Cosmological bounces in spatially flat FRW spacetimes in metric f ( R ) gravity Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Dept. of Physics, IIT Kanpur 27/01/2015 1 1 JCAP10(2014)009 Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Talk outline Brief introduction to cosmological bounce, f ( R ) gravity. Motivations to discuss bounce in f ( R ) gravity. Friedmann equations and bouncing conditions in f ( R ) gravity. Analyzing a typical bouncing scenario in an R + R 2 gravity. Analyzing the evolution of the scalar perturbation through bounce in such a scenario. Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Introductions Cosmological bounce is a paradigm proposed to avoid the singularity at the beginning of the universe. Scale factor decreases,reaches a certain nonvanishing minimum,and then increase again. H b = 0 , ˙ H b > 0. f ( R ) theories are modified gravity theories which include corrections to GR for high or low values of R . d 4 x √− gf ( R ) + S m 1 � f ( R ) action : S = 2 κ G µν ≡ R µν − 1 2 g µν R = f ( R ) − Rf ′ ( R ) + ∇ µ ∇ ν f ′ ( R ) − g µν � f ′ ( R ) f ′ ( R ) ( T µν + g µν κ ) 2 κ κ f ′ ( R ) > 0 for positive gravitational coupling Unlike GR, here T = 0 � R = 0(in general); A hidden d.o.f. is in play ! Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Motivations and main focus GR : Bounce possible only for k=+1 Friedmann universe. f ( R ) : Bounce possible for both k=+1,0. At early times R was high, so corrections to GR are likely. I will focus on R + α R 2 gravity with α < 0. I will resort to radiation background( ω = 1 3 ) and flat spatial section( k = 0 Friedmann universe). Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
The metric and the equations Maximally symmetric FLRW spacetime ds 2 = − dt 2 + a 2 ( t )[ dr 2 1 − kr 2 + r 2 d Ω 2 ] For an ideal fluid T µ ν = diag ( − ρ, p , p , p ), FLRW equations for k = 0 : 3 H 2 = − 3 H ˙ Rf ′′ ( R ) ρ eff ≡ Rf ′ − f f ′ ( R ) ( ρ + ρ eff ), κ 2 κ κ H + 3 H 2 = R 2 f ′′′ +2 H ˙ ˙ Rf ′′ + ¨ 2 ˙ Rf ′′ − Rf ′ − f − κ f ′ ( R ) ( p + p eff ), P eff ≡ 2 κ κ ρ eff , p eff ; Originates NOT from some other type of matter component, but from the modified geometry of space-time itself !!! Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Bounce Conditions in f(R) gravity We assume validity of the WEC in the matter sector : ρ M ≥ 0 , ρ + p > 0 ρ b + R b f ′ b − f b = 0 (for k = 0) 2 κ For k = 0 both matter bounce and matterless bounce is possible depending on the form of f ( R ). Matterless bounce is possible iff ( Rf ′ − f ) has a positive root (e.g. f ( R ) = R + α R + β R 2 ; α < 0 , 0 < α 2 < 3 β ). Also then f ′′′ is not identically zero. f ( R ) = R + α R n (for any n ≥ 2) : Only matter bounce possible and that too for α < 0. Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Einstein frame picture of f ( R ) gravity � 2 κ ln f ′ ( R ) ; V ( ϕ ) = Rf ′ − f 3 g µν = f ′ ( R ) g µν ; ϕ = ˜ 2 κ f ′ 2 The extra d.o.f. recast as a scalar field directly coupled to matter . � √ √ ˜ t = f ′ dt , ˜ a = f ′ a p ρ = ˜ f ′ 2 , ˜ ρ p = f ′ 2 In Einstein frame the theory becomes GR with the matter field and the scalar field . The dynamical equations are usual GR Friedmann equations and the KG equation for the scalr field. Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Solving for bounce in Einstein frame Dynamical equations : ′′ + 3 ˜ ′ + V ,ϕ = � κ ϕ H ϕ 6 (1 − 3 ω )˜ ρ ′ + � κ ′ ˜ ρ + 3 ˜ ρ ˜ 6 (1 − 3 ω ) ϕ H ˜ ρ (1 + ω ) = 0 ′ = k ′ 2 + ˜ ˜ 2 ( ϕ ρ (1 + ω )) H a 2 − κ ˜ Equations for initial conditions : √ F ( ˜ H − � κ → ˜ H b = � κ ′ ) − ′ H = 6 ϕ 6 ϕ b ′ 2 + V ( ϕ ) + ˜ H 2 = κ ˜ 3 ( 1 2 ϕ ρ ) − → ˜ ρ b = − V ( ϕ ) b To solve the system for k = 0, we need to put by hand only ϕ b , ϕ ′ b . Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
A typical symmetric bounce for f ( R ) = R + α R 2 , ( α = − 10 12 in Planck units) , k = 0 , ω = 1 3 : Jordan frame ° H t L•= 9 100 H 2 H t L , ° H a H t L 1.385 1.4 µ 10 - 13 1.2 µ 10 - 13 1.380 1. µ 10 - 13 1.375 8. µ 10 - 14 1.370 6. µ 10 - 14 4. µ 10 - 14 1.365 2. µ 10 - 14 t t - 1 µ 10 6 - 500000 500000 1 µ 10 6 - 1 µ 10 6 - 500000 500000 1 µ 10 6 Bounce in Jordan frame. The era before and after the bounce can be approximated by a ’ deflationary ’ and ’ inflationary ’ era. A comparison: α > 0(Starobinsky model) ⇒ Vacuum dominated ; α < 0 ⇒ Matter driven inflation Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
A typical symmetric bounce for f ( R ) = R + α R 2 , ( α = − 10 12 in Planck units) , k = 0 , ω = 1 3 : Einstein frame : 100 H è 2 H t è L , ° H è £ H t è L•> è H t è L a 4. µ 10 - 13 1.000 0.995 3. µ 10 - 13 0.990 0.985 2. µ 10 - 13 0.980 0.975 1. µ 10 - 13 0.970 è è t t - 1 µ 10 6 - 500000 500000 1 µ 10 6 - 1 µ 10 6 - 500000 500000 1 µ 10 6 No bounce in the Einstein frame! For k = 0, considering there is a bounce in the Jordan frame, there can never be an analogous bounce in the Einstein frame . Interestingly, for k = +1 , there can be simultaneous bounce in both frames iff ˙ F ( t = 0) = 0 , ¨ F ( t = 0) > 0. Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Scalar metric perturbations for k = 0 We use conformal time η : d η = dt / a = d ˜ t / ˜ a Scalar perturbed FRW metric in Jordan frame ds 2 = a 2 ( η )[ − (1 + 2Φ) d η + (1 − 2Ψ) δ ij dx i dx j ] Φ , Ψ : 2 gauge invariant Bardeen potentials in Jordan frame. Scalar perturbed FRW metric in Einstein frame s 2 = ˜ a 2 ( η )[ − (1 + 2˜ Φ) d η + (1 − 2˜ Φ) δ ij dx i dx j ] d ˜ ˜ Φ : 1 gauge invariant Bardeen potential in Einstein frame. 3 ( F 2 / F ′ a )[( a / F )˜ Φ = − 2 Φ] ′ 3 (1 / FF ′ a )( aF 2 ˜ Ψ = 2 Φ) ′ Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Perturbation equations for k = 0 in the Einstein frame Perturbation equation in GR, but for both scalar field and hydrodynamic matter together!! ′′ −∇ 2 ˜ ′′ ′ + a 2 s (˜ s ( ˜ 0 )+ ˜ s )]˜ c 2 Φ)+[2 c 2 H− ϕ ρ 0 (1+ c 2 s )(1 − 3 c 2 � κ Φ 0 6 ˜ Φ ϕ ′ ϕ ′ 0 ′′ ˜ s ) � κ [2( ˜ ′ − ˜ s )]˜ H ϕ 0 ) c 2 a 2 ˜ ρ 0 (1+ c 2 H 0 (1 − 3 c 2 2 (1 − c 2 s +˜ s ) − κ Φ = 0 H 0 6 ϕ ′ ϕ ′ For background comprised of only one type of hydrodynamic matter, adiabaticity of the perturbations are preserved from Jordan frame to Einstein frame. The equation involves the term ϕ ′′ ϕ ′ ; φ ′ (0) = 0 ; Is the equation singular at η = 0 ?? ; Nope! it is a removable singularity!! So the equation is well defined throughout. Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
Some special cases of the perturbation equation No matter : ′′ − ∇ 2 ˜ � � ′ + � � ′ − ˜ �� ′′ ′′ ˜ ˜ ˜ ˜ ˜ H − ϕ H ϕ Φ Φ + 2 0 Φ 2 H 0 Φ = 0 ϕ ′ ϕ ′ 0 0 Usual perturbation equation in presence of a scalar field ω = 1 3 ′′ − ∇ 2 ˜ ′ + [2( ˜ ′ − ˜ ′′ ′′ ˜ Φ + 2( ˜ 0 )˜ ρ 0 ]˜ 0 ) − 4 κ a 2 ˜ H − ϕ H ϕ Φ 0 Φ H 0 3 ˜ Φ = 0 ϕ ′ ϕ ′ Only the coefficient of the 0 th order term modified by a single term c 2 s = ω = 0 ′ + � � � ˜ ˜ ′ ˜ 3 Φ H − ϕ Φ = 0 0 2 κ No info about the matter content!! Usually one takes c 2 s − → 0 but � = 0 Author : Saikat Chakraborty Co-author : Niladri Paul, Kaushik Bhattacharya Cosmological bounces in spatially flat FRW spacetimes in metric
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