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arXiv:1112.2995v1 [astro-ph.CO] 13 Dec 2011

Prepared for submission to JCAP

A gradient expansion for cosmological

backreaction

Kari EnqvistaShaun HotchkissaGerasimos Rigopoulosb

aUniversity of Helsinki and Helsinki Institute of Physics, P.O.Box 64, FIN-00014, University

of Helsinki, Finland

bInstitute for Theoretical Particle Physics and Cosmology, RWTH Aachen, D - 52056, Ger-

many

Abstract. We address the issue of cosmological backreaction from non-linear structure for-

mation by constructing an approximation for the time evolved metric of a dust dominated

universe based on a gradient expansion.Our metric begins as a perturbation of a flat

Friedmann-Robertson-Walker state described by a nearly scale invariant, Gaussian, power-

law distribution, and evolves in time until non-linear structures have formed. After describing

and attempting to control for certain complications in the implementation of this approach,

this metric then forms a working model of the universe. We numerically calculate the evo-

lution of the average scale factor in this model and hence the backreaction. We argue that,

despite its limitations, this model is more realistic than previous models that have confronted

the issue of backreaction. We find that the effects of backreaction in this model can be as

large as 5−10% of the background. This suggests that a proper understanding of the effects

of backreaction could be important for precision cosmology. However, in the most realistic

implementations of the model, the backreaction is never large enough to be responsible for

the observed accelerated expansion of the universe.

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Contents

1Introduction

1

2 The inhomogeneous CDM Universe as a series in spatial gradients

2.1Hamilton-Jacobi for CDM

2.2 The spherical case

2.3Potential complications for application to backreaction

2.4Backreaction in the synchronous gauge

2

3

8

9

10

3Numerical implementation

3.1Setting up the initial gravitational potential Φ(x)

3.2Dealing with the complications and presenting the backreaction

3.2.1The almost UV divergence

3.2.2Fixing the metric

3.2.3 Two definitions of Q and a

3.2.4Backreaction with artificial local isotropy

3.2.5Backreaction with the squared metric

3.2.6What our model universes look like

11

11

13

13

14

16

17

20

23

4Discussion

4.1Potential improvements to the model

4.2Conclusions

26

26

27

5Acknowledgements28

A The squared metric29

B Fixing the metric continuously29

1Introduction

The issue of cosmological backreaction has fueled a lively debate in the literature. The ques-

tion is: can inhomogeneities in our universe backreact and affect the average dynamics within

our causal horizon such that the observed acceleration can be attributed to their influence?

If the answer is positive then the mystery of the cosmological constant will have found a

solution requiring no new physics but which will nevertheless demonstrate the subtlety of

gravitational phenomena. Even if the answer is negative, backreaction will be operative

at some level due to the non-linear nature of gravity and its effects may be visible in the

new generation of cosmological observations. Either way, calculating its magnitude is an

important cosmological question.

The problem with assessing the magnitude of this backreaction lies with the complexities

of the non-linear structures forming under gravity in the late universe. For example, using

second order perturbation theory falls short of providing an answer since the effect is expected

to emerge in the non-linear regime where perturbation theory breaks down. Attempts to

model the non-linear structures involving toy models of voids and overdensities may be

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useful for developing intuition but are nevertheless simplistic. N-body simulations imply

that the effect is small but they are Newtonian with small backreaction by construction;

the backreaction is a purely relativistic effect. A strong argument for the smallness of the

effect uses the fact that even with non-linear overdensities, the local metric perturbations

and peculiar velocities are still much smaller than unity (away from black holes) and that a

perturbative FRW framework should therefore hold. However, counterarguments have been

put forward involving subtleties in the choice of background employed in these treatments.

For references finding a a small backreaction and validating the perturbative nature of the

metric in the real universe see [2], [3], [4], [5] while arguments for significant backreaction can

be found in [6], [7], [8], [9]. For recent reviews of the subject with more extensive references

see [10–20].

In this paper we discuss cosmological backreaction in a novel manner, with a fully rel-

ativistic framework for a universe that begins as a perturbed Friedmann-Robertson-Walker,

Ω = 1, CDM universe. To be precise, we employ a gradient expansion to express the metric

as a series of terms with an increasing number of spatial gradients and coefficients which

are functions of proper time. The initial conditions are the standard adiabatic and Gaussian

post-inflationary primordial perturbations. We use the synchronous gauge: our coordinate

lines comove with CDM particles and our time hypersurfaces are labeled by their proper time.

Of course the gradient series has to be truncated and thus does not capture the developing

non-linearities entirely realistically. However, we argue that it provides a well motivated

model for the true geometry which can be made increasingly accurate in principle. Further-

more, it extends into the non-linear regime, describing the collapse of initial over-densities

and the rarefaction of initial under-densities which go on to form the voids dominating the

cosmological volume.

Here is the outline of the rest of the paper: In the following section we obtain the series

solution for the metric in a gradient expansion using a Hamilton-Jacobi formulation. This ap-

proach, first developed in e.g. [21] and [22], simplifies the calculation significantly compared

to a more straightforward expansion of the standard Einstein equations. Our treatment fol-

lows a slightly different logical development. Then, in section 3, we apply these results to the

backreaction problem by numerically evaluating the evolution of the average scale factor and

the backreaction parameter Q. We find that, up to certain qualifications which we explain,

backreaction leads to non-negligible deviations from the unperturbed background model. It

appears insufficient to lead to acceleration but our results indicate that the effect might be

relevant, indeed crucial, for precision cosmology at the percent level and the exploration of

the dark sector. We summarize and discuss our findings in section 4 where future directions

are also laid out.

2 The inhomogeneous CDM Universe as a series in spatial gradients

The study of the backreaction of cosmological inhomogeneities indicates that relevant effects,

if any, will necessarily be in the relativistic and non-linear regime, in the realm beyond cos-

mological perturbation theory. In this regime deviations from a homogeneous FRW Universe

can be studied using an expansion in spatial gradients. Furthermore, calculations beyond

the lowest quasi-homogeneous order can be drastically simplified via the application of such

a gradient expansion in the Hamilton-Jacobi formulation of gravitational dynamics. We

present this formulation below and use it to obtain the first terms in the gradient series for

the metric. We then compare with an exact spherically symmetric solution to gauge the

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accuracy of the approximation and develop some intuition. The section ends with a descrip-

tion of how to apply this gradient expansion for evaluating the backreaction of cosmological

inhomogeneities by controlling a number of complications that arise in such an application.

2.1Hamilton-Jacobi for CDM

The Hamilton-Jacobi approach requires a Hamiltonian formulation which in turn requires an

action. The action for gravity and a non-relativistic matter fluid (dust) can be written as

S =

?

d4x√−g

?1

2κ

(4)R −1

2ρ(gµν∂µχ∂νχ + 1)

?

(2.1)

where χ is a potential for the 4-velocity of the fluid, Uµ= −gµν∂νχ, and ρ, the energy

density, acts as a Lagrange multiplier whose variation ensures that UµUµ= −1. Variation

wrt χ gives the continuity equation ∇µ(ρ∂µχ) = 0, while variation of the dust part of the

action wrt gµνgives the usual energy momentum tensor Tµν= ρUµUν. Gravity is described

by the standard Einstein-Hilbert term.

Let us now use the ADM decomposition for the metric and develop a Hamiltonian

formalism. The metric is written as

g00= −N2+ hijNiNj,g0i= γijNj,gij= γij,(2.2)

with the inverse

g00= −1

N2,g0i=Ni

N2,gij= γij−NiNj

N2

.(2.3)

By defining the canonical momenta

πij≡

πχ≡δS

δS

δ˙ γij

δ ˙ χ= ρ√γ

=

√γ

2κ

Eij− γijE

N

,(2.4)

?

1 + γij∂iχ∂jχ,(2.5)

with

Eij=1

2˙ γij− ∇(iNj),E = hijEij,(2.6)

we can bring the action to the canonical form

S =

?

d4x

?

πχ∂χ

∂t+ πij∂γij

∂t

− NU − NiUi

?

(2.7)

where

U =2κ

√γ

?

πijπij−π2

2

?

−

√γ

2κR + πχ?

1 + γij∂iχ∂jχ(2.8)

and

Ui= −2∇kπk

i+ πχ∂iχ(2.9)

As is well known [23], the action (2.7) does not define a conventional Hamiltonian system

since solutions to the equations of motion must set the “Hamiltonian” to zero, U = 0, a fact

which reflects the time reparametrization invariance of General Relativity. One can however

obtain a conventional Hamiltonian formulation by using as a time parameter one of the scalar

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fields of the system [23]; in this case the obvious choice is to use the hypersurfaces of χ as

time hypersurfaces, setting∂χ

∂t= 1, N = 1 and ∂iχ = 0. Thus, the metric takes the form

ds2= −dt2+ γij(t,x)dxidxj

(2.10)

and the spatial coordinate lines comove with the matter. We then impose the energy con-

straint U = 0 and from this equation determine the (non-zero) πχwhich now plays the role

of the Hamiltonian density: −πχ≡ H. In particular

U = 0 ⇒ H =2κ

√γπijπkl

?

γikγjl−1

2γijγkl

?

−

√γ

2κR,

(2.11)

and the action becomes

S =

?

dtd3x

?

πij∂γij

∂t

− H + 2Ni∇kπki

?

. (2.12)

In this form it defines a constrained Hamiltonian system where the canonical momentum πij

is constrained to be covariantly conserved

∇kπki= 0. (2.13)

Let us now apply the Hamilton-Jacobi approach to the Hamiltonian system (2.12).

Writing

πij=

δS

δγij

(2.14)

we obtain the Hamilton-Jacobi equation

∂S

∂t+

?

d3x

?2κ

√γ

δS

δγij

δS

δγkl

?

γikγjl−1

2γijγkl

?

−

√γ

2κR

?

= 0, (2.15)

which is a single partial differential equation for S as a functional of γijand a function of t.

Once S[t,γij] is determined the metric can be obtained from

∂γij

∂tδγkl

=

2

√γ

δS

(2γikγjl− γijγkl)(2.16)

Let us now turn to the remaining constraint (2.13). It will be automatically satisfied if

S =?d3x√γ F(t,γij) where F(t,γij) is a scalar function of the metric making S invariant

a covariantly conserved tensor which will thus satisfy (2.13). In 3 dimensions all information

about the spacetime curvature is contained in the Ricci tensor and F can be written as

[21, 22]

F = −2H(t) + J(t)R + L1(t)R2+ L2(t)RijRij+ ... ,

a series in powers of the 3-D Ricci curvature involving an increasing number in gradients of

γij. H(t) will turn out to be the Hubble rate. Note that under our assumptions this form

is essentially unique. The Hamilton-Jacobi equation (2.15) can now be solved separately for

under 3-D diffeomorphisms. Indeed, the variation of such a functional wrt the metric will yield

(2.17)

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