Averaging Robertson-Walker Cosmologies Iain A. Brown Institut f ur - - PowerPoint PPT Presentation

▶
averaging robertson walker cosmologies
SMART_READER_LITE
LIVE PREVIEW

Averaging Robertson-Walker Cosmologies Iain A. Brown Institut f ur - - PowerPoint PPT Presentation

Jul 29, 2023 160 likes •464 views

Averaging Robertson-Walker Cosmologies Iain A. Brown Institut f ur Theoretische Physik, Universit at Heidelberg Backreaction from Perturbations, J. Behrend, IB and G. Robbers, JCAP 0801 013 Averaging Robertson-Walker

slide-1
SLIDE 1

1 / 23

Averaging Robertson-Walker Cosmologies

Iain A. Brown

Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg

“Backreaction from Perturbations”, J. Behrend, IB and G. Robbers, JCAP 0801 013 ‘Averaging Robertson-Walker Cosmologies”, IB, G. Robbers and J. Behrend, in preperation

Cosmo 08, Madison, 25th August 2008

slide-2
SLIDE 2

Motivation

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

2 / 23

slide-3
SLIDE 3

Standard Cosmology

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

3 / 23

■

Copernican Principle + CMB observations ⇒ Universe homogeneous and isotropic.

slide-4
SLIDE 4

Standard Cosmology

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

3 / 23

■

Copernican Principle + CMB observations ⇒ Universe homogeneous and isotropic.

■

Robertson-Walker cosmology: foliate spacetime with maximally-symmetric three-spaces

—

Line element: ds2 = −dt2 + a2(t)δijdxidxj

—

Friedmann equation: (˙

a/a)2 = (8πG/3)ρ + Λ/3

—

Raychaudhuri equation: ¨

a/a = −(4πG/3)(ρ + p) + Λ/3

—

Perturb metric with O(ǫ) ≈ 10−5

slide-5
SLIDE 5

Standard Cosmology

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

3 / 23

■

Copernican Principle + CMB observations ⇒ Universe homogeneous and isotropic.

■

Robertson-Walker cosmology: foliate spacetime with maximally-symmetric three-spaces

—

Line element: ds2 = −dt2 + a2(t)δijdxidxj

—

Friedmann equation: (˙

a/a)2 = (8πG/3)ρ + Λ/3

—

Raychaudhuri equation: ¨

a/a = −(4πG/3)(ρ + p) + Λ/3

—

Perturb metric with O(ǫ) ≈ 10−5

■

We have assumed the existence of an average and added perturbations

slide-6
SLIDE 6

Averaging in Cosmology

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

4 / 23

■

An implicit averaging in cosmology transfers local equations to global cosmology; should be made explicit

■

∂tρ = ∂t ρ ⇒ Na¨

ıve EFE for assumed averages does not reflect a true average of small-scale physics.

slide-7
SLIDE 7

Averaging in Cosmology

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary

4 / 23

■

An implicit averaging in cosmology transfers local equations to global cosmology; should be made explicit

■

∂tρ = ∂t ρ ⇒ Na¨

ıve EFE for assumed averages does not reflect a true average of small-scale physics.

■

We should be using

Gµν(gµν) = 8πG Tµν + Λ gµν

instead of

Gµν(gµν) = 8πG Tµν + Λ gµν .

■

“Backreaction” may not be dark energy, but all cosmological models should be properly averaged

■

Aim: Express Buchert equations in general form, apply to range of perturbed Robertson-Walker models from radiation domination to present day.

slide-8
SLIDE 8

Backreaction

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

5 / 23

slide-9
SLIDE 9

Formalism: 3+1 Split

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

6 / 23

■

Line-element: ds2 = −α2dt2 + hijdxidxj

■

Fluids (Λ, φ, b, CDM, γ, ν): ̺ = nµnνTµν, ji = −nµTiµ, Sij = Tij

■

Perfect fluids, Tµν = (ρ + p)uµuν + pgµν:

̺ = nµnνTµν = ρ (nµuµ)2 + p

  • (nµuµ)2 − 1
  • ,

S = T i

i = 3p + (ρ + p) uiui .

slide-10
SLIDE 10

Formalism: Buchert Averaging

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

7 / 23

■

Select average

A = 1 V

  • D

A √ hd3x,

Define averaged “scale factor” and Hubble rate by

3HD = 3 ˙ aD aD = ˙ V V = − 1 V

  • D

αK √ hd3x = − αK = H ,

■

Buchert equations:

˙ aD aD 2 = 8πG 3

  • α2̺
  • + Λ

3

  • α2

− 1 6 (QD + RD) ¨ aD aD = −4πG 3

  • α2(̺ + S)
  • + Λ

3

  • α2

+ 1 3 (QD + PD)

slide-11
SLIDE 11

Formalism: Modifications to Standard Cosmology

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

8 / 23

■

Kinematical “backreaction”:

QD =

  • α2

K2 − Ki

jKj i

  • − 2

3 αK2

■

Dynamical “backreaction”: PD = ˙

αK +

  • αDiDiα
  • ■

Curvature contribution: RD =

  • α2R
  • ■

Deviation from average density and pressure:

3T (a)

D

8πG =

  • α2̺(a)
  • − ρ(a),

3S(a)

D

4πG =

  • α2S(a)
  • − S(a)
slide-12
SLIDE 12

Formalism: Modifications to Standard Cosmology

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

9 / 23

■

The Buchert equations can then be written as

˙ aD aD 2 = 8πG 3

  • a

ρ(a) + Λ 3 + 8πG 3 ρeff, ¨ aD aD = −4πG 3

  • a
  • ρ(a) + S(a)
  • + Λ

3 − 4πG 3

  • ρeff + Seff
  • with effective correction fluid

8πG 3 ρeff =

  • a

T (a)

D

+

  • α2 − 1

Λ 3 − 1 6 (QD + RD) , 16πGpeff = 4

  • a

S(a)

D − 2Λ

  • α2 − 1
  • + 1

3 (RD − 3QD − 4PD) , weff = −1 3 RD − 3QD − 4PD + 12

a S(a) D − 6Λ

  • α2 − 1
  • RD + QD − 6

a T (a) D

− 2Λ α2 − 1 .

slide-13
SLIDE 13

Link to Perturbation Theory

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

10 / 23

■

Identify ADM and Newtonian co-ordinates (c.f. Mukhanov et. al.)

ds2 = −(1+2Ψ)dt2+a2(t)(1−2Φ)δijdxidxj = −α2dt2+hijdxidxj

■

aD(t) is “observational”, a(t) is “physical” – drawback of re-averaging

assumed average (Kolb, Marra, Matarrese 08; IB, Behrend, Robbers 08)

slide-14
SLIDE 14

Link to Perturbation Theory

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

10 / 23

■

Identify ADM and Newtonian co-ordinates (c.f. Mukhanov et. al.)

ds2 = −(1+2Ψ)dt2+a2(t)(1−2Φ)δijdxidxj = −α2dt2+hijdxidxj

■

aD(t) is “observational”, a(t) is “physical” – drawback of re-averaging

assumed average (Kolb, Marra, Matarrese 08; IB, Behrend, Robbers 08)

■

Quickly find

˙ aD aD = ˙ a a −

  • ˙

Φ (1 + 2Φ)

slide-15
SLIDE 15

Link to Perturbation Theory: Backreaction Terms

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

11 / 23

■

Kinematical and dynamical backreactions:

QD = 6

  • ˙

Φ2 −

  • ˙

Φ 2 , PD = 1 a2

  • ∇2Ψ − (∇Ψ)2 + 2Φ∇2Ψ − (∇Φ) · (∇Ψ)
  • +3 ˙

a a

  • ˙

Ψ − 2Ψ ˙ Ψ

  • − 3
  • ˙

Ψ ˙ Φ

  • ■

Curvature correction:

RD = 2 a2

  • 2∇2Φ + 3(∇Φ)2 + 4(2Φ + Ψ)∇2Φ
  • .
slide-16
SLIDE 16

Link to Perturbation Theory: Backreaction Terms

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

12 / 23

■

Fluid corrections:

TD = 8πG 3 ρ

  • δ + 2Ψ + (1 + w)a2v2 + 2Ψδ
  • ,

SD = 4πG 3 ρ

  • 3c2

sδ + 6wΨ + (1 + w)a2v2 + 6c2 sΨδ

slide-17
SLIDE 17

Link to Perturbation Theory: Backreaction Terms

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary

12 / 23

■

Fluid corrections:

TD = 8πG 3 ρ

  • δ + 2Ψ + (1 + w)a2v2 + 2Ψδ
  • ,

SD = 4πG 3 ρ

  • 3c2

sδ + 6wΨ + (1 + w)a2v2 + 6c2 sΨδ

  • ■

Note: alternative gauges – uniform density to simplify TD and SD, uniform curvature to remove RD, synchronous gauge to remove PD.

QD cannot be entirely removed except in EdS matter domination.

slide-18
SLIDE 18

Numerical Study

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

13 / 23

slide-19
SLIDE 19

Ergodic Averaging

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

14 / 23

■

Boltzmann codes are 1-d, averages are 3-d, so take D large enough to employ ergodic principle

slide-20
SLIDE 20

Ergodic Averaging

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

14 / 23

■

Boltzmann codes are 1-d, averages are 3-d, so take D large enough to employ ergodic principle

■

Corrections to standard case to be evaluated with cmbeasy:

QD = 6

  • Pψ(k)
  • ˙

Φ

  • 2

(dk/k) etc.

■

Integration domain k ∈ (1/η, 100Mpc−1)

slide-21
SLIDE 21

ΛCDM and EdS: Friedmann and Raychaudhuri Equations

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

15 / 23

1 10 100 1000 1+z 1e-08 1e-07 1e-06 1e-05 Modification ΛCDM, Friedmann ΛCDM, Raychaudhuri EdS, Friedmann EdS, Raychaudhuri

■

Impact at recombination ∼ 10−8 – potentially observable with Planck?

■

Boosts with Halofit not significant

slide-22
SLIDE 22

ΛCDM: w

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

16 / 23

0.01 0.1 1 10 100 z 0.01 weff

■

For z >∼ 100 find increase up to weff ≈ 0.2 at z ≈ 8000

slide-23
SLIDE 23

Quintessence Cosmology

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

17 / 23

■

Models tested

—

Early dark energy parameterisation

—

Exponential potential

—

Inverse power-law potential

■

Still linear anaylsis ⇒ still expect small impacts on the observed evolution

■

Expect weff to increase with dark matter perturbations – so weff clearest discriminant

slide-24
SLIDE 24

Early Dark Energy

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

18 / 23

■

Rough model of early dark energy; Ωz=0

φ

= 0.7, Ωz=∞

φ

= 0.05, w0 = −0.95

■

Very similar to ΛCDM; larger at present day, smaller at peak

1 10 100 1+z

  • 2e-05
  • 1e-05

1e-05 Modification Friedmann Raychaudhuri 0.01 0.1 1 10 100 z 0.01 0.015 weff

slide-25
SLIDE 25

Inverse Power Law Potential

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

19 / 23

■

Inverse-Power Law Potential (Ratra-Peebles); Ωφ = 0.118,

Ωb = 0.046, Ωc = 0.837, n = −2

■

Similar to but smaller than EdS for these parameters

1 10 100 1+z

  • 8e-05
  • 4e-05

4e-05 8e-05 Modification Friedmann Raychaudhuri 0.01 0.1 1 10 100 z 0.01 0.015 weff

slide-26
SLIDE 26

Exponential Potential

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

20 / 23

■

Exponential Potential; Ωφ = 0.200, Ωb = 0.041, Ωm = 0.759

■

Similar to inverse power law

1 10 100 1+z

  • 1e-05

1e-05 Modification Friedmann Raychaudhuri 0.01 0.1 1 10 100 z 0.01 0.015 weff

slide-27
SLIDE 27

Equations of State

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Potential Exponential Potential Equations of State Summary

21 / 23

0.01 0.1 1 10 100 z 0.01 0.015 weff

■

weff > 0 – as before acts against acceleration

■

But: this includes quintessence perturbations!

■

These differences far too small to observe, but smaller-scale study looks vital

slide-28
SLIDE 28

Summary

Motivation Backreaction Numerical Study Summary Summary

22 / 23

slide-29
SLIDE 29

Summary

Motivation Backreaction Numerical Study Summary Summary

23 / 23

■

Have expressed Buchert equations in multifluid form easily incorporated into general Boltzmann codes for wide variety of models

■

Differing linear models barely change impact on Friedmann equation;

  • n Raychaudhuri equation it’s similar and remains ∼ 10−5

■

Impact at recombination is close to observable anisotropies ⇒ possible chance of detection?

—

ΛCDM: weff ≈ 0.007

—

Early dark energy: weff ≈ 0.009

—

Exponential: weff ≈ 0.014

—

Inverse power law: weff ≈ 0.016

■

Equation of state from quintessence perturbations > −1: is there a problem with clustering quintessence models? Small scale study is needed (c.f. Wetterich ’02).

■

CMB observables?

■

Non-Linear models

■

Modified averaging procedure (Behrend/Nachtmann?)