Consistent Parameterization of Modified Gravity

Transcription

1 arxiv Consistent Parameterization of Modified Gravity Tessa Baker Oxford University

2 Outline The Parameterized Post-Friedmann form. An alternative construction for modified gravity. Hidden assumptions - why the parameterization matters. Effects on observables. Conclusions.

3 The PPF Approach (Hu & Sawicki, 2007) PPF = Parameterized Post-Friedmann formalism. A framework to test for deviations from GR. Parameterize around the perturbed field equations: modified Poisson eqn. : 2 Φ =4πG eff a 2 ρ M M slip relation : simplest choice : Φ = η(a, k)ψ This slip relation is often taken to apply on all scales.

4 An Alternative Approach Write field equations as δg µν =8πG 0 a 2 δt µν + δu µν Consider the case of metric theories - any action built from curvature invariants. Expand Uμν in terms of Bardeen variables, where ˆΓ = 1 ( ˆΦ + H ˆΨ) k a 2 δu 0 0 = a 2 δu 0 i = a 2 δu i i = a 2 δu i j = N 2 n=0 N 2 n=0 N 1 n=0 N 1 n=0 k 2 n ( A n ˆΦ(n) + F nˆγ(n) ) k 1 n ( B n ˆΦ(n) + I nˆγ(n) ) k 2 n ( C n ˆΦ(n) + J nˆγ(n) ) k n ( D n ˆΦ(n) + K nˆγ(n) ) functions of cosmological background (a, ρ...)

5 Building a Parameterization (Skordis, 2009) Perturbed Bianchi identity: δ ( µ U µ ν ) = 0... ]ˆΦ +... ] ˆΦ ]ˆΓ +... ] ˆΓ +... =0 Coefficients must vanish (otherwise obtain source-free evolution equations). Leads to constraint equations for A0, B0, etc.

6 Building a Parameterization (Skordis, 2009) Perturbed Bianchi identity: δ ( µ U µ ν ) = 0... ]ˆΦ +... ] ˆΦ ]ˆΓ +... ] ˆΓ +... =0 Coefficients must vanish (otherwise obtain source-free evolution equations). Leads to constraint equations for A0, B0, etc. Results: G eff = G 0 1 g(a, k) where g(a, k) = 1 2 ( A 0 +3 H ) k B 0 g(a, k) = B 1 2k 2 ( 3(H 2 Ḣ)+k2) + H k D 1

7 The Two Parameterizations g(a, k) = B ( 1 2k 2 3(H 2 Ḣ)+k2) + H k D 1 (TB, Ferreira, Skordis & Zuntz, 2011)

8 The Two Parameterizations g(a, k) = B ( 1 2k 2 3(H 2 Ḣ)+k2) + H k D 1 (TB, Ferreira, Skordis & Zuntz, 2011) Simple slip relation ˆΦ = η(a, k) ˆΨ D1= 0 Need B1 0 U θ = k 2 B 0 ˆΦ + kb1 ˆΦ +... Higher-order theory

9 The Two Parameterizations g(a, k) = B ( 1 2k 2 3(H 2 Ḣ)+k2) + H k D 1 (TB, Ferreira, Skordis & Zuntz, 2011) Simple slip relation ˆΦ = η(a, k) ˆΨ D1= 0 Need B1 0 U θ = k 2 B 0 ˆΦ + kb1 ˆΦ +... Higher-order theory More complex slip relation ˆΦ ˆΨ g(a, k) =ζ(a, k)ˆφ+ H ˆΦ Can have B1 = 0 Second-order theory

10 The Two Parameterizations g(a, k) = B ( 1 2k 2 3(H 2 Ḣ)+k2) + H k D 1 (TB, Ferreira, Skordis & Zuntz, 2011) Simple slip relation ˆΦ = η(a, k) ˆΨ D1= 0 Need B1 0 U θ = k 2 B 0 ˆΦ + kb1 ˆΦ +... Higher-order theory More complex slip relation ˆΦ ˆΨ g(a, k) =ζ(a, k)ˆφ+ H ˆΦ Can have B1 = 0 Second-order theory However, the PPF form works on quasistatic scales: 2 ˆΦ =4πGeff a 2 ρ M M ˆΦ = η(a, k) ˆΨ

11 Example: f(r) gravity in PPF format Poisson equation: 2k 2 Φ =8π G 0 a 2 f ρ M M +3 ] R Φ + HΨ f R f R f RRδR f R 3 ( ) H 2 Ḣ + k 2] Slip relation: Φ Ψ = f RR f R δr Apply the quasistatic limit Φ = η(a, k)ψ η(a, k) = 3 + 2Q 3 + 4Q where Q =3 k2 a 2 f RR f R = ( λc λ ) 2 G eff (a, k) G 0 = 1 f R 1+ 1 η ( f R H f R k 2 ) + 3 k 2 (Ḣ H2 ) ( 1 1 η ) ] 1

12 Problems on the Horizon A two-function PPF parameterization works well on quasistatic scales. BUT The simple slip relation cannot be extended to larger scales. (unless you are specifically interested in a higher-derivative theory) Correct treatment of horizon scales is important for calculation of ISW and P(k) in Einstein-Boltzmann codes. Yoo, See also: Bonvin & Durrer 2011, Challinor & Lewis 2011, Wands & Slosar 2009.

13 Effects on the CMB Late-time modified gravity affects the ISW plateau.

14 Effects on the CMB (2nd-order case) Lensing deflection angle power spectrum with ACT data: (Das et al., 2011)

15 Growth of Structure (2nd-order case) Matter power spectrum Growth rate: f(k, z) = d ln M (k, a) d lna

16 Summary A parameterized framework can be used to constrain modified gravity without recourse to a specific theory. The simple slip relation cannot be extended to larger scales without specific implications for the order of the theory. Theories with extra fields require a more complex parameterization. The choice of parameterization leads to differences in observables, which will in turn affect constraints on model parameters/functions. Details in arxiv tessa.baker@astro.ox.ac.uk