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EFTCAMB: exploring Large Scale Structure observables with viable dark energy and modified gravity models Noemi Frusciante Instituto de Astrof sica e Ci encias do Espa co, Faculdade de Ci encias da Universidade de Lisboa, Portugal

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SLIDE 1

EFTCAMB: exploring Large Scale Structure

  • bservables 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.

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SLIDE 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,

[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

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SLIDE 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: SEFT =

  • d4x√−g

m2 2 (1 + Ω(t))R + Λ(t) − c(t)δg 00 +M4

2(t)

2 (δg 00)2 − ¯ M3

1(t)

2 δg 00δK − ¯ M2

2(t)

2 (δK)2 − ¯ M2

3(t)

2 δK µ

ν δK ν µ +

ˆ M2(t) 2 δg 00δR + m2

2(t)hµν∂µg 00∂νg 00

  • + Sm[χi, gµν],

where e.g. δA = A − A(0), A(0) background value in FLRW

  • M2

2 = − ¯

M2

3 = 2 ˆ

M2 and m2

2 = 0: Horndeski (and all the models

belonging to them);

  • M2

2 + ¯

M2

3 = 0 and m2 2 = 0 : Beyond Horndeski class of models;

  • m2

2 = 0: Lorentz violating theories (e.g. low-energy Hoˇ

rava gravity).

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SLIDE 4

Extensions

  • Additional linear operators

¯ m5(t) 2 δRδK , λ1(t)(δR)2 , λ2(t)δRµ

ν δRν µ ,

λ3(t)δRhµν∇µ∂νg00 , λ4(t)hµν∂µg00∇2∂νg00 , λ5(t)hµν∇µR∇νR , λ6(t)hµν∇µRij∇νRij , λ7(t)hµν∂µg00∇4∂νg00 , λ8(t)hµν∇2R∇µ∂νg00 [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

M4

3(t)(δg00)3 ,

M1(t)(δK)3 , M3

1(t)(δg00)2δK ,

M2

4(t)δg00(δK)2 ,

M2

5(t)(δg00)2δR ,

M2

6(t)δg00δK µ ν δK ν µ ,

M2(t)δK ν

µδK µ λ δK λ ν ,

M3(t)δKδK ν

µδK µ ν

M4(t)δg00δRδK , M5(t)δg00δK µ

ν δRν µ ,

m2

3(t)hµν(∂µg00∂νg00)δg00

[ NF, G. Papadomanolakis, JCAP 1712 (2017) no.12, 014 ]

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SLIDE 5

General Mapping

Let us introduce the ADM metric: ds2 = −N2dt2 + hij(dxi + Nidt)(dxj + Njdt) , a general Lagrangian can be written as follows: L = L(N, R, S, K, Z, U, Z1, Z2, α1, α2, α3, α4, α5; t) , where S = KµνK µν , Z = RµνRµν , U = RµνK µν , Z1 = ∇iR∇iR , Z2 = ∇iRjk∇iRjk , α1 = aiai , α2 = ai∆ai , α3 = R∇iai , α4 = ai∆2ai , α5 = ∆R∇iai,

[R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D 23, no. 13, 1443008 (2015)]

  • Write the general action
  • d4x√−gL in unitary gauge and expand it

up to second order in perturbations;

  • Write the EFT action in ADM notation;
  • Compare the two actions.
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SLIDE 6

General Mapping

Ω(t) = 2 m2 E − 1, c(t) = 1 2( ˙ F − LN) + (H ˙ E − ¨ E − 2E ˙ H), Λ(t) = ¯ L + ˙ F + 3HF − (6H2E + 2 ¨ E + 4H ˙ E + 4 ˙ HE) , ¯ M2

2(t) = −A − 2E,

M4

2(t) = 1

2

  • LN + LNN

2

  • − c

2, ¯ M3

1(t) = −B − 2 ˙

E, ¯ M2

3(t) = −2LS + 2E,

m2

2(t) = Lα1

4 , ¯ m5(t) = 2C, ˆ M2(t) = D, λ1(t) = G 2 , λ2(t) = LZ, λ3(t) = Lα3 2 , λ4(t) = Lα2 4 , λ5(t) = LZ1, λ6(t) = LZ2, λ7(t) = Lα4 4 , λ8(t) = Lα5 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 ]

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SLIDE 7

Example: Minimally coupled quintessence

The action with the scalar field φ: Sφ =

  • d4x√−g

m2 2 R − 1 2∂νφ∂νφ − V (φ)

  • ,

Apply unitary gauge and ADM formalism ⇓ Sφ =

  • d4x√−g
  • m2

2

  • R + S − K 2

+ 1 N2 ˙ φ2

0(t)

2 − V (φ0)

  • ,

Apply the general mapping recipe ⇓ Ω(t) = 0, c(t) = ˙ φ2 2 , Λ(t) = ˙ φ2 2 − V (φ0).

[NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 ]

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SLIDE 8

Stability conditions

Let us consider the following second order action for more than one scalar fields S(2) = 1 (2π)3

  • d3kdta3

˙

  • χtA ˙
  • χ − k2

χtG χ − ˙

  • χtB

χ − χtM χ

  • ,

where χt = (φ1, φ2, ...). In order to avoid instabilities one has to demand:

  • no-Ghost condition: positive kinetic term;
  • no-Gradient condition: c2

s,i > 0,

  • no-tachyonic instability: assure the Hamiltonian to be bounded from

below, then, we demand |µi(t, 0)| H2.

[A. De Felice, NF and G. Papadomanolakis, JCAP 1703 (2017) no.03, 027 ]

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SLIDE 9

Stability conditions for the tensor modes

The EFT action for tensor modes can be written as ST (2)

EFT =

1 (2π)3

  • d3kdt a3 AT(t)

8

  • (˙

hT

ij )2 − c2 T(t, k)

a2 k2(hT

ij )2

  • ,

with AT(t) = m2

0(1 + Ω) − ¯

M2

3 ,

c2

T(t, k) = ¯

c2

T(t) − 8

λ2 k2

a2 + λ6 k4 a4

m2

0(1 + Ω) − ¯

M2

3

, ¯ c2

T(t) =

m2

0(1 + Ω)

m2

0(1 + Ω) − ¯

M2

3

, Stability conditions

  • no-Ghost instability: AT > 0,
  • No gradient instability: positive speed of propagation c2

T > 0.

[NF, G. Papadomanolakis and A. Silvestri, JCAP 1607 (2016) no.07, 018 ]

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SLIDE 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: c2

s,d = 0 ,

(3c2

s − 1)ρr

  • ρd
  • c2

s (F3F 2 1 + 3F 2 2 F1) − 2a2F 2 2 G11

  • − 4B2

12F2 2

−16c2

s B2 13F 2 2 ρd = 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]

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SLIDE 11

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
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SLIDE 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 Hoˇ rava gravity, Covariant Galileon, f(R)- Hu & Sawicki (soon), Reparametrized Horndeski (RPH);

  • Built-in: several choices for EFT functions & wDE(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]

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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]

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Constraining power of viability conditions

  • 0.5
  • 0.80
  • 0.996
  • 0.997
  • 0.998
  • 0.999
  • 1.000
  • 1.001
  • 0.85
  • 0.90
  • 0.95
  • 1.00
  • 1.05
  • 0.6
  • 0.8
  • 1.0
  • 1.2
  • 1.4
  • 1.5
  • 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:

  • w0 ∈ (−1, −0.94) 95%C.L.

Planck+WP+BAO,

  • w0 ∈ (−1, −0.9997) 95%C.L.

Planck+WP+BAO+lensing.

[M. Raveri, B. Hu, NF, A. Silvestri, PRD 90 (2014) 4, 043513 ]

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SLIDE 15

After GW170817

Horndeski action reduces to SrH =

  • d4x√−g [K(φ, X) + G3(φ, X)φ + G4(φ)R] ,

[P. Creminelli and F. Vernizzi, Phys. Rev. Lett. 119, 251302 (2017) ]

In terms of EFT functions we only have: Ω, ¯ M3

1, M4 2

101 102

ℓ

0.2 0.4 0.6 0.8

|∆C TT

ℓ

| σℓ

M2

  • 1.2
  • 1.0
  • 0.8
  • 0.6
  • 0.4
  • 0.2

w0

  • 0.8
  • 0.4

0.4 0.8

wa

γ0

1 = 0.1

γ0

1 = −0.1

γ0

1 = 0

Unstable

IF ¯ M3

1 = 0 →M4 2 = 0

[In preparation: NF, S. Peirone, N. Lima, S. Cansas]

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SLIDE 16

Can we trust quasi static approximation?

QS approximation:

µ(a, k) = 1 1 + Ω 1 + M2(a) a2

k2

g1(a) + M2(a) a2

k2

, Σ(a, k) = 1 2(1 + Ω) 1 + g2(a) + M2(a) a2

k2

g1(a) + M2(a) a2

k2

. [In preparation: NF, S. Peirone, N. Lima, S. Cansas]

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SLIDE 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, Hoˇ rava, 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.

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SLIDE 18