cosmology near the general relativity limit

Transkript

1 Thirteenth Marcel Grossmann Meeting - MG3 Stockholm, 6 July 202 (I) Scalar-tensor and (II) Multiscalar-tensor cosmology near the general relativity limit Laur Järv University of Tartu, Estonia (I) LJ, Piret Kuusk, Margus Saal Phys Rev D8: (200), Phys Lett B694: -5 (200), Phys Rev D (202) (II) LJ, Piret Kuusk, Erik Randla forthcoming

2 (I) Scalar-tensor gravity (STG) One scalar field Ψ non-minimally coupled to gravity, Brans-Dicke like parametrization, Jordan frame S = 2κ 2 d 4 x [ g ΨR(g µν ) ω(ψ) ] Ψ ρ Ψ ρ Ψ 2κ 2 V (Ψ) +S m (g µν, χ mat ) Family of theories, each pair ω(ψ) and V (Ψ) specifies a theory Variable gravitational constant set by the dynamical scalar field, 8πG = κ2 Ψ, assume 0 < Ψ < Assume positive energy density: 2ω(Ψ) + 3 0, V (Ψ) 0 Paradigmatic example of a modified gravity theory Eg comes from higher dimensions, braneworlds, effective field theory approach to dark energy; several proposed modifications to Einstein s general relativity can be cast in the form of STG, or contain STG as a subsector Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 2/6

3 The limit of general relativity Observations point towards a specific corner of the solutions space: CMB: Grec Gnow G now 005, hence Ψ CMB: 2ω(Ψ rec) Solar System PPN: Solar System PPN: 2ω(Ψ now)+3 The limit of general relativity : Assume ( ) d 0, dψ 2ω(Ψ) + 3 Ψ dω dψ (2ω(Ψ now)+3) 2 (2ω(Ψ now)+4) 0 4 2ω(Ψ ) + 3 = 0, Ψ = 0 2ω(Ψ) + 3 is differentiable at Ψ Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 3/6

4 Outline We study flat (k = 0) FLRW STG cosmology with arbitrary ω(ψ) and V (Ψ) in the potential dominated and dust matter dominated regimes Derive the approximate equations that govern the dynamics near the limit of general relativity (Nonlinear, nonautonomous system!) Find the general analytic form of solutions for these equations, ie Ġ Ψ(t), H(t), can also compute w eff (t), G, etc (Complete classification!) Argue that the full and approximate phase spaces are in qualitative agreement (Can trust the results!) Determine the conditions on a STG for its solutions to dynamically converge to the GR limit (Select viable theories!) Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 4/6

5 Scalar-tensor cosmology Flat FLRW, barotropic matter fluid p = wρ H 2 = H Ψ Ψ + Ψ 2 6 κ2 ω(ψ) + Ψ2 3 ρ Ψ + κ2 V (Ψ) 3 Ψ, 2Ḣ + 3H2 = 2H Ψ Ψ Ψ 2 Ψ ω(ψ) 2 Ψ2 Ψ κ2 κ2 wρ + Ψ Ψ V (Ψ), Ψ = 3H Ψ 2ω(Ψ) + 3 2κ 2 + 2ω(Ψ) + 3 dω(ψ) dψ Ψ 2 + [ 2V (Ψ) Ψ κ 2 2ω(Ψ) + 3 ], dv (Ψ) dψ ( 3w) ρ ρ = 3H (w + ) ρ Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 5/6

6 Remark: STG ja GR cosmology In the general realtivity limit get the usual GR Friedmann equations with a cosmological constant (V (Ψ) = Λ) H 2 = H Ψ Ψ + Ψ 2 6 κ2 ω(ψ) + Ψ2 3 ρ Ψ + κ2 V (Ψ) 3 Ψ, 2Ḣ + 3H2 = 2H Ψ Ψ Ψ 2 Ψ ω(ψ) 2 Ψ2 Ψ κ2 κ2 wρ + Ψ Ψ V (Ψ), Ψ = 3H Ψ 2ω(Ψ) + 3 2κ 2 + 2ω(Ψ) + 3 dω(ψ) dψ Ψ 2 + [ 2V (Ψ) Ψ κ 2 2ω(Ψ) + 3 ], dv (Ψ) dψ ( 3w) ρ ρ = 3H (w + ) ρ Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 6/6

7 Approximation near the GR limit (ρ = 0 case) Consider small deviations (x ẋ) Ψ(t) = Ψ + x(t), Ψ(t) = ẋ(t) Expand in series, keeping the leading terms, obtain an approximate equation ẍ = C ẋ + C 2 x + ẋ 2 2x, () where C ± 2ω(Ψ ) + 3 0, A d ( dψ 2ω(Ψ) + 3 ( 3κ 2 V (Ψ ), C 2 2κ 2 A 2V (Ψ) Ψ encode the behavior of the functions ω and V near this point ) Ψ, ) dv (Ψ) dψ Ψ Ψ Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 7/6

8 Solutions to the approximate equation (ρ = 0 case) The solutions of () in cosmological time fall into three classes, depending on C C 2 + 2C 2: exponential, linear-exponential, or oscillating, ] 2 e [M Ct e 2 t C M 2 e 2 t C, if C > 0, Ψ(t) = Ψ ± e Ct [M t M 2 ] 2, if C = 0, 2 e [N Ct sin( 2 t C ) N 2 cos( 2 C )] t, if C < 0 Here M, M 2, N, N 2 are constants of integration (determined by initial conditions) Also H(t) = C 3 ± Eg oscillating solutions oscillate across the phantom divide line (w eff = ) Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 8/6

9 Classification of phase portraits (Ψ, Ψ) C > 0 C > 0 C = 0 C < 0

10 Classification of phase portraits (Ψ, Ψ )

11 Approximation near the GR limit (V = 0 case) Again consider small deviations x(t), h(t), Ψ(t) = Ψ + x(t), H(t) = H (t) + h(t) where H (t) = 2 3(t t s ) is the Hubble parameter corresponding to Ψ Expand in series, keeping the leading terms, obtain approximate equations ẍ = ẋ 2 2x 3H ẋ + 3A Ψ H x 2, (2) ḣ + 3H h = ( + ) ẋ 2 4Ψ 2A Ψ x + H ẋ 3 2Ψ 2 A H x 2 (3) Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit /6

12 Solutions to the approximate equation (V = 0 case) The solutions fall again into three classes, depending on D A Ψ : polynomial, logarithmic, or oscillating, ( D ) t M t 2 M 2 t D 2 2, if D > 0, Ψ(t) = Ψ ± t (M ln t M 2 ) 2, if D = 0, [ ( ) ( 2 D D t N sin 2 ln t N 2 cos 2 ln t)], if D < 0, where M, M 2, N, N 2 are constants of integration (determined by initial conditions) Also H(t) = 2 [ ± t ] 3t ( ) Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 2/6

13 Conditions for GR to be an attractor Observational constraints are naturally satisfied, if the GR limit is an attractor, ie solutions dynamically converge towards it GR limit exists, if Ψ, such that 2ω(Ψ ) + 3 = 0 Attractor in the dust matter dominated era ( d dψ 2ω(Ψ) + 3 ) Ψ Ψ < 0 Attractor in the potential dominated era [ ] Ψ dv (Ψ) V (Ψ ) > 0, < 2V (Ψ) dψ Ψ Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 3/6

14 (II) Multiscalar-tensor gravity (MSTG) N non-minimally coupled scalar fields Φ a, Jordan frame S = 2κ 2 d 4 x g ( FR Z ab g µν µ Φ a ν Φ b 2κ 2 U ) +S(g µν, χ matter ), where F = F(Φ, Φ 2,, Φ N ), Z ab = Z ab (Φ, Φ 2,, Φ N ), U = U(Φ, Φ 2,, Φ N ) are arbitrary functions Can use N redefinitions of the scalar fields to cast the theory in the form where only one of the scalar fields, Ψ, is non-minimally coupled, while the others, φ i, are minimally coupled to gravity, S = 2κ 2 d 4 x g ( ΨR Z ij ρ φ i ρ φ j ω ) Ψ ρψ ρ Ψ 2κ 2 U +S m (g µν, χ m ), however F = F (φ, φ 2,, φ N, Ψ), Z ab = Z ab (φ, φ 2,, φ N, Ψ), U = U(φ, φ 2,, φ N, Ψ)) Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 4/6

15 Two fields case (and ρ = 0) Now expand Ψ(t) = Ψ + x(t), φ(t) = φ + x(t) around 2ω(Ψ, φ ) + 3 = 0, Z(Ψ, φ ) = 0 Derive approximate equations hard to solve generally For instance, if φ Ψ(t) = Ψ ± Ω(t) = ( 2ω + 3 ) Ψ,φ = φ ( e mt aω(t) dt + k ) 2 k2 J n+(ξ) e C t J n(ξ) 4e (m+c)t aω(t) dt, ξ = 2Dk3 2C e C t, ( ) = Z Ψ,φ φ U Ψ,φ = 0 t = Ψ ±D e Ct ( M e Ct M 2 e Ct) 2 m = C 2 + 2C2, n = m 2C, a = 2D k 3 C, C 2, C, D, D constants that specify MSTG, k, k 2, k 3, M, M2 integration constants, J n Bessel function Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 5/6

16 Summary (I) We studied flat FLRW STG cosmology with arbitrary ω(ψ) and V (Ψ) in the potential dominated and dust matter dominated regimes Determined the conditions on a STG for its solutions to dynamically converge to the GR limit Can use to select viable theories Found the general analytic form of solutions near the GR limit, ie Ġ Ψ(t), H(t), can also compute w eff (t), G, etc Complete classification Can use to compare with actual expansion history (cosmography, statefinder diagnostic, etc), Can use as the background for the growth of cosmological perturbations (parameterized post-friedmannian formalism, etc) (II) We also studied MSTG The same methods can be applied here, analytic results are harder to obtain, but still possible in some cases Laur Järv Scalar-tensor and Multiscalar-tensor cosmology near the general relativity limit 6/6