EFTCAMB: exploring Large Scale Structure observables with viable dark energy and modified gravity models

Transcription

1 EFTCAMB: exploring Large Scale Structure observables with viable dark energy and modified gravity models Noemi Frusciante Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Portugal 9th Feb 018, GC018, YITP, Kyoto Univ.

2 Test gravity on cosmological scales Observations: extra component Dark Energy Pletora of Dark Energy & Modified Gravity models focus on models with one extra scalar DoF Model independent parametrizations to test gravity: Growth functions: µ and γ, Parametrized Post Friedmann framework, Effective Field Theory of Cosmic Acceleration, Horndeski and beyond parametrizations, {µ, γ}, Horndeski and bh EFT [Silvestri et al. PRD 87, (013)] [Baker et al., PRD 87, (013)] [Gubitosi et al. JCAP 130 (013) 03 Bloomfield et al. JCAP 1308 (013) 010] [Bellini & Sawicki, JCAP 1407 (014) 050 Gleyzes et al. JCAP 150 (015) 018 NF et al. JCAP 1607 (016) no.07, 018 ]

3 EFT for dark energy and modified gravity: the action Operators are time-dependent spatial diffeomorphisms invariants; Unitary gauge: the extra scalar d.o.f. does not appear directly; The action: S EFT = d 4 x g + M4 (t) M 3 (t) (δg 00 ) M 3 1 (t) { m 0 00 (1 + Ω(t))R + Λ(t) c(t)δg δg 00 δk M (t) (δk) δk ν µ δkµ ν + ˆM } (t) δg 00 δr + m (t)h µν µ g 00 ν g 00 + S m [χ i, g µν ], where e.g. δa = A A (0), A (0) background value in FLRW M = M 3 = ˆM and m = 0: Horndeski (and all the models belonging to them); M + M 3 = 0 and m = 0 : Beyond Horndeski class of models; m 0: Lorentz violating theories (e.g. low-energy Hořava gravity).

4 Extensions Additional linear operators m 5 (t) δrδk, λ 1 (t)(δr), λ (t)δr µ ν δr ν µ, λ 3 (t)δrh µν µ νg 00, λ 4 (t)h µν µg 00 νg 00, λ 5 (t)h µν µr νr, λ 6 (t)h µν µr ij νr ij, λ 7 (t)h µν µg 00 4 νg 00, λ 8 (t)h µν R µ νg 00 [J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, JCAP 1308, 05 (013) NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (016) no.07, 018 ] beyond the linear order M 4 3 (t)(δg 00 ) 3, M 1 (t)(δk) 3, M 3 1 (t)(δg 00 ) δk, M 4 (t)δg 00 (δk), M 5 (t)(δg 00 ) δr, M 6 (t)δg 00 δk µ ν δk ν µ, M (t)δk ν µ δk µ λ δk λ ν, M 3(t)δKδK ν µ δk µ ν M 4 (t)δg 00 δrδk, M 5 (t)δg 00 δk µ ν δr ν µ, m 3 (t)hµν ( µg 00 νg 00 )δg 00 [ NF, G. Papadomanolakis, JCAP 171 (017) no.1, 014 ]

5 General Mapping Let us introduce the ADM metric: ds = N dt + h ij (dx i + N i dt)(dx j + N j dt), a general Lagrangian can be written as follows: where L = L(N, R, S, K, Z, U, Z 1, Z, α 1, α, α 3, α 4, α 5 ; t), S = K µν K µν, Z = R µν R µν, U = R µν K µν, Z 1 = i R i R, Z = i R jk i R jk, α 1 = a i a i, α = a i a i, α 3 = R i a i, α 4 = a i a i, α 5 = R i a i, [R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D 3, no. 13, (015)] Write the general action d 4 x gl in unitary gauge and expand it up to second order in perturbations; Write the EFT action in ADM notation; Compare the two actions.

6 General Mapping Ω(t) = m 0 E 1, c(t) = 1 ( F L N ) + (H E Ë E H), Λ(t) = L + F + 3HF (6H E + Ë + 4HE + 4HE), M (t) = A E, M 4 (t) = 1 ( L N + L ) NN c, M3 1 (t) = B E, M 3 (t) = L S + E, m (t) = L α 1 4, m 5(t) = C, ˆM (t) = D, λ 1 (t) = G, λ (t) = L Z, λ 3 (t) = L α 3, λ 4(t) = L α 4, λ 5(t) = L Z1, λ 6 (t) = L Z, λ 7 (t) = L α 4 4, λ 8(t) = L α 5. where A, B, C, D, E, F, G are combinations of terms obtained by deriving the Lagrangian w.r.t. the main variables. [NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (016) no.07, 018 ]

7 Example: Minimally coupled quintessence The action with the scalar field φ: S φ = d 4 x [ m g 0 R 1 ] ν φ ν φ V (φ), S φ = Apply unitary gauge and ADM formalism d 4 x { m [ 0 g R + S K ] } + 1 φ 0 (t) N V (φ 0 ), Apply the general mapping recipe Ω(t) = 0, c(t) = φ 0, Λ(t) = φ 0 V (φ 0). [NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (016) no.07, 018 ]

8 Stability conditions Let us consider the following second order action for more than one scalar fields S () = 1 ( (π) 3 d 3 kdta 3 χ t A χ k χ t G χ χ ) t B χ χ t M χ, where χ t = (φ 1, φ,...). In order to avoid instabilities one has to demand: no-ghost condition: positive kinetic term; no-gradient condition: c s,i > 0, no-tachyonic instability: assure the Hamiltonian to be bounded from below, then, we demand µ i (t, 0) H. [A. De Felice, NF and G. Papadomanolakis, JCAP 1703 (017) no.03, 07 ]

9 Stability conditions for the tensor modes The EFT action for tensor modes can be written as S T () EFT = 1 (π) 3 d 3 kdt a 3 A [ ] T (t) 8 (ḣt ij ) c T (t, k) a k (hij T ) with, A T (t) = m 0(1 + Ω) M 3, k ct (t, k) = c T λ k a + λ 4 (t) 8 6 a 4 m0 (1 + Ω) M 3 c T m0 (1 + Ω) (t) = m0 (1 + Ω) M, 3, Stability conditions no-ghost instability: A T > 0, No gradient instability: positive speed of propagation ct > 0. [NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (016) no.07, 018 ]

10 The parameters space Matter fields: in general do not affect the no-ghost and speed conditions, only one exception: beyond Horndeski. In matter the speeds of propagation of the three DoFs are: c s,d = 0, (3c s 1)ρ r [ ρd ( c s (F 3 F 1 + 3F F 1 ) a F G 11 ) 4B 1 F ] 16c s B 13F ρ d = 0 for Horndeski: they completely decouple. they change the no-tachyonic conditions. [(in vacuum) NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (016) no.07, 018 (in matter) A. De Felice, NF, G. Papadomanolakis, JCAP 1703 (017) no.03, 07]

11 EFTCAMB website: B. Hu, M. Raveri, NF, A. Silvestri, PRD 89 (014) , M. Raveri, B. Hu, NF, A. Silvestri, PRD 90 (014)

12 EFTCAMB & EFTCosmoMC EFTCAMB evolves the full scalar and tensor perturbative equations without relying on QSA; EFTCAMB is compatible with massive neutrinos; Built-in models: designer-f(r), minimally couple quintessence, low-energy Hořava gravity, Covariant Galileon, f(r)- Hu & Sawicki (soon), Reparametrized Horndeski (RPH); Built-in: several choices for EFT functions & w DE (a); Built-in: Stability requirements viability priors for EFTCosmoMC; EFTCosmoMC: exploration of the parameter space performing comparison with several cosmological data sets; Validated with other EB codes, agreement at sub-percent level [Bellini et al., Phys.Rev. D97 (018) no., 0350]

13 The threefold face of EFTCAMB Model Background Mapping Perturbations PURE EFT / FULL MAPPING / / Other Parametrizations / / Built-in: ; To be implemented:. Numerical Notes: B. Hu, M. Raveri, NF, A. Silvestri, arxiv: [astro-ph.im]

14 Constraining power of viability conditions Designer f(r) on wcdm: w 0 ( 1, 0.94) 95%C.L. Planck+WP+BAO, w 0 ( 1, ) 95%C.L. Planck+WP+BAO+lensing. [M. Raveri, B. Hu, NF, A. Silvestri, PRD 90 (014) 4, ]

15 After GW Horndeski action reduces to S rh = d 4 x g [K(φ, X ) + G 3 (φ, X ) φ + G 4 (φ)r], [P. Creminelli and F. Vernizzi, Phys. Rev. Lett. 119, 5130 (017) ] In terms of EFT functions we only have: Ω, M 1 3, M Cl TT σl 0.4 M w a 0 γ1 0 = 0.1 γ1 0 = 0.1 γ1 0 = 0 Unstable l w 0 IF M M4 0 [In preparation: NF, S. Peirone, N. Lima, S. Cansas]

16 [In preparation: NF, S. Peirone, N. Lima, S. Cansas] Can we trust quasi static approximation? QS approximation: µ(a, k) = Ω 1 + M (a) a k g 1 (a) + M (a) a k, Σ(a, k) = g (a) + M (a) a k. (1 + Ω) g 1 (a) + M (a) a k

17 Conclusion In the context of DE/MG models it is important to have: a model independent parametrization: EFT approach for cosmic acceleration; a model independent parametrization preserving a DIRECT LINK with the most popular models used in cosmology (f(r), Galileon, Hořava, etc..); a general recipe for mapping any MG theory in EFT language; general stability requirements in vacuum and in matter (when exploring cosmological models); a Einstein-Boltzman code for DE/MG: EFTCAMB/EFTCosmoMC; built-in module which enforces the stability requirements Viability Priors.

18