arXiv:1112.2995v1 [astro-ph.CO] 13 Dec 2011
Prepared for submission to JCAP
A gradient expansion for cosmological
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-
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.
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
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
4.1Potential improvements to the model
A The squared metric29
B Fixing the metric continuously29
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
– 1 –
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 , , ,  while arguments for significant backreaction can
be found in , , , . For recent reviews of the subject with more extensive references
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
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.  and , 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
– 2 –
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
2ρ(gµν∂µχ∂νχ + 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
By defining the canonical momenta
δ ˙ χ= ρ√γ
1 + γij∂iχ∂jχ,(2.5)
2˙ γij− ∇(iNj),E = hijEij,(2.6)
we can bring the action to the canonical form
− NU − NiUi
2κR + πχ?
1 + γij∂iχ∂jχ(2.8)
As is well known , 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
– 3 –
fields of the system ; 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
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
and the action becomes
− H + 2Ni∇kπki
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).
we obtain the Hamilton-Jacobi equation
= 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
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
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
– 4 –
other features of the model. However, when comparing the model to precise measurements,
perhaps 10% of 10% is no longer an ignorably small quantity.
Let us close by summarizing our methodology and the conclusions we can draw from it.
We have approximated the evolving inhomogeneous synchronous gauge metric of a CDM
Ω = 1 universe with realistic initial conditions using a gradient expansion, and studied the
backreaction on the average dynamics. This gradient expansion is an approximation to the
true metric of such a universe which goes beyond the description of standard perturbation
theory. Instead of being limited by the smallness of the inhomogeneities, it has a temporal
range of validity set by the initial local 3-curvature. For timescales t > tcon the gradient
expansion does not converge anymore. This does not pose a real problem for studying
regions with positive initial curvature which collapse at around that timescale and eventually
virialize. However, it does present limitations in following the late time evolution of under-
dense regions with negative initial curvature. Nevertheless, if the late time evolution of such
regions is fixed in some prescribed manner, this approach offers a way to construct a realistic
model even for the late-time metric of such a universe.
Our methodology improves over previous approaches in that it is fully relativistic, goes
beyond perturbation theory and does not treat the universe as a collection of unconnected
under-dense and over-dense structures with high symmetry. Of course it does require mod-
eling of the late time behaviour of the under-dense regions which is not naturally captured
by the gradient expansion but such modeling can be simply parameterized and possibly in-
formed by observation. We have attempted to achieve this in a simple manner in this paper
but it can still be made more realistic in a number of ways.
Given the approximations mentioned, we computed the backreaction of inhomogeneities
in this model metric and found it to be insufficient to cause acceleration. However, our
qualitative results indicate that backreaction may constitute a percent effect and is thus
highly relevant in future considerations of precision cosmology. Of course more realism is
needed before definite statements can be made regarding the backreaction in our universe.
There is another aspect in which such a model would aid in understanding the effects
of inhomogeneities for cosmology. We have focused on the average dynamics but what we
actually see is light propagating through the inhomogeneous universe and not the average
scale factor. One could question whether the latter is at all relevant for our observations.
To really assess the impact of inhomogeneities on observations one should trace light rays
through the inhomogeneous spacetime and determine what observers would see [27, 28]. This
is easy to do in a box described by our metric. In particular we can compute what happens
to the redshift of photons or the travel time though over-dense and under-dense regions. The
great advantage of our approach is that this is easily calculable within our model. Therefore,
even irrespective of whether our model describes the universe entirely accurately, we can
answer the question of whether a large ∆a in the synchronous gauge corresponds to a large
shift in the time it takes a photon to traverse the universe and/or the redshift it experiences
as it does the traversing. We will return to this issue and further improvements to our model
in future work.
– 27 –
KE is supported by the Academy of Finland grant 218322. SH is supported by the Academy
of Finland grant 131454. GR is supported by the Gottfried Wilhelm Leibniz programme of
the Deutsche Forschungsgemeinschaft.
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A The squared metric
The metric (2.33) is the result of a gradient expansion approximation to the Einstein equa-
tions for a dust dominated universe. It has however the undesirable feature that this ex-
pression eventually becomes negative in regions of positive curvature. To overcome this, the
authors of  advocated the following fix: they took the square root of the original metric
to the desired gradient order and then squared it again. More concretely, in our case this
procedure yields for the 1st and 2nd order metric
where Bijis now
Note that now there are higher order terms which ensure that the metric is always positive.
It was argued in  that such a procedure improves convergence and in fact can re-
produce exactly the Szekeres solution studied there. When applied to our spherical test case
it actually does worse than the straightforward gradient expansion as can be seen from fig-
ure A. However, its use for studying backreaction has merits, as argued in the main text.
Furthermore, we see that the (less important for backreaction) collapsing regions are still
modeled relatively well while the behaviour of the expanding regions at two gradients gets
regulated since they eventually slow down. In this configuration our recipe for fixing the
squared metric would result in the collapsing region shrinking down to zero volume (and
hence be irrelevant for backreaction) while the expanding region would be made to follow the
background evolution at t ∼ 14. This corresponds to a volume ratio between the expanding
region and the background Vexp/Vbg∼ 3.5.
BFixing the metric continuously
To avoid sharp features appearing in Q(aD) (and to a lesser extent Q) it is advantageous
to smoothly impose the fix of the expanding regions to the background. For the model
that consists of spherically symmetric regions, introduced in section 3.2.4, the asymptotic
state that is required is written in eq.(3.13). We choose to to impose the following smooth
– 29 –
0 102030 4050
Figure 8. The ”squared metric” for the same configuration as in figure 1. The solid line is the exact
solution, the dotted line is the zero gradient approximation for the metric (flat FRW or ”separate
universe approximation”), the dashed line is the 1st order metric (2 gradients) squared and the dash-
dotted line is the 2nd order metric (4 gradients) squared. The difference in the behaviour compared
to the straightforward gradient expansion for large times is obvious.
γii(t) = γ1/3
˙ γii(t∗) −4γ1/3
˙ γii(t∗) −4γ1/3
˙ γii(t) =4γ1/3
˙ γii(t∗) −4γ1/3
imposed for all t > t∗, which asymptotically approaches eq.(3.13). Note, γ = γ(t∗,x∗).
Remember that for the model of spherically symmetric regions, γ11= γ22= γ33and γij= 0
for all i ?= j. Note that at t = t∗, γii= γ1/3and ˙ γii= ˙ γii(t∗) as is required for the transition
to be smooth. Although ¨ γii(and therefore ¨ aD) will not be continuous during this transition,
it is well-defined. By increasing α the transition can be made sharper. This will result in
sharper features in Q(aD). By making α smaller the transition can be made to be more
gradual. However this has the effect of increasing the final constant term in the expression
for γiiin eq.(B.1) making the grid point take much longer to reach the asymptotic state.
For the model introduced in section 3.2.5, the issue of smoothness in the transition is
not as important. That model uses the squared metric and all of the components in Φ,ij. For
that model, we actually impose the fix to the background at precisely the moment when ˙ γ
at some grid point is the same as the background. That is, the moment when the gridpoint
stops expanding faster than the background. We still apply the fix described in (B.1), with
the additional conditions that γij = ˙ γij = 0 for all i ?= j (and all t > t∗). This instantly
forces the gridpoint into a spherically symmetric state, which is obviously not a smooth or
even continuous transition; however γ and ˙ γ do remain continuous and therefore so do aD
and ˙ aD.
For the two lower panels of figure 2 we used α = 1. For figure 4 we usd α = 50. For
figure 5 we used α = 20. For figure 6 we used α = 5. And for figure 7 we used α = 1. The
much larger values of α used for the model of section 3.2.5 is a result of the transition already
– 30 –
being smooth. That is, using eq.(B.1) instead of eq.(3.13) does not improve the situation.
Setting α ≫ 1 effectively makes the transition instantaneous. However setting α ≫ 1 for the
model of section 3.2.4 makes Q(aD) much more noisy.
– 31 –