Matter Bounce Scenario in F(T) gravityffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/HARO_Jaume.pdf · Jaume Haro and Jaume Amoros´ Matter Bounce Scenario in F(T)gravity. F(T)gravity in

F (T ) gravity in flat FLRW geometryMatter Bounce Scenario

Perturbations in F (T ) gravityMBS for different potentials: numeric analysis

Matter Bounce Scenario in F (T ) gravity

Jaume Haro and Jaume Amoros

Departament de Matematica Aplicada IUniversitat Politecnica de Catalunya

Frontiers in Fundamental PhysicsMarseille, 2014

Jaume Haro and Jaume Amoros Matter Bounce Scenario in F (T ) gravity



Introduction

F (T ) gravity in flat FLRW geometryWeitzenbock space-timeFriedmann equation in F (T ) gravity (flat FLRW geometry)Relation with Loop Quantum Cosmology (flat FLRW geometry)

Matter Bounce Scenario (MBS)MBS as an alternative to inflationThe simplest model: dynamicsProperties of the simplest model

Perturbations in F (T ) gravityDynamical equations for perturbationsPower spectrum and tensor/scalar ratio in MBSNumerical results

MBS for different potentials: numeric analysisMatching with a power law or plateau potentialMatching with a quintessence potencial




Introduction








Introduction








Introduction








Introduction








F (T ) gravity in flat FLRW geometryWeitzenbock space-timeFriedmann equation in F (T ) gravity (flat FLRWgeometry)Relation with Loop Quantum Cosmology (flatFLRW geometry)




Weitzenbock space-time

Teleparallelism is based in Weitzenbock space-time.Global system of four orthonormal vector fields ei in thetangent vector bundle.Covariant derivative ∇ that defines absolute parallelism withrespect the global basis ei, that is, ∇ei = 0.

Properties of Weitzenbock space-time.

The connection is metric, i.e., it satisfies ∇g = 0.Is curvature-free (Riemann tensor vanishes) but has torsion!!!.The main invariant is the scalar torsion, namely T .For a flat FLRW geometry is given by T = −6H2.




























































Friedmann eq. in F (T ) gravity (flat FLRW geometry)

Lagrangian: LT = V(F (T ) + LM ).V = a3 is the volume.LM is the matter Lagrangian density.

Hamiltonian: HT =(

2T dF (T )dT − F (T ) + ρ

)V.

ρ is the energy density.

Modified Friedmann equation: The Hamiltonian constrainHT = 0 leads to the equation (a curve in the plane (H, ρ))

ρ = −2dF (T )

dTT + F (T ) ≡ G(T ).

Inverse problem: Given a curve of the form ρ = G(T ) thecorresponding F (T ) theory is

F (T ) = −√−T2

∫G(T )

T√−T

dT .






Hamiltonian: HT =(

2T dF (T )dT − F (T ) + ρ

)V.



ρ = −2dF (T )

dTT + F (T ) ≡ G(T ).


F (T ) = −√−T2

∫G(T )

T√−T

dT .






Hamiltonian: HT =(

2T dF (T )dT − F (T ) + ρ

)V.



ρ = −2dF (T )

dTT + F (T ) ≡ G(T ).


F (T ) = −√−T2

∫G(T )

T√−T

dT .






Hamiltonian: HT =(

2T dF (T )dT − F (T ) + ρ

)V.



ρ = −2dF (T )

dTT + F (T ) ≡ G(T ).


F (T ) = −√−T2

∫G(T )

T√−T

dT .






Hamiltonian: HT =(

2T dF (T )dT − F (T ) + ρ

)V.



ρ = −2dF (T )

dTT + F (T ) ≡ G(T ).


F (T ) = −√−T2

∫G(T )

T√−T

dT .




Relation with LQC (flat FLRW geometry)

Curve (Ellipse): Friedmann equation in LQC

H2 =ρ

3

(1− ρ

ρc

), ellipse in the plane (H, ρ).

Splitting in two branches:ρ = G−(T ) (branch with H < 0).ρ = G+(T ) (branch with H > 0).

G±(T ) = ρc2

(1±

√1 + 2T

ρc

).

We obtain the model [Amoros et al., PRD 87, [arXiv:1108.0893]]

F±(T ) = ±√−T ρc

2arcsin

(√−2Tρc

)+G±(T ).






H2 =ρ

3

(1− ρ

ρc



G±(T ) = ρc2

(1±

√1 + 2T

ρc

).


F±(T ) = ±√−T ρc

2arcsin

(√−2Tρc

)+G±(T ).






H2 =ρ

3

(1− ρ

ρc



G±(T ) = ρc2

(1±

√1 + 2T

ρc

).


F±(T ) = ±√−T ρc

2arcsin

(√−2Tρc

)+G±(T ).






H2 =ρ

3

(1− ρ

ρc



G±(T ) = ρc2

(1±

√1 + 2T

ρc

).


F±(T ) = ±√−T ρc

2arcsin

(√−2Tρc

)+G±(T ).




The Matter Bounce Scenario (MBS)MBS as an alternative to inflationThe simplest model: dynamicsProperties of the simplest model




MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description

It depicts, at early times, a matter dominated Universe in thecontracting phase, after the bounce it enters in the expanding onewhere it matches with the hot Friedmann Universe.

Horizon problem solved: All the parts of the Universe are incausal contact at bouncing time.

Flatness problem improved: The spatial curvature decreasesin the contracting phase at the same rate as it increases in theexpanding one.

Power spectrum: The power spectrum of cosmologicalperturbations is scale invariant.








































The simplest model: dynamics

Matter dominated Universe: Dynamics

a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field

Potential of the simplest model: V (ϕ) = 2ρce−√

3ϕ

(1+e−√

3ϕ)2.

Dynamical equation: ¨ϕ+ 3H± ˙ϕ+ ∂V (ϕ)∂ϕ = 0.

Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.






a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field


3ϕ

(1+e−√

3ϕ)2.


Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.






a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field


3ϕ

(1+e−√

3ϕ)2.


Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.






a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field


3ϕ

(1+e−√

3ϕ)2.


Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.






a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field


3ϕ

(1+e−√

3ϕ)2.


Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.






a(t) =

(3

4ρct

2 + 1

)1/3

.

Matter scalar field


3ϕ

(1+e−√

3ϕ)2.


Analytic solution:

ϕ(t) =2√3

ln

(√3

4ρct+

√3

4ρct2 + 1

)⇐⇒ a(t) =

(3

4ρct

2 + 1

)1/3

.




Phase Portrait of the model V (ϕ) = 2ρce−√

3ϕ

(1+e−√

3ϕ)2

The Universe bounces when it reaches black curves defined by ρ = ρc.

The point (0, 0) is a saddle point, red (resp. green) curves are theinvariant curves in the contracting (resp. expanding) phase.

The blue curve corresponds to an orbit.

Before (resp. after) the bounce the blue curve do not cut the red (resp.green) curves.





3ϕ

(1+e−√

3ϕ)2









3ϕ

(1+e−√

3ϕ)2









3ϕ

(1+e−√

3ϕ)2









3ϕ

(1+e−√

3ϕ)2








Properties of the simplest model

The analytic solution is a repeler (attractor) at early (late) times.At early times, all the orbits depict a matter-dominated Universe,in the contracting phase.After the bounce it enters in the expanding phase and finishbeing, once again, matter dominated.

Potencials of the form V(ϕ) ∼ ρce−√

3|ϕ| when |ϕ| → ∞ has thisproperty.

This dynamics doen’t explain the current acceleration of the universe

Introducing a cosmological constant.

Matching V(ϕ) ∼ ρce−√

3|ϕ| with a quintessence potential. Forexample with V(ϕ) = ρc

ϕ40

ϕ4+ϕ40

or V(ϕ) ∼ ρce−√

3(1+ω)|ϕ| whereω = −1.10± 0.14 (WMAP observational data).












ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40














ϕ40

ϕ4+ϕ40










Dynamical equations for perturbations

Longitudinal gauge: ds2 = (1 + 2Φ)dt2 − a2(1− 2Φ)dx2.

Matter Lagrangian density LM = 12 ϕ

2 − |∇V (ϕ)|2 − V (ϕ).

Dynamical equations ζ ′′S(T ) − c2s∇2ζS(T ) +

Z′S(T )

ZS(T )ζ ′S(T ) = 0,

[Cai et al., Class.Quant.Grav. 28, [arXiv:1104.4349]] where

c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH

, with Ω = 1− 2ρρc.

ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .

Comparison with Holonomy Corrected LQC[Cailleteau et al., PRD 86, [arXiv:1206.6736]]

c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.







2 − |∇V (ϕ)|2 − V (ϕ).


Z′S(T )

ZS(T )ζ ′S(T ) = 0,


c2s = |Ω|arcsin

(2√

3ρcH)

2√

3ρcH


ZS = a2|Ω| ˙ϕ2

c2sH2 and ZT =

a2c2s|Ω| .


c2s = Ω, ZS =a2 ˙ϕ2

H2and ZT =

a2

Ω.




Power spectrum and tensor/scalar ratio in MBS

Bunch-Davies vacuum: ζT (η) =√

3ζS(η) = e−ikη√2ka

(1− i

kη

)when η → −∞

Long wavelength approximation:

ζS(T )(η) = AS(T )(k) +BS(T )(k)

∫ η

−∞

dη

ZS(T )(η).

Matching: For modes well outside the Hubble radius k2η2 1

AS(k) =AT (k)√

3= −

√8

3

k3/2

ρc, BS(k) =

√3BT (k) = i

√3

8

ρc2k3/2

.

Power spectrum:

PS(T )(k) =k3

2π2|BS(T )(k)|2

∣∣∣∣∫ ∞−∞

dt

a(t)ZS(T )(t)

∣∣∣∣2 .Jaume Haro and Jaume Amoros Matter Bounce Scenario in F (T ) gravity






(1− i

kη

)when η → −∞


ζS(T )(η) = AS(T )(k) +BS(T )(k)

∫ η

−∞

dη

ZS(T )(η).


AS(k) =AT (k)√

3= −

√8

3

k3/2

ρc, BS(k) =

√3BT (k) = i

√3

8

ρc2k3/2

.

Power spectrum:

PS(T )(k) =k3

2π2|BS(T )(k)|2

∣∣∣∣∫ ∞−∞

dt

a(t)ZS(T )(t)







(1− i

kη

)when η → −∞


ζS(T )(η) = AS(T )(k) +BS(T )(k)

∫ η

−∞

dη

ZS(T )(η).


AS(k) =AT (k)√

3= −

√8

3

k3/2

ρc, BS(k) =

√3BT (k) = i

√3

8

ρc2k3/2

.

Power spectrum:

PS(T )(k) =k3

2π2|BS(T )(k)|2

∣∣∣∣∫ ∞−∞

dt

a(t)ZS(T )(t)







(1− i

kη

)when η → −∞


ζS(T )(η) = AS(T )(k) +BS(T )(k)

∫ η

−∞

dη

ZS(T )(η).


AS(k) =AT (k)√

3= −

√8

3

k3/2

ρc, BS(k) =

√3BT (k) = i

√3

8

ρc2k3/2

.

Power spectrum:

PS(T )(k) =k3

2π2|BS(T )(k)|2

∣∣∣∣∫ ∞−∞

dt

a(t)ZS(T )(t)







(1− i

kη

)when η → −∞


ζS(T )(η) = AS(T )(k) +BS(T )(k)

∫ η

−∞

dη

ZS(T )(η).


AS(k) =AT (k)√

3= −

√8

3

k3/2

ρc, BS(k) =

√3BT (k) = i

√3

8

ρc2k3/2

.

Power spectrum:

PS(T )(k) =k3

2π2|BS(T )(k)|2

∣∣∣∣∫ ∞−∞

dt

a(t)ZS(T )(t)




Tensor/scalar ratio:

r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.

Calculations with the analytic solution

Power spectrum in F (T ) gravity: [Haro, JCAP 11,[arXiv:1309.0352]] PS(k) = 16

9ρcρplC2,

(C = 1− 132 + 1

52 − ... = 0.915965... Catalan’s constant).Power spectrum in holonomy corrected LQC: [Wilson-Ewing,JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9ρcρpl

.

Tensor/scalar ratio in F (T ) gravity: r = 3(Si(π/2)C

)2 ∼= 6.7187,

where Si(x) ≡∫ x

0sin yy dy is the Sine integral function.

Tensor/scalar ratio in holonomy corrected LQC: r ∼= 0.





r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.



9ρcρplC2,

(C = 1− 132 + 1


9ρcρpl

.


)2 ∼= 6.7187,








r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.



9ρcρplC2,

(C = 1− 132 + 1


9ρcρpl

.


)2 ∼= 6.7187,








r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.



9ρcρplC2,

(C = 1− 132 + 1


9ρcρpl

.


)2 ∼= 6.7187,








r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.



9ρcρplC2,

(C = 1− 132 + 1


9ρcρpl

.


)2 ∼= 6.7187,








r =1

3

(∫∞−∞

1a(t)ZT (t)dt∫∞

−∞1

a(t)ZS(t)dt

)2

.



9ρcρplC2,

(C = 1− 132 + 1


9ρcρpl

.


)2 ∼= 6.7187,







Numerical resultsNumerical results

[Haro and Amoros, [arXiv:1403.6396]]

Figure: Tensor/scalar ratio for different orbits in function of the bouncing valueof ϕ. First picture for F (T ) gravity and second for holonomy corrected LQC.




Minimum value of the power spectrum in F (T ) gravityPS(k) ∼= 40× 10−3 ρc

ρplobtained at ϕ ∼= −0.9892.

Minimum value of the power spectrum in holonomycorrectecd LQC PS(k) ∼= 23× 10−3 ρc


Matching with BICEP2 dataIn F (T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncingwhen ϕ belongs in [−1.162,−1.144] ∪ [1.144, 1.162].In holonomy corrected LQC Never.

Matching with Planck’s dataIn F (T ) gravity r ≤ 0.11 is realized by solutions bouncing when ϕbelongs in [−1.205,−1.17] ∪ [1.17, 1.205].In holonomy corrected LQC Always.










































































MBS for different potentials: numericanalysis

Matching with a power lawMatching with a quintessence potencialMatching with a plateau potencial




Matching with a power law potentialMatching with a quadratic potencial

Figure: First picture: shape of ρce−√

3|ϕ| matched with ρc(ϕ− ϕ0)2. Secondpicture: phase portrait.




Matching with a quadratic potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing valueof ϕ. In first picture for F (T ) gravity, and in the second one for holonomycorrected LQC.




Matching with a quartic potencial

Figure: First picture: shape of ρce−√

3|ϕ| matched with ρc(ϕ− ϕ0)4. Secondpicture: phase portrait.




Matching with a quartic potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing valueof ϕ, for a potential obtained matching ρce−

√3|ϕ| with ρc(ϕ− ϕ0)4. In first

picture for F (T ) gravity, and in the second one for holonomy corrected LQC.




Matching with a plateua potentialMatching with a plateau potencial

Figure: Shape and phase space portrait of a potential obtained matching theexponential potential ρce−

√3|ϕ| with a plateau potential.




Matching with a plateau potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing valueof ϕ, for a potential obtained matching ρce−

√3|ϕ| with a plateau potential. In

first picture for F (T ) gravity, and in the second one for holonomy correctedLQC.




Matching with a quintessence potentialMatching with a quintessence potencial

Figure: Shape and phase space portrait of a potential that has matterdomination at early times in the contracting phase and quintessence at latetimes in the expanding phase.




Matching with a quintessence potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing valueof ϕ. In the first picture for F (T ) gravity, and in the second one for holonomycorrected LQC.


Documents

Matter Bounce Scenario in F(T) gravityffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/HARO_Jaume.pdf · Jaume Haro and Jaume Amoros´ Matter Bounce Scenario in F(T)gravity. F(T)gravity in