Matter Bounce Scenario in F ( T ) gravity Jaume Haro and Jaume Amor - - PowerPoint PPT Presentation

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Matter Bounce Scenario in F ( T ) gravity Jaume Haro and Jaume Amor - - PowerPoint PPT Presentation

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F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Matter Bounce Scenario in F ( T ) gravity Jaume Haro and Jaume Amor os Departament de Matem` atica

slide-1
SLIDE 1

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matter Bounce Scenario in F(T ) gravity

Jaume Haro and Jaume Amor´

  • s

Departament de Matem` atica Aplicada I Universitat Polit` ecnica de Catalunya

Frontiers in Fundamental Physics Marseille, 2014

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-2
SLIDE 2

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Introduction

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann 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 inflation The simplest model: dynamics Properties of the simplest model

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

MBS for different potentials: numeric analysis

Matching with a power law or plateau potential Matching with a quintessence potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-3
SLIDE 3

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Introduction

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann 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 inflation The simplest model: dynamics Properties of the simplest model

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

MBS for different potentials: numeric analysis

Matching with a power law or plateau potential Matching with a quintessence potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-4
SLIDE 4

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Introduction

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann 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 inflation The simplest model: dynamics Properties of the simplest model

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

MBS for different potentials: numeric analysis

Matching with a power law or plateau potential Matching with a quintessence potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-5
SLIDE 5

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Introduction

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann 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 inflation The simplest model: dynamics Properties of the simplest model

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

MBS for different potentials: numeric analysis

Matching with a power law or plateau potential Matching with a quintessence potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-6
SLIDE 6

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Introduction

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann 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 inflation The simplest model: dynamics Properties of the simplest model

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

MBS for different potentials: numeric analysis

Matching with a power law or plateau potential Matching with a quintessence potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-7
SLIDE 7

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

F(T ) gravity in flat FLRW geometry

Weitzenb¨

  • ck space-time

Friedmann equation in F(T ) gravity (flat FLRW geometry) Relation with Loop Quantum Cosmology (flat FLRW geometry)

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-8
SLIDE 8

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-9
SLIDE 9

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-10
SLIDE 10

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-11
SLIDE 11

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-12
SLIDE 12

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-13
SLIDE 13

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-14
SLIDE 14

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-15
SLIDE 15

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-16
SLIDE 16

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Weitzenb¨

  • ck space-time

Teleparallelism is based in Weitzenb¨

  • ck space-time.

Global system of four orthonormal vector fields {ei} in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis {ei}, that is, ∇ei = 0. Properties of Weitzenb¨

  • ck 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-17
SLIDE 17

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 constrain HT = 0 leads to the equation (a curve in the plane (H, ρ)) ρ = −2dF(T ) dT T + F(T ) ≡ G(T ). Inverse problem: Given a curve of the form ρ = G(T ) the corresponding F(T ) theory is F(T ) = − √ −T 2

  • G(T )

T √ −T dT .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-18
SLIDE 18

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 constrain HT = 0 leads to the equation (a curve in the plane (H, ρ)) ρ = −2dF(T ) dT T + F(T ) ≡ G(T ). Inverse problem: Given a curve of the form ρ = G(T ) the corresponding F(T ) theory is F(T ) = − √ −T 2

  • G(T )

T √ −T dT .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-19
SLIDE 19

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 constrain HT = 0 leads to the equation (a curve in the plane (H, ρ)) ρ = −2dF(T ) dT T + F(T ) ≡ G(T ). Inverse problem: Given a curve of the form ρ = G(T ) the corresponding F(T ) theory is F(T ) = − √ −T 2

  • G(T )

T √ −T dT .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-20
SLIDE 20

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 constrain HT = 0 leads to the equation (a curve in the plane (H, ρ)) ρ = −2dF(T ) dT T + F(T ) ≡ G(T ). Inverse problem: Given a curve of the form ρ = G(T ) the corresponding F(T ) theory is F(T ) = − √ −T 2

  • G(T )

T √ −T dT .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-21
SLIDE 21

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 constrain HT = 0 leads to the equation (a curve in the plane (H, ρ)) ρ = −2dF(T ) dT T + F(T ) ≡ G(T ). Inverse problem: Given a curve of the form ρ = G(T ) the corresponding F(T ) theory is F(T ) = − √ −T 2

  • G(T )

T √ −T dT .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-22
SLIDE 22

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 ) = ρc

2

  • 1 ±
  • 1 + 2T

ρc

  • .

We obtain the model [Amor´

  • s et al., PRD 87, [arXiv:1108.0893]]

F±(T ) = ±

  • −T ρc

2 arcsin

  • −2T

ρc

  • + G±(T ).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-23
SLIDE 23

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 ) = ρc

2

  • 1 ±
  • 1 + 2T

ρc

  • .

We obtain the model [Amor´

  • s et al., PRD 87, [arXiv:1108.0893]]

F±(T ) = ±

  • −T ρc

2 arcsin

  • −2T

ρc

  • + G±(T ).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-24
SLIDE 24

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 ) = ρc

2

  • 1 ±
  • 1 + 2T

ρc

  • .

We obtain the model [Amor´

  • s et al., PRD 87, [arXiv:1108.0893]]

F±(T ) = ±

  • −T ρc

2 arcsin

  • −2T

ρc

  • + G±(T ).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-25
SLIDE 25

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 ) = ρc

2

  • 1 ±
  • 1 + 2T

ρc

  • .

We obtain the model [Amor´

  • s et al., PRD 87, [arXiv:1108.0893]]

F±(T ) = ±

  • −T ρc

2 arcsin

  • −2T

ρc

  • + G±(T ).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-26
SLIDE 26

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The Matter Bounce Scenario (MBS)

MBS as an alternative to inflation The simplest model: dynamics Properties of the simplest model

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-27
SLIDE 27

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description It depicts, at early times, a matter dominated Universe in the contracting phase, after the bounce it enters in the expanding one where it matches with the hot Friedmann Universe. Horizon problem solved: All the parts of the Universe are in causal contact at bouncing time. Flatness problem improved: The spatial curvature decreases in the contracting phase at the same rate as it increases in the expanding one. Power spectrum: The power spectrum of cosmological perturbations is scale invariant.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-28
SLIDE 28

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description It depicts, at early times, a matter dominated Universe in the contracting phase, after the bounce it enters in the expanding one where it matches with the hot Friedmann Universe. Horizon problem solved: All the parts of the Universe are in causal contact at bouncing time. Flatness problem improved: The spatial curvature decreases in the contracting phase at the same rate as it increases in the expanding one. Power spectrum: The power spectrum of cosmological perturbations is scale invariant.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-29
SLIDE 29

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description It depicts, at early times, a matter dominated Universe in the contracting phase, after the bounce it enters in the expanding one where it matches with the hot Friedmann Universe. Horizon problem solved: All the parts of the Universe are in causal contact at bouncing time. Flatness problem improved: The spatial curvature decreases in the contracting phase at the same rate as it increases in the expanding one. Power spectrum: The power spectrum of cosmological perturbations is scale invariant.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-30
SLIDE 30

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description It depicts, at early times, a matter dominated Universe in the contracting phase, after the bounce it enters in the expanding one where it matches with the hot Friedmann Universe. Horizon problem solved: All the parts of the Universe are in causal contact at bouncing time. Flatness problem improved: The spatial curvature decreases in the contracting phase at the same rate as it increases in the expanding one. Power spectrum: The power spectrum of cosmological perturbations is scale invariant.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-31
SLIDE 31

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS as an alternative to inflation

The Matter Bounce Scenario: Brief description It depicts, at early times, a matter dominated Universe in the contracting phase, after the bounce it enters in the expanding one where it matches with the hot Friedmann Universe. Horizon problem solved: All the parts of the Universe are in causal contact at bouncing time. Flatness problem improved: The spatial curvature decreases in the contracting phase at the same rate as it increases in the expanding one. Power spectrum: The power spectrum of cosmological perturbations is scale invariant.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-32
SLIDE 32

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-33
SLIDE 33

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-34
SLIDE 34

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-35
SLIDE 35

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-36
SLIDE 36

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-37
SLIDE 37

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

The simplest model: dynamics

Matter dominated Universe: Dynamics a(t) = 3 4ρct2 + 1 1/3 . Matter scalar field Potential of the simplest model: V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 4ρct2 + 1 1/3 .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-38
SLIDE 38

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Phase Portrait of the model V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 the invariant 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-39
SLIDE 39

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Phase Portrait of the model V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 the invariant 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-40
SLIDE 40

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Phase Portrait of the model V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 the invariant 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-41
SLIDE 41

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Phase Portrait of the model V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 the invariant 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-42
SLIDE 42

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Phase Portrait of the model V ( ¯ ϕ) = 2ρc

e−

√ 3 ¯ ϕ

(1+e−

√ 3 ¯ ϕ)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 the invariant 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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-43
SLIDE 43

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-44
SLIDE 44

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-45
SLIDE 45

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-46
SLIDE 46

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-47
SLIDE 47

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-48
SLIDE 48

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-49
SLIDE 49

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-50
SLIDE 50

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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 finish being, once again, matter dominated. Potencials of the form V(¯ ϕ) ∼ ρce−

√ 3|¯ ϕ| when |¯

ϕ| → ∞ has this property. 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. For

example with V(¯ ϕ) = ρc

¯ ϕ4 ¯ ϕ4+¯ ϕ4

0 or V(¯

ϕ) ∼ ρce−√

3(1+ω)|¯ ϕ| where

ω = −1.10 ± 0.14 (WMAP observational data).

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-51
SLIDE 51

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Perturbations in F(T ) gravity

Dynamical equations for perturbations Power spectrum and tensor/scalar ratio in MBS Numerical results

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-52
SLIDE 52

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-53
SLIDE 53

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-54
SLIDE 54

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-55
SLIDE 55

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-56
SLIDE 56

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-57
SLIDE 57

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-58
SLIDE 58

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Dynamical equations for perturbations

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

2 ˙

ϕ2 − |∇V (ϕ)|2 − V (ϕ). Dynamical equations ζ′′

S(T ) − c2 s∇2ζS(T ) + Z′

S(T )

ZS(T ) ζ′ S(T ) = 0,

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

c2

s = |Ω| arcsin

  • 2
  • 3

ρc H

  • 2
  • 3

ρc H

, with Ω = 1 − 2ρ

ρc .

ZS = a2|Ω| ˙

¯ ϕ2 c2

sH2

and ZT =

a2c2

s

|Ω| .

Comparison with Holonomy Corrected LQC [Cailleteau et al., PRD 86, [arXiv:1206.6736]] c2

s = Ω,

ZS = a2 ˙ ¯ ϕ2 H2 and ZT = a2 Ω .

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-59
SLIDE 59

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

−∞

d¯ η 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 ρc 2k3/2 . Power spectrum: PS(T )(k) = k3 2π2 |BS(T )(k)|2

  • ∞

−∞

dt a(t)ZS(T )(t)

  • 2

.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-60
SLIDE 60

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

−∞

d¯ η 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 ρc 2k3/2 . Power spectrum: PS(T )(k) = k3 2π2 |BS(T )(k)|2

  • ∞

−∞

dt a(t)ZS(T )(t)

  • 2

.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-61
SLIDE 61

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

−∞

d¯ η 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 ρc 2k3/2 . Power spectrum: PS(T )(k) = k3 2π2 |BS(T )(k)|2

  • ∞

−∞

dt a(t)ZS(T )(t)

  • 2

.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-62
SLIDE 62

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

−∞

d¯ η 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 ρc 2k3/2 . Power spectrum: PS(T )(k) = k3 2π2 |BS(T )(k)|2

  • ∞

−∞

dt a(t)ZS(T )(t)

  • 2

.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-63
SLIDE 63

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

−∞

d¯ η 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 ρc 2k3/2 . Power spectrum: PS(T )(k) = k3 2π2 |BS(T )(k)|2

  • ∞

−∞

dt a(t)ZS(T )(t)

  • 2

.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-64
SLIDE 64

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-65
SLIDE 65

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-66
SLIDE 66

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-67
SLIDE 67

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-68
SLIDE 68

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-69
SLIDE 69

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Tensor/scalar ratio: r = 1 3 ∞

−∞ 1 a(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

9 ρc ρpl C2,

(C = 1 − 1

32 + 1 52 − ... = 0.915965... Catalan’s constant).

Power spectrum in holonomy corrected LQC: [Wilson-Ewing, JCAP 03, [arXiv:1211.6269]] PS(k) = π2

9 ρc ρpl .

Tensor/scalar ratio in F(T ) gravity: r = 3

  • Si(π/2)

C

2 ∼ = 6.7187, where Si(x) ≡ x

sin y y dy is the Sine integral function.

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-70
SLIDE 70

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Numerical results

Numerical results [Haro and Amor´

  • s, [arXiv:1403.6396]]

Figure: Tensor/scalar ratio for different orbits in function of the bouncing value

  • f ¯

ϕ. First picture for F(T ) gravity and second for holonomy corrected LQC.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-71
SLIDE 71

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-72
SLIDE 72

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-73
SLIDE 73

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-74
SLIDE 74

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-75
SLIDE 75

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-76
SLIDE 76

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-77
SLIDE 77

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-78
SLIDE 78

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

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

ρpl obtained at ¯

ϕ ∼ = −0.9892. Minimum value of the power spectrum in holonomy correctecd LQC PS(k) ∼ = 23 × 10−3 ρc

ρpl obtained at ¯

ϕ ∼ = −0.9870. Matching with BICEP2 data

In F(T ) gravity r = 0.20+0.07

−0.05 is realized by solutions bouncing

when ¯ ϕ belongs in [−1.162, −1.144] ∪ [1.144, 1.162]. In holonomy corrected LQC Never.

Matching with Planck’s data

In F(T ) gravity r ≤ 0.11 is realized by solutions bouncing when ¯ ϕ belongs in [−1.205, −1.17] ∪ [1.17, 1.205]. In holonomy corrected LQC Always.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-79
SLIDE 79

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

MBS for different potentials: numeric analysis

Matching with a power law Matching with a quintessence potencial Matching with a plateau potencial

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-80
SLIDE 80

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a power law potential

Matching with a quadratic potencial

Figure: First picture: shape of ρce−

√ 3|ϕ| matched with ρc(ϕ − ϕ0)2. Second

picture: phase portrait.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-81
SLIDE 81

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a quadratic potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing value

  • f ¯

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-82
SLIDE 82

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a quartic potencial

Figure: First picture: shape of ρce−

√ 3|ϕ| matched with ρc(ϕ − ϕ0)4. Second

picture: phase portrait.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-83
SLIDE 83

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a quartic potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing value

  • f ¯

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.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-84
SLIDE 84

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a plateua potential

Matching with a plateau potencial

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

√ 3|ϕ| with a plateau potential.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-85
SLIDE 85

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a plateau potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing value

  • f ¯

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 corrected LQC.

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-86
SLIDE 86

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a quintessence potential

Matching with a quintessence potencial

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity

slide-87
SLIDE 87

F (T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F (T ) gravity MBS for different potentials: numeric analysis

Matching with a quintessence potencial

Figure: Tensor/scalar ratio for different orbits in function of the bouncing value

  • f ¯

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

Jaume Haro and Jaume Amor´

  • s

Matter Bounce Scenario in F (T ) gravity