Equation of state of dark energy. Phys. Rev. D 91, (2015)

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1 Equation of state of dark energy in f R gravity The University of Tokyo, RESCEU K. Takahashi, J. Yokoyama Phys. Rev. D 91, (2015)

2 Motivation Many modified theories of gravity have been considered f(r) gravity one of the simplest generalizations of GR There are some f R models which are viable on both cosmological and local scales EoS of dark energy: P DE = wρ DE w = 1 in ΛCDM model w 1 in f R theories Important for distinguishing models. Observational constraint (Kowalski et al. (2008)) SN+BAO+CMB 1 + w z<0.5 < 0.1 Fifth force must be small (Brax et al. (2008)) local constraint 1 + w Ω DE < 10 4 Extremely small! We argue that this is incorrect 2/20

3 f(r) gravity Action Einstein frame S = M 2 Pl 2 d 4 x gf(r) + S m (g μν, Ψ) g μν g μν = f R g μν matter field f R F R e 2βφ/M Pl, β 1 6 S = d 4 x g M 2 Pl 2 R 1 2 φ 2 V(φ) + S m (e 2βφ/M Pl g μν, Ψ) GR V φ = 1 RF f 16πG F 2 scalar field φ couples to matter fifth force = β M Pl φ 3/20

4 EoM in the Einstein frame Einstein field equation G μν = 8πG μ φ ν φ g μν 1 2 φ 2 + V φ + T μν m Klein-Gordon equation φ = V φ + β M Pl ρ m e βφ/m Pl The dynamics of φ are governed by an effective potential V eff φ V φ + ρ m e βφ/m Pl Depends on local matter densities 4/20

5 Chameleon mechanism small ρ large ρ large φ min small φ min ρ 1 < ρ 2 φ 1 min > φ 2 min m 1 < m 2 5/20

6 Thin-shell solution Thin-shell solution scalar field configuration around a uniform spherical object It played a crucial role in the previous work thin-shell outside ρ = ρ b φ = φ b R s R c inside ρ = ρ c φ = φ c 6/20

7 Thin-shell solution General form of the thin-shell solution δφ φ φ b surface 0 R s R c r thin-shell φ c φ b ρ c ρ b Thin-shell parameter 1 > ε th = R c R s R c fifth force Newtonian force It is a solution of the Poisson equation (static assumption): 2 φ = V eff φ 7/20

8 Thin-shell solution Functional form δφ c, r < R s δφ φ φ b δφ = βρ c r 2 3M Pl 2 + R 3 s r 3 2 R s 2 + δφ c, R s < r < R c βρ 3 c R c ε 3M th Pl r e m b r R c, r > R c ε th M Pl β δφ c R c 2 ρ c R c R s R c 8/20

9 EoS of dark energy Einstein eqs. (background) H 2 = 8πG ρ m 3 F + FV(φ b) 2 a a + H2 = 8πGV φ + 2 (Effective) EoS of Dark Energy 3M Pl + 2β H φ M b 8πG eff,0 Pl 3 2 φ b 2 2β M Pl P DE ρ DE = w φ + 2H φ b ρ m + ρ DE 8πG eff,0 P DE G eff,0 G F w Ω DE = ρ DE + P DE ρ cr = 2β 3M Pl φ b H φ b H M Pl φ b 2 H 2 + F 0 F 1 Ω m Since φ b HΔφ, we get Δφ: variation of φ b in the last Hubble time 1 + w Ω DE Ο β M Pl Δφ 9/20

10 EoS of dark energy 1 + w DE Ω DE Ο β M Pl Δφ Δφ: variation of φ b from time t to t 0 t: past time at which z 1 Relation between the density and the minimum of the effective potential some celestial object e.g. galaxy cluster ρ c > ρ b t > ρ b t 0 φ c < φ b t < φ b t 0 β Δφ < β δφ M Pl M c t 0 Pl δφ c t 0 φ c φ b t 0 Consider an object with thin shell β M Pl δφ c t 0 < Φ N ε th < w DE Ω DE < Φ N Models with large 1 + w cannot have a thin shell?? 10/20

11 What is wrong? In a cosmological situation, the exterior solution of the Poisson equation does not satisfy the original Klein-Gordon equation since δφ 3H δφ + 2 a 2 δφ m b 2 δφ = 0 2 a 2 δφ O m b 2 δφ δφ, H δφ O H 2 δφ same order for models with m b O H Note that the interior solution need not be changed since m c H. For example, Starobinsky s model f R = R + λr s 1 + R R s 2 n 1 has m b H for small n, λ. Actually, it is in such models that w deviates appreciably from 1. For n = 2 and λ = 1, m b /H 3.1 and w /20

12 Our work Give a counterexample for the previous work Assume the background spacetime evolves as in wcdm model with w 1, and solve the scalar field equation around a spherical object. 2 steps: Construct the solution in w = 1 (de Sitter) case. Construct the solution in w 1 case perturbatively, up to first order in ε 1 + w. Conformal time η is used as time variable in order that ε 0 limit is well-defined. a t t 2 3ε, ε > 0 2 t rip t 3ε, ε < 0 t η a η η ε η t dt a t ε 0 a ds t e Ht t η a ds η η 1 12/20

13 w = 1 case Klein-Gordon equation outside the object δφ out 2Hδφ out + 2 m 2 b a 2 δφ out = 0 Find a solution which is smoothly connected to the interior solution δφ c, ar < R s δφ in = βρ c ar 2 3M Pl 2 + R s 3 ar 3 2 R s 2 + δφ c, R s < ar < R c If we make an ansatz δφ η, r = φ Har, using some one variable function φ u, the KG equation becomes an ODE: d 2 φ out u du 2 + 4u2 2 dφ out u u u 2 + m 2 b φ out u 1 du H u 2 1 = 0 with the following boundary conditions: φ out = φ in and u φ out = u φ in at u = HR c φ out 0 as u η H a a = ah r ar in the original Thin-shell solution u Har 13/20

14 w = 1 case The exterior solution is obtained as δφ out = βρ 2 cr c ε th HR c g α Har M Pl ε th M Pl β δφ c R c 2 ρ c R c R s R c where g α u φ α 2 u 2 Γ 3 + 2iα 4 Γ 1 + 2iα 4 φ α 1 u = 2 F iα 4 φ α 2 u = 1 u 2 F iα 4 Γ 3 2iα 4 Γ 1 2iα 4, 3 2iα 4, 1 2iα 4 ; 3 2 ; u2, ; 1 2 ; u2 φ α 1 u α m b H F 1 : hypergeometric function The constant before φ α 1 u is chosen so that g α u does not diverge at u = 1. 14/20

15 w = 1 case General form of the solution δφ φ φ b 0 R s R c surface horizon H 1 ar δφ c The ratio between the fifth force and the Newtonian force is of order ε th as in an ordinary thin-shell solution. Small fifth force! 15/20

16 w 1 case We choose the solution in w = 1 case δφ out = βρ 2 cr c ε th HR c g α Hr as a zeroth-order solution. M Pl Perturbative expansion with respect to ε = 1 + w δφ out = βρ 2 cr c ε th HR c g α Hr + εa η, r M Pl O(1) should be O(1) checked later KG equation for the perturbative part ε A + 2HA 2 m 2 b a 2 A = ε η 2 2C φ 3 Hrg α Hr 2C φ g α Hr where C φ is determined once a model is fixed. φ b φ b C φ η ε 16/20

17 w 1 case Again, assume a solution in the form of A η, r B(u) where u Hr = Har The perturbed KG equation is rewritten as an ODE d 2 B u du 2 + 4u2 2 db u u u 2 1 du + m 2 b B u H u 2 = j u 1 j u 2C φ 3 ug α u + 2C φ g α u u 2 1 The homogeneous solutions are already known the solutions in w = 1 case The inhomogeneous solutions can be obtained by the method of variation of parameters! 17/20

18 w 1 case Basis of the homogeneous solutions B B 1 1 u u φ φ 1 1 α α u u B 2 u g α u Inhomogeneous solution B u = C 1 B 1 u + C 2 B 2 u B 1 u We require that B does not diverge at u = 1 (ar = H 1 ) B = 0 at u = HR c (ar = R c ) 0 u db 2 W B 1 du B db 1 2 du du B u 2 W j + B 2 u du B 1 0 W j B p = B 1 u 1 u du B 2 W j + B 2 u u HR c du B 1 W j B 1 HR c B 2 HR c 1 HR c du B 2 W j 18/20

19 w 1 case The form of the solution (written again) δφ out = βρ 2 cr c ε th HR c g α Hr + εa η, r M Pl A η, r B u, u Hr = Har Plots of B p for various parameters of Starobinsky s model ε 1 + w Here again the fifth force is small. O 1 for all parameters 19/20

20 Summary The effective EoS parameter w deviates from 1 in f R gravity. In the previous work, the thin-shell solution was naively used to a cosmological situation and it was concluded that w must be extremely close to 1. This is incorrect because the time derivative becomes important in a cosmological scale. We took time derivative into consideration and constructed a scalar configuration with small fifth force in the case where w deviates appreciably from 1. Models with 1 + w O(0.1) can not be excluded by the fifth-force constraint. 20/20