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
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 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