EFTCAMB: exploring Large Scale Structure observables with viable - PowerPoint PPT Presentation

EFTCAMB: exploring Large Scale Structure observables with viable dark energy and modified gravity models Noemi Frusciante Instituto de Astrof´ ısica e Ciˆ encias do Espa¸ co, Faculdade de Ciˆ encias da Universidade de Lisboa, Portugal 9th Feb 2018, GC2018, YITP, Kyoto Univ.

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 , [Silvestri et al. PRD 87, 104015 (2013)] • Parametrized Post Friedmann framework, [Baker et al. , PRD 87, 024015 (2013)] • Effective Field Theory of Cosmic Acceleration, [Gubitosi et al. JCAP 1302 (2013) 032 Bloomfield et al. JCAP 1308 (2013) 010] • Horndeski and beyond parametrizations , [Bellini & Sawicki, JCAP 1407 (2014) 050 Gleyzes et al. JCAP 1502 (2015) 018 NF et al. JCAP 1607 (2016) no.07, 018 ] { µ, γ } , Horndeski and bH ⇒ EFT

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: d 4 x √− g � m 2 � 2 (1 + Ω( t )) R + Λ( t ) − c ( t ) δ g 00 0 S EFT = ¯ ¯ + M 4 M 3 M 2 2 ( t ) 1 ( t ) 2 ( t ) ( δ g 00 ) 2 − δ g 00 δ K − ( δ K ) 2 2 2 2 � ¯ ˆ M 2 M 2 ( t ) 3 ( t ) δ K µ ν δ K ν δ g 00 δ R + m 2 2 ( t ) h µν ∂ µ g 00 ∂ ν g 00 − µ + + S m [ χ i , g µν ] , 2 2 where e.g. δ A = A − A (0) , A (0) background value in FLRW M 2 and m 2 2 = − ¯ 3 = 2 ˆ • M 2 M 2 2 = 0: Horndeski (and all the models belonging to them); 2 + ¯ • M 2 M 2 3 = 0 and m 2 2 = 0 : Beyond Horndeski class of models; • m 2 2 � = 0: Lorentz violating theories (e.g. low-energy Hoˇ rava gravity).

Extensions • Additional linear operators m 5 ( t ) ¯ λ 1 ( t )( δ R ) 2 , λ 2 ( t ) δ R µ ν δ R ν δ R δ K , µ , 2 λ 3 ( t ) δ R h µν ∇ µ ∂ ν g 00 , λ 4 ( t ) h µν ∂ µ g 00 ∇ 2 ∂ ν g 00 , λ 6 ( t ) h µν ∇ µ R ij ∇ ν R ij , λ 5 ( t ) h µν ∇ µ R∇ ν R , λ 7 ( t ) h µν ∂ µ g 00 ∇ 4 ∂ ν g 00 , λ 8 ( t ) h µν ∇ 2 R∇ µ ∂ ν g 00 [J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, JCAP 1308, 025 (2013) NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 ] • beyond the linear order 3 ( t )( δ g 00 ) 3 , M 1 ( t )( δ K ) 3 , M 4 M 3 1 ( t )( δ g 00 ) 2 δ K , 4 ( t ) δ g 00 ( δ K ) 2 , M 2 M 2 5 ( t )( δ g 00 ) 2 δ R , M 2 6 ( t ) δ g 00 δ K µ ν δ K ν µ , µ δ K µ M 4 ( t ) δ g 00 δ R δ K , M 2 ( t ) δ K ν λ δ K λ M 3 ( t ) δ K δ K ν µ δ K µ ν , ν M 5 ( t ) δ g 00 δ K µ ν δ R ν m 2 3 ( t ) h µν ( ∂ µ g 00 ∂ ν g 00 ) δ g 00 µ , [ NF, G. Papadomanolakis, JCAP 1712 (2017) no.12, 014 ]

General Mapping Let us introduce the ADM metric: ds 2 = − N 2 dt 2 + h ij ( dx i + N i dt )( dx j + N j dt ) , a general Lagrangian can be written as follows: L = L ( N , R , S , K , Z , U , Z 1 , Z 2 , α 1 , α 2 , α 3 , α 4 , α 5 ; t ) , where S = K µν K µν , Z = R µν R µν , U = R µν K µν , Z 1 = ∇ i R∇ i R , Z 2 = ∇ i R jk ∇ i R jk , α 1 = a i a i , α 2 = a i ∆ a i , α 3 = R∇ i a i , α 4 = a i ∆ 2 a i , α 5 = ∆ R∇ i a i , [R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D 23 , no. 13, 1443008 (2015)] d 4 x √− gL in unitary gauge and expand it • Write the general action � up to second order in perturbations; • Write the EFT action in ADM notation; • Compare the two actions.

General Mapping Ω( t ) = 2 c ( t ) = 1 E − 2 E ˙ 2( ˙ F − L N ) + ( H ˙ E − ¨ E − 1 , H ) , m 2 0 Λ( t ) = ¯ L + ˙ F + 3 H F − (6 H 2 E + 2 ¨ E + 4 H ˙ E + 4 ˙ ¯ M 2 H E ) , 2 ( t ) = −A − 2 E , 2 ( t ) = 1 � L N + L NN � − c ¯ 1 ( t ) = −B − 2 ˙ ¯ M 4 M 3 M 2 2 , E , 3 ( t ) = − 2 L S + 2 E , 2 2 λ 1 ( t ) = G 2 ( t ) = L α 1 ˆ m 2 M 2 ( t ) = D , 4 , m 5 ( t ) = 2 C , ¯ 2 , λ 3 ( t ) = L α 3 λ 4 ( t ) = L α 2 λ 2 ( t ) = L Z , λ 5 ( t ) = L Z 1 , 2 , 4 , λ 7 ( t ) = L α 4 λ 8 ( t ) = L α 5 λ 6 ( t ) = L Z 2 , 4 , 2 . 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 (2016) no.07, 018 ]

Example: Minimally coupled quintessence The action with the scalar field φ : � m 2 d 4 x √− g 2 R − 1 � � 0 2 ∂ ν φ∂ ν φ − V ( φ ) S φ = , Apply unitary gauge and ADM formalism ⇓ � ˙ � d 4 x √− g m 2 φ 2 � + 1 0 ( t ) 0 R + S − K 2 � � S φ = − V ( φ 0 ) , N 2 2 2 Apply the general mapping recipe ⇓ ˙ ˙ φ 2 φ 2 0 0 Ω( t ) = 0 , c ( t ) = 2 , Λ( t ) = 2 − V ( φ 0 ) . [NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 ]

Stability conditions Let us consider the following second order action for more than one scalar fields 1 � S (2) = d 3 kdta 3 � � χ t A ˙ ˙ χ − ˙ χ − k 2 � χ t G � χ t B � χ t M � � � � χ − � χ , (2 π ) 3 χ t = ( φ 1 , φ 2 , ... ) . where � In order to avoid instabilities one has to demand: • no-Ghost condition: positive kinetic term; • no-Gradient condition: c 2 s , i > 0 , • no-tachyonic instability: assure the Hamiltonian to be bounded from below, then, we demand | µ i ( t , 0) | � H 2 . [A. De Felice, NF and G. Papadomanolakis, JCAP 1703 (2017) no.03, 027 ]

Stability conditions for the tensor modes The EFT action for tensor modes can be written as ij ) 2 − c 2 1 � d 3 kdt a 3 A T ( t ) � T ( t , k ) � S T (2) (˙ h T k 2 ( h T ij ) 2 EFT = , (2 π ) 3 a 2 8 with A T ( t ) = m 2 0 (1 + Ω) − ¯ M 2 3 , λ 2 k 2 a 2 + λ 6 k 4 c 2 c 2 a 4 T ( t , k ) = ¯ T ( t ) − 8 , 0 (1 + Ω) − ¯ m 2 M 2 3 m 2 0 (1 + Ω) c 2 ¯ T ( t ) = , 0 (1 + Ω) − ¯ m 2 M 2 3 Stability conditions • no-Ghost instability: A T > 0, • No gradient instability: positive speed of propagation c 2 T > 0. [NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 ]

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 2 s , d = 0 , (3 c 2 c 2 s ( F 3 F 2 1 + 3 F 2 2 F 1 ) − 2 a 2 F 2 − 4 B 2 2 � � � � s − 1) ρ r ρ d 2 G 11 12 F 2 − 16 c 2 s B 2 13 F 2 2 ρ d = 0 for Horndeski: they completely decouple. • they change the no-tachyonic conditions. [(in vacuum) NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 (in matter) A. De Felice, NF, G. Papadomanolakis, JCAP 1703 (2017) no.03, 027]

EFTCAMB website: http://www.eftcamb.org/ B. Hu, M. Raveri, NF, A. Silvestri, PRD 89 (2014) 103530, M. Raveri, B. Hu, NF, A. Silvestri, PRD 90 (2014) 043513

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 Hoˇ rava 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 (2018) no.2, 023520]

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:1405.3590[astro-ph.IM]

Constraining power of viability conditions -0.5 -0.80 -0.6 -0.996 -0.85 -0.997 -0.8 -0.90 -0.998 -1.0 -0.95 -0.999 -1.2 -1.00 -1.000 -1.4 -1.05 -1.5 -1.001 -0.5 -0.3 -0.1 0.0 0.1 0.3 0.5 -0.025 0.0 0.025 0.05 0.075 -4.5 -3.5 -2.75 -2.0 -1.0 Designer f(R) on wCDM: • w 0 ∈ ( − 1 , − 0 . 94) 95% C . L . Planck+WP+BAO , • w 0 ∈ ( − 1 , − 0 . 9997) 95% C . L . Planck+WP+BAO+lensing . [M. Raveri, B. Hu, NF, A. Silvestri, PRD 90 (2014) 4, 043513 ]

After GW170817 Horndeski action reduces to d 4 x √− g [ K ( φ, X ) + G 3 ( φ, X ) � φ + G 4 ( φ ) R ] , � S rH = [P. Creminelli and F. Vernizzi, Phys. Rev. Lett. 119, 251302 (2017) ] In terms of EFT functions we only have: Ω, ¯ M 3 1 , M 4 2 0.8 γ 0 1 = 0 . 1 M2 0.8 γ 0 1 = − 0 . 1 γ 0 1 = 0 0.4 Unstable 0.6 | ∆ C TT | w a ℓ σ ℓ 0 0.4 -0.4 0.2 -0.8 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 10 1 10 2 w 0 ℓ IF ¯ M 3 1 � = 0 → M 4 2 � = 0 [In preparation: NF, S. Peirone, N. Lima, S. Cansas]

Can we trust quasi static approximation? QS approximation: 1 + M 2 ( a ) a 2 1 + g 2 ( a ) + M 2 ( a ) a 2 1 1 k 2 k 2 µ ( a , k ) = , Σ( a , k ) = . g 1 ( a ) + M 2 ( a ) a 2 g 1 ( a ) + M 2 ( a ) a 2 1 + Ω 2(1 + Ω) k 2 k 2 [In preparation: NF, S. Peirone, N. Lima, S. Cansas]