Backreaction as an explanation for Dark Energy?

with some remarks on cosmological perturbation theory

James M. Bardeen

University of Washington

The Very Early Universe 25 Years On

Cambridge, December 17, 2007

3+1 Approach to Cosmological Perturbations

Ref: J. M. Bardeen in Cosmology and Particle Physics, ed. Li-Zhi Fang and A. Zee (Gordon and Breach 1988)

Background homogeneity and isotropy:

First-order perturbed geometry for scalar perturbations:

Metric: ds2 =−N02dt2 +a2 dr2 / 1−Cr2( )+ r2dΩ2( ),

Expansion rate: H ≡&a

aN0

,

Perturbations completely in terms of spatially gauge invariant variables.

Lapse N =N0 1+α( ),

Spatial curvature 3Rji =

2Ca2 δ j

i − Δϕ +4Ca2 ϕ

⎛⎝⎜

⎞⎠⎟δ j

i −DiDjϕ ,

Total energy-momentum tensor:

Time gauge transformations:

Energy density E =E0 +ε,Momentum density J i =Diψ ,

Stress tensor Sji = P0 +π( )δ j

i + DiDj −13δ j

iΔ⎛⎝⎜

⎞⎠⎟σ.

Extrinsic curvature

K =K ii =−3H +κ,

K ji −

13δ j

iK =− DiDj −13δ j

iΔ⎛⎝⎜

⎞⎠⎟χ,

κ =−3 &ϕ / N0 −Hα( )−Δχ.

%α =α − N0T( )g/ N0 , %ϕ = ϕ − H N0T( ),

%χ = χ − N0T( ), %κ =κ + 3 &HT + N0ΔT ,

%ε = ε − &E0T = ε + 3H E0 + P0( ) N0T( ), %ψ =ψ + E0 + P0( ) N0T( ),

%π = π − &P0T = π +dP0

dE0

3H E0 + P0( ) N0T( ), %σ = σ .

Gauge choices:

The gauge transformation to a particular gauge is singular if and when the coefficient of T in the gauge condition vanishes. If so, the gauge is badly behaved there.

Gauge-invariant variables:Combine two or more variables to make a gauge-invariant combination, e.g.,

In most cases the physical meaning of a gauge-invariant variable is gauge-dependent.

%α =0 (synchronous), %ϕ = 0 (uniform curvature), %χ = 0 (zero shear),

%κ = 0 (uniform expansion), %ε = 0 (uniform energy density), %ψ = 0 (comoving).

ΦA ≡ α − &χ / N0 , ΦH ≡ ϕ − Hχ , Vzs ≡ψ

E0 + P0

+ χ ,

ε c ≡ ε − 3Hψ , ϕ c ≡ ϕ +H

E0 + P0

ψ , ε zs = ε + 3H E0 + P0( )χ ,

ς ≡ ϕ +ε

3 E0 + P0( ), η ≡ π −

dP0

dE0

ε , etc.

Background evolution equations:

First-order perturbation equations:

supplement with other matter and/or field evolution equations as appropriate.

H 2 =

8π3

GE0 −Ca2 , &H / N0 =−4πG E0 + P0( )+

Ca2 , &E0 =−3 E0 + P0( )H .

spatial metric evolution &ϕ / N0 =Hα − κ +Δχ( ) / 3;

Einstein equations

Δ +3Ca2

⎛⎝⎜

⎞⎠⎟ϕ +Hκ =−4πGε, Δ +

3Ca2

⎛⎝⎜

⎞⎠⎟χ +κ =−12πGψ ,

&κ / N0 +2Hκ + Δ + &H / N0( )α =−4πG ε + 3π( ),

DiDj −13δ j

iΔ⎛⎝⎜

⎞⎠⎟

&χ / N0 +Hχ −α −ϕ −8πGσ( ) =0;

local energy-momentum conservation

&ε / N0 + 3H ε +π( )+ E0 +P0( ) 3Hα −κ( )+ Δψ =0,

&ψ / N0 + 3Hψ + E0 + P0( )α +π +23

Δ +3Ca2

⎛⎝⎜

⎞⎠⎟σ =0;

Example of a single scalar field:

WΦ=dV

dΦ, Φ = Φ0 +φ;

background 1

N0

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

g

+ 3H&Φ0

N0

+dV

dΦ= 0;

first-order

1

N0

&φ

N0

⎛

⎝⎜⎞

⎠⎟

g

+ 3H&φ

N0

− Δφ +d 2V

dΦ2 φ =2α

N0

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

g

+&α &Φ0

N02 +κ

&Φ0

N0

;

energy-momentum tensor

E0 =1

2

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

2

+V , P0 =1

2

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

2

−V , E0 + P0 =&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

2

;

ε =&Φ0

&φ

N02 −

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

2

α +dV

dΦφ, ψ = −

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟φ,

π =&Φ0

&φ

N02 −

&Φ0

N0

⎛

⎝⎜⎞

⎠⎟

2

α −dV

dΦφ, σ = 0.

Solution Strategies

• There is no particular virtue in using gauge-invariant variables to carry out a calculation. They do facilitate transforming results from one gauge to another.

• Be carefult to avoid gauge singularities, such as arise in the comoving gauge if and when . The synchronous gauge is good in this respect and greatly simplifies the matter/field dynamical equations.

• Be careful in choosing a gauge and in choosing which of the redundant Einstein equations and matter evolution equations to use, to ensure that the the problem is well posed numerically, without near cancellations between large terms in the equations or in extracting the physics. This is mainly an issue when .

• While certainly not the only “conserved quantity” when , the gauge-invariant variable introduced in BST is perhaps best suited as a measure of the overall amplitude of the perturbation. In the long wavelength limit any change in is of order the average over an e-folding of expansion of the non-adiabatic stress perturbation ( and/or (k/a)2σ) divided by E0+P0.

E0 +P0 =0

k / aH <<1

k / aH <<1

Backreaction and Dark Energy

The Claim (Buchert, Celerier, Rasanen, Kolb et al, Wiltshire, etc.):

The average expansion in a locally inhomogeneous universe behaves differently than expected from the Friedmann equation based on the large scale average energy density. Due to the non-linearity of the Einstein equations spatial averaging and solving the Einstein equations do not commute (Ellis).

Observations of the CMB radiation indicate that the primordial amplitude of perturbations, the amplitude of curvature potential fluctuations, which in a matter-dominated universe correspond to time-independent fluctuations in the Newtonian potential on scales small compared to the Hubble radius, is very small, about 10-5. However, density perturbations grow and become non-linear, first on smaller scales, and at present on scales the order of 100 Mpc, leading to formation of structure in the universe. Can non-linearity in the density cause the average expansion to deviate enough from background Einstein-deSitter model to convert the Einstein-deSitter deceleration into the effective acceleration inferred from the high-Z Type Ia supernovae magnitude-redshift relation?

Counter-arguments (Ishibashi and Wald, Flanagan):

The local dynamics of a matter-dominated universe should be Newtonian to a good approximation, as long as potential perturbations and peculiar velocities are non-relativistic, which they are both from direct observation and as inferred from the CMB anisotropy. Since Newtonian gravity is linear, averaging and evolution do commute in the Newtonian limit and should commute to a good approximation in general relativity. Any relativistic corrections should be much to small to turn Einstein-deSitter deceleration into an effective acceleration.

In very local regions, where black holes are forming, etc., deviations from Newton gravity may be large, but by Birkhoff’s theorem in GR longer range gravitational interactions should be independent of the internal structure of compact objects.

Simulations based on local Newtonian dynamics and a global zero-curvature CDM model with acceleration seem to give a very good account of all observations of large scale structure as well as the supernovae data.

The Buchert equations (see Buchert gr-qc/0707.2153):

Exact GR equations constraining the evolution of averaged quantities assuming a zero-pressure dust energy-momentum tensor. Averaging is weighted by proper volume on hypersurfaces orthogonal to the dust worldlines. Define:

The equations are indeterminate. They say nothing about the time dependence of QD and whether QD can become large enough to make the average expansion accelerate. Also, these equations become invalid once the dust evolves to form caustics, which generically happens as the density perturbations become large.

aD t( ) ∝VD t( )1/3 , ρD=

ρ t1( )aD

3 =MD

VD

, QD ≡23

θ − θ D( )2−2 σ 2

D.

Equations:

&aD

aD

⎛

⎝⎜⎞

⎠⎟

2

=8πG3

ρD−16

QD −16

3RD,

&&aD

aD

=−4πG3

ρD+13QD,

1

aD6

ddt

aD6QD( )+

1aD

2

ddt

aD2 3R

D( ) =0.

Lemaitre-Tolman-Bondi (LTB) Models

Spherically symmetric (zero pressure) dust,

metric ds2 =−dt2 +b t,r( )2 dr2 +R t,r( )2 dΩ2 ,

&R= r2k r( )+2m r( )

R, b=

′RΓ r( )

, Γ = 1+ r2k r( ).

Choose comoving radius coordinate r such that m r( ) =29

r3.

With u≡32

k R / r ,

94

k 3/2 t−t0 r( )( ) =u 1+u2 −sinh−1u k> 0

sin−1u−u 1−u2 k< 0

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪≅23

u3 1±15

u2 +K⎛⎝⎜

⎞⎠⎟ u<<1,

and setting t0 r( ) =0, b=1Γ

Rr+

r ′kk

trR+94

k−Rr

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥.

The scalar curvature is 3R=−2krR

⎛⎝⎜

⎞⎠⎟2

−22k+ r ′k

bΓrR

⎛⎝⎜

⎞⎠⎟.

Initial Conditions

In cosmological perturbation theory with an Einstein-deSitter background the primordial amplitude of the curvature potential perturbation in a comoving gauge is the same as the gauge-invariant amplitude . If the background scale factor is S(t) = t2/3 consistent with S = R/r as t 0 in the LTB solution,

3R=−4

r2S2 r2 ′( )′ =−2

r2S2 r3k( )′ ⇒ k=2r

′ .

Consider the class of models with

=−a 1−cr2( ) 1−r2( )2, k r( ) =4a 2 +c−3cr2( ) 1−r2( ), 0 ≤r ≤1,

and r( ) =k r( ) =0, r >1.

With k 0( ) > 0 the matter expands more rapidly and becomes underdense near the center. If c>1 there is an outer region which expands more slowly than Einstein-deSitter, part of which becomes overdense. If a void develops near the center, a caustic must eventually develop away from the center sooner or later. Where the caustic forms, the density becomes infinite and the dust solution

breaks down. Any discontinuity in k r( ) would imply a caustic or a separation is present right from t=0, which is why we force continuity at r =1.

Dust Shell Evolution

Once a caustic forms, assume all matter flowing into it stays in an infinitesmally thin shell.

The shell is characterized by its circumferential radius Rsh as a function of

its own proper time τ, its internal "rest mass" msh, and its position at a

given τ in the interior and exterior LTB spacetimes, t± τ( ), r± τ( ). Note

that R±=Rsh, but m+ −m−=29

r+3 −r−

3( ) ≠msh. The Israel junction

conditions give the equations

dRsh

dτ⎛⎝⎜

⎞⎠⎟2

=m+ −m−

msh

⎛

⎝⎜⎞

⎠⎟

2

−1+m−+m+

R+

msh

2R⎛⎝⎜

⎞⎠⎟2

,

dmsh

dτ=23

r+2

Γ+

dt+dτ

dr+dτ

−r−

2

Γ−

dt−dτ

dr−dτ

⎡

⎣⎢

⎤

⎦⎥,

b±

dr±dτ

=

m+ −m−

msh

mmsh

2R⎛

⎝⎜⎞

⎠⎟

2

−Γ±2

Γ±dRsh

dτ+ &R±

m+ −m−

msh

mmsh

2R⎛

⎝⎜⎞

⎠⎟

, dt±dτ

= 1+ b±

dr±dτ

⎛⎝⎜

⎞⎠⎟2

.

Application to the Nambu-Tanimoto model (gr-qc/0507057), which is still cited as evidence for getting acceleration out of backreaction:

Problems:• Shell crossing starts immediately at t = 0, so the full LTB solution is never valid.• Assuming a surface layer shell forms at the interface, the outer LTB region is

completely swallowed up by the shell before it starts to recollapse.• Volume averaging over the LTB regions makes no sense, since most of the

mass ends up in the shell, and a completely empty region opens up between the outer LTB region and the EdS region. Averaging over the LTB regions has nothing to do with an average cosmological expansion.

• The shell does start to expand significantly faster than the EdS region once the outer LTB region is swallowed, but this is a smaller deceleration, not an acceleration.

• All of the dynamics is Newtonian to a very good approximation once t >> 1. The LTB regions deviate from EdS expansion only at t >> 1, if |k|r2 << 1.

• Genuinely relativistic back-reaction effects are completely negligible.

Two uniform curvature LTB regions are combined, an inner region (0 ≤r ≤r0 ) with k(r) =k1 > 0 and an outer region with k(r) =k2 < 0. These are embedded in an EdS model for r >1. As the outer region starts to collapse, they find a volume average accelerated expansion in the Buchert sense.

Conclusions

• Exact GR calculations indicate that non-linear backreaction modifying average expansion rates is completely insignificant in our universe. Newtonian gravity is a perfectly adequate description of dynamics on sub-horizon scales (but clearly evident only in a Newtonian gauge).

• A close to horizon-scale perturbation close to spherically symmetric about our location could modify the supernova magnitude-redshift relation to mimic dark energy, but the primordial perturbation amplitude would have to be ~ a thousand times larger than than what is seen in CMB anisotropy (e.g. Biswas, et al 2006, Vanderveld, et al 2006).

• Effects of inhomogeneities on light propagation (weak lensing) would in principle dim distant sources on average, but estimates by Bonvin, et al (2006) and Vanderveld, et al (2007) indicate that the effect is much too small to mimic apparent acceleration.

• Various modifications of GR remain on the table, but are they any less contrived than a cosmological constant?