillating universe is bounded inthe range of (T0 TS) for gs isin (minusg2r 12 0) We then ap-

proximate the period as T sim g12r

We have found an oscillating FLRW universe becausewe have ldquonegativerdquo energy of ldquostiff matterrdquo which comesfrom the higher curvature term The condition for anoscillating universe is rewritten in terms of the originalcoupling constants as

g3 + 3g2 gt 0 (326)

minus (g3 + 3g2)2

4le 9g5 + 3g6 + g7 lt 0 (327)

B Λ gt 0 (ǫ = 1)

In this case the potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 minus a6 minus gr3a2 minus gs

3

]

(328)

For each value of K we depict the fate of the universe inFig 3 which depends on the values of gr and gs

(a) K = 1

(b) K = minus1

FIG 3 Phase diagram of spacetimes for Λ gt 0 The oscillat-ing universe is found only for the case of K = 1 The staticuniverses (Ss and Su) exist on the boundaries Γ1K(plusmn) Wealso find dynamical universes with an asymptotically staticspacetime Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or Su rArr BC

on Γ1minus1

We find non-singular evolution of the universe (BounceOscillation or Static) as well as the universe with a cosmo-logical singularity (BB rArr BC BB rArr dS or dS rArr BC)

Except for the case of BB rArr BC and a static universethe expanding universe approaches de Sitter spacetime(exponentially expanding universe) because of a positivecosmological constant Λ The oscillating universe existsif and only if K = 1 and the following conditions aresatisfied

gr gt 0 (329)

g [11](minus)s (gr) le gs

lt 0

le g[11](+)s (gr)

(330)

where g[11](plusmn)s is defined by Eq (39) with ǫ = 1K = 1

This condition gives the constraint on gs as minus19 le gs lt0 Note that in the limit of gr ≪ 1 (ie Λ rarr 0) werecover the condition (317)The boundaries of two different phases of spacetimes

consist of the gr-axis and two curves (Γ11(plusmn)) for K = 1or one curve (Γ1minus1(+)) for K = minus1 Those boundary

curves Γ1K(plusmn) are defined by gs = g[1K](plusmn)s (gr)

A stable static universe exist on the boundary curveΓ11(minus) while unstable static universes appear on theboundary curves Γ1plusmn1(+) For K = 1 there are twotypes of static universes (stable and unstable) corre-sponding to two curves Γ11(minus) and Γ11(+) respectivelywhich coincide at gr = 1 and gs = minus19 In the branchesof unstable static universes (Γ1K(+)) we also find dy-namical universes with an asymptotically static space-time Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or SurArr BC on Γ1minus1(+)The period T of an oscillating universe is calculated by

T = 2

int amax

amin

daradic

minus2U(a) (331)

where T = Tℓ and amax and amin are the maximumand minimum radii of the oscillating universe We shallevaluate the period near the boundaries of the parameterrange of oscillating universes (the light-orange region inFig 3(a)) We first show the potential U(a) for three(near-) boundary values of gs in Fig 4For the case with an unstable static universe (the

dashed blue curve) (Γ11(+) with gs lt 0) the larger dou-ble root of the equation of U(a) = 0 is given by

aS = a[11](+)S =

radic

1

3

(

1 +radic

1minus gr

)

(332)

while the smaller root is

aT = a[11](+)T =

radic

1

3

(

1minus 2radic

1minus gr

)

(333)

which corresponds to a turning radius at a bounce Theperiod T diverges in the limit of a static universe becauseamax = aS is the double root

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

3

and ΛW = minus(3λ minus 1)(micro2κ2) In the detailed balancecase micro and ω are two free parametersIn what follows we adopt the unit of κ2 = 1(M2

PL = 1)for brevity

III FLRW UNIVERSE IN HORAVA-LIFSHITZ

GRAVITY

We discuss an isotropic and homogeneous vacuum uni-verse in Horava-Lifshitz gravity Note that such a vac-

uum spacetime is not realized in general relativity Wewill extend our analysis to anisotropic spacetime (Bianchicosmology) in the separate paper

Assuming a FLRW spacetime which metric is givenby

ds2 = minusdt2 + a2(

dr2

1minusKr2+ r2dΩ2

)

(31)

with K = 0 or plusmn1 We find the Friedmann equation as

H2 +2

(3λminus 1)

K

a2=

2

3(3λminus 1)

[

Λ +gda3

+gra4

+gsa6

]

(32)

where H = aa

gd = 8C

gr = 6(g3 + 3g2)K2

gs = 12(9g5 + 3g6 + g7)K3 (33)

A constant C may appear from the projectability condi-tion and could be ldquodark matterrdquo[5] For a flat universe(K = 0) the higher curvature terms do not give anycontribution and then the dynamics is almost trivialHence in this paper we discuss only non-flat universe(K = plusmn1)If λ = 1 we find a usual Friedmann equation for an

isotropic and homogeneous universe in GR with a cos-mological constant dust radiation and stiff matter Ifgd gr and gs are non-negative such a spacetime givesa conventional FLRW universe model However sincethose coefficients come from higher curvature terms theirpositivity is not guaranteed Rather some of them couldbe negative As a result we find an unconventional cos-mological scenario which we shall discuss here In whatfollows we assume that λ gt 13 but do not fix it to beunityIn this paper we assume C = 0 just for simplicity The

Friedmann equation is written as

1

2a2 + U(a) = 0 (34)

where

U(a) =1

3λminus 1

[

K minus Λ

3a2 minus gr

3a2minus gs

3a4

]

(35)

Since the scale factor a changes as a particle with zeroenergy in this ldquopotentialrdquo U the condition U(a) le 0gives the possible range of a when the universe evolvesSo we can classify the ldquomotionrdquo of the universe by thesigns of K and Λ and by the values of gr and gs Note

that in the case with the detailed balance condition wehave

gr = 6(g3 + 3g2) = minus 3micro2

2(3λminus 1)lt 0 for λ gt 13

gs = 12(9g5 + 3g6 + g7)K = 0 (36)

It is some special case of our analysis although its dy-namics will be completely different from generic casesbecause gs vanishesWe find mainly the following four types of the FLRW

universe

(1) [BB rArrBC ] Suppose U(a) le 0 for a isin (0 aT ] andthe equality is true only when a = aT A spacetimestarts from a big bang (BB) and expands but it even-tually turns around at a = aT to contract finding a bigcrunch (BC ) aT is a scale factor when the universe turnsaround from expansion to contraction

(2)[BB rArr infin or infin rArr BC ] If U(a) lt 0 for any posi-tive values of a a spacetime starts from a big bang andexpands forever or its time reversal (A spacetime con-tracts to a big crunch) As for the asymptotic spacetimewe find a prop t [MilneM ] (K = minus1) for Λ = 0 while

a prop exp(radic

Λ3 t) [de SitterdS ] for Λ gt 0 We denotethem as BB rArr M and BB rArr dS respectively For thecontracting cases we describe them as M rArr BC anddS rArr BC respectively

(3) [Bounce] If U(a) le 0 for a isin [aT infin) and the equalityholds only when a = aT a spacetime initially contractsfrom an infinite scale and it eventually turns around ata finite scale aT and expands forever The asymptoticspacetimes are the same as the case (2) M and dS

(4) [Oscillation] If U(a) le 0 for a isin [amin amax] and theequality holds only when a = amin and a = amax aspacetime oscillates between two finite scale factorsFor some specific values (or specific relations) of gr and

gs which divides two different phases of spacetimes we

4

find a static universe (S)(5) [S ] A spacetime is static with a constant scale factoraS if U(aS) = 0 and Uprime(aS) = 0There are two types of static universes one is stable

(Ss) and the other is unstable (Su) When we have an un-stable static universe we also find the following types ofdynamical universes with a static spacetime as an asymp-totic state as well(6) [Su rArr infin or infin rArrSu] If U(a) le 0 for a isin [aS infin) andthe equality holds only at aS a spacetime starts from astatic state in the infinite past and expands forever or itinitially contracts from an infinite scale and eventuallyreach a static state in the infinite future We then haveSu rArr dS M or dS M rArrSu

(7) [BBrArr Su or Su rArr BC ] If U(a) le 0 for a isin (0 aS ] andthe equality holds only at aS A spacetime starts from abig bang and expands to a static state with a finite scaleaS or its time reversal (A spacetime contracts from astatic state to a big crunch)

(8) [Su rArr Bounce rArr Su] If U(a) le 0 for a isin [aS aT ] (ora isin [aT aS]) and the equality holds only at aS and aT aspacetime starts from a static state in the infinite pastand expands (or contracts) It eventually bounces at afinite scale aT and then reach a static state again in theinfinite futureFor the case of Λ 6= 0 introducing the curvature scale

ℓ which is defined by

Λ

3=

ǫ

ℓ2 (37)

where ǫ = plusmn1 we can rescale the variables and rewritethe ldquopotentialrdquo U by the rescaled variables as

U(a) =1

3λminus 1

[

K minus ǫa2 minus gr3a2

minus gs3a4

]

(38)

where a = aℓ gr = grℓ2 and gs = gsℓ

4 Using thispotential and variables we can discuss the fate of theuniverse without specifying the value of ΛA static universe will appear if we find a solution a =

aS(gt 0) which satisfies U(aS) = 0 and Uprime(aS) = 0 IfΛ 6= 0 (ǫ = plusmn1) it happens if there is a relation betweengr and gs which is defined by

gs = g [ǫK](plusmn)s (gr)

=1

9ǫ2

[

2K minus 3ǫKgr plusmn 2(1minus ǫgr)32]

(39)

This gives the curve ΓǫK(plusmn) on the gr-gs plane whichgives the boundary between two different phases of space-time The radius of a static universe is given by

aS = a[ǫK](plusmn)S =

radic

1

3ǫ

[

K plusmnradic

1minus ǫgr

]

(310)

if it is real and positive Here plusmn correspond to the curvesΓǫ(plusmn) Since U(a) = 0 is the cubic equation with respect

to a2 and a2 = a2S is the double root we have the thirdroot which is given by

aT = a[ǫK](plusmn)T =

radic

1

3ǫ

[

K ∓ 2radic

1minus ǫgr

]

(311)

where the universe turns around (or bounces) To existsuch a point it must be real and positiveIf Λ = 0 we find

gs = minusK

12g2r (312)

which is found from (39) in the limit of ǫ = 0 The cor-responding curve on the gr-gs plane is denoted by Γ0K The radius is given by

aS = a[0K]S =

radic

gr6K

(313)

assuming Kgr gt 0Note that our classification depends just on gr and gs

(or gr and gs) apart from K and Λ Since gr and gs aregiven by g2 g3 g5 g6 and g7 but do not include g4 g8and g9 the fate of the universe is classified only by theconditions on the coupling constants of higher curvatureterms but not on those of their derivatives such as nablajR ℓ

kIn the case with the detailed balance conditions we

find Λ lt 0 from Eq (28) and then obtain from Eq(36)

gr = minus94 gs = 0 (314)

Now we shall discuss what kind of spacetimes are real-ized under which conditions in the following three casesseparately [A Λ = 0 B Λ gt 0 and C Λ lt 0]

A Λ = 0

If a cosmological constant is absent the ldquopotentialrdquo iswritten as

U(a) =1

(3λminus 1)a4

[

Ka4 minus gr3a2 minus gs

3

]

(315)

In Fig 1 we show the fate of the universe which dependson the values of gr and gs For the case ofK = 1 there aretwo types of spacetime phases One is BB rArr BC and theother is an oscillating universe In fact if gr gt 0 gs lt 0and g2r + 12gs gt 0 we find the scale factor a is boundedin a finite range as (0 lt) amin le a le amax (lt infin) where

a2min equiv 1

6

[

gr minusradic

g2r + 12gs

]

a2max equiv 1

6

[

gr +radic

g2r + 12gs

]

(316)

which gives an oscillating universe The condition for anoscillating universe is written as

gr gt 0 minus g2r12

le gs lt 0 (317)

5

(a) K = 1

(b) K = minus1

FIG 1 Phase diagram of spacetimes for Λ = 0 The os-cillating universe is found only for the case of K = 1 Thestable and unstable static universes (Ss and Su) exist on theboundary Γ01 and Γ0minus1 respectively On Γ0minus1 we also finddynamical universes with an asymptotically static spacetimeSu rArr BC Su rArr M BBrArr Su or MrArr Su

which is shown in Fig 1(a) by ldquoOscillationrdquo (the light-orange colored region) in the gs-gr plane The equality inEq (317) which is the curve Γ01 gives a static universe

with the scale factor a = aS =radic

gr6

For the case of K = minus1 we find three types of space-time phases BB rArr M (or M rArr BC) BB rArr BC andBounce (see Fig1(b)) On the boundary curve Γ0minus1which is defined by Eq (312) ie gs = g2r 12 (gr lt 0)we find an unstable static universe Su and the dynamicaluniverses with an asymptotically static spacetime Su rArrBC Su rArr M BBrArr Su or MrArr Su

The bounce universe is found if gs lt 0 or gs = 0 withgr lt 0 which is shown by rdquoBouncerdquo (the light-green re-gion) The radius at a turning point aT is given by

aT =

radic

1

6

(

minusgr +radic

g2r minus 12gs

)

(318)

Next we shall evaluate the period of an oscillating uni-verse in the case of K = 1 The solution for Eq (34) isgiven by

tminus tmax = minusint a

amax

daradic

minus2U(a)

= amax

radic

3λminus 1

2E (φ[a] k) (319)

where E(φ k) is the elliptic integral of the second kind

which is defined by

E(φ k) =

int φ

0

dθradic

1minus k2 sin2 θ (320)

k and φ[a] are given by

k =

radic

a2max minus a2min

amax (321)

φ[a] = sinminus1

(

a2max minus a2

a2max minus a2min

)

(322)

The period T is given by

T = 2(tmin minus tmax) = 2amax

radic

3λminus 1

2E (k) (323)

where E(k) is the complete elliptic integral of the secondkind defined by E(k) = E(π2 k)In order to evaluate the period we consider some lim-

iting cases which are the boundaries of the region ofOscillation In Fig 2 we show the potential U(a) by theblue curve for one boundary curve Γ01 which is givenby gs = minusg2r 12 It gives a stable static universe withthe scale factor aS We also show the potential near theother boundary of Oscillation (the positive gr-axis) by thedashed orange curve Choosing for example gr = 1 andgs = minus0001 we find an oscillating univese with the scalefactor a isin [00316705 0576481] Since these two poten-

FIG 2 The potential U(a) for a stable static universe andan oscillating universe near the gr-axis The ldquocouplingrdquo con-stants are chosen as gr = 1 and gs = minus112 on Γ01 for astatic universe which radius is shown by aS = 1

radic6 We

also show the case with gr = 1 and gs = minus0001 for an oscil-lating universe which maximum and minimum radii are givenby amax = 0576481 and amin = 00316705 respectively

tials give the limiting cases we find that 0 lt amin le aSand aS le amax lt

radic2 aS for an oscillating universe

In the limit of a static universe (near Γ01) we find theperiod TS as

TS = π

radic

(

3λminus 1

2

)

gr6 (324)

while in the other boundary limit (gs rarr 0) we obtain

T0 =

radic

(

3λminus 1

2

)

4gr3

(325)

6

From these evaluations giving the value of gr we findthe period T of any oscillating universe is bounded inthe range of (T0 TS) for gs isin (minusg2r 12 0) We then ap-

proximate the period as T sim g12r

We have found an oscillating FLRW universe becausewe have ldquonegativerdquo energy of ldquostiff matterrdquo which comesfrom the higher curvature term The condition for anoscillating universe is rewritten in terms of the originalcoupling constants as

g3 + 3g2 gt 0 (326)

minus (g3 + 3g2)2

4le 9g5 + 3g6 + g7 lt 0 (327)

B Λ gt 0 (ǫ = 1)

In this case the potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 minus a6 minus gr3a2 minus gs

3

]

(328)

For each value of K we depict the fate of the universe inFig 3 which depends on the values of gr and gs

(a) K = 1

(b) K = minus1

FIG 3 Phase diagram of spacetimes for Λ gt 0 The oscillat-ing universe is found only for the case of K = 1 The staticuniverses (Ss and Su) exist on the boundaries Γ1K(plusmn) Wealso find dynamical universes with an asymptotically staticspacetime Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or Su rArr BC

on Γ1minus1

We find non-singular evolution of the universe (BounceOscillation or Static) as well as the universe with a cosmo-logical singularity (BB rArr BC BB rArr dS or dS rArr BC)

Except for the case of BB rArr BC and a static universethe expanding universe approaches de Sitter spacetime(exponentially expanding universe) because of a positivecosmological constant Λ The oscillating universe existsif and only if K = 1 and the following conditions aresatisfied

gr gt 0 (329)

g [11](minus)s (gr) le gs

lt 0

le g[11](+)s (gr)

(330)

where g[11](plusmn)s is defined by Eq (39) with ǫ = 1K = 1

This condition gives the constraint on gs as minus19 le gs lt0 Note that in the limit of gr ≪ 1 (ie Λ rarr 0) werecover the condition (317)The boundaries of two different phases of spacetimes

consist of the gr-axis and two curves (Γ11(plusmn)) for K = 1or one curve (Γ1minus1(+)) for K = minus1 Those boundary

curves Γ1K(plusmn) are defined by gs = g[1K](plusmn)s (gr)

A stable static universe exist on the boundary curveΓ11(minus) while unstable static universes appear on theboundary curves Γ1plusmn1(+) For K = 1 there are twotypes of static universes (stable and unstable) corre-sponding to two curves Γ11(minus) and Γ11(+) respectivelywhich coincide at gr = 1 and gs = minus19 In the branchesof unstable static universes (Γ1K(+)) we also find dy-namical universes with an asymptotically static space-time Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or SurArr BC on Γ1minus1(+)The period T of an oscillating universe is calculated by

T = 2

int amax

amin

daradic

minus2U(a) (331)

where T = Tℓ and amax and amin are the maximumand minimum radii of the oscillating universe We shallevaluate the period near the boundaries of the parameterrange of oscillating universes (the light-orange region inFig 3(a)) We first show the potential U(a) for three(near-) boundary values of gs in Fig 4For the case with an unstable static universe (the

dashed blue curve) (Γ11(+) with gs lt 0) the larger dou-ble root of the equation of U(a) = 0 is given by

aS = a[11](+)S =

radic

1

3

(

1 +radic

1minus gr

)

(332)

while the smaller root is

aT = a[11](+)T =

radic

1

3

(

1minus 2radic

1minus gr

)

(333)

which corresponds to a turning radius at a bounce Theperiod T diverges in the limit of a static universe becauseamax = aS is the double root

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

4

find a static universe (S)(5) [S ] A spacetime is static with a constant scale factoraS if U(aS) = 0 and Uprime(aS) = 0There are two types of static universes one is stable

(Ss) and the other is unstable (Su) When we have an un-stable static universe we also find the following types ofdynamical universes with a static spacetime as an asymp-totic state as well(6) [Su rArr infin or infin rArrSu] If U(a) le 0 for a isin [aS infin) andthe equality holds only at aS a spacetime starts from astatic state in the infinite past and expands forever or itinitially contracts from an infinite scale and eventuallyreach a static state in the infinite future We then haveSu rArr dS M or dS M rArrSu

(7) [BBrArr Su or Su rArr BC ] If U(a) le 0 for a isin (0 aS ] andthe equality holds only at aS A spacetime starts from abig bang and expands to a static state with a finite scaleaS or its time reversal (A spacetime contracts from astatic state to a big crunch)

(8) [Su rArr Bounce rArr Su] If U(a) le 0 for a isin [aS aT ] (ora isin [aT aS]) and the equality holds only at aS and aT aspacetime starts from a static state in the infinite pastand expands (or contracts) It eventually bounces at afinite scale aT and then reach a static state again in theinfinite futureFor the case of Λ 6= 0 introducing the curvature scale

ℓ which is defined by

Λ

3=

ǫ

ℓ2 (37)

where ǫ = plusmn1 we can rescale the variables and rewritethe ldquopotentialrdquo U by the rescaled variables as

U(a) =1

3λminus 1

[

K minus ǫa2 minus gr3a2

minus gs3a4

]

(38)

where a = aℓ gr = grℓ2 and gs = gsℓ

4 Using thispotential and variables we can discuss the fate of theuniverse without specifying the value of ΛA static universe will appear if we find a solution a =

aS(gt 0) which satisfies U(aS) = 0 and Uprime(aS) = 0 IfΛ 6= 0 (ǫ = plusmn1) it happens if there is a relation betweengr and gs which is defined by

gs = g [ǫK](plusmn)s (gr)

=1

9ǫ2

[

2K minus 3ǫKgr plusmn 2(1minus ǫgr)32]

(39)

This gives the curve ΓǫK(plusmn) on the gr-gs plane whichgives the boundary between two different phases of space-time The radius of a static universe is given by

aS = a[ǫK](plusmn)S =

radic

1

3ǫ

[

K plusmnradic

1minus ǫgr

]

(310)

if it is real and positive Here plusmn correspond to the curvesΓǫ(plusmn) Since U(a) = 0 is the cubic equation with respect

to a2 and a2 = a2S is the double root we have the thirdroot which is given by

aT = a[ǫK](plusmn)T =

radic

1

3ǫ

[

K ∓ 2radic

1minus ǫgr

]

(311)

where the universe turns around (or bounces) To existsuch a point it must be real and positiveIf Λ = 0 we find

gs = minusK

12g2r (312)

which is found from (39) in the limit of ǫ = 0 The cor-responding curve on the gr-gs plane is denoted by Γ0K The radius is given by

aS = a[0K]S =

radic

gr6K

(313)

assuming Kgr gt 0Note that our classification depends just on gr and gs

(or gr and gs) apart from K and Λ Since gr and gs aregiven by g2 g3 g5 g6 and g7 but do not include g4 g8and g9 the fate of the universe is classified only by theconditions on the coupling constants of higher curvatureterms but not on those of their derivatives such as nablajR ℓ

kIn the case with the detailed balance conditions we

find Λ lt 0 from Eq (28) and then obtain from Eq(36)

gr = minus94 gs = 0 (314)

Now we shall discuss what kind of spacetimes are real-ized under which conditions in the following three casesseparately [A Λ = 0 B Λ gt 0 and C Λ lt 0]

A Λ = 0

If a cosmological constant is absent the ldquopotentialrdquo iswritten as

U(a) =1

(3λminus 1)a4

[

Ka4 minus gr3a2 minus gs

3

]

(315)

In Fig 1 we show the fate of the universe which dependson the values of gr and gs For the case ofK = 1 there aretwo types of spacetime phases One is BB rArr BC and theother is an oscillating universe In fact if gr gt 0 gs lt 0and g2r + 12gs gt 0 we find the scale factor a is boundedin a finite range as (0 lt) amin le a le amax (lt infin) where

a2min equiv 1

6

[

gr minusradic

g2r + 12gs

]

a2max equiv 1

6

[

gr +radic

g2r + 12gs

]

(316)

which gives an oscillating universe The condition for anoscillating universe is written as

gr gt 0 minus g2r12

le gs lt 0 (317)

5

(a) K = 1

(b) K = minus1

FIG 1 Phase diagram of spacetimes for Λ = 0 The os-cillating universe is found only for the case of K = 1 Thestable and unstable static universes (Ss and Su) exist on theboundary Γ01 and Γ0minus1 respectively On Γ0minus1 we also finddynamical universes with an asymptotically static spacetimeSu rArr BC Su rArr M BBrArr Su or MrArr Su

which is shown in Fig 1(a) by ldquoOscillationrdquo (the light-orange colored region) in the gs-gr plane The equality inEq (317) which is the curve Γ01 gives a static universe

with the scale factor a = aS =radic

gr6

For the case of K = minus1 we find three types of space-time phases BB rArr M (or M rArr BC) BB rArr BC andBounce (see Fig1(b)) On the boundary curve Γ0minus1which is defined by Eq (312) ie gs = g2r 12 (gr lt 0)we find an unstable static universe Su and the dynamicaluniverses with an asymptotically static spacetime Su rArrBC Su rArr M BBrArr Su or MrArr Su

The bounce universe is found if gs lt 0 or gs = 0 withgr lt 0 which is shown by rdquoBouncerdquo (the light-green re-gion) The radius at a turning point aT is given by

aT =

radic

1

6

(

minusgr +radic

g2r minus 12gs

)

(318)

Next we shall evaluate the period of an oscillating uni-verse in the case of K = 1 The solution for Eq (34) isgiven by

tminus tmax = minusint a

amax

daradic

minus2U(a)

= amax

radic

3λminus 1

2E (φ[a] k) (319)

where E(φ k) is the elliptic integral of the second kind

which is defined by

E(φ k) =

int φ

0

dθradic

1minus k2 sin2 θ (320)

k and φ[a] are given by

k =

radic

a2max minus a2min

amax (321)

φ[a] = sinminus1

(

a2max minus a2

a2max minus a2min

)

(322)

The period T is given by

T = 2(tmin minus tmax) = 2amax

radic

3λminus 1

2E (k) (323)

where E(k) is the complete elliptic integral of the secondkind defined by E(k) = E(π2 k)In order to evaluate the period we consider some lim-

iting cases which are the boundaries of the region ofOscillation In Fig 2 we show the potential U(a) by theblue curve for one boundary curve Γ01 which is givenby gs = minusg2r 12 It gives a stable static universe withthe scale factor aS We also show the potential near theother boundary of Oscillation (the positive gr-axis) by thedashed orange curve Choosing for example gr = 1 andgs = minus0001 we find an oscillating univese with the scalefactor a isin [00316705 0576481] Since these two poten-

FIG 2 The potential U(a) for a stable static universe andan oscillating universe near the gr-axis The ldquocouplingrdquo con-stants are chosen as gr = 1 and gs = minus112 on Γ01 for astatic universe which radius is shown by aS = 1

radic6 We

also show the case with gr = 1 and gs = minus0001 for an oscil-lating universe which maximum and minimum radii are givenby amax = 0576481 and amin = 00316705 respectively

tials give the limiting cases we find that 0 lt amin le aSand aS le amax lt

radic2 aS for an oscillating universe

In the limit of a static universe (near Γ01) we find theperiod TS as

TS = π

radic

(

3λminus 1

2

)

gr6 (324)

while in the other boundary limit (gs rarr 0) we obtain

T0 =

radic

(

3λminus 1

2

)

4gr3

(325)

6

From these evaluations giving the value of gr we findthe period T of any oscillating universe is bounded inthe range of (T0 TS) for gs isin (minusg2r 12 0) We then ap-

proximate the period as T sim g12r

We have found an oscillating FLRW universe becausewe have ldquonegativerdquo energy of ldquostiff matterrdquo which comesfrom the higher curvature term The condition for anoscillating universe is rewritten in terms of the originalcoupling constants as

g3 + 3g2 gt 0 (326)

minus (g3 + 3g2)2

4le 9g5 + 3g6 + g7 lt 0 (327)

B Λ gt 0 (ǫ = 1)

In this case the potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 minus a6 minus gr3a2 minus gs

3

]

(328)

For each value of K we depict the fate of the universe inFig 3 which depends on the values of gr and gs

(a) K = 1

(b) K = minus1

FIG 3 Phase diagram of spacetimes for Λ gt 0 The oscillat-ing universe is found only for the case of K = 1 The staticuniverses (Ss and Su) exist on the boundaries Γ1K(plusmn) Wealso find dynamical universes with an asymptotically staticspacetime Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or Su rArr BC

on Γ1minus1

We find non-singular evolution of the universe (BounceOscillation or Static) as well as the universe with a cosmo-logical singularity (BB rArr BC BB rArr dS or dS rArr BC)

Except for the case of BB rArr BC and a static universethe expanding universe approaches de Sitter spacetime(exponentially expanding universe) because of a positivecosmological constant Λ The oscillating universe existsif and only if K = 1 and the following conditions aresatisfied

gr gt 0 (329)

g [11](minus)s (gr) le gs

lt 0

le g[11](+)s (gr)

(330)

where g[11](plusmn)s is defined by Eq (39) with ǫ = 1K = 1

This condition gives the constraint on gs as minus19 le gs lt0 Note that in the limit of gr ≪ 1 (ie Λ rarr 0) werecover the condition (317)The boundaries of two different phases of spacetimes

consist of the gr-axis and two curves (Γ11(plusmn)) for K = 1or one curve (Γ1minus1(+)) for K = minus1 Those boundary

curves Γ1K(plusmn) are defined by gs = g[1K](plusmn)s (gr)

A stable static universe exist on the boundary curveΓ11(minus) while unstable static universes appear on theboundary curves Γ1plusmn1(+) For K = 1 there are twotypes of static universes (stable and unstable) corre-sponding to two curves Γ11(minus) and Γ11(+) respectivelywhich coincide at gr = 1 and gs = minus19 In the branchesof unstable static universes (Γ1K(+)) we also find dy-namical universes with an asymptotically static space-time Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or SurArr BC on Γ1minus1(+)The period T of an oscillating universe is calculated by

T = 2

int amax

amin

daradic

minus2U(a) (331)

where T = Tℓ and amax and amin are the maximumand minimum radii of the oscillating universe We shallevaluate the period near the boundaries of the parameterrange of oscillating universes (the light-orange region inFig 3(a)) We first show the potential U(a) for three(near-) boundary values of gs in Fig 4For the case with an unstable static universe (the

dashed blue curve) (Γ11(+) with gs lt 0) the larger dou-ble root of the equation of U(a) = 0 is given by

aS = a[11](+)S =

radic

1

3

(

1 +radic

1minus gr

)

(332)

while the smaller root is

aT = a[11](+)T =

radic

1

3

(

1minus 2radic

1minus gr

)

(333)

which corresponds to a turning radius at a bounce Theperiod T diverges in the limit of a static universe becauseamax = aS is the double root

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

5

(a) K = 1

(b) K = minus1

FIG 1 Phase diagram of spacetimes for Λ = 0 The os-cillating universe is found only for the case of K = 1 Thestable and unstable static universes (Ss and Su) exist on theboundary Γ01 and Γ0minus1 respectively On Γ0minus1 we also finddynamical universes with an asymptotically static spacetimeSu rArr BC Su rArr M BBrArr Su or MrArr Su

which is shown in Fig 1(a) by ldquoOscillationrdquo (the light-orange colored region) in the gs-gr plane The equality inEq (317) which is the curve Γ01 gives a static universe

with the scale factor a = aS =radic

gr6

For the case of K = minus1 we find three types of space-time phases BB rArr M (or M rArr BC) BB rArr BC andBounce (see Fig1(b)) On the boundary curve Γ0minus1which is defined by Eq (312) ie gs = g2r 12 (gr lt 0)we find an unstable static universe Su and the dynamicaluniverses with an asymptotically static spacetime Su rArrBC Su rArr M BBrArr Su or MrArr Su

The bounce universe is found if gs lt 0 or gs = 0 withgr lt 0 which is shown by rdquoBouncerdquo (the light-green re-gion) The radius at a turning point aT is given by

aT =

radic

1

6

(

minusgr +radic

g2r minus 12gs

)

(318)

Next we shall evaluate the period of an oscillating uni-verse in the case of K = 1 The solution for Eq (34) isgiven by

tminus tmax = minusint a

amax

daradic

minus2U(a)

= amax

radic

3λminus 1

2E (φ[a] k) (319)

where E(φ k) is the elliptic integral of the second kind

which is defined by

E(φ k) =

int φ

0

dθradic

1minus k2 sin2 θ (320)

k and φ[a] are given by

k =

radic

a2max minus a2min

amax (321)

φ[a] = sinminus1

(

a2max minus a2

a2max minus a2min

)

(322)

The period T is given by

T = 2(tmin minus tmax) = 2amax

radic

3λminus 1

2E (k) (323)

where E(k) is the complete elliptic integral of the secondkind defined by E(k) = E(π2 k)In order to evaluate the period we consider some lim-

iting cases which are the boundaries of the region ofOscillation In Fig 2 we show the potential U(a) by theblue curve for one boundary curve Γ01 which is givenby gs = minusg2r 12 It gives a stable static universe withthe scale factor aS We also show the potential near theother boundary of Oscillation (the positive gr-axis) by thedashed orange curve Choosing for example gr = 1 andgs = minus0001 we find an oscillating univese with the scalefactor a isin [00316705 0576481] Since these two poten-

FIG 2 The potential U(a) for a stable static universe andan oscillating universe near the gr-axis The ldquocouplingrdquo con-stants are chosen as gr = 1 and gs = minus112 on Γ01 for astatic universe which radius is shown by aS = 1

radic6 We

also show the case with gr = 1 and gs = minus0001 for an oscil-lating universe which maximum and minimum radii are givenby amax = 0576481 and amin = 00316705 respectively

tials give the limiting cases we find that 0 lt amin le aSand aS le amax lt

radic2 aS for an oscillating universe

In the limit of a static universe (near Γ01) we find theperiod TS as

TS = π

radic

(

3λminus 1

2

)

gr6 (324)

while in the other boundary limit (gs rarr 0) we obtain

T0 =

radic

(

3λminus 1

2

)

4gr3

(325)

6

From these evaluations giving the value of gr we findthe period T of any oscillating universe is bounded inthe range of (T0 TS) for gs isin (minusg2r 12 0) We then ap-

proximate the period as T sim g12r

We have found an oscillating FLRW universe becausewe have ldquonegativerdquo energy of ldquostiff matterrdquo which comesfrom the higher curvature term The condition for anoscillating universe is rewritten in terms of the originalcoupling constants as

g3 + 3g2 gt 0 (326)

minus (g3 + 3g2)2

4le 9g5 + 3g6 + g7 lt 0 (327)

B Λ gt 0 (ǫ = 1)

In this case the potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 minus a6 minus gr3a2 minus gs

3

]

(328)

For each value of K we depict the fate of the universe inFig 3 which depends on the values of gr and gs

(a) K = 1

(b) K = minus1

FIG 3 Phase diagram of spacetimes for Λ gt 0 The oscillat-ing universe is found only for the case of K = 1 The staticuniverses (Ss and Su) exist on the boundaries Γ1K(plusmn) Wealso find dynamical universes with an asymptotically staticspacetime Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or Su rArr BC

on Γ1minus1

We find non-singular evolution of the universe (BounceOscillation or Static) as well as the universe with a cosmo-logical singularity (BB rArr BC BB rArr dS or dS rArr BC)

Except for the case of BB rArr BC and a static universethe expanding universe approaches de Sitter spacetime(exponentially expanding universe) because of a positivecosmological constant Λ The oscillating universe existsif and only if K = 1 and the following conditions aresatisfied

gr gt 0 (329)

g [11](minus)s (gr) le gs

lt 0

le g[11](+)s (gr)

(330)

where g[11](plusmn)s is defined by Eq (39) with ǫ = 1K = 1

This condition gives the constraint on gs as minus19 le gs lt0 Note that in the limit of gr ≪ 1 (ie Λ rarr 0) werecover the condition (317)The boundaries of two different phases of spacetimes

consist of the gr-axis and two curves (Γ11(plusmn)) for K = 1or one curve (Γ1minus1(+)) for K = minus1 Those boundary

curves Γ1K(plusmn) are defined by gs = g[1K](plusmn)s (gr)

A stable static universe exist on the boundary curveΓ11(minus) while unstable static universes appear on theboundary curves Γ1plusmn1(+) For K = 1 there are twotypes of static universes (stable and unstable) corre-sponding to two curves Γ11(minus) and Γ11(+) respectivelywhich coincide at gr = 1 and gs = minus19 In the branchesof unstable static universes (Γ1K(+)) we also find dy-namical universes with an asymptotically static space-time Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or SurArr BC on Γ1minus1(+)The period T of an oscillating universe is calculated by

T = 2

int amax

amin

daradic

minus2U(a) (331)

where T = Tℓ and amax and amin are the maximumand minimum radii of the oscillating universe We shallevaluate the period near the boundaries of the parameterrange of oscillating universes (the light-orange region inFig 3(a)) We first show the potential U(a) for three(near-) boundary values of gs in Fig 4For the case with an unstable static universe (the

dashed blue curve) (Γ11(+) with gs lt 0) the larger dou-ble root of the equation of U(a) = 0 is given by

aS = a[11](+)S =

radic

1

3

(

1 +radic

1minus gr

)

(332)

while the smaller root is

aT = a[11](+)T =

radic

1

3

(

1minus 2radic

1minus gr

)

(333)

which corresponds to a turning radius at a bounce Theperiod T diverges in the limit of a static universe becauseamax = aS is the double root

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

6

From these evaluations giving the value of gr we findthe period T of any oscillating universe is bounded inthe range of (T0 TS) for gs isin (minusg2r 12 0) We then ap-

proximate the period as T sim g12r

We have found an oscillating FLRW universe becausewe have ldquonegativerdquo energy of ldquostiff matterrdquo which comesfrom the higher curvature term The condition for anoscillating universe is rewritten in terms of the originalcoupling constants as

g3 + 3g2 gt 0 (326)

minus (g3 + 3g2)2

4le 9g5 + 3g6 + g7 lt 0 (327)

B Λ gt 0 (ǫ = 1)

In this case the potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 minus a6 minus gr3a2 minus gs

3

]

(328)

For each value of K we depict the fate of the universe inFig 3 which depends on the values of gr and gs

(a) K = 1

(b) K = minus1

FIG 3 Phase diagram of spacetimes for Λ gt 0 The oscillat-ing universe is found only for the case of K = 1 The staticuniverses (Ss and Su) exist on the boundaries Γ1K(plusmn) Wealso find dynamical universes with an asymptotically staticspacetime Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or Su rArr BC

on Γ1minus1

We find non-singular evolution of the universe (BounceOscillation or Static) as well as the universe with a cosmo-logical singularity (BB rArr BC BB rArr dS or dS rArr BC)

Except for the case of BB rArr BC and a static universethe expanding universe approaches de Sitter spacetime(exponentially expanding universe) because of a positivecosmological constant Λ The oscillating universe existsif and only if K = 1 and the following conditions aresatisfied

gr gt 0 (329)

g [11](minus)s (gr) le gs

lt 0

le g[11](+)s (gr)

(330)

where g[11](plusmn)s is defined by Eq (39) with ǫ = 1K = 1

This condition gives the constraint on gs as minus19 le gs lt0 Note that in the limit of gr ≪ 1 (ie Λ rarr 0) werecover the condition (317)The boundaries of two different phases of spacetimes

consist of the gr-axis and two curves (Γ11(plusmn)) for K = 1or one curve (Γ1minus1(+)) for K = minus1 Those boundary

curves Γ1K(plusmn) are defined by gs = g[1K](plusmn)s (gr)

A stable static universe exist on the boundary curveΓ11(minus) while unstable static universes appear on theboundary curves Γ1plusmn1(+) For K = 1 there are twotypes of static universes (stable and unstable) corre-sponding to two curves Γ11(minus) and Γ11(+) respectivelywhich coincide at gr = 1 and gs = minus19 In the branchesof unstable static universes (Γ1K(+)) we also find dy-namical universes with an asymptotically static space-time Su rArr dS or Su rArr BC on Γ11(+)(gs ge 0) Su rArr dS

or Su rArr Bounce rArr Su on Γ11(+)(gs lt 0) Su rArr dS or SurArr BC on Γ1minus1(+)The period T of an oscillating universe is calculated by

T = 2

int amax

amin

daradic

minus2U(a) (331)

where T = Tℓ and amax and amin are the maximumand minimum radii of the oscillating universe We shallevaluate the period near the boundaries of the parameterrange of oscillating universes (the light-orange region inFig 3(a)) We first show the potential U(a) for three(near-) boundary values of gs in Fig 4For the case with an unstable static universe (the

dashed blue curve) (Γ11(+) with gs lt 0) the larger dou-ble root of the equation of U(a) = 0 is given by

aS = a[11](+)S =

radic

1

3

(

1 +radic

1minus gr

)

(332)

while the smaller root is

aT = a[11](+)T =

radic

1

3

(

1minus 2radic

1minus gr

)

(333)

which corresponds to a turning radius at a bounce Theperiod T diverges in the limit of a static universe becauseamax = aS is the double root

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

7

FIG 4 The potential U(a) for a stable and unstable staticuniverses (the solid blue and the dashed blue) and that foran oscillating universe near gr-axis (orange) The constantsare chosen as gr = 08 and gs = minus00643206 on Γ11(minus) andminus00245683 on Γ11(+) for static universes which radii aregiven by aS and gr = 07 and gs = minus0001 for an oscillatinguniverse which maximum and minimum radii are given byamax and amin respectively We also find Su rArr Bounce rArr Suwhich bounce radius is given by amin

While near a stable static universe (the solid bluecurve) (Γ11(minus)) the period is finite and is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

1minus (1minus gr)12

3(1minus gr)12

]12

(334)

asymp(

3λminus 1

2

)12

times

πradic6g12r (gr ≪ 1)

πradic3

1

(1minus gr)14(gr asymp 1)

The period TS changes from 0 toinfin along the static curveΓ11(minus)

The radius of this stable static universe is given by

aS = a[11](minus)S which is the smaller root of the equation

of U(a) = 0 The larger root aT = a[11](+)S corresponds

to a turning radius of a bounce universe which is shownby aT in Fig 4

There is another boundary limit ie gs rarr 0minus In thislimit we find the roots of U(a) = 0 as

a21 asymp 0 (335)

a22 asymp 1

2

(

1minusradic

1minus 4

3gr

)

(336)

a23 asymp 1

3

(

1 +

radic

1minus 4

3gr

)

(337)

Since the largest root (a3) corresponds to a turning radiusaT of a bounce universe the oscillation range is [a1 a2]and then the period is evaluated approximately by

T0 = 2

int a2

0

daradic

minus2U(a) (338)

The period is then given by

T0 =

(

3λminus 1

2

)12

times 2 sinhminus1

1minus(

1minus 43 gr)

12

2(

1minus 43 gr)

12

12

asymp(

3λminus 1

2

)12

times

2radic3g

12r (gr ≪ 1)

ln

radic3

(

34 minus gr

)12

(gr asymp 34 )

(339)

The period T0 also changes from 0 to infin along the gr-axisgs = 0 (0 lt gr lt 34)

We summarize our result as T sim g12r when gr ≪ 1 but

it diverges near Γ11(+) on which we have the unstablestatic universe

C Λ lt 0 (ǫ = minus1)

The potential is given by

U(a) =1

(3λminus 1)a4

[

Ka4 + a6 minus gr3a2 minus gs

3

]

(340)

We summarize our result in Fig 5

(a) K = 1

(b) K = minus1

FIG 5 Phase diagram of spacetimes for Λ lt 0 The os-cillating universe is found for both K = plusmn1 The staticuniverse exists on the boundary Γminus11(minus) (K = 1) and onΓminus1minus1(plusmn) (K = minus1) In the branch of unstable static uni-verse on Γminus1minus1(+) we also find dynamical universes with anasymptotically static spacetime BB rArr Su Su rArr BC or Su rArrBouncerArr Su

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

8

In this case if gs gt 0 we find a big bag and a bigcrunch singularities (BB rArr BC) except for a small re-gion in K = minus1 If gs lt 0 however we always find anoscillating universe if the solution existsThe conditions for an oscillating universe is shown by

the light-orange region in Fig 5 which is given by thefollowing inequalitiesFor K = 1

gr gt 0

g[minus11](minus)s (gr) le gs lt 0 (341)

and for K = minus1

g[minus1minus1](minus)s (gr) le gs lt 0 with gr ge 0

g[minus1minus1](minus)s (gr) le gs le g

[minus1minus1](+)s (gr) with gr lt 0

(342)

In the limit of gr ≪ 1 (ie Λ rarr 0) for K = 1 we recoverthe condition (317)The boundary of the range of oscillating universe is

given by the positive gr-axis and Γminus11(minus) forK = 1 andΓminus1minus1(plusmn) for K = minus1 On those boundaries Γminus1K(plusmn)

which are defined by gs = g[minus11](minus)s (gr) (K = 1) and gs =

g[minus1minus1](plusmn)s (gr) (K = minus1) we find a stable and unstablestatic universesThe period of an oscillating universe is given by

Eq(331) We again evaluate its value near the boundarycurves (Γminus1K(minus)) and the positive gr-axis The poten-tials U(a) for the (near-) boundary values of gs are shownin Fig 6 (K = 1) and Figs 7 and 8 (K = minus1)

FIG 6 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) in the caseof K = 1 We set gr = 08 and gs = minus00477674 on Γminus11 fora static universe with the radius aS = 0337461 and gr = 08and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 0466615 and amin =0035439 respectively

Note that the period diverges in the limit of an un-stable static universe (on Γminus1minus1(+)) where we find theradius of a static universe by aS = a1

a21 =1

3

(

1minusradic

1 + gr

)

(343)

The turning point is given by aT = a2 where

a22 =1

3

(

1 + 2radic

1 + gr

)

(344)

FIG 7 The potential U(a) for a stable static universe (blue)and an oscillating universe near gr-axis (orange) for K = minus1We set gr = 02 and gs = minus0581008 on Γminus1minus1 for a staticuniverse which radius is given by aS = 0835752 and gr = 02and gs = minus0001 for an oscillating universe which maximumand minimum radii are given by amax = 103075 and amin =00683656 respectively

FIG 8 The potential U(a) for a stable and unstable staticuniverses (blue and red respectively) and an oscillating uni-verse on gr-axis (dashed orange) for K = minus1 We set gr =minus05 and gs = minus0134123 on Γminus1minus1(minus) and gs = 00230119on Γminus1minus1(+) for static universes which radius is given byaS = 0754344 and gr = minus05 and gs = 0 for an oscillat-ing universe which maximum and minimum radii are givenby amax = 0888074 and amin = 0459701 respectively Wealso find Su rArr Bounce rArr Su which bounce radius is given byaT = 0897072

Near a stable static universe (Γminus11(minus) and Γminus1minus1(minus))the period is evaluated as

TS =

(

3λminus 1

2

)12

times π

[

(1 + gr)12 minusK

3(1 + gr)12

]12

(345)

which approaches a constant

TS asymp πradic3

(

3λminus 1

2

)12

(346)

when gr ≫ 1

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

9

Near the lower bound of gr we find

TS asymp πradic3

(

3λminus 1

2

)12

times

g12rradic2

rarr 0 (as gr rarr 0 for K = 1)

(1 + gr)minus14 rarr infin (as gr rarr minus1 for K = minus1)

(347)

Hence the period TS changes from 0 to a finite value(346) along the curve Γminus11(minus) for K = 1 while from infinto at the same finite value along the curve Γminus1minus1(minus)

The radius of a static universe is given by

aS = a[minus1K](minus)S =

radic

1

3

(

radic

1 + gr minusK)

(348)

In the case of gr lt minus34 with K = minus1 there is anotherzero point of U(a) which gives a maximum turning pointof BB rArr BC ie

aT = a[minus1minus1](minus)T =

radic

1

3

(

1minus 2radic

1 + gr

)

(349)

Near gr-axis we find the solutions of the equationU(a) = 0 as

a2plusmn =1

2

(

minusK plusmnradic

1 +4

3gr

)

(350)

as well as a0 asymp 0 We have a maximum radius amax = a+and find that the minimum radius amin is almost zero forgr gt 0 because a2minus lt 0 but in the case of K = minus1for minus34 lt gr lt 0 we find a finite minimum radiusamin = aminus

Using those values we evaluate the period as

T0 =

(

3λminus 1

2

)12

secminus1

radic

1 +4

3gr (351)

for K = 1 and

T0 =

(

3λminus 1

2

)12

times

π minus secminus1

radic

1 +4

3gr (gr ge 0)

π (minus34 lt gr lt 0)

for K = minus1 The period T0 also changes from 0 to infinalong the gr-axis gs = 0 (0 lt gr lt 34)

In the case with the detailed balance condition sinceΛ lt 0 gr = minus94 gs = 0 we do not find any FLRW so-lution If we include matter fluid the result will changeFor example if we have ldquoradiationrdquo fluid which energydensity is proportional to aminus4 we should shift the valueof gr Then if minus34 le gr lt 0 we find an oscillating uni-verse for K = minus1 which period is π[(3λminus 1)2]12 Theequality (gr = minus34) gives a static universe

IV TOWARD MORE REALISTIC

COSMOLOGICAL MODEL

In the Horava-Lifshitz gravity without the detailed bal-ance condition we find a variety of phase structures ofvacuum spacetimes depending on the coupling constantsgr and gs as well as the spatial curvature K and a cosmo-logical constant Λ Note that there is no vacuum FLRWsolution in the case with the detailed balance conditionWe summarize our result in Table I We have obtainedan oscillating spacetime as well as a bounce universe for awide range of coupling constants We have also evaluatedthe period of the oscillating universe

K = 1 K = minus1

lowast Oscillation

lowast dSlArrrArrBounce lowast dSlArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

lowast BB rArr dS (dS rArr BC) lowast BB rArr dS (dS rArr BC)

Λ gt 0 Γ11(plusmn) lowast Su Ss Γ1minus1(+) lowast Su

lowast BBrArrSu (Su rArr BC) lowast BBrArrSu (Su rArr BC)lowast Su rArr dS (dSrArr Su) lowast Su rArr dS (dSrArr Su)lowast Su lArrrArr Bounce

lowast Oscillation lowast M lArrrArrBounce

lowast BB rArrBC lowast BB rArrBC

Λ = 0 lowast BB rArr M (M rArr BC)

Γ01 Γ0minus1 lowast Su

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su rArr M (M rArr Su)

lowast Oscillation lowastOscillation

lowast BB rArrBC lowast BB rArrBC

Λ lt 0 Γminus11(minus) Γminus1minus1(plusmn) lowast Su Ss

lowast Ss lowast BB rArr Su (Su rArr BC)lowast Su lArrrArr Bounce

TABLE I Summary What type of spacetime is possible foreach Λ and each K Non-singular universes are shown by thecolored letters (an oscillating universe and dynamical space-times evolving in a finite scale range by red static universesby blue dynamical spacetimes evolving from or to an asymp-totically infinite scale by green) dS BB BC Su Ss and M

denote de Sitter space a big bang a big crunch an unstablestatic universe a stable static universe and Milne universerespectively

In our analysis we assume that the integration constantC from the projectability condition vanishes If C 6= 0one may find a different story In fact if gs = 0 and gr lt 0just as the case with the detailed balance condition wewill find the similar vacuum solutions to the present ones

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

10

because C and gr without gs-term play the similar rolesto those of gr and gs in the present model For examplewe obtain an oscillating universe for large C(gt 0) withgs = 0 gr lt 0 Λ = 0 and K = 1 This avoidance ofa singularity is however caused by the negative ldquoradi-ationrdquo density from the higher curvature terms Henceif one includes the conventional radiation then the ef-fective gr becomes positive as we will show below andas a result the universe will inevitably collapse to a big-crunch singularity Furthermore if radiation field evolvesas aminus6 in the UV limit[23] the inclusion of such radiationwill kill the possibility of singularity avoidance by ldquodarkrdquoradiation

As we have evaluated the oscillation period and am-plitude are expected to be the Planck scale or the scaleℓ defined by a cosmological constant Λ unless the cou-pling constants are unnaturally large Hence it cannotbe a cyclic universe which period is macroscopic such asthe age of the universe

In order to find more realistic universe we have toinclude some other components which we shall discusshere First of all one may claim inclusion of matter fluidWhen we include a dust fluid (P = 0) the conventionalradiation (P = ρ3) and stiff matter (P = ρ) we cantreat such a case just by replacing the constant gd grand gs with

gd = 8C + gdust

gr = 6(g3 + 3g2) + grad

gs = 12(9g5 + 3g6 + g7)K + gstiff (41)

where gdust grad and gstiff which come from real dustfluid radiation and stiff matter are positive constantsIn this case the present analysis is still valid If gradis large enough just as our universe a maximum scalarfactor amax of the the oscillating universe will becomelarge (see for example Eq (316)) and then it can be acyclic universe

If the equation of state is still given by P = wρ(w=constant) the analysis is straightforward When wehave other types of matter fields eg a scalar field witha potential the analysis will be more complicated Thephase space analysis may be appropriate for the case witha scalar field [63]

From our present analysis one may speculate the fol-lowing ldquorealisticrdquo scenario for the early stage of the uni-verse Suppose a closed universe is created from ldquonoth-ingrdquo initially in an oscillating phase (see Fig 9) [64 65]Such a universe may be very small and oscillating be-tween two radii (amin and amax) with a time scale ℓ Ifwe have a positive cosmological constant (Λ gt 0) thereexists a potential barrier as shown in Fig 9

After numbers of oscillations the universe may quan-tum mechanically tunnel to a bounce point aT Thenthe universe will expand to de Sitter phase because apositive cosmological constant finding the universe in a

FIG 9

macroscopic scale1 Furthermore one can refine this sce-nario if there exists a scalar field which is responsiblefor inflation instead of a cosmological constant Beforetunneling we may find the similar scenario to the aboveone After tunneling the potential of the scalar fieldwill behaves as a cosmological constant in a slow-rollingperiod We will find an exponential expansion of the uni-verse after tunneling However inflation will eventuallyend and the energy of the scalar field is converted to thatof conventional matter fluid via a reheating of the uni-verse We find a big bang universe Since the universeis closed but the scale factor has lower bound becauseof negative ldquostiff matterrdquo we will find a macroscopicallylarge cyclic universe after all To confirm such a scenariowe should analyze the dynamics of the universe with aninflaton field in detail The work is in progressWe also have another extension of the present FLRW

spacetime to anisotropic one It may be interesting andimportant not only to study the dynamics of Bianchispacetime [66 67] but also to analyze the stability of theFLRW universe against anisotropic perturbations[68]

Acknowledgments

We would like to thank Yuko Urakawa for valuablecomments and discussions This work was partially sup-ported by the Grant-in-Aid for Scientific Research Fundof the JSPS (No22540291) and for the Japan-UK Re-search Cooperative Program and by the Waseda Univer-sity Grants for Special Research Projects

Appendix A stability of a flat background and the

coupling constants

In this Appendix we discuss the conditions on the cou-pling constants by which gravitons are perturbatively sta-

1 After we have written up this paper we have found [60] in whicha cosmological transition scenario from a static (or an oscillating)universe to an inflationary stage was discussed They assumethat the equation of state changes in time which mechanism isnot specified

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

11

ble From the perturbation analysis around a flat back-ground we obtain the dispersion relation for the usualhelicity-2 polarizations of the graviton [17]

ω2TT(plusmn) = minusg1k

2 + g3k4

M2PL

plusmn g4k5

M3PL

+ g9k6

M4PL

(A1)

The stability both in the IR and UV regimes requires

g1 lt 0 g9 gt 0 (A2)

By a suitable rescaling of time we then set g1 = minus1As a result of the reduced symmetry (24) the longitu-

dinal degree of freedom of the graviton appears and itsstability is more subtle First of all the longitudinal gravi-ton is plagued with ghost instabilities for 13 lt λ lt 1 [1]The dispersion relation for the longitudinal mode turnsout to be [17]

(

3λminus 1

λminus 1

)

ω2L = g1k

2 + (8g2 + 3g3)k4

M2PL

+(minus8g8 + 3g9)k6

M4PL

(A3)

We see that the sound speed squared is negative in the IRif g1 lt 0 and λ gt 1 which implies that the longitudinalgraviton is unstable in the IR [36] However this factitself does not necessarily mean that the theory suffersfrom pathologies because whether or not an instabilityreally causes a trouble depends upon its time scale [27]Moreover there is an attempt to improve the behaviorof the longitudinal graviton by promoting N to an ~x-dependent function and adding terms constructed fromthe 3-vector partiNN in the Lagrangian [35]2 It can beshown that the non-projectable Horava gravity thus ex-tended appropriately does not plagued with instabilitiesof the longitudinal gravitons [35] In light of these sub-tleties we do not consider the stability of the longitudinalsector furthermore while we do require the stability forthe usual helicity-2 polarizations of the gravitonNote that the detailed balance condition satisfies g1 lt

0 and g9 gt 0

Appendix B quantum tunneling from an oscillating

universe

In the case of K = 1 and Λ gt 0 we have a bouncinguniverse as well as an oscillating universe These two so-lutions are separated by a finite potential wall as we seein Fig 9 Hence we expect quantum tunneling from anoscillating universe to an exponentially expanding uni-verse In this Appendix we shall evaluate the tunnelingprobability

2 Obviously in this case the Hamiltonian constraint is imposedlocally and the additional dust-like component does not appearin the Friedmann equation

First we consider the normalized Euclidean metric

ds2 = dτ2 + b2(τ )dΣ2K=1 (B1)

which satisfies the following equation

bprime2 minus 2U(b) = 0 (B2)

where the prime denotes the derivative with respect tothe Euclidean time τ and the potential U is written as

2U(b) =2

3λminus 2

1

b4

[

minus(b2 minus b2max)(b2 minus b2min)(b

2 minus b2T )]

(B3)

The variables with a tilde are normalized ones by use ofthe scale length ℓ =

radic

3Λ just as in the text The

bounce solution b(τ ) is obtained by integraton of Eq(B2) The Euclidean action is given by

SE = 3(3λminus 1)ℓ

int

dτd3xb

[

1

2bprime2 + U(b)

]

(B4)

Using Eq (B2) we find the action SE as

SE = 3(3λminus 1)ℓ2V3

int

dbb

radic

2U(b) (B5)

where V3 = 2π2 is the volume of a unit three sphereIntroducing u by

b2 = b2T (1minus k2u2) (B6)

where k2 = (b2T minus b2max)b2T (lt 1) We then find

SE =12π2ℓ2

κ2(b2T minus b2max)

2(b2T minus b2min)12

timesint 1

0

u2du

1minus k2u2

radic

(1minus u2)(1minusm2u2) (B7)

where m2 = (b2T minus b2max)(b2T minus b2min)(lt 1)

It can be easily evaluated in the limit of a static uni-

verse ie gs = g[11](minus)s (gr) Using bmax asymp bmin asymp bS we

find

SE =12π2ℓ2

κ2(b2T minus b2S)

52

times(

3minus 2k2

3k4minus 1minus k2

k5tanhminus1 k

)

(B8)

where k =radic

b2T minus b2SbT Since b2T = (1 + 2radic1minus gr)3

and b2T minus b2S =radic1minus gr we find

SE =4π2ℓ2

κ2(1minus gr)

14

times[

1minus (1 + 2radic1minus gr)

12(1 minusradic1minus gr)radic

3(1minus gr)14tanhminus1 k

]

(B9)

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

[10] M R Setare arXiv09090456 [hep-th] M R Setareand M Jamil JCAP 1002 010 (2010) [arXiv 10011251[hep-th]]

[11] Y Piao Phys Lett B 681 1 (2009) [arXiv09044117[hep-th]]

[12] X Gao arXiv09044187 [hep-th] X Gao Y WangR Brandenberger and A Riotto Phys Rev D 81083508 (2010) [arXiv09053821 [hep-th]]

[13] B Chen S Pi and J Tang JCAP 0908 007 (2009)[arXiv09052300 [hep-th]]

[14] R Cai B Hu and H Zhang Phys Rev D 80 041501(2009) [arXiv09050255 [hep-th]]

[15] K Yamamoto T Kobayashi and G Nakamura PhysRev D 80 063514 (2009) [arXiv09071549 [astro-phCO]]

[16] C Bogdanos and E N Saridakis Class Quant Grav27 075005 (2010) [arXiv09071636 [hep-th]]

[17] A Wang and R Maartens Phys Rev D 81 024009(2010) [arXiv09071748 [hep-th]]

[18] Y Lu and Y Piao arXiv09073982 [hep-th][19] T Kobayashi Y Urakawa and M Yamaguchi JCAP

0911 015 (2009) [arXiv09081005 [astro-phCO]]T Kobayashi Y Urakawa and M Yamaguchi JCAP1004 025 (2010) [arXiv10023101 [hep-th]]

[20] A Wang D Wands and R Maartens JCAP 1003 013(2010) [arXiv09095167 [hep-th]]

[21] X Gao Y Wang W Xue and R Brandenberger JCAP1002 020 (2010) [arXiv09113196 [hep-th]]

[22] J Gong S Koh and M Sasaki Phys Rev D 81 084053(2010) [arXiv10021429 [hep-th]]

[23] S Mukohyama K Nakayama F Takahashi andS Yokoyama Phys Lett B 679 6 (2009) [arXiv09050055 [hep-th]]

[24] M Park JCAP 1001 001 (2010) [arXiv09064275 [hep-th]]

[25] S Dutta and E N Saridakis JCAP 1001 013 (2010)[arXiv09111435 [hep-th]] S Dutta and E N SaridakisJCAP 1005 013 (2010) [arXiv10023373 [hep-th]]

[26] S Maeda S Mukohyama and T Shiromizu Phys RevD 80 123538 (2009) [arXiv09092149 [astro-phCO]]

[27] K Izumi and S Mukohyama Phys Rev D 81 044008(2010) [arXiv09111814 [hep-th]]

[28] J Greenwald A Papazoglou and A WangarXiv09120011 [hep-th]

[29] D Orlando and S Reffert Class Quant Grav 26155021(2009) [arXiv09050301 [hep-th]]

[30] C Charmousis G Niz A Padilla and P M Saffin JHEP0908 070 (2009) [arXiv09052579 [hep-th]]

[31] M Li and Y Pang JHEP 0908 015 (2009) [arXiv09052751 [hep-th]]

[32] G Calcagni Phys Rev D 81 044006 (2010)[arXiv09053740 [hep-th]]

[33] D Blas O Pujolas and S Sibiryakov JHEP 0910 029(2009) [arXiv09063046 [hep-th]]

[34] S Mukohyama JCAP 0909 005 (2009) [arXiv09065069 [hep-th]]

[35] D Blas O Pujolas and S Sibiryakov arXiv09093525[hep-th]

12

with

k2 =3radic1minus gr

1 + 2radic1minus gr

(B10)

The tunneling probability is given by P sim eminusSE We show the behavior of SE in Fig 10 We find

P sim exp

[

minus(20minus 40)times(

ℓ

ℓPL

)2]

sim exp

[

minus(60minus 120)times(

m4PL

ρvac

)]

(B11)

except for two limiting cases gr sim 1 in which SE van-ishes and gr sim 0 in which SE diverges In the for-mer case the potential barrier vanishes giving a hightunneling probability while in the latter case the po-tential barrier diverges giving zero tunneling probability

If the vacuum energy (or potential) just after tunnel-ing is the Planck scale the probability is evaluated asP sim eminus(60minus120) which is very small but finite

FIG 10

[1] P Horava Phys Rev D 79 084008 (2009) [arXiv09013775 [hep-th]]

[2] G Calcagni JHEP 0909 112 (2009) [arXiv09040829[hep-th]]

[3] E Kiritsis and G Kofinas Nucl Phys B 821 467 (2009)[arXiv09041334 [hep-th]]

[4] R Brandenberger Phys Rev D 80 043516 (2009)[arXiv09042835 [hep-th]] R H Brandenberger [arXiv10031745 [hep-th]]

[5] S Mukohyama Phys Rev D 80 064005 (2009) [arXiv09053563 [hep-th]]

[6] T Takahashi and J Soda Phys Rev Lett 102 231301(2009) [arXiv09040554 [hep-th]]

[7] S Mukohyama JCAP 0906 001 (2009) [arXiv09042190 [hep-th]]

[8] E N Saridakis Eur Phys J C 67 229 (2010) [arXiv09053532 [hep-th]] M Jamil and E N SaridakisarXiv10035637 [physicsgen-ph]

[9] C Appignani R Casadio and S ShankaranarayananJCAP 1004 006 (2010) [arXiv09073121 [hep-th]]

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