i scalar-tensor and ii multiscalar-tensor cosmology near ... · the limit of general relativity...

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?


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!)


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


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


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.


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).


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)


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(. . .)

]


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 .


(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)).


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.


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.


i scalar-tensor and ii multiscalar-tensor cosmology near ... · the limit of general relativity...

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