linear perturbations in eddington-inspired born...
TRANSCRIPT
Linear perturbations in Eddington-inspired
Born-Infeld gravity
Yu-Xiao Liu (4��)
Institute of Theoretical Physics, Lanzhou University
Based on: Ke Yang, Xiao-Long Du, YXL, arXiv:1307.2969.
ICTS, USTC Sep 5, 2013
ITP,LZU Linear perturbations of EiBI Gravity
Outline
1. Introduction to EiBI gravity
2. Linear Perturbations
3. The stability of the perturbations
4. Conclusion and discussion
ITP,LZU Linear perturbations of EiBI Gravity
1. Introduction to EiBI gravity
1915: General Relativity
It provides precise descriptions to a variety of phenomena in
our Universe for almost a century.
It also suffers various troublesome theoretical problems: dark
matter/energy, nonrenormalization, singularity...
The the Einstein-Hilbert action is
SEH[g] =
∫d4x√−g [R(g)− 2Λ] . (1)
ITP,LZU Linear perturbations of EiBI Gravity
1. Introduction to EiBI gravity
Modified Gravity
Scalar-tensor (Brans-Dicke) gravity
Einstein-Aether gravity
F(R) gravity and general higher-order theories,
Horava-Lifschitz gravity
Galileons
Ghost Condensates
Models of extra dimensions: KK, ADD, RS, DGP
Born-Infeld Gravity
Bimetric theories
· · ·ITP,LZU Linear perturbations of EiBI Gravity
1924, Eddington proposed a purely affine gravity.
Eddington gravity
SEdd(Γ) =1
8πG
2
κ
∫d4x√−|κRµν(Γ)|, (2)
⇓ (gµν = κRµν)
gµν;λ = 0, i.e., Γλµν =1
2gλρ(gρµ,ν + gρν,µ − gµν,ρ)
⇓ (Λ = 1κ)
Rµν = Λgµν (3)
Eddington’s theory is totally equivalent to the GR with Λ.
HOWEVER, it is incomplete because matter is not included.
ITP,LZU Linear perturbations of EiBI Gravity
Consider the Palatini action for gravity with Λ
SP[g,Γ] =
∫d4x
(√−g gµνRµν(Γ)− 2Λ
), (4)
Eliminating the connection using its own EoM gives
SEH[g] =
∫d4x√−g [R(g)− 2Λ] , (5)
If Λ 6= 0, eliminating the metric yields [Annals Phys. 162(1985)31]
SEdd[Γ] =2
Λ
∫d4x√−|Rµν(Γ)|. (6)
SP[g,Γ] is called the Parent action, while the Einstein-Hilbert
action and Eddington’s action are its daughters.
SEH[g] and SEdd[Γ] are said to be dual to each other, and in
many respects they are equivalent.
ITP,LZU Linear perturbations of EiBI Gravity
The vector Born-Infeld theory (1934 Born, Infeld)
SVBI =
∫d4x√−|gµν + Fµν |. (7)
The Born-Infeld gravity [Deser and Gibbons, CQG 15, L35 (1998)]
SBI[gµν ] =
∫d4x√−|gµν − l2Rµν +Xµν(R)|. (8)
Xµν(R) must be chosen such that the action is free of ghost.
For the vector BI theory, the EoM is of second order.
For the spin two theory this is not automatic and requires the
addition of Xµν(R).
ITP,LZU Linear perturbations of EiBI Gravity
Inspired by the Eddington and BI gravity theories, a new
theory was put forward by Banados and Ferreira [PRL 105 (2010)
011101].
Eddington Inspired Born-Infeld (EiBI) gravity
SEiBI[g,Γ,Φ] =2
κ
∫d4x
(√−|gµν + κRµν(Γ)| − λ
√−|gµν |
)+SM (g,Γ,Φ), (9)
where the dimensionless parameter λ must be nonvanishing.
ITP,LZU Linear perturbations of EiBI Gravity
By varying the action with respect to the metric simply gives√−|gµν + κRµν |
[(gµν + κRµν)−1
]αβ=√−|gµν |
(λgαβ − κTαβ
),
(10)
By introducing an auxiliary metric
qµν = gµν + κRµν , (11)
the variation to the connection simply gives us qµν;σ = 0, i.e., Γ is
just the Christoffel symbol of the auxiliary metric.
Then Eq. (10) is rewritten as√−|qµν | qαβ = λ
√−|gµν | gαβ − κ
√−|gµν | Tαβ, (12)
Eqs. (11) and (12) and matter field equations form a
complete set of equations of the theory.ITP,LZU Linear perturbations of EiBI Gravity
Some properties of the EiBI gravity:
When κR� g, SEiBI → SEdd.
When κR� g, the EiBI action reproduces the GR with Λeff in
lowest order approximation:
SEiBI ≈1
2κ
∫dnx
√−|gµν |
(R− 2Λeff +
κ
4RR− κ
2RµνR
νµ
)+SM [g,Γ,Φ], (13)
where Λeff ≡ (λ− 1)/κ.
In the nonrelativistic limit, the EiBI theory gives the modified
Poisson equation
∇2Φ = −1
2ρ− κ
4∇2ρ. (14)
So, it reproduces Einstein gravity precisely within the vacuum
but deviates from it in the presence of source.
ITP,LZU Linear perturbations of EiBI Gravity
Homogeneous and isotropic Universe
ds2 = −dt2 + a2(t)d~x · d~x (15)
With coupling to an ideal fluid Tµν = (p+ ρ)uµuν + pgµν , the
Friedmann equation is given by
H2 =2
3
G
F 2, (16)
F = 1− 3κ(ρT + pT )(1− ω − bκρT − κpT )
(1 + κρT )(1− κpT ),
G =1
κ[1 + 2U − 3
U
V],
U = (1 + κρT )−1/2(1− κpT )3/2,
V = a2(1 + κρT )1/2(1− κpT )1/2, (17)
with ρT = ρ+ Λκ and pT = p− Λ
κ .
ITP,LZU Linear perturbations of EiBI Gravity
Focus on the evolution of the scale factor at early times.
Assume radiation domination: ρT = ρ, pt = p = 13ρ.
Define ρ = κρ, the Friedmann equation becomes
3H2(ρ) =
[ρ− 1 +
√(1 + ρ)(3− ρ)3
3√
3
](1 + ρ)(3− ρ)2
κ(3 + ρ2)2.
For small ρ we recover the conventional Friedmann universe,
H2 ≈ ρ/3.
But at high density there is a stationary point H2 = 0
at ρ = 3 (ρ = 3/κ) for κ > 0, and
at ρ = −1 (ρ = −1/κ) for κ < 0.
ITP,LZU Linear perturbations of EiBI Gravity
The Hubble rate H2 as a function of energy density ρ
The new stationary points correspond to a maximum
density ρB (ρB = 3/κ for κ > 0 and ρB = −1/κ for κ < 0)
ITP,LZU Linear perturbations of EiBI Gravity
The scale factor a as a function of time t
The maximum density ρB corresponds to a minimum
length aB in cosmology.
Thus the universe may be entirely singularity free.
ITP,LZU Linear perturbations of EiBI Gravity
arXiv:0801.4103.
ITP,LZU Linear perturbations of EiBI Gravity
arXiv:1204.1691.
ITP,LZU Linear perturbations of EiBI Gravity
arXiv:1210.1521.
Motivation:
Do the linear scalar and vector perturbations stable in
Eddington regime?
So, we study the full linear perturbations of a
homogeneous and isotropic spacetime in the EiBI gravity.
ITP,LZU Linear perturbations of EiBI Gravity
2. Linear Perturbations of EiBI cosmology
2.1 The perturbed metrics
The background space-time metric is
ds2 = gµνdxµdxν
= −dt2 + a2(t)δijdxidxj . (18)
The background auxiliary metric is
ds′2 = qµνdxµdxν
= −X2(t)dt2 + a2(t)Y 2(t)δijdxidxj . (19)
ITP,LZU Linear perturbations of EiBI Gravity
The perturbed space-time metric is
ds2 = gµνdxµdxν = (gµν +Hµν)dxµdxν
= (−1 + h00(x))dt2 + a2(t)(δij + hij(x))dxidxj
+2h0i(x)dtdxi. (20)
The perturbed auxiliary metric is
ds′2 = qµνdxµdxν = (qµν + Πµν)dxµdxν
= X2(t)(−1 + γ00(x))dt2 + a2(t)Y 2(t)(δij + γij(x))dxidxj
+2Y 2(t)γ0i(x)dtdxi. (21)
h ≡ ηµνhµν , γ ≡ ηµνhµν . (22)
ITP,LZU Linear perturbations of EiBI Gravity
2.2 The background field equation
The 1st field equation reads√−|qµν |qµν = λ
√−|gµν |gµν − κ
√−|gµν |Tµν , (23)
Y 3
X= λ+ κρ, (24)
XY = λ− κP. (25)
The 2nd field equation
qµν = gµν + κRµν , (26)
X2 = 1 + 3κ[a
a+Y
Y− a
a
X
X+ 2
a
a
Y
Y− X
X
Y
Y], (27)
Y 2 = 1 + κY 2
X2(a
a+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y− X
X
Y
Y+Y
Y+ 2
Y 2
Y 2).
ITP,LZU Linear perturbations of EiBI Gravity
2.3 The energy-momentum tensor
Here the matter is the perfect fluid,
Tµν = Pgµν + (P + ρ)uµuν , (28)
and the static observer is uµ = (1, 0, 0, 0) with gµνuµuν = −1.
The 1st order perturbation of the energy-momentum tensor is
δT 00 = δρ+ ρh00, (29)
δT i0 = −a−2ρh0i + a−2(P + ρ)δui, (30)
δT ij = a−2δPδij − a−2Phij . (31)
ITP,LZU Linear perturbations of EiBI Gravity
2.4 The first perturbed field equation
Perturbing the first field equation√−|qµν |qµν = λ
√−|gµν |gµν − κ
√−|gµν |Tµν , (32)
we can get
γ00 = h00 +κδρ
2(λ+ κρ)+
3κδP
2(λ− κP ),
γ0i = h0i −(P + ρ)
(λ+ κρ)κδui,
γij = hij + [κδρ
2(λ+ κρ)− κδP
2(λ− κP )]δij . (33)
ITP,LZU Linear perturbations of EiBI Gravity
2.5 The perturbed Ricci tensor
Perturbing the Ricci tensor
Rµν = ∂λΓλµν − ∂νΓλλµ + ΓλλαΓαµν − ΓαµλΓλαν . (34)
we have
δR00 = −1
2
X2
Y 2a−2∂i∂iγ00 −
3
2(a
a+Y
Y)∂0γ00 −
1
2∂0∂0γii
−(a
a− 1
2
X
X+Y
Y)∂0γii + a−2∂0∂iγi0
+a−2(2Y
Y− X
X)∂iγi0,
δR0i =Y 2
X2[a
a+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+Y
Y+ 2
Y 2
Y 2− X
X
Y
Y]γi0
−[a
a+Y
Y]∂iγ00 +
1
2∂0∂jγij −
1
2∂0∂iγjj −
1
2a−2∂j∂jγi0
+1
2a−2∂i∂jγj0,
ITP,LZU Linear perturbations of EiBI Gravity
δRij =a2Y 2
X2[a
a+Y
Y+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+ 2
Y 2
Y 2− X
X
Y
Y]
×[γ00δij + γij ] +1
2
a2Y 2
X2[a
a+Y
Y]∂0γ00δij
+1
2
a2Y 2
X2[3a
a− X
X+ 3
Y
Y]∂0γij −
1
2
Y 2
X2[a
a− X
X+ 3
Y
Y]∂iγj0
−1
2
Y 2
X2[a
a− X
X+ 3
Y
Y]∂jγi0 +
1
2
a2Y 2
X2[a
a+Y
Y]∂0γkkδij
−Y2
X2[a
a+Y
Y]∂kγk0δij +
1
2∂i∂jγ00 +
1
2
a2Y 2
X2∂0∂0γij
+1
2∂k∂jγki +
1
2∂k∂iγkj −
1
2∂k∂kγij −
1
2∂i∂jγkk
−1
2
Y 2
X2∂0∂iγj0 −
1
2
Y 2
X2∂0∂jγi0. (35)
ITP,LZU Linear perturbations of EiBI Gravity
2.6 The perturbed field equation
With the scalar-vector-tensor decomposition of the
perturbed metric hµν and δui
h00 = −E, hi0 = ∂iF +Gi,
hij = Aδij + ∂i∂jB + ∂jCi + ∂iCj +Dij ,
δui = ∂iδu+ δUi,
where ∂iCi = ∂iGi = ∂iδUi = 0, ∂iDij = 0, and Dii = 0,
the 2nd perturbed field equation
δqµν = δgµν + κδRµν (36)
can be calculated as
ITP,LZU Linear perturbations of EiBI Gravity
00-component:
1
2
X2
Y 2a−2∇2E + 3(
a
a+Y
Y− a
a
X
X+ 2
a
a
Y
Y− X
X
Y
Y)E
+3
2(a
a+Y
Y)E − 1
2[3A+∇2B]− (
a
a− 1
2
X
X+Y
Y)[3A+∇2B]
+a−2∇2F + a−2(2Y
Y− X
X)∇2F − κ
4a−2X
2
Y 2
∇2δρ
λ+ κρ
−3κ
4a−2X
2
Y 2
∇2δP
λ− κP −3κ
4∂0∂0[
δρ
λ+ κρ] +
3κ
4∂0∂0[
δP
λ− κP ]
−3κ
4(3a
a+ 3
Y
Y− X
X)∂0[
δρ
λ+ κρ]− 3κ
4(a
a+Y
Y+X
X)∂0[
δP
λ− κP ]
−1
2[1 + 3κ(
a
a+Y
Y− a
a
X
X+ 2
a
a
Y
Y− X
X
Y
Y)][
δρ
λ+ κρ+
3δP
λ− κP ]
−κa−2∂0[P + ρ
λ+ κρ∇2δu]− κa−2(2
Y
Y− X
X)P + ρ
λ+ κρ∇2δu = 0. (37)
ITP,LZU Linear perturbations of EiBI Gravity
i0-component:
∂i
{[a
a+Y
Y]E − ∂iA−
κ
2∂0[
∂iδρ
λ+ κρ] +
κ
2∂0[
∂iδP
λ− κP]
−κ2
[a
a+Y
Y](
∂iδρ
λ+ κρ+
3∂iδP
λ− κP) +
P + ρ
λ+ κρ∂iδu
}+
1
2∇2Ci −
1
2a−2∇2Gi +
κ
2a−2 P + ρ
λ+ κρ∇2δUi +
P + ρ
λ+ κρδUi
= 0. (38)
ITP,LZU Linear perturbations of EiBI Gravity
ij-component:{− a2Y 2
X2[a
a+Y
Y+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+ 2
Y 2
Y 2− X
X
Y
Y]E
−1
2
a2Y 2
X2[a
a+Y
Y]E +
1
2
a2Y 2
X2A− 1
2∇2A
− Y2
X2[a
a+Y
Y]∇2F +
1
2
a2Y 2
X2[3a
a+ 3
Y
Y− X
X]A
+1
2
a2Y 2
X2[a
a+Y
Y][3A+∇2B] +
κ
4
a2Y 2
X2∂0∂0[
δρ
λ+ κρ]
−κ4
a2Y 2
X2∂0∂0[
δP
(λ− κP )]− κ
4[∇2δρ
λ+ κρ− ∇
2δP
λ− κP ]
+κ
2
a2Y 2
X2[a
a+Y
Y+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+ 2
Y 2
Y 2− X
X
Y
Y]
×[ δρ
λ+ κρ+
3δP
λ− κP ] +κ
4
a2Y 2
X2[7a
a+ 7
Y
Y− X
X]∂0[
δρ
λ+ κρ]
−1
2a2[
δρ
λ+ κρ− δP
λ− κP ] + κY 2
X2[a
a+Y
Y]P + ρ
λ+ κρ∇2δu
−κ4
a2Y 2
X2[3a
a+ 3
Y
Y− X
X]∂0[
δP
λ− κP ]
}δij
ITP,LZU Linear perturbations of EiBI Gravity
∂i∂j
{− 1
2E − 1
2A+
1
2
a2Y 2
X2B − Y 2
X2F + κ
δP
λ− κP
+1
2
a2Y 2
X2[3a
a+ 3
Y
Y− X
X]B − Y 2
X2[a
a− X
X+ 3
Y
Y]F
+κY 2
X2∂0[
P + ρ
λ+ κρδu] + κ
Y 2
X2[a
a− X
X+ 3
Y
Y]P + ρ
λ+ κρδu
}+1
2
a2Y 2
X2[3a
a+ 3
Y
Y− X
X][∂iCj + ∂jCi]
+1
2
a2Y 2
X2[∂iCj + ∂jCi]−
1
2
Y 2
X2[∂iGj + ∂jGi]
−1
2
Y 2
X2[a
a− X
X+ 3
Y
Y][∂iGj + ∂jGi] +
κ
2
Y 2
X2
P + ρ
λ+ κρ[∂iδUj + ∂jδUi]
+κ
2
Y 2
X2
P + ρ
λ+ κρ[∂iδUj + ∂jδUi]−
κ2
2
Y 2
X2
ρ(P + ρ)
(λ+ κρ)2[∂iδUj + ∂jδUi]
+κ
2
Y 2
X2[a
a− X
X+ 3
Y
Y]P + ρ
λ+ κρ[∂iδUj + ∂jδUi]
−1
2∇2Dij +
1
2
a2Y 2
X2Dij +
1
2
a2Y 2
X2[3a
a+ 3
Y
Y− X
X]Dij = 0.
ITP,LZU Linear perturbations of EiBI Gravity
A. Scalar modes:
The 00-component of the perturbed equation (36) gives
1
2
X2
Y 2a−2∇2E + 3(
a
a+Y
Y− a
a
X
X+ 2
a
a
Y
Y− X
X
Y
Y)E +
3
2(a
a+Y
Y)E
−1
2(3A+∇2B)− (
a
a− 1
2
X
X+Y
Y)(3A+∇2B) + a−2∇2F
+a−2(2Y
Y− X
X)∇2F − κ
4a−2X
2
Y 2
∇2δρ
λ+ κρ− 3κ
4a−2X
2
Y 2
∇2δP
λ− κP
−3κ
4∂0∂0
δρ
λ+ κρ− 3κ
4(3a
a+ 3
Y
Y− X
X)∂0
δρ
λ+ κρ+
3κ
4∂0∂0
δP
λ− κP
−1
2[1 + 3κ(
a
a+Y
Y− a
a
X
X+ 2
a
a
Y
Y− X
X
Y
Y)](
δρ
λ+ κρ+
3δP
λ− κP )
−3κ
4(a
a+Y
Y+X
X)∂0
δP
λ− κP − κa−2∂0[
P + ρ
λ+ κρ∇2δu]
−κa−2(2Y
Y− X
X)P + ρ
λ+ κρ∇2δu = 0. (39)
ITP,LZU Linear perturbations of EiBI Gravity
The part of ij-component of (36) proportional to δij gives
−a2Y 2
X2(a
a+Y
Y+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+ 2
Y 2
Y 2− X
X
Y
Y)E +
1
2
a2Y 2
X2A
−1
2
a2Y 2
X2(a
a+Y
Y)E − 1
2∇2A− Y 2
X2(a
a+Y
Y)∇2F
+1
2
a2Y 2
X2(3a
a+ 3
Y
Y− X
X)A+
1
2
a2Y 2
X2(a
a+Y
Y)(3A+∇2B)
+κ
4
a2Y 2
X2∂0∂0
δρ
λ+ κρ− κ
4
a2Y 2
X2∂0∂0
δP
λ− κP
−κ4(∇2δρ
λ+ κρ− ∇
2δP
λ− κP )− 1
2a2(
δρ
λ+ κρ− δP
λ− κP )
+κ
2
a2Y 2
X2(a
a+Y
Y+ 2
a2
a2− a
a
X
X+ 6
a
a
Y
Y+ 2
Y 2
Y 2− X
X
Y
Y)
×( δρ
λ+ κρ+
3δP
λ− κP ) +κ
4
a2Y 2
X2(7a
a+ 7
Y
Y− X
X)∂0
δρ
λ+ κρ
−κ4
a2Y 2
X2(3a
a+ 3
Y
Y− X
X)∂0
δP
λ− κP + κY 2
X2(a
a+Y
Y)P + ρ
λ+ κρ∇2δu
= 0. (40)
ITP,LZU Linear perturbations of EiBI Gravity
The part of i0-component of (36) with the form ∂iS (where
S is any scalar) gives
(a
a+Y
Y)E − A− κ
2∂0
δρ
λ+ κρ+κ
2∂0
δP
λ− κP+P + ρ
λ+ κρδu
−κ2
(a
a+Y
Y)(
δρ
λ+ κρ+
3δP
λ− κP) = 0. (41)
The part of ij-component of (36) of the form ∂i∂jS gives
−1
2E − 1
2A+
1
2
a2Y 2
X2B +
1
2
a2Y 2
X2(3a
a+ 3
Y
Y− X
X)B − Y 2
X2F
−Y2
X2(a
a− X
X+ 3
Y
Y)F + κ
δP
λ− κP+ κ
Y 2
X2∂0(
P + ρ
λ+ κρδu)
+κY 2
X2(a
a− X
X+ 3
Y
Y)P + ρ
λ+ κρδu = 0. (42)
ITP,LZU Linear perturbations of EiBI Gravity
The 0-component of the perturbed conservation equation
gives
δρ+ 3a
a(δρ+ δP ) +
1
2(P + ρ)(3A+∇2B)
−a−2(P + ρ)∇2(F − δu) = 0. (43)
The part of i-component of the perturbed conservation equation of
the form ∂iS (where S is any scalar) gives
δP +1
2(P + ρ)E + (P + ρ)δu+ (P + ρ)δu+ 3
a
a(P + ρ)δu = 0.(44)
ITP,LZU Linear perturbations of EiBI Gravity
B. Vector modes:
The part of i0-component of the perturbed equation (36)
with the form Vi (where Vi is any vector satisfying ∂iVi = 0) gives
1
2∇2Ci −
1
2a−2∇2Gi +
κ
2a−2 P + ρ
λ+ κρ∇2δUi +
P + ρ
λ+ κρδUi = 0.(45)
The part of i-component of the perturbed conservation equation
with the form Vi (where Vi is any vector satisfying ∂iVi = 0) gives
(P + ρ)δUi + (P + ρ)δUi + 3a
a(P + ρ)δUi = 0. (46)
ITP,LZU Linear perturbations of EiBI Gravity
The part of ij-component of (36) with the form ∂iVj (where
Vj is any vector satisfying ∂jVj = 0) gives
1
2
a2Y 2
X2Cj +
1
2
a2Y 2
X2(3a
a+ 3
Y
Y− X
X)Cj −
1
2
Y 2
X2Gj
−1
2
Y 2
X2(a
a− X
X+ 3
Y
Y)Gj +
κ
2
Y 2
X2∂0(
P + ρ
λ+ κρδUj)
+κ
2
Y 2
X2(a
a− X
X+ 3
Y
Y)P + ρ
λ+ κρδUj = 0. (47)
ITP,LZU Linear perturbations of EiBI Gravity
B. Tensor modes:
The part of ij-component of (36) with the form of a
transverse-traceless tensor is
−∇2Dij +a2Y 2
X2Dij +
a2Y 2
X2(3a
a+ 3
Y
Y− X
X)Dij = 0. (48)
ITP,LZU Linear perturbations of EiBI Gravity
3. The stability of the perturbations
The perturbed equations involve 7 scalars modes
E,F,A,B, δρ, δP, δu, 3 transverse vectors modes Ci, Gi, δUi,
and 1 transverse-traceless tensor mode Dij .
For scalar perturbed modes we work in the Newtonian gauge,
i.e., we set B = F = 0 in the linear perturbed equations.
For vector mode, we fix the gauge freedom to eliminate Ci.
Finally, with the state equation P = ωρ, then δP = ωδρ, all
the remaining perturbed modes are solvable.
The dominant component in Eddington regime is the highly
relativistic ideal gas (ω = 1/3, δP = 13δρ) and the
cosmological constant can be neglected (λ = 1).
ITP,LZU Linear perturbations of EiBI Gravity
3.1 The case κ > 0
For κ > 0, the approximate background solution near the
maximum density (t→ −∞) is given by
[Escamilla-Rivera, Banados, and Ferreira, PRD 85(2012)087302],
[Scargill, Banados, and Ferreira, PRD 86 (2012) 103533]
a = aB[1 + e
√83κ
(t−t0)],
X = U12 = 2e
34
√83κ
(t−t0),
Y = V12 = 2e
14
√83κ
(t−t0). (49)
ITP,LZU Linear perturbations of EiBI Gravity
3.1 The case κ > 0
A. Scalar perturbations
For scalar perturbations, the solution (t→ −∞) is given by
A ' 2c1(xi)e74b(t−t0), (50)
E ' −2c1(xi)e114b(t−t0), (51)
δρ ' −12
κc1(xi)e
74b(t−t0), (52)
δu ' 4
7bc1(xi)e
74b(t−t0) + c2(xi). (53)
Therefore, the scalar perturbations vanish when the universe
approaches the maximum density and we conclude that the scalar
perturbations are stable in the Eddington regime.
ITP,LZU Linear perturbations of EiBI Gravity
3.1 The case κ > 0
B. Transverse vector modes
For transverse vector modes, from Eqs. (46) and (47), we
have
Gi ' c3(xi), (54)
δUi ' c4(xi). (55)
The vector perturbations are also stable in the Eddington regime.
ITP,LZU Linear perturbations of EiBI Gravity
3.1 The case κ > 0
C. Transverse-traceless tensor mode
The transverse-traceless tensor mode is given by Eq. (48),
with the approximate solution (49), the dominant part simply gives
Dij = 0. (56)
So it gives that
Dij ∝ m(x)t+ n(x). (57)
When the universe approaches the maximum density (t→ −∞),
the tensor perturbation is divergent, it causes an instability in the
Eddington regime.
ITP,LZU Linear perturbations of EiBI Gravity
3.2 The case κ < 0
For κ < 0, the approximate background solution near the
maximum density (t→ 0) is given by
[Escamilla-Rivera, Banados, and Ferreira, PRD 85(2012)087302],
[Scargill, Banados, and Ferreira, PRD 86 (2012) 103533]
a = aB
(1− 2
3κ|t|2), (58)
X =2√3
(−κ/2)1/4 |t|−12 , (59)
Y =2√3
(−2/κ)1/4 |t|12 . (60)
ITP,LZU Linear perturbations of EiBI Gravity
3.2 The case κ < 0
The perturbations are approximately given by
A ' κ
2C1(xi) |t|
32 , E ' κ2
16C1(xi)|t|−
12 , (61)
δρ ' C1(xi) |t|32 , δu ' −κ
2
16C1(xi) |t|
12 , (62)
δU ' C2(xi), Gi ' C3(xi)|t|−2 +2
3C2(xi), (63)
Dij ' C4(xi)|t|−1 + C5(xi). (64)
Therefore, scalar, vector and tensor modes will all cause
instabilities in the Eddington regime in this case.
ITP,LZU Linear perturbations of EiBI Gravity
Conclusion and discussion
We studied the full linear perturbations in the radiation era of
a homogeneous and isotropic spacetime in EiBI theory.
For κ > 0, the scalar and transverse vector modes are stable,
while the tensor mode is unstable in Eddington regime.
For κ < 0, all the scalar, vector and tensor modes cause
instabilities in Eddington regime.
It may necessary to consider the nonlinear perturbations.
ITP,LZU Linear perturbations of EiBI Gravity
Conclusion and discussion
In Einstein theory, in the radiation era, ( aa)2 = ρ3 , a = a0
√t.
The linear perturbed modes are
E = −A ' d1t− 3
2 + d2, (65)
δu ' d1t− 1
2 − d2
2t, (66)
δρ ' 3
2d1t− 7
2 − 3
4d2t−2, (67)
Gi ' d3t− 1
2 , δUi ' d4t12 , (68)
Dij ' d5t− 1
2 + d6. (69)
All modes are all unstable in early universe.
ITP,LZU Linear perturbations of EiBI Gravity
Conclusion and discussion
Thus the EiBI cosmology with κ > 0 presents as an interesting
theory with stable scalar and vector modes.
This was also demonstrated in the following references:
Pani, Cardoso, Delsate, Compact stars in eddington inspired gravity,
PRL 107 (2011) 031101;
P. Avelino and R. Ferreira, Bouncing eddington-inspired born-infeld
cosmologies: an alternative to inflation?, PRD 86 (2012) 041501;
P. Pani, T. Delsate, and V. Cardoso, Eddington-inspired born-infeld
gravity. phenomenology of non-linear gravity-matter coupling, PRD
85 (2012) 084020,
where the EiBI theory with positive κ shows more well
properties than negative one.
ITP,LZU Linear perturbations of EiBI Gravity
Conclusion and discussion
For a 5D braneworld model with
SEiBI =1
κ
∫d5x
(√−|gPQ + κRPQ(Γ)| − λ
√−|gPQ|
)−∫d5x√−|gPQ|
(1
2gMN∂Mφ∂Nφ+ V (φ)
),
ds2 = a2(y)ηµνdxµdxν + dy2,
We found in [PRD 85(2012)124053] a brane solution for κ > 0
and the TT tensor perturbation is stable.
ITP,LZU Linear perturbations of EiBI Gravity
Thank you !
ITP,LZU Linear perturbations of EiBI Gravity