Using the Post-Newtonian Formalism to Understand Theories of Gravity in Cosmology

Using the post-Newtonian formalism to understand theories of gravity in cosmology

Viraj A A Sanghai

Dalhousie University, Halifax, Canada

6th March, 2018 T. Clifton and VAAS Based on arXiv: 1803.01157 (today!) VAAS and T. Clifton Based on arXiv: 1610.08039, Class. Quan. Grav. 34 (2017) 065003 VAAS, P. Fleury and T. Clifton Based on arXiv: 1705.02328, JCAP 07 (2017) 028 .

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 1 / 27 In memory of Jonathon Gleason

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 2 / 27 Outline

1 Parameterizing theories of gravity in cosmology Post-Newtonian formalism Parameterized post-Newtonian Formalism Slip and gravitational constant parameter Parameterized post-Newtonian Cosmology Results

2 Inhomogeneities in cosmology Building a post-Newtonian cosmology Ray-tracing and Hubble diagrams in Post-Newtonian Cosmology Results

3 Summary and Future Work

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 3 / 27 Parameterizing theories of gravity in cosmology Post-Newtonian formalism Post-Newtonian formalism

In the limit of slow motions (v c) and weak gravitational ﬁelds (Φ 1), we use an expansion parameter

|v| ≡ 1, c where v = v α is the 3-velocity associated with the matter ﬁelds. The orders of smallness of other quantities are ρ v 2 ∼ ∼ U ∼ Π ∼ 2 p

(Will, 1993)

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 4 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Formalism Parameterized post-Newtonian (PPN) Formalism

At leading order the PPN metric is given by

2 2 µ ν ds ≡ (−1 + 2αU)dt + (1 + 2γU)δµνdx dx ,

where U ∼ 2 is the Newtonian gravitational potential. 2 ∇ U ≡ −4πGρM . We assume U has asymptotically ﬂat solutions. α and γ are constant PPN parameters. (Will, 1993)

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 5 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Formalism Applicable scales of PPN

Credit: Adapted from NASA / JPL images Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 6 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Formalism Cosmological scales

Credit: Adapted from SDSS / Millenium Simulation images

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 7 / 27 Parameterizing theories of gravity in cosmology Slip and gravitational constant parameter Slip ζ and gravitational constant parameter µ

For a perturbed FLRW background " # (dx2 + dy 2 + dz2) ds2 = a2 −(1 − 2Φ)dτ 2 + (1 + 2Ψ) 1 22 1 + 4 κr General parameterization 1 4πG ∇2Ψ − H2Φ − HΨ˙ + κΨ = − µ δρ a2 3 3 and 1 4πG ∇2Φ + 2H˙ Φ + HΦ˙ + Ψ¨ + HΨ˙ = − µ(1 − ζ) δρ a2 , 3 3

Small-scale quasi static regime (Amendola et al. 2008) ∇2Ψ ∇2Φ µ = − and ζ = 1 − . 4πG δρ a2 ∇2Ψ

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 8 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Weak Fields to Cosmology arXiv: 1803.01157

Coordinate Transformations: perturbed FLRW → perturbed Minkowski 1 1 Φ = h − H˙ r 2 2 00 2 1 1 Ψ = δµνh + H2 + κ r 2 , 6 µν 4

where the post-Newtonian metric is given by gab = ηab + hab. Post-Newtonian potentials 1 h = 2αU + α r˜2 and δµνh = 6γU + γ r˜2 , 00 3 c µν c where U is the Newtonian gravitational potential.

Note: Now α, αC , γ and γC are functions of time only.

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 9 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

ΛCDM arXiv: 1610.08039

The PPNC parameters are given by

α = γ = 1,

αC = Λ,

Λ γ = − . C 2

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 10 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Scalar-tensor theories of gravity arXiv: 1610.08039

The Lagrangian of such a theory is given by 1 ω(φ) ab L = φR − g φ;aφ − 2φΛ(φ) + Lm(ψ, g ) , 16πG φ ;b ab The PPNC parameters are given by 2ω + 4 1 2ω + 2 1 α = , γ = 2ω + 3 φ¯ 2ω + 3 φ¯ ¯ ¯¨ 2ω + 4 X 4πGρI 2ω + 2 X 12πGpI ¯ ω(φ) ¯˙2 φ αC = − + − + Λ(φ) − φ − 2ω + 3 φ¯ 2ω + 3 φ¯ φ¯2 φ¯ I I 1 ω0φ¯˙2 + + φ¯Λ0(φ¯) , 2ω + 3 2φ¯

0 2ω + 2 X 4πGρI 1 X 12πGpI ω ¯˙2 ¯ 0 ¯ γC = − − + φ + φΛ (φ) 2ω + 3 φ¯ 2ω + 3 φ¯ 2φ¯ I I ω(φ¯) 2ω + 1 φ¯¨ − φ¯˙2 − Λ(φ¯) − . 4φ¯2 4ω + 6 2φ¯

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 11 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Other examples arXiv: 1610.08039

Dark energy models like quintessence 1 ρ = φ˙2 + V (φ) I 2 1 p = φ˙2 − V (φ) I 2 Vector-tensor theories of gravity

1 a a b ab a;b L = R+ωAaA R+ηA A R −F F +τA A +Lm(ψ, g ) 16πG ab ab a;b ab

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 12 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Background Cosmological Expansion arXiv: 1803.01157

Acceleration equation

2 4πG αc a H˙ = − α hρi a2 + 3 3 Constraint equation

2 8πG 2γc a H2 = γ hρi a2 − − κ 3 3 Additional constraint

dγ dγ 4πGhρi = α + 2γ + C α − γ + C C d ln a d ln a

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 13 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Small-scale cosmological perturbations arXiv: 1803.01157

Poisson-like equations in an FLRW background

∇2Φ = −4πGa2αδρ , ∇2Ψ = −4πGa2γδρ .

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 14 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Horizon-scale perturbations arXiv: 1803.01157

Parameterizing the large-scale pertubations

δ −H2Φ − HΨ˙ + κΨ = − H2 − H˙ + κ 3 and ! δ 2HH˙ − H¨ 2H˙ Φ + HΦ˙ + Ψ¨ + HΨ˙ = 3 H

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 15 / 27 Parameterizing theories of gravity in cosmology Parameterized post-Newtonian Cosmology

Horizon-scale perturbations arXiv: 1803.01157

Parameterizing the large-scale pertubations

4πG 1 1 −H2Φ − HΨ˙ + κΨ = − δρ a2 γ − γ0 + γ0 3 3 12πGhρi c and 4πG 2 1 0 1 0 2H˙ Φ + HΦ˙ + Ψ¨ + HΨ˙ = − δρ a α − α + α c . 3 3 12πGhρi

where primes denote d/d ln a.

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 16 / 27 Parameterizing theories of gravity in cosmology Results

Main Results arXiv: 1803.01157

Slip ζ and gravitational constant parameter µ Small-scale limit

lim µ = γ k→∞ α lim ζ = 1 − , k→∞ γ Large-scale limit

1 0 1 0 lim µ = γ − γ + γc k→0 3 12πGhρi

0 0 α − α /3 + αc /12πGhρi lim ζ = 1 − 0 0 . k→0 γ − γ /3 + γc /12πGhρi

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 17 / 27 Parameterizing theories of gravity in cosmology Results

Illustration arXiv: 1803.01157

0.000

-0.001 1 -0.002 μ - , ζ

-0.003

-0.004

10-5 10-4 0.001 0.010 0.100 k/ Mpc -1 −1 −4 −1 Figure: The small-scale (k & 0.01 Mpc ) and large-scale (k . 10 Mpc ) limits of the ζ (red) and µ (blue) parameters at the present time, connected by an interpolating tanh function (dotted). Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 18 / 27 Parameterizing theories of gravity in cosmology Results Compare PPNC with other approaches

Eﬀective Field Theory (EFT) approaches (Bellini et al. 2014) Parameterize the action directly - valid on linear scales.

In general contains 5 independent functions of time αM , αB , αT , αK , αH for perturbations.

Parameterized post-Friedmann (PPF) approaches (Baker et al. 2011) Parameterize the ﬁeld equations directly Contains a large number of space-time dependent functions

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 19 / 27 Inhomogeneities in cosmology Building a post-Newtonian cosmology

2 Inhomogeneities in cosmology Building a post-Newtonian cosmology Ray-tracing and Hubble diagrams in Post-Newtonian Cosmology Results

3 Summary and Future Work

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 20 / 27 Inhomogeneities in cosmology Building a post-Newtonian cosmology

Building a post-Newtonian cosmology Arxiv:1503.08747

Figure: This ﬁgure was produced using an image from D. J. Croton et al., 2005

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 21 / 27 Inhomogeneities in cosmology Ray-tracing and Hubble diagrams in Post-Newtonian Cosmology Illustrations

(a) Point Mass (b) Homogeneous Halo

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 22 / 27 Inhomogeneities in cosmology Results

Hubble Diagrams Arxiv: 1705.02328

0.3 45 galaxy simulation

44 halo simulation 0.2 43 ΛCDMΩ m=0.22

42 0.3 μ 0.1 μ - 41 galaxy simulation 40 0.0 halo simulation 39 ΛCDM Ωm=0.3 38 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z z

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 23 / 27 Inhomogeneities in cosmology Results Comparison with Weinberg-Kibble-Lieu Theorem Arxiv: 1705.02328

Z 2 1 2 ¯2 hDA(λ)iΩ ≡ DA dΩ = DA(λ) . 4π λ=cst

galaxy simulation halo simulation

0.20 〈DA〉Ω/DA-1 0.00000 2 2 〈DA 〉Ω/DA -1 -0.00005 0.15 -0.00010

-0.00015 0.10 -0.00020

0.05 -0.00025 〈DA〉Ω/DA-1 2 -0.00030 2 〈DA 〉Ω/DA -1 0.00 -0.00035 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

-H0λ -H0λ

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 24 / 27 Inhomogeneities in cosmology Results Comparison with Stochastic Lensing Formalism Arxiv: 1705.02328

galaxy simulation 0.12 0.00000 empty beam 0.10 -0.00005 stochastic lensing

1 -0.00010 -

0.08 1 A - A D / -0.00015 〉 D / A

0.06 〉 A D 〈

D -0.00020 〈 0.04 -0.00025 halo simulation 0.02 0.00030 - stochastic lensing 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z z

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 25 / 27 Inhomogeneities in cosmology Results Comparison with Numerical Relativity Arxiv: 1705.02328

25 EdS ○ geodesic A

20 ΛCDM △ geodesic B ○ ○ Milne ○ △ 15 △ ○ △

L EBA ○ △ D

0 ○ △

H ○ △ 10 ○ △ △ ○△ ○△ ○△ 5 ○△ ○△ ○△ ○△ ○△ ○△ ○△ 0 ○△ ○△ 0 1 2 3 4 5 6 z

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Figure 2: Top: the paths of geodesics A and B in one of the BHL cells. The geodesics run close to the symmetry loci at all times. Middle: photon redshift as a function of the coordinate distance from the source. Bottom: luminosity distance as a function of the coordinate distance from the source. The error bars are indicated by shaded regions (when not visible, they are included in the width of the curves).

– 15 – Summary and Future Work Summary and Future Work

We have constructed a parametrization that requires only 4 functions of time to parameterize a large class of metric theories of gravity in cosmology. In GR, we have calculated the eﬀect of small-scale inhomogeneities on observables in these type of models. Apply it to more general distributions of matter Constrain the parameters using observations T. Clifton and VAAS Based on arXiv: 1803.01157 VAAS and T. Clifton Based on arXiv: 1610.08039, Class. Quan. Grav. 34 (2017) 065003 VAAS, P. Fleury and T. Clifton Based on arXiv: 1705.02328, JCAP 07 (2017) 028 .

Viraj A A Sanghai (Dal) Post-Newtonian Cosmology 27 / 27