Can Backreaction Mimic Dark Energy?

with comments on using large scale inhomogeneity to modify the DL-z relation

James M. Bardeen

University of Washington

CIfAR/Linde Fest

Stanford, March 6, 2008

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 metric deviations from Robertson-Walker are linear and the local dynamics of a matter-dominated universe are Newtonian to an excellent approximation almost everywhere, as long as the Newtonian potential perturbations and peculiar velocities are non-relativistic, which is true 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 too 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 are independent of the internal structure of compact objects.

Simulations based on local Newtonian dynamics and a global zero-curvature CDM model with acceleration 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 in a comoving domain D:

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

, D t1 aD

3 MD

VD, QD

2

3

D 2 2 2

D.

Equations:

&aDaD

2

8G

3

D

1

6QD

1

63R

D,

&&aDaD

4G3

D

1

3QD ,

1

aD6

d

dtaD

6QD 1

aD2

d

dtaD

2 3RD 0.

Lemaitre-Tolman-Bondi (LTB) Models

Spherically symmetric (zero pressure) dust,

metric ds2 dt 2 b t,r 2 dr2 R t,r 2 d2 ,

&R r2k r 2m r R

, bR

r , 1 r2k r .

Choose a comoving radius coordinate r such that m r 2

9r 3.

With u 3

2k R / r ,

9

4k

3/2t t0 r u 1u2 sinh 1u k 0

sin 1u u 1 u2 k 0

2

3u3 1

1

5u2 K

u 1,

and setting t0 r 0, b1

R

rr kktr

R

9

4k

R

r

.

The hypersurface scalar curvature is 3R 2kr

R

2

22k r kb

r

R

.

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,

3R4

r2S2 r2 2

r2S2 r3k k

2

r .

Consider the class of models with

a 1 cr2 1 r2 2 , k r 4a 2 c 3cr2 1 r2 , 0r 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 , its internal "rest mass" msh , and its position at a

given in the interior and exterior LTB spacetimes, t , r . Note

that RRsh , but m m 2

9r

3 r3 msh . The Israel junction

conditions give the equations

dRsh

d

2

m mmsh

2

1m mR

msh

2R

2

,

dmsh

d

2

3

r2

dtddrdr

2

dtddrd

,

bdrd

m mmsh

mmsh

2R

2

2

dRsh

d &R

m mmsh

mmsh

2R

, dtd 1 b

drd

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 (0r 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.

Large Scale Voids

A carefully constructed large scale void centered on our location in the universe can modify the DL-Z relation in such a way as to roughly mimic a homogeneous Lambda CDM model, with more rapid expansion in the local universe modifying redshifts along the past light cone instead of accelerating expansion of a homogeneous universe. Several authors have explored such models, and some have made claims that they can be made to work with not wildly unreasonable assumptions (see, e.g., Iguchi, et al astro-ph/0112419, Vanderveld, et al astro-ph/0602476, Garfinkle gr-qc/0605088, Biswas, et al astro-ph/0606703, Alexander, et al astro-ph/0712.0370).

LTB Models

Integrate the geodesic equation and the optical scalar equation for sources on the backward light cone from an observer at the origin to find, for each source, the redshift Z and the area A of the light beam at the observer.

The luminosity distance is DL (1 Z )A1/2 with A br 2 near the source.

Model 1: k r a

1 r2 r12 r2, a 0.075, r1 0.1, Z 1.0 at r 0.95.

Model 2: k r a

r12 r2 1 r2

, a 0.1, r1 0.1, Z 1.0 at r 0.80.

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.

• Direct evidence for dark energy from baryon acoustic oscillations and the integrated Sachs-Wolfe effect is becoming more convincing.