Cosmology with non-minimal derivative coupling

Kazan Federal University, Kazan, Russia 8th Spontaneous Workshop on Cosmology Institut d Etude Scientifique de Cargèse, Corsica May 13, 2014

Plan Plan Scalar fields: minimal and nonminimal coupling to gravity Horndeski theory Scalar fields with nonminimal derivative coupling Cosmological models with nonminimal derivative coupling Perturbations Summary Based on Sushkov, PRD 80, 103505 (2009) Saridakis, Sushkov, PRD 81, 083510 (2010) Sushkov, PRD 85, 123520 (2012) Skugoreva, Sushkov, Toporensky, PRD 88, 083539 (2013)

Scalar fields minimally coupled to gravity S = d 4 x g [ L GR + L S ] L GR gravitational Lagrangian general relativity; L GR = R square gravity; L GR = R + cr 2 f(r)-theories; L GR = f(r) etc... L S scalar field Lagrangian; ordinary STT; L S = ɛ( φ) 2 2V (φ) ɛ = +1 canonical scalar field ɛ = 1 phantom or ghost scalar field with negative kinetic energy V (φ) potential of self-action K-essence; L s = K(X) [X = ( φ) 2 ] etc...

Scalar fields nonminimally coupled to gravity Bergmann-Wagoner-Nordtvedt scalar-tensor theories S = d 4 x g [ f(φ)r h(φ)( φ) 2 2V (φ) ] f(φ)r = nonminimal coupling between φ and R

Scalar fields nonminimally coupled to gravity Bergmann-Wagoner-Nordtvedt scalar-tensor theories S = d 4 x g [ f(φ)r h(φ)( φ) 2 2V (φ) ] f(φ)r = nonminimal coupling between φ and R Conformal transformation to the Einstein frame (Wagoner, 1970): g µν = f(φ)g µν ; dφ dψ = f fh + 3 2 ( df dφ ) 2 1/2 S = d 4 x [ ] g R ɛ( ψ) 2 2U(ψ) ψ = new scalar field U(ψ) = new effective potential [ ( ) ] 2 ɛ = sign fh + 3 df 2 dφ

Scalar fields nonminimally coupled to gravity General scalar-tensor theories S = d 4 x g [ F (φ, R) ( φ) 2 2V (φ) ] F (φ, R) = generalized nonminimal coupling between φ and R

Scalar fields nonminimally coupled to gravity General scalar-tensor theories S = d 4 x g [ F (φ, R) ( φ) 2 2V (φ) ] F (φ, R) = generalized nonminimal coupling between φ and R Conformal transformation to the Einstein frame (Maeda, 1989): g µν = Ω 2 Ω 2 g µν ; 16π F (φ, R) R S = d 4 x g [ ] R 16π h(φ)ψ 1 ( φ) 2 3 32π ψ 2 ( ψ) 2 + U(φ, ψ) ψ Ω 2 = new (second!) scalar field U(φ, ψ) = new effective potential

Scalar fields nonminimally coupled to gravity Some remarks: A nonminimal scalar field is conformally equivalent to the minimal one possessing some effective potential V (φ) A behavior of the scalar field is encoded in the potential V (φ) The potential V (φ) is a very important ingredient of various cosmological models: a slowly varying potential behaves like an effective cosmological constat providing one or more than one inflationary phases. An appropriate choice of V (φ) is known as a problem of fine tuning of the cosmological constant.

Nonminimal derivative coupling generalization S = d 4 x g [ R g µν φ,µ φ,ν 2V (φ) ] F (φ, R, R µν,... ) nonminimal coupling generalization! K(φ, φ,µ, φ ;µν,..., R, R µν,... ) nonminimal derivative coupling generalization!

Horndeski (Galileon) theory In 1973, Horndeski derived the action of the most general scalar-tensor theories with second-order equations of motion G.Horndeski, IJTP 10, 363 (1974) Horndeski Lagrangian: L = i L i L 2 =K(φ, X), X = 1 2 ( φ)2 L 3 =G 3 (φ, X) φ, L 4 =G 4 (φ, X)R + G 4,X (φ, X) [ ( φ) 2 ( µ ν φ) 2], L 5 =G 5 (φ, X)G µν µ ν φ + 1 6 G 5,X(φ, X) [ ( φ) 3 3( φ)( µ ν φ) 2 + ( µ ν φ) 3], Notice: Generally, there are no conformal transformations which transform the Horndeski theory to the Einstein frame!

Theory with nonminimal kinetic coupling Action: S = d 4 x { } R g 8π [ɛgµν + κg µν ]φ,µ φ,ν 2V (φ) Field equations: G µν = 8π [ T µν + κθ µν ], [ɛg µν + κg µν ] µ ν φ = V φ T µν =ɛ µ φ ν φ 1 2 ɛg µν( φ) 2 g µν V (φ), Θ µν = 1 2 µφ ν φ R + 2 α φ (µ φr α ν) 1 2 ( φ)2 G µν + α φ β φ R µανβ + µ α φ ν α φ µ ν φ φ + g µν [ 1 2 α β φ α β φ + 1 2 ( φ)2 α φ β φ R αβ] Notice: The field equations are of second order!

Cosmological models: General formulas Ansatz: Sushkov, 2009; Saridakis, Sushkov, 2010 ds 2 = dt 2 + a 2 (t)dx 2, φ = φ(t) Field equations: a(t) cosmological factor, H = ȧ/a Hubble parameter 3H 2 = 4π φ 2 ( ɛ 9κH 2) + 8πV (φ), 2Ḣ + 3H2 = 4π φ [ ( 2 ɛ + κ 2Ḣ + 3H2 + 4H φ φ 1)] + 8πV (φ), [ (ɛ 3κH 2 )a 3 φ] = a 3 dv (φ) dφ

Cosmological models: General formulas Ansatz: Sushkov, 2009; Saridakis, Sushkov, 2010 ds 2 = dt 2 + a 2 (t)dx 2, φ = φ(t) Field equations: a(t) cosmological factor, H = ȧ/a Hubble parameter 3H 2 = 4π φ 2 ( ɛ 9κH 2) + 8πV (φ), 2Ḣ + 3H2 = 4π φ [ ( 2 ɛ + κ 2Ḣ + 3H2 + 4H φ φ 1)] + 8πV (φ), [ (ɛ 3κH 2 )a 3 φ] = a 3 dv (φ) dφ V (φ) const = φ = C a 3 (ɛ 3κH 2 )

Cosmological models: I. No kinetic coupling Trivial model without kinetic coupling (κ = 0) S = d 4 x [ ] R g 8π ( φ)2

Cosmological models: I. No kinetic coupling Trivial model without kinetic coupling (κ = 0) S = d 4 x [ ] R g 8π ( φ)2 Solution: a 0 (t) = t 1/3 ; φ 0 (t) = 1 2 3π ln t ds 2 0 = dt 2 + t 2/3 dx 2 t = 0 is an initial singularity

Cosmological models: II. No free kinetic term Model without free kinetic term (ɛ = 0) S = d 4 x [ ] R g 8π κgµν φ,µ φ,ν

Cosmological models: II. No free kinetic term Solution: Model without free kinetic term (ɛ = 0) S = d 4 x [ ] R g 8π κgµν φ,µ φ,ν a(t) = t 2/3 ; φ(t) = ds 2 0 = dt 2 + t 4/3 dx 2 t 2 3π κ, κ < 0 t = 0 is an initial singularity

Cosmological models: III. Ordinary scalar field without potential Model for an ordinary scalar field (ɛ = 1) without potential (V = 0) S = d 4 x { } R g 8π (gµν + κg µν )φ,µ φ,ν

Cosmological models: III. Ordinary scalar field without potential Model for an ordinary scalar field (ɛ = 1) without potential (V = 0) S = d 4 x { } R g 8π (gµν + κg µν )φ,µ φ,ν Asymptotic for t : a(t) a 0 (t) = t 1/3 ; φ(t) φ 0 (t) = 1 2 3π ln t Notice: At large times the model with κ 0 has the same behavior like that with κ = 0

Cosmological models: III. Ordinary scalar field without potential Asymptotics for early times The case κ < 0: a t 0 t 2/3 ; t φ t 0 2 3π κ ds 2 t 0 = dt 2 + t 4/3 dx 2 t = 0 is an initial singularity The case κ > 0: a t e Hκt ; φ t Ce t/ κ ds 2 t = dt 2 + e 2Hκt dx 2 de Sitter asymptotic with H κ = 1/ 9κ

Cosmological models: III. Ordinary scalar field without potential Plots of α = ln a in case κ 0, ɛ = 1, V = 0. (a) κ < 0; κ = 0; 1; 10; 100 De Sitter asymptotics: α(t) = t/3 κ (b) κ > 0; κ = 0; 1; 10; 100 Notice: In the model with nonmnimal kinetic coupling one get de Sitter phase without any potential!

Cosmological models: IV. Phantom scalar field without potential Plots of α(t) in case κ < 0, ɛ = 1, V = 0 (a) (9k 2 ) 1 < α 2 < (3k 2 ) 1 (b) α 2 > (3k 2 ) 1 De Sitter asymptotics: α 1(t) = t/ 9 κ (dash-dotted), α 2(t) = t/ 3 κ (dotted).

Cosmological models: V. Constant potential Models with the constant potential V (φ) = Λ/8π = const S = d 4 x [ ] R 2Λ g [ɛg µν + κg µν ]φ,µ φ,ν 8π

Cosmological models: V. Constant potential Models with the constant potential V (φ) = Λ/8π = const S = d 4 x [ ] R 2Λ g [ɛg µν + κg µν ]φ,µ φ,ν 8π There are two exact de Sitter solutions: I. α(t) = H Λ t, φ(t) = φ 0 = const, II. α(t) = t, φ(t) = 3κHΛ 2 1 3 κ 8πκ H Λ = Λ/3 1/2 t,

Cosmological models: V. Constant potential Plots of α(t) in case κ > 0, ɛ = 1, V = Λ (a) H 2 Λ < α2 < 1/9κ (b) 1/9κ < α 2 < 1/3κ < H 2 Λ De Sitter asymptotics: α 1(t) = H Λt (dashed), α 2(t) = t/ 9κ (dash-dotted), α 3(t) = t/ 3κ (dotted).

Cosmological models: V. Constant potential Plots of α(t) in cases κ > 0, ɛ = 1 and κ < 0, ɛ = 1 (a) κ < 0, ɛ = 1 (b) κ > 0, ɛ = 1 De Sitter asymptotic: α 1(t) = H Λt (dashed).

Cosmological models: V. Constant potential Plots of α(t) in case κ < 0, ɛ = 1 (a) H 2 Λ < 1/9 κ < α 2 < 1/3 κ (b) H 2 Λ < 1/9 κ < 1/3 κ < α 2 (c) α 2 < H 2 Λ < 1/9 κ (d) α 2 < 1/9 κ < H 2 Λ De Sitter asymptotics: α 1(t) = H Λt (dashed), α 2(t) = t/ 9κ (dash-dotted), α 3(t) = t/ 3κ (dotted).

Cosmological models: VI. Power-law potential S = d 4 x { R g 8π [ g µν + κg µν] } φ,µ φ,ν m 2 φ 2 Asymptotics: H t 1/ 9κ(1 + 1 2 κm2 ) primary inflation H <t< 1/ ( ) 1 3κ 1 ± 6 κm2 secondary inflation H t 2 [ ] sin 2mt 1 oscillatory asymptotic or graceful exit 3t 2mt from inflation

Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) 123520 S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π

Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) 123520 S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π Stress-energy tensor: T µν = diag(ρ, p, p, p) Field equations: 3H 2 = 4π φ 2 ( 1 9κH 2) + Λ + 8πρ, 2Ḣ + 3H2 = 4π φ [ ( 2 1 + κ 2Ḣ + 3H2 + 4H φ φ 1)] + Λ 8πp [ (1 3κH 2 )a 3 φ] = 0

Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) 123520 S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π Stress-energy tensor: T µν = diag(ρ, p, p, p) Field equations: 3H 2 = 4π φ 2 ( 1 9κH 2) + Λ + 8πρ, 2Ḣ + 3H2 = 4π φ [ ( 2 1 + κ 2Ḣ + 3H2 + 4H φ φ 1)] + Λ 8πp [ (1 3κH 2 )a 3 φ] = 0 = φ = C a 3 (1 3κH 2 )

Realistic cosmological scenario: Master equation Perfect fluid: ρ i = ρ i0 a, p 3(1+wi) i = w i ρ i Density parameters: Ω mi0 = ρ i0, Ω φ0 = C2 /2, Ω Λ0 = Λ ρ cr ρ cr 8πρ cr ρ cr = 3H0 2 /8π Modified Friedmann equation: H 2 = H 2 0 [ Ω Λ0 + i Constraint for parameters: ] Ω mi0 a + Ω φ0(1 9κH 2 ) 3(1+wi) a 6 (1 3κH 2 ) 2 Ω Λ0 + i Ω mi0 + Ω φ0(1 9κH 2 0 ) (1 3κH 2 0 )2 = 1

Realistic cosmological scenario: Simple model No coupling: κ = 0; No cosmological constant: Λ = 0 Pressureless matter: p = 0; ρ = ρ 0 /a 3 H 2 = H 2 0 [ Ωm0 a 3 + Ω ] φ0 a 6 ; Ω m0 + Ω φ0 = 1 Scale factor a(t) a(t) (t t ) δ Hubble parameter H(a) H = ȧ/a H(a) at a 0 H(a) 0 at a Acceleration parameter q(a) q = äa/ȧ 2 q is negative Ω φ0 = 0; 0.25; 0.5; 0.75; 1, and Ω m0 = 1 Ω φ0

Realistic cosmological scenario: Simple model with cosmological constant No coupling: κ = 0 H 2 = H 2 0 [ Ω Λ0 + Ω m0 a 3 + Ω ] φ0 a 6 ; Ω Λ0 + Ω m0 + Ω φ0 = 1 Scale factor a(t) a(t) (t t ) 1/3 at t t a(t) e H Λt at t Hubble parameter H(a) H = ȧ/a H(a) at a 0 H(a) H Λ at a Acceleration parameter q(a) q = äa/ȧ 2 q changes its sign Ω φ0 = 0.23, Ω Λ0 = 0; 0.07; 0.27; 0.47; 0.73, and Ω m0 = 1 Ω φ0 Ω Λ0

Realistic cosmological scenario with kinetic coupling κ > 0 H 2 = H 2 0 [ Ω Λ0 + Ω m0 a 3 + Ω φ0(1 9κH 2 ] ) a 6 (1 3κH 2 ) 2 Universal asymptotic: H H κ = 1/ 9κ at a 0 Notice: The asymptotic H H κ at early cosmological times is only determined by the coupling parameter κ and does not depend on other parameters!

Realistic cosmological scenario: Numerical solutions Scale factor a(t) Hubble parameter H(a) Acceleration parameter q

Realistic cosmological scenario: Estimations H κ t f 60 e-folds t f 10 35 sec the end of initial inflationary stage H κ = 1/ 9κ 6 10 36 sec 1 κ 10 74 sec 2 or l κ = κ 1/2 10 27 cm

Realistic cosmological scenario: Estimations H κ t f 60 e-folds t f 10 35 sec the end of initial inflationary stage H κ = 1/ 9κ 6 10 36 sec 1 κ 10 74 sec 2 or l κ = κ 1/2 10 27 cm H 0 70 (km/sec)/mpc 10 18 sec 1 Present Hubble parameter γ = 3κH0 2 10 109 Extremely small! [ H 2 = H0 2 Ω Λ0 + Ω m0 a 3 + Ω φ0(1 9κH 2 ] ) a 6 (1 3κH 2 ) 2 Ω Λ0 + Ω m0 + Ω φ0 1 Ω Λ0 = 0.73, Ω φ0 = 0.23, Ω m0 = 0.04 q 0 = 0.25

Perturbations Perturbed FRW metric: g µν = ḡ µν + h µν background metric: ḡ 00 = 1, ḡ i0 = ḡ 0i = 0, ḡ ij = a 2 (t)δ ij metric perturbations (Newtonian gauge): h 00 = Φ, h 0i = h i0 = 0, h ij =a 2 (Ψδ ij + i C j + j C i + Θ ij ) Perturbed SET: δt 00 =ρφ + δρ, δt i0 = (ρ + p)δu i, δt ij =ph ij + a 2 δ ij δp Perturbed scalar field: φ (x, t) = φ 0 (t) + δφ (x, t)

Perturbations: Perturbed equations Perturbed Einstein equations P 1 Φ + P 2 Ψ + P3 2 Ψ + P 4 δ φ + P 5 2 δφ + P 6 δρ = 0, Q 1 Φ + Q 2 Φ + Q3 2 Φ + Q 4 Ψ + Q5 Ψ + Q6 2 Ψ +Q 7 δ φ + Q 8 δ φ + Q 9 2 δφ + Q 10 δp = 0, R 1 i Φ + R 2 i Ψ + R3 i δφ + R 4 i δ φ + R 5 2 Ċ i + R 6 δu i = 0, S 1 i j Φ + S 2 i j Ψ + S 3 i j δφ ( ) ( ) +S 4 i Ċ j + j Ċ i + S 5 i Cj + j Ci +S 6 θij + S 7 θij + S 8 2 θ ij = 0 P i, Q j, R k, S l coefficients depending on unperturbed values P 1 = 8π (ρ + 92 κh2 φ ) 2, and so on...

Tensor perturbations Two polarizations: θ ij θ +, θ Equation for tensor modes (1 + 4πκ φ ( 2 ) θ + 3H + 4πκ(2 φ φ + 3H φ ) 2 k ) θ 2 + a 2 (1 4πκ φ 2 )θ = 0 The case 4πκ φ 2 1: The case 4πκ φ 2 1: θ + θ + 3H θ + k2 a 2 θ = 0 ( 2 φ ) φ + 3H θ k2 a 2 θ = 0

Conclusions The non-minimal kinetic coupling provides an essentially new inflationary mechanism which does not need any fine-tuned potential. At early cosmological times the coupling κ-terms in the field equations are dominating and provide the quasi-de Sitter behavior of the scale factor: a(t) e Hκt with H κ = 1/ 9κ and κ 10 74 sec 2 (or l κ κ 1/2 10 27 cm) The model provides a natural mechanism of epoch change without any fine-tuned potential.

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