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

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 Aplicada I Universitat Polit` ecnica de Catalunya Frontiers in Fundamental Physics Marseille, 2014 Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock 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´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock 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´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock 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´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock 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´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock 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´ os Matter Bounce Scenario in F ( T ) gravity

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¨ ock space-time Friedmann equation in F ( T ) gravity (flat FLRW geometry) Relation with Loop Quantum Cosmology (flat FLRW geometry) Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity

F ( T ) gravity in flat FLRW geometry Matter Bounce Scenario Perturbations in F ( T ) gravity MBS for different potentials: numeric analysis Weitzenb¨ ock space-time Teleparallelism is based in Weitzenb¨ ock space-time. Global system of four orthonormal vector fields { e i } in the tangent vector bundle. Covariant derivative ∇ that defines absolute parallelism with respect the global basis { e i } , that is, ∇ e i = 0 . Properties of Weitzenb¨ ock 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 = − 6 H 2 . Jaume Haro and Jaume Amor´ os Matter Bounce Scenario in F ( T ) gravity