i scalar-tensor and ii multiscalar-tensor cosmology near ... · the limit of general relativity...
TRANSCRIPT
Thirteenth Marcel Grossmann Meeting - MG13Stockholm, 6 July 2012
(I)Scalar-tensor and (II)Multiscalar-tensorcosmology near the general relativity limit
Laur JarvUniversity of Tartu, Estonia
(I) LJ, Piret Kuusk, Margus SaalPhys. Rev. D81: 104007 (2010), Phys. Lett. B694: 1-5 (2010),
Phys. Rev. D85 064013 (2012)
(II) LJ, Piret Kuusk, Erik Randla forthcoming
(I) Scalar-tensor gravity (STG)
One scalar field Ψ non-minimally coupled to gravity, Brans-Dicke likeparametrization, Jordan frame
S =1
2κ2
∫d4x√−g[
ΨR(gµν)− ω(Ψ)
Ψ∇ρΨ∇ρΨ− 2κ2V (Ψ)
]+Sm(gµν , χmat)
I Family of theories, each pair ω(Ψ) and V (Ψ) specifies a theory.
I Variable gravitational “constant” set by the dynamical scalar field,
8πG = κ2
Ψ , assume 0 < Ψ <∞.
I Assume positive energy density: 2ω(Ψ) + 3 ≥ 0, V (Ψ) ≥ 0.
Paradigmatic example of a modified gravity theory. E.g. comes fromhigher dimensions, braneworlds, effective field theory approach to darkenergy; several proposed modifications to Einstein’s general relativity canbe cast in the form of STG, or contain STG as a subsector.
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 2/16
The limit of general relativity
Observations point towards a specific corner of the solutions space:
I CMB: |Grec−Gnow|Gnow
. 0.05 , hence Ψ 1.
I CMB: 12ω(Ψrec)+3 . 7× 10−2
.
I Solar System PPN: 12ω(Ψnow)+3 . 7× 10−4
.
I Solar System PPN:| dωdΨ |
(2ω(Ψnow)+3)2(2ω(Ψnow)+4) . 10−4
.“The limit of general relativity”:
1
2ω(Ψ?) + 3= 0 , Ψ? = 0
Assume
d
dΨ
(1
2ω(Ψ) + 3
)Ψ?
6= 0 ,1
2ω(Ψ) + 3is differentiable at Ψ?
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 3/16
Outline
We study flat (k = 0) FLRW STG cosmology with arbitrary ω(Ψ) andV (Ψ) in the potential dominated and dust matter dominated regimes..
I Derive the approximate equations that govern the dynamics near thelimit of general relativity. (Nonlinear, nonautonomous system!).
I Find the general analytic form of solutions for these equations, i.e.
Ψ(t), H(t), can also compute weff (t), GG , etc. (Complete
classification!).
I Argue that the full and approximate phase spaces are in qualitativeagreement. (Can trust the results!).
I Determine the conditions on a STG for its solutions to dynamicallyconverge to the GR limit. (Select viable theories!)
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 4/16
Scalar-tensor cosmology
Flat FLRW, barotropic matter fluid p = wρ
H2 = −H Ψ
Ψ+
1
6
Ψ2
Ψ2ω(Ψ) +
κ2
3
ρ
Ψ+κ2
3
V (Ψ)
Ψ,
2H + 3H2 = −2HΨ
Ψ− 1
2
Ψ2
Ψ2ω(Ψ)− Ψ
Ψ− κ2
Ψwρ+
κ2
ΨV (Ψ) ,
Ψ = −3HΨ− 1
2ω(Ψ) + 3
dω(Ψ)
dΨΨ2 +
κ2
2ω(Ψ) + 3(1− 3w) ρ
+2κ2
2ω(Ψ) + 3
[2V (Ψ)−Ψ
dV (Ψ)
dΨ
],
ρ = −3H (w + 1) ρ
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 5/16
Remark: STG ja GR cosmology
In the general realtivity limit get the usual GR Friedmann equations witha cosmological constant (V (Ψ) = Λ).
H2 = −H Ψ
Ψ+
1
6
Ψ2
Ψ2ω(Ψ) +
κ2
3
ρ
Ψ+κ2
3
V (Ψ)
Ψ,
2H + 3H2 = −2HΨ
Ψ− 1
2
Ψ2
Ψ2ω(Ψ)− Ψ
Ψ− κ2
Ψwρ+
κ2
ΨV (Ψ) ,
Ψ = −3HΨ− 1
2ω(Ψ) + 3
dω(Ψ)
dΨΨ2 +
κ2
2ω(Ψ) + 3(1− 3w) ρ
+2κ2
2ω(Ψ) + 3
[2V (Ψ)−Ψ
dV (Ψ)
dΨ
],
ρ = −3H (w + 1) ρ
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 6/16
Approximation near the GR limit (ρ = 0 case)
Consider small deviations (x ∼ x)
Ψ(t) = Ψ? + x(t), Ψ(t) = x(t) .
Expand in series, keeping the leading terms, obtain an approximateequation
x = −C1 x + C2 x +x2
2x, (1)
where
1
2ω(Ψ?) + 3≡ 0 , A? ≡
d
dΨ
(1
2ω(Ψ) + 3
) ∣∣∣Ψ?
,
C1 ≡ ±
√3κ2V (Ψ?)
Ψ?, C2 ≡ 2κ2A?
(2V (Ψ)− dV (Ψ)
dΨΨ
) ∣∣∣Ψ?
encode the behavior of the functions ω and V near this point.
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 7/16
Solutions to the approximate equation (ρ = 0 case)
The solutions of (1) in cosmological time fall into three classes, dependingon C ≡ C 2
1 + 2C2: exponential, linear-exponential, or oscillating,
Ψ(t) = Ψ?±
e−C1t
[M1e
12 t√
C −M2e− 1
2 t√
C]2
, if C > 0 ,
e−C1t [M1t −M2]2, if C = 0 ,
e−C1t[N1 sin( 1
2 t√|C |)− N2 cos( 1
2 t√|C |)
]2
, if C < 0 .
Here M1,M2,N1,N2 are constants of integration (determined by initialconditions)..Also
H(t) =C1
3± . . . .
E.g. oscillating solutions oscillate across the “phantom divide” line(weff = −1).
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 8/16
Classification of phase portraits (Ψ, Ψ)
C > 0 C > 0 C = 0 C < 0
Classification of phase portraits (Ψ, Ψ)
C > 0 C > 0 C = 0 C < 0
Approximation near the GR limit (V = 0 case)
Again consider small deviations x(t), h(t),
Ψ(t) = Ψ? + x(t) , H(t) = H?(t) + h(t) .
where H?(t) = 23(t−ts ) is the Hubble parameter corresponding to Ψ?.
.Expand in series, keeping the leading terms, obtain approximate equations
x =x2
2x− 3H?x + 3A?Ψ?H
2?x , (2)
h + 3H?h = − 1
4Ψ?
(1 +
1
2A?Ψ?
)x2
x+
1
2Ψ?H?x −
3
2A?H
2?x (3)
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 11/16
Solutions to the approximate equation (V = 0 case)
The solutions fall again into three classes, depending on D ≡ 1 + 83A?Ψ?:
polynomial, logarithmic, or oscillating,
Ψ(t) = Ψ?±
1t
(M1t
√D
2 −M2t−
√D
2
)2
, if D > 0 ,1t (M1 ln t −M2)2
, if D = 0 ,
1t
[N1 sin
(√|D|2 ln t
)− N2 cos
(√|D|2 ln t
)]2
, if D < 0 ,
where M1,M2,N1,N2 are constants of integration (determined by initialconditions). Also
H(t) =2
3t
[1± 1
t(. . .)
]
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 12/16
Conditions for GR to be an attractor
Observational constraints are naturally satisfied, if the GR limit is anattractor, i.e. solutions dynamically converge towards it.
I GR limit exists, if ∃Ψ?, such that
1
2ω(Ψ?) + 3= 0 .
I Attractor in the dust matter dominated era
d
dΨ
(1
2ω(Ψ) + 3
) ∣∣∣∣∣Ψ?
Ψ? < 0 .
I Attractor in the potential dominated era
V (Ψ?) > 0 ,
[Ψ
2V (Ψ)
dV (Ψ)
dΨ
]Ψ?
< 1 .
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 13/16
(II) Multiscalar-tensor gravity (MSTG)
N non-minimally coupled scalar fields Φa, Jordan frame
S =1
2κ2
∫d4x√−g(FR −Zabg
µν∂µΦa∂νΦb − 2κ2U)+S(gµν , χmatter) ,
where F = F(Φ1,Φ2, . . . ,ΦN ), Zab = Zab(Φ1,Φ2, . . . ,ΦN ),U = U(Φ1,Φ2, . . . ,ΦN ) are arbitrary functions..Can use N redefinitions of the scalar fields to cast the theory in the formwhere only one of the scalar fields, Ψ, is non-minimally coupled, while theothers, φi , are minimally coupled to gravity,
S =1
2κ2
∫d4x√−g(
ΨR − Zij∂ρφi∂ρφj − ω
Ψ∂ρΨ∂ρΨ− 2κ2U
)+Sm(gµν , χm) ,
however F = F (φ1, φ2, . . . , φN−1,Ψ), Zab = Zab(φ1, φ2, . . . , φN−1,Ψ),U = U(φ1, φ2, . . . , φN−1,Ψ)).
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 14/16
Two fields case (and ρ = 0)
Now expand Ψ(t) = Ψ? + x(t) , φ(t) = φ? + x(t) around
1
2ω(Ψ?, φ?) + 3= 0 ,
1
Z (Ψ?, φ?)= 0 .
Derive approximate equations . . . hard to solve generally . . ..For instance, if
∂
∂φ
(1
2ω + 3
) ∣∣∣Ψ?,φ?
=∂
∂φ
(1
Z
) ∣∣∣Ψ?,φ?
=∂
∂φU∣∣Ψ?,φ?
= 0
Ψ(t) = Ψ?±(∫
emt−aΩ(t)dt + k1
)2k2
4e(m+C1)t−aΩ(t)
t→∞= Ψ?±D eC1t
(M1e
Ct −M2e−Ct)2
Ω(t) =
∫Jn+1(ξ)
eC1t Jn(ξ)dt , ξ =
2D1k3
2C1eC1t, m =
√C 2
1 + 2C2 , n =m
2C1, a =
√2D1k3
C1,C2,C ,D1,D constants that specify MSTG, k1, k2, k3,M1,M2 integration constants, Jn Bessel
function.
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 15/16
Summary
(I) We studied flat FLRW STG cosmology with arbitrary ω(Ψ) and V (Ψ)in the potential dominated and dust matter dominated regimes..
I Determined the conditions on a STG for its solutions to dynamicallyconverge to the GR limit.
I Can use to select viable theories.
.
I Found the general analytic form of solutions near the GR limit, i.e.
Ψ(t), H(t), can also compute weff (t), GG , etc. Complete
classification.
I Can use to compare with actual expansion history (cosmography,statefinder diagnostic, etc.),
I Can use as the background for the growth of cosmologicalperturbations (parameterized post-Friedmannian formalism, etc).
.(II) We also studied MSTG. The same methods can be applied here,analytic results are harder to obtain, but still possible in some cases.
Laur Jarv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 16/16