Backreaction of cosmological perturbations in covariant macroscopic gravity

Aseem Paranjape*

Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai-400 005, India(Received 24 June 2008; published 15 September 2008)

The problem of corrections to Einstein’s equations arising from averaging of inhomogeneities (back-

reaction) in the cosmological context has gained considerable attention recently. We present results of

analyzing cosmological perturbation theory in the framework of Zalaletdinov’s fully covariant macro-

scopic gravity. We show that this framework can be adapted to the setting of cosmological perturbations in

a manner which is free from gauge related ambiguities. We derive expressions for the backreaction which

can be readily applied in any situation (not necessarily restricted to the linear perturbations considered

here) where the metric can be brought to the perturbed Friedmann-Lemaıtre-Robertson-Walker form. In

particular, these expressions can be employed in toy models studying nonlinear structure formation, and

possibly also in N-body simulations. Additionally, we present results of example calculations which show

that the backreaction remains negligible well into the matter dominated era.

DOI: 10.1103/PhysRevD.78.063522 PACS numbers: 98.80.�k, 04.20.�q, 98.80.Jk

I. INTRODUCTION

It is known [1] that, in order to apply the equations ofEinstein’s general relativity at the large length scales ofinterest in cosmology, it is necessary to first perform asmoothing or averaging operation, which will generatenontrivial corrections in these nonlinear equations. Thereis a debate in the literature concerning two basic questionsregarding this smoothing operation: (a) How does oneobtain consistent and observationally relevant variablesassociated with the corrections due to averaging or ‘‘back-reaction,’’ and (b) what is the magnitude of thisbackreaction?

This discussion has had a long history [2–4], although inrecent times attention has been focused on two promisingcandidates for a consistent nonperturbative averaging pro-cedure, namely, the spatial averaging of scalars due toBuchert [5] and the covariant approach due toZalaletdinov [6,7]. Perhaps owing to its appealing simplic-ity of implementation, the approach due to Buchert hasattracted a significant amount of attention both in thecontext of cosmological perturbation theory [8–10] andwith regards fully nonlinear calculations [11–13], althoughvery interesting results have been derived usingZalaletdinov’s covariant procedure as well [14,15]. Themost fascinating aspect of these studies has been the pos-sibility that the phenomenon of dark energy [9,13] and alsodark matter [15] might be attributed to the backreactionfrom averaging (see Ref. [16] for a recent review of thesubject).

The physical relevance of the averaged variables definedin Buchert’s formalism has been questioned in the litera-ture [17]. It has been argued [17,18] that effects of inho-mogeneities which only perturbatively affect the (on

average homogeneous) metric of the universe cannot leadto effects large enough to account for the inferred accel-eration of the universe from, say supernovae observations[19]. This has been countered by the argument [13] thatperturbation theory may break down during the epoch offully nonlinear structure formation. Recently, it was shown[20] in the context of a specific model of spherical collapse,that an explicit coordinate transformation could be found,which brought the metric to the perturbed Friedmann-Lemaıtre-Robertson-Walker (FLRW) form,

ds2 ¼ �ð1þ 2’Þd�2 þ a2ð�Þð1� 2 Þd~x2; (1)

satisfying all conditions required for a perturbation formal-ism in ’ and to hold, even in the regime of fully non-linear collapse.In such a context, it is important to have a consistent

formalism, free from issues such as gauge artifacts, inwhich one has derived expressions for the backreactionwhich can be applied in a straightforward manner to anymodel in which the metric can be brought to the form (1).Since the Buchert framework by construction is bestadapted to coordinates comoving with the matter, mostapplications using perturbation theory in this frameworkhave focused on the synchronous and comoving gauge,although recently Behrend et al. [10] have also performedcalculations in the conformal Newtonian gauge. However,as pointed out elsewhere [20], calculations using theBuchert framework in perturbation theory necessarilyface the ambiguity of dealing with two scale factors: oneis the scale factor with which the perturbed FLRW metricis defined, and the other is the volume averaged scale factordefined by Buchert [5], and it is not clear which of thesescale factors is the observationally relevant one.In this paper we shall deal with the covariant framework

of Zalaletdinov’s macroscopic gravity (MG), and its spatialaveraging limit proposed in [21]. We will see that here, one*aseem@tifr.res.in

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has a well-defined averaged metric with a correspondinguniquely defined scale factor. Further, we will argue thatthe structure of MG allows us to completely specify aconsistent, observationally relevant averaging operationadapted for perturbations to the FLRW geometry. The finalexpressions obtained for the backreaction can be applieddirectly (or with some straightforward modifications), toany situation in which the metric can be brought to theform (1) or a suitable generalization thereof.

The organization of this paper is as follows: Sec. IIpresents a brief recap of the basic ideas underlying MGand some useful expressions from standard cosmologicalperturbation theory (PT), together with a proposal forpractically estimating the effect of the backreaction onthe time evolution of the FLRW scale factor. Section IIIpresents details of the MG averaging procedure adapted tocosmological PT, including general expressions for theleading order backreaction terms, with a discussion ofgauge related issues and the definition of the averagingoperator. The heart of the paper is in Sec. IV, where wederive final expressions for the backreaction, both in realspace and Fourier space, which can be directly utilized inmodel calculations. While these expressions use a fewsimplifying restrictions, these can be lifted if necessaryin a completely straightforward manner. Section V con-tains example calculations in first order PT, which showthat the magnitude of the backreaction is, as expected,negligible compared to the homogeneous energy densityof matter in the radiation dominated era and for a signifi-cant part of the matter dominated era. We conclude inSec. VI with some final comments. Throughout the paper,lower case Latin indices a; b; c . . . will refer to spacetimeindices 0, 1, 2, 3, and upper case Latin indices A; B; C; . . .to spatial indices 0, 1, 2, 3. The speed of light c is set tounity, and a prime refers to a derivative with respect toconformal time unless stated otherwise.

II. COVARIANT MACROSCOPIC GRAVITY (MG)AND COSMOLOGY

A. Recap of MG formalism

In this section we present a rapid overview of the gen-erally covariant averaging formalism of macroscopic grav-ity (MG), developed by Zalaletdinov and co-workers[6,7,22,23], and its spatial averaging limit proposed in[21]. The reader is referred to these papers for more details.

The MG formalism uses a bilocal operator W a0j ðx0; xÞ,

called the coordination bivector, to define a covariantspacetime averaging operation on tensors in a spacetimemanifold M. [The prime here is being used to distinguishtwo different spacetime points, and must not be confusedwith a derivative with respect to conformal time.] Thevarious properties that this operator must satisfy, and aproof of its existence can be found in Refs. [6,7,22]. Theform of the coordination bivector used in all MG calcu-

lations is

W a0b ðx0; xÞ ¼

@xa

@xjVP

��������x0

@xjVP@xb

��������x; (2)

where xjVP refers to a coordinate system in which the metricdeterminant is constant. Such coordinate systems arecalled volume preserving coordinates (VPCs). Not surpris-ingly, the entire formalism of MG simplifies considerablyin VPC systems, in which the coordination bivector re-duces to the Kronecker delta

W a0j ðx0; xÞjVPC ¼ �aj : (3)

We will see later that VPC systems play an important rolein consistently setting up cosmological perturbation theoryin the context of MG.The average of a tensor Pab over a finite spacetime

domain � is given by

�PabðxÞ ¼ h ~PabiðxÞ ¼1

V�

Z�d4x0

ffiffiffiffiffiffiffiffiffi�g0

q~Pabðx0; xÞ;

V� ¼Z�d4x0

ffiffiffiffiffiffiffiffiffi�g0

q;

(4)

where ~Pabðx0; xÞ is the bilocal extension of Pab defined as

~P abðx0; xÞ ¼ W a

i0 ðx; x0ÞPi0j0 ðx0ÞW j0

b ðx0; xÞ: (5)

The averaging operation, when appropriately applied to the

connection �abc on M, gives an averaged connection ��abcwhich is taken to be the connection on an averaged mani-

fold �M. In other words, the connection on �M satisfies thecondition

h~�abci ¼ ��abc: (6)

The metric Gab associated with the averaged connectioncan be assumed to be the average of the inhomogeneousmetric gab onM, i.e. Gab ¼ �gab. (We will see later that inthe perturbative setting this amounts to a very naturalcondition on the perturbations.) Averaging the Einsteinequations on M leads to the equations satisfied by theaveraged metric, which can be written as

Eab ¼ 8�GNTab þ ðgravÞTab: (7)

Here Eab is the Einstein tensor constructed from the metric

Gab and its inverse Gab, Tab is the averaged energy-

momentum tensor, GN is Newton’s gravitational constant,

and ðgravÞTab is a tensorial correlation object which acts like

an effective gravitational energy-momentum tensor. Forbrevity we shall omit its detailed definition, referring thereader to Refs. [6,7], and only note that its broad structurecan be symbolically represented as

ðgravÞT � h~�2i � h~�i2; (8)

where ~� symbolically denotes the bilocal extension of the

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Christoffel connection on M. In general, the total energy-

momentum tensor ð8�GNTab þ ðgravÞTabÞ is covariantly con-

served,

ð8�GNTab þ ðgravÞTabÞ;a ¼ 0; (9)

with the semicolon denoting the covariant derivative with

respect to the averaged geometry ( ��abc and Gab).In Ref. [21] it was argued that in the cosmological

context, it is essential to consider a spatial averaging limitof the covariant averaging used in MG. The simplest wayto see this is to note that the homogeneous and isotropicFLRW spacetime must be left invariant under the averag-ing operation, and this is only possible if the averaging istuned to the uniquely defined spatial slices of constantcurvature in the FLRW spacetime. A related statementalso applies to perturbation theory in general. The appli-cation of the MG formalism to perturbation theory hasbeen discussed in Refs. [7,23], which also compare theresults of MG in the perturbative scenario, with thoseobtained by Isaacson [24] in the study of the short wave-length limit of gravitational waves. A crucial observation[23] is that, what one normally refers to as the ‘‘back-ground’’ in perturbation theory, must be defined by anaveraging procedure. In particular, the background mustremain invariant under the averaging operation. InRef. [21] no a priori assumption was made about theform of the inhomogeneous metric, and hence certainassumptions had to be made about the choice of spatialslicing used to define the spatial averaging. In the case athand, namely, when the inhomogeneous metric is taken tobe a perturbation around the FLRWmetric, we will see thatthe situation simplifies to some extent and a consistentspatial averaging operation can be identified.

Throughout this paper we will assume that the metric ofthe universe is a perturbation around the FLRW metricgiven by

ds2 ¼ a2ð�Þð�d�2 þ �ABdxAdxBÞ: (10)

Here a is the scale factor and � is the conformal timecoordinate related to cosmic time � by the differentialrelation

d� ¼ að�Þd�: (11)

In Eq. (10) we have allowed the spatial metric to have thegeneral form a2�AB where �AB is the metric of a 3-space ofconstant curvature. For the calculations in this paper wewill assume a flat FLRW background in coordinates suchthat �AB ¼ �AB; however, for future reference we shallpresent certain expressions in terms of the more generalspatial metric.

Assuming the averaged metric to have the form (10), andthe averaged energy-momentum tensor to have the form

Tab ¼ ð�þ pÞ �va �vb þ p�ab; (12)

where �va is the timelike 4-vector which defines the homo-geneous spatial slices of the FLRW spacetime, and �ð�Þand pð�Þ are, respectively, the homogeneous energy den-sity and pressure corresponding to the averaged matterdistribution, the modified cosmological equations obtainedby averaging a general (i.e. not necessarily perturbedFLRW) spacetime can be shown to reduce to the following[see Eq. (87) of Ref. [21] ]:

�1

a

da

d�

�2 ¼ 8�GN

3�� 1

6½P ð1Þ þ Sð1Þ�; (13a)

1

a

d2a

d�2¼ � 4�GN

3ð�þ 3pÞ þ 1

3½P ð1Þ þ P ð2Þ þ Sð2Þ�;

(13b)

where the combinations ðP ð1Þ þ Sð1ÞÞ and ðP ð1Þ þ P ð2Þ þSð2ÞÞ are generally covariant scalars defined by the rela-tions [see Eq. (88) of Ref. [21], with f ¼ a]

P ð1Þ ¼ 1

a2½h~�A0A~�B0Bi � h~�A0B~�B0Ai � 6H 2�; (14a)

Sð1Þ ¼ h~gJKi½h~�AJB~�BKAi � h~�AJA~�BKBi�; (14b)

P ð2Þ þ P ð1Þ ¼ � 1

a2h~�A0A~�0

00i � h~gJKih~�0JA~�A0Ki þ 6H 2

a2;

(14c)

Sð2Þ ¼ 1

a2h~�A00~�0

A0i þ h~gJKih~�0J0~�AKAi; (14d)

where we have defined

H ¼ 1

a

da

d�� a0

a; (15)

and accounted for the fact that, in general, the average ofthe inverse inhomogeneous metric need not equal theinverse metric Gab. However, we will soon see that in theperturbative setting we can in fact set these two tensors tobe equal via a very natural condition on the perturbations.The index 0 in Eq. (14) refers to the conformal time �. Theaveraging in Eq. (14) is assumed to be a spatial averagingin an unspecified spatial slicing in the inhomogeneousmanifold M; in Sec. III we will specify the averagingprocedure more exactly.In addition, the following ‘‘cross-correlation’’ con-

straints must also be satisfied by the inhomogeneities[see Eq. (89) of Ref. [21] ]

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1

a2½h~�0

0A~�BB0i � h~�0

0B~�BA0i� þ h~gJKi½h~�0

JB~�BAKi � h~�0

JA~�BBKi� ¼ 0 (16a)

1

a2½h~�A00~�BB0i � h~�B00~�AB0i� þ h~gJKi½h~�AJB~�B0Ki � h~�AJ0~�BBKi� ¼ 0 (16b)

1

a2½h~�AB0~�m0mi � h~�Am0~�m0Bi� þ h~gJKi½h~�AJm~�mKBi � h~�AJB~�mKmi� ¼ �AB

�� 1

3ðP ð2Þ þ Sð2Þ � Sð1ÞÞ þ 4H 2

a2

�; (16c)

where the lower case indexm in the last equation runs overall spacetime indices 0, 1, 2, 3.

B. Cosmological perturbations and gaugetransformations

For ready reference, in this subsection we present ex-pressions for the metric, its inverse, and the Christoffelconnection in first order cosmological PT, in an arbitrary,unfixed gauge. The notation we use is similar to that usedin Ref. [25]. Wewill also give expressions for the first ordergauge transformations of the perturbation functions (seee.g. Ref. [26]).

The first order perturbed FLRW metric in an arbitrarygauge can be written as

ds2 ¼ a2ð�Þ½�ð1þ 2’Þd�2 þ 2!AdxAd�

þ ðð1� 2 Þ�AB þ �ABÞdxAdxB�: (17)

The functions ’ and are scalars under spatial coordinatetransformations. The functions !A and �AB can be decom-posed as follows:

!A ¼ @A!þ !A; �AB ¼ DAB�þ 2rðA�BÞ þ �AB;

(18)

where the parentheses indicate symmetrization; DAB is thetracefree second derivative defined by

DAB � rArB � ð1=3Þ�ABr2; r2 � �ABrArB;

(19)

with rA the covariant spatial derivative compatible with�AB; and !A, �A, and �AB satisfy

rA!A ¼ 0 ¼ rA�

A; rA�AB ¼ 0 ¼ �AA; (20)

where spatial indices are raised and lowered using �AB andits inverse �AB. From their definitions it is clear that ’, ,!, and � each correspond to one scalar degree of freedom,the transverse 3-vectors !A and �A each correspond to twofunctional degrees of freedom, and the transverse tracefree3-tensor �AB corresponds also to two functional degrees offreedom. This totals to 10 degrees of freedom, of which 4are coordinate degrees of freedom which can be arbitrarilyfixed, which is what one means by a gauge choice. Forexample, the conformal Newtonian or longitudinal orPoisson gauge [26–28] is defined by the conditions

! ¼ 0 ¼ �; �A ¼ 0: (21)

For the metric (17) we have at first order,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgp ¼ a4ð�Þð1þ ’� 3 Þ: (22)

The inverse of metric (17), correct to first order, has thecomponents

g00 ¼ � 1

a2ð1� 2’Þ; g0A ¼ 1

a2!A;

gAB ¼ 1

a2ðð1þ 2 Þ�AB � �ABÞ:

(23)

Denoting H ¼ ða0=aÞ, the prime denoting a derivativewith respect to conformal time �, the first order accurateChristoffel symbols are

�000 ¼ H þ ’0;

�00A ¼ @A’þH!A;

�A00 ¼ @A’þ!A0 þH!A;

�0AB ¼ ðH � 0 � 2H ð’þ ÞÞ�AB �rðA!BÞ

þ 12�

0AB þH�AB;

�A0B ¼ ðH � 0Þ�AB þ 12ðrB!

A �rA!BÞ þ 12�

A0B ;

�ABC ¼ ð3Þ ��ABC � ð�AB@C þ �AC@B � �BC@A Þ

�H!A�BC þ 12ðrC�

AB þrB�

AC �rA�BCÞ;

(24)

where ð3Þ ��ABC denotes the Christoffel connection associatedwith the homogeneous 3-metric �AB.Gauge transformations.—While the concept of gauge

transformations can be described in a rather sophisticatedlanguage using pullback operators between manifolds [26],for our purposes it suffices to implement a gauge trans-formation using the simpler notion of an infinitesimalcoordinate transformation (also known as the ‘‘passive’’point of view) [29]. Hence, denoting the coordinates andperturbation functions in the new gauge with a tilde (i.e. ~xa,

~’, ~!A, and so on), we have

~x a ¼ xa þ �aðxÞ; xa ¼ ~xa � �a; (25)

where the infinitesimal 4-vector �a can be decomposed as

�a ¼ ð�0; �AÞ ¼ ð; @Aþ dAÞ; (26)

where and are scalars and dA is a transverse 3-vectorsatisfying rAd

A ¼ 0.It is then easy to show that, if this transformation is

assumed to change the metric (17) by changing only theperturbation functions but leaving the background intact (aso-called ‘‘steady’’ coordinate transformation), then the

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old perturbations and the new are related by [26]

’ ¼ ~’þ 0 þH; ¼ ~ � 13r2�H;

! ¼ ~!� þ 0; !A ¼ ~!A þ dA0;

� ¼ ~�þ 2; �A ¼ ~�A þ dA; �AB ¼ ~�AB:

(27)

The last equality shows that the transverse tracefree tensorperturbations are gauge invariant. They correspond togravitational waves.

C. Time evolution of the background:An iterative approach

Before we move on to deriving formulas for the corre-lation terms (14) in terms of perturbation functions in themetric, there is one issue which merits discussion. Thecosmological perturbation setting, together with the para-digm of averaging, presents us with a rather peculiarsituation. On the one hand, the time evolution of the scalefactor is needed in order to solve the equations satisfied bythe perturbations. Indeed, the standard practice is to fix thetime evolution of the background once and for all, and touse this in solving for the evolution of the perturbations. Onthe other hand, the evolution of the perturbations (i.e.—theinhomogeneities) is needed to compute the correlationterms appearing in Eq. (13). Until these terms are known,the behavior with time of the scale factor cannot be deter-mined; and until we know the scale factor as a function oftime, we cannot solve for the perturbations. Note that thisis a generic feature independent of all details of the aver-aging procedure.

It would appear therefore that we have reached animpasse. To clear this hurdle, one can try the followingiterative approach: Symbolically denote the background asa, the inhomogeneities as ’, and the correlation objects asC. Note that a, ’, and C all refer to functions of time. Westart with a chosen background, say a standard flat FLRWbackground with radiation, baryons, and cold dark matter(CDM), and solve for the perturbations in the usual way,without accounting for the correlation terms C. In otherwords, for this ‘‘zeroth iteration,’’ we artificially set C to

zero and obtain að0Þ and ’ð0Þ using the standard approach(see e.g. Ref. [30]). Clearly, since the ‘‘true’’ background(say a�) satisfies Eq. (13) with a nonzero C, we have in

general að0Þ � a�. Now, using the solution ’ð0Þ, we can

calculate the zeroth iteration correlation objects Cð0Þ byapplying the prescription to be developed later in this

paper. As a first correction to the solution að0Þ, we now

solve for a new background að1Þ, with the known functions

Cð0Þ acting as sources in Eq. (13). This first iteration will

then yield a solution ’ð1Þ for the inhomogeneities, and

hence a new set of correlation terms Cð1Þ, and this proce-dure can be repeatedly applied. (See, however, the firstpaper in Ref. [14] for an alternative approach exclusively

using averaged quantities in solving the full MG equa-tions.) Pictorially,

að0Þ ! ’ð0Þ ! Cð0Þ ! að1Þ ! ’ð1Þ ! � � � : (28)

As for convergence, if perturbation theory is in fact a goodapproximation to the real universe, then one can expect thatthe correlation terms will tend to be small compared toother background objects, and will therefore not affect thebackground significantly at each iteration, leading to rapidconvergence. On the other hand, if the correlation terms arelarge, this procedure may not converge and one mightexpect a breakdown of the perturbative picture itself. Wewill see that, in the linear regime of cosmological pertur-bation theory, the correlation terms do in fact remainnegligibly small.

III. THE AVERAGING OPERATION AND GAUGERELATED ISSUES

In this section, we will describe the details of the MG(spatial) averaging procedure adapted to the setting ofcosmological PT.

A. Volume preserving (VP) gauges and the correlationscalars

It will greatly simplify the discussion if we start withsymbolic calculations which allow us to see the broadstructure of the objects we are after. Since the correlationobjects in Eq. (13) depend only on derivatives of themetric, we will primarily deal with metric fluctuations;matter perturbations will only come into play when solvingfor the actual dynamics of the system. Before dealing withthe issue of which gauge to choose in order to set thecondition (6), we will show that irrespective of this choice,the leading order contribution to the correlations requiresknowledge of only first order perturbation functions.We will use the following symbolic notation:(i) Inhomogeneous connection: �(ii) FLRW connection: �F(iii) Perturbation in the connection: �� � �� �F ¼

��ð1Þ þ ��ð2Þ þ . . .

(iv) Coordination bivector: W � 1þ �W ¼1þ �Wð1Þ þ �Wð2Þ þ � � �

(v) Bilocal extension of the connection: ~�(vi) Inhomogeneous part of the bilocal extension of the

connection: f�� � ~�� �F ¼ f��ð1Þ þ f��ð2Þ þ � � �(vii) Correlation object: ðgravÞT

The integer superscripts denote the order of perturbation.The form of the coordination bivector arises from the factthat in perturbation theory, in the spatial averaging limit, atransformation from an arbitrary gauge to a VP one can beachieved by an infinitesimal coordinate transformation. Bya VP gauge we mean a gauge in which the metric deter-minant is independent of the spatial coordinates to the

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relevant order in PT, but may be a function of time. It canbe shown that such a function of time (which will typicallybe some power of the scale factor), is completely consistentwith all definitions and requirements of MG in the spatialaveraging limit. An easy way of seeing this is to note that inany averaged quantity, the metric determinant appears intwo integrals, one in the numerator and the other in thedenominator (which gives the normalizing volume). In the‘‘thin time slicing’’ approximation we are using to definethe averaging, any overall time dependent factor in themetric determinant therefore cancels out. Also, a fullyvolume preserving coordinate system can clearly be ob-tained from any VP gauge as defined above, by a suitablerescaling of the time coordinate. It is not hard to show thatin the thin time slicing approximation, this gives the same

coordination bivector W a0b ðx0; xÞ as the VP gauge defini-

tion above.To see that first order perturbations are sufficient to

calculate ðgravÞT to leading order, we only have to notethat the background connection �F satisfies

h�Fi ¼ �F; (29)

and that the structure of ðgravÞT is given by Eq. (8). ðgravÞTthen reduces to

TðgravÞ ¼ hf��2i � hf��i2; (30)

which is exact. Clearly, the correlation is quadratic in the

perturbation as expected, and hence to leading order, f��above can be replaced by f��ð1Þ

.Equations (29) and (30) treat the averaging operation at

a conceptual level only. To make progress however, wealso need to prescribe how to practically impose the aver-aging assumption

h~�i ¼ �F i:e: hf��i ¼ 0; (31)

in any given perturbative context. This requires some dis-cussion since, for example, the bilocal extension of the

connection ~� has the structure

~� ¼ �W þW�1ð@þ @0ÞW; (32)

where @ is a derivative at x and @0 a derivative at x0. (Thereader is referred to Refs. [6,7] for the detailed expression.Suffice it to note that this structure ensures that the aver-aged connection has the correct transformation properties.)The actual MG averaging operation in general is thereforea rather involved procedure. Additionally, it is also neces-sary to address certain gauge related issues.

To clarify the situation, let us start with a fictitioussetting in which the geometry has exactly the flat FLRWform, with no physical perturbations. Clearly, if we work inthe standard comoving coordinates in which the metric �ABof Eq. (10) is simply �AB ¼ �AB, then since these coordi-nates are volume preserving in the sense described above,the coordination bivector becomes trivial. The averaging

involves a simple integration over 3-space, and we caneasily see that Eq. (29) is explicitly recovered.Now suppose that we perform an infinitesimal coordi-

nate transformation, after imposing Eq. (29). Since theaveraging operation is covariant, then from the point ofview of a general coordinate transformation, both sides ofEq. (29) will be affected in the same way. However,suppose that we had performed the transformation beforeimposing Eq. (29). In the language of cosmological PT, wewould then be dealing with some ‘‘pure gauge’’ perturba-tions around the fixed, spatially homogeneous background.If we did not know that these perturbations were puregauge, we might naively construct the nontrivial coordina-tion bivector for this metric, compute the bilocal extensionof the connection according to Eq. (32) and try to imposeEq. (31). This would be incorrect since these perturbationswere arbitrarily generated and need not average to zero (forexample they could be positive definite functions). In orderto maintain consistency, it is then necessary to ensure inpractice that the averaging condition (31) is applied only togauge invariant fluctuations (which is rather obvious inhindsight).There is another problem associated with the structure of

the coordination bivector, even when there are real, gaugeinvariant inhomogeneities present. Note from Eq. (2) thatthe coordination bivector has the structure

W ¼ @x

@xV

��������x0

@xV@x

��������x; (33)

where x denotes the coordinates we are working in and xV aset of VPCs. In perturbation theory (in the spatial averag-ing limit) we will have, at leading order,

x ¼ xV � �; xV ¼ xþ �; (34)

where � symbolically denotes an infinitesimal 4-vectordefining the transformation, and hence

ð@xVÞ=ð@xÞ ¼ 1þ @� (35)

and so on, which gives us

W ¼ 1� ð@�Þjx0 þ ð@�Þjx þ � � � ¼ 1þ �Wð1Þ þ � � � :(36)

Now when we compute a quantity such as h�F�Wð1Þiwhich appears in the expression (32) for h~�i, we will beleft with a fluctuating ( ~x-dependent) term of the form�Fðh@�i � @�Þ, where ~x denotes the 3 spatial coordinates.Hence, if we try to impose Eq. (31) we will ultimately beleft with equations of the type

hfið ~xÞ � fð ~xÞ ¼ 0 (37)

for some functions derived from the inhomogeneitieswhich we have collectively denoted f. In other words,consistency would seem to demand that the inhomogene-ities vanish in this coordinate system, which is of coursenot desirable.

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It therefore appears that we are forced to impose Eq. (31)in a volume preserving gauge, since by definition, only insuch a gauge will we have W ¼ 1 exactly. We emphasizethat this is a purely practical aspect related to defining theaveraging operation, and is completely decoupled from,e.g. the choice of gauge made when studying the timeevolution of perturbations. We are in no way breakingthe usual notion of gauge invariance by choosing an aver-aging operator. The conditions Eq. (37) now reduce to theform

hfVPCið ~xÞ ¼ 0 (38)

which are far more natural than Eq. (37). The averagingcondition is now unambiguous, but depends on a choice ofthe VP gauge which defines the averaging operation, anissue we shall discuss in the next subsection. For now, allwe can assert is that this VP gauge must be such that in theabsence of gauge invariant fluctuations, it must reduce tothe standard comoving (volume preserving) coordinates ofthe background geometry as in Eq. (10). This of course issimply the statement that the VP gauge must be welldefined and must not contain any residual degrees offreedom.

The averaging operation now takes on an almost trivialform—to leading order it is easy to show that for anyquantity fð�; ~xÞ (with or without indices), the average off in a VP gauge in the spatial averaging limit, is given by

hfið�; ~xÞ ¼ 1

VL

ZV ð ~xÞ

d3yfð�; yÞ; (39)

where the integral is over a spatial domain V ð ~xÞ with aconstant volume VL. The spatial coordinates are the co-moving coordinates of the background metric, and at lead-ing order the boundaries of V ð ~xÞ can be specified in astraightforward manner as, e.g.,

V ð ~xÞ ¼ f ~yjxA � L=2< yA < xA þ L=2; A ¼ 1; 2; 3:g;(40)

where L is a comoving scale over which the averaging isperformed (in which case VL ¼ L3). The averaging defi-nition can be written more compactly in terms of a windowfunction WLð ~x; ~yÞ as

hfið�; ~xÞ ¼Zd3yWLð ~x; ~yÞfð�; ~yÞ;

Zd3yWLð ~x; ~yÞ ¼ 1;

(41)

where WLð ~x; ~yÞ vanishes everywhere except in the regionV ð ~xÞ, with the integrals now being over all space. Thisexpression will come in handy when working in Fourierspace, as we shall do in later sections.

A couple of comments are in order at this stage. First, wehave not specified the magnitude of the averaging scale L.The general philosophy is that this scale must be largeenough that a single averaging domain encompasses sev-

eral realizations of the random inhomogeneous fluctua-tions, and small enough that the observable universecontains a large number of averaging domains. However,as we will show later in Sec. IV, if one is ultimatelyinterested in quantities which are formally averaged overan ensemble of realizations of the universe (as is usuallydone in interpreting observations), then the actual value ofthe averaging scale becomes irrelevant.This brings us to the second issue. The above discussion

is valid only in the situation where there are no fluctuationsat arbitrarily large length scales, since in the presence ofsuch fluctuations the averaging condition (31) loses mean-ing (in such a situation it would be impossible to isolate thebackground from the perturbation by an averaging opera-tion on any finite length scale). Indeed, we shall see amanifestation of this restriction in Sec. IV, where thecorrelation scalars will be seen to diverge in the presenceof a nonzero amplitude at arbitrarily large scales, of thepower spectrum of metric fluctuations.We will end this subsection by explicitly writing out the

averaging condition in an ‘‘unfixed VP’’ gauge, to bedefined below, and also writing the correlation terms ap-pearing in Eq. (13) in this gauge. As we can see fromEq. (22), the basic condition to be satisfied by a VP gaugeis

~’ ¼ 3 ~ : (42)

Hereafter, all VP gauge quantities will be denoted using atilde. This should not be confused with the similar notationthat was used for the bilocal extension in Sec. II A, which

will not be needed in the rest of the paper. ~’ and ~ are thescalar potentials appearing in the perturbed FLRW metric(17). The single condition (42) leaves 3 degrees of freedomto be fixed, in order to completely specify the VP gaugeone is working with. The MG formalism by itself does notprescribe a method to choose a particular VPC system; infact this freedom of choice of VPCs is an inherent part ofthe formalism. We shall return to this issue in the nextsubsection. For now we define the ‘‘unfixed VP (uVP)gauge’’ by the single requirement (42), with 3 unfixeddegrees of freedom, and present the expressions for theaveraging condition and the correlation scalars, with thischoice.It is straightforward to determine the consequences of

requiring Eq. (6) to hold, with the right-hand side corre-sponding to the FLRW connection in conformal coordi-nates, and remembering that the coordination bivector (inthe spatial averaging limit) is now trivial [see Eq. (3)].Together with some additional reasonable requirements,namely

hr2si ¼ 0 ¼ hr2@Asi; (43)

for any scalar sð�; ~xÞ, the averaging condition in the uVPgauge reduces to

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h ~ i ¼ 0; h@A ~ i ¼ 0 ¼ h ~ 0i; h ~!Ai ¼ 0 ¼ h ~!0Ai;

h~�0ABi ¼ 0; hrC ~�

ABi þ hrB ~�

ACi � hrA ~�BCi ¼ 0;

hrA ~!Bi ¼ hrB ~!Ai ¼ H h~�ABi; (44)

where we have used the expressions in Eq. (24) with theuVP condition (42). We will also make the additionalreasonable requirement that

h~�ABi ¼ 0; (45)

and then it is easy to see that the perturbed FLRW metric(17) and its inverse (23), in the uVP gauge, both onaveraging reduce to their respective homogeneous counter-parts, namely

hgabi ¼ gðFLRWÞab ; hgabi ¼ gabðFLRWÞ: (46)

Using these results, the expressions (14) simplify to give,in the uVP gauge,

P ð1Þ ¼ 1

a2

�6hð ~ 0Þ2i þ hr½A ~!B�r½A ~!B�i

� 1

4h~�0

AB ~�AB0i

�; (47a)

Sð1Þ ¼ 1

a2

��10h@A ~ @A ~ i � 2h@A ~ rB ~�

ABi

þ 1

4hrB ~�ACð2rA ~�BC �rB ~�ACÞi

�; (47b)

P ð1Þ þ P ð2Þ ¼ 1

a2

�6hð ~ 0Þ2i � 24H h ~ 0 ~ i � h ~ 0r2 ~!i

þ 1

2h~�0

ABrA ~!Bi

� 1

4h~�0

ABð~�AB0 þ 2H ~�ABÞi�; (47c)

Sð2Þ ¼ 1

a2½3h ~!A0@A ~ i þH h ~!A ~!0

Ai�; (47d)

where square brackets denote antisymmetrization.

B. Choice of VP gauge

In this subsection we will prescribe a choice for the VPgauge which defines the averaging operation. In general,the class of volume preserving coordinate systems for anyspacetime, is very large (see Ref. [22] for a detailedcharacterization). We have so far managed to pare itdown by requiring that the VP gauge we choose shouldreduce to the standard FLRW coordinates in the absence offluctuations. It turns out to be somewhat difficult to gobeyond this step, since there does not appear to be anyunambiguously clear guiding principle governing thischoice. We will therefore motivate a choice for the VPgauge based on certain details of cosmological PT whichone knows from the standard treatments of the subject.

In particular, we shall make use of certain nice proper-ties of the conformal Newtonian or longitudinal or Poisson

gauge, which is defined by the conditions (21) [26,27](henceforth we shall refer to this gauge as the cN gaugefor short). Since this gauge is well defined and has noresidual degrees of freedom, all the nonzero perturbationfunctions in the cN gauge, namely’, , !A, and �AB in thenotation of Sec. II B, are equal to gauge invariant objects.This is trivially true for �AB, as seen in the last equation in(27). For the rest, note that in any arbitrary unfixed gauge,the following combinations are gauge invariant at firstorder:

�B ¼ ’þ 1

a@�

�a

�!� 1

2�0��;

�B ¼ �H�!� 1

2�0�þ 1

6r2�;

VA ¼ !A � �0A;

(48)

which can be easily checked using Eq. (27), and in the cNgauge, !, �, and �A all vanish. Here �B and �B are theBardeen potentials [31] (up to a sign), and�B in particularhas the physical interpretation of giving the gauge invariantcurvature perturbation, which is the quantity on whichinitial conditions are imposed post inflation [32].Additionally, it is also known that the cN gauge for the

metric remains stable even during structure formation,when matter inhomogeneities have become completelynonlinear. (See Ref. [17] for an intuitive description ofwhy this is so, and Ref. [20] for an explicit demonstrationin a toy model of structure formation.) We believe that thisis a strong argument in favor of using the cN gauge todefine a VP gauge which will then define the averagingoperation in the perturbative context. This will ensure thatthis ‘‘truncated’’ averaging operation, defined for first or-der PT, will remain valid at leading order even during thenonlinear epochs of structure formation.To implement this in practice, consider a transformation

from the cN gauge to the uVP gauge defined by Eq. (42).The transformation equations (27) reduce to

0 þ 4Hþr2 ¼ ’� 3 ; ~ ¼ 13’� 0 �H;

~! ¼ � 0; ~!A ¼ !A � dA0; ~� ¼ �2;

~�A ¼ �dA; ~�AB ¼ �AB: (49)

Recall that to completely specify a VP gauge, we need tofix 3 degrees of freedom in the uVP gauge. Our require-ment regarding the ‘‘well-defined’’-ness of the VP gauge,forces us to set dA ¼ 0, and to choose and such thatthey vanish in the case where ’ ¼ 0 ¼ .This has fixed 2 degrees of freedom, in addition to the

condition (42) which is just the definition of the uVPgauge, and has hence not yielded a uniquely specified VPgauge. To do this, we shall make the following additionalrequirement. Since we are dealing with a spatial averaging,it seems reasonable to require that the VP gauge being usedto define the averaging should be ‘‘as close as possible’’ to

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the cN gauge in terms of time slicing, and for this reasonwe shall set the function to zero.

To summarize, the VP gauge chosen is defined in termsof the gauge transformation functions �a ¼ ð; @Aþ dAÞbetween the cN gauge and the VP gauge, by the followingrelations:

¼ 0 ¼ dA; (50)

and

~’ ¼ 3 ~ ¼ ’; (51a)

r2 ¼ ’� 3 ; (51b)

~! ¼ �0; ~� ¼ �2; (51c)

~�A ¼ 0; (51d)

~!A ¼ !A; ~�AB ¼ �AB; (51e)

where the function is restricted not to contain any non-trivial solution of the homogeneous (Laplace) equationr2 ¼ 0.

Having made this choice for the VP gauge, we are nowassured that all averaged quantities which we compute aregauge invariant: our choice ensures that the averagingprocedure does not introduce any pure gauge modes, andthe philosophy of steady coordinate transformations en-sures that all background objects are, by assumption, un-affected by gauge transformations. In particular, thecorrelation objects in Eq. (14) are all gauge invariant.This is different from the gauge invariance conditionsderived in the first paper of Ref. [4], where the backgroundwas also taken to change under gauge transformations atsecond order in the perturbations. It is at present not clearhow these results are related to ours.

Note that all these arguments are valid at first order inPT, which is sufficient for our present purposes. A consis-tent treatment at second order would require more work,although as long as one is interested only in the leadingorder effect, these arguments are expected to go through.

IV. THE CORRELATION SCALARS

With the VP gauge choice defined by Eq. (51), it isstraightforward to rewrite the correlation objects inEq. (47) (which are in the uVP gauge) in terms of theperturbation functions in the cN gauge. We will restrict thesubsequent calculations in this paper to the case wherethere are no transverse vector perturbations, i.e.,

! A ¼ 0; (52)

in the cN gauge. This is a reasonable choice since suchvector perturbations, even if they are excited in the initialconditions, decay rapidly and do not source the otherperturbations at first order [30].

In addition, for simplicity (and to keep this paper con-cise) we will choose to ignore the gauge invariant tensorperturbations as well,

� AB ¼ 0: (53)

It will be an interesting exercise to account for the effectsof gravitational waves in the correlation scalars, howeverwe will leave this to future work. Thus, the results pre-sented here apply only to scalar perturbations.In terms of the scalar perturbations in the cN gauge, for a

flat FLRW background, the correlation objects (47) reduceto

P ð1Þ ¼ 1

a2½2hð 0Þ2iþ hð’0 � 0Þ2i

� hðrArB0ÞðrArB0Þi�; (54a)

Sð1Þ ¼ � 1

a2½6h@A @A iþ h@Að’� Þ@Að’� Þi

� hðrArBrCÞðrArBrCÞi�; (54b)

P ð1Þ þP ð2Þ ¼ 1

a2½h’0ð’0 � 0Þi� 2H fh’0’i

� h 0 iþ h 0ð’� Þiþ h ð’0 � 0Þiþ hðrArBÞðrArB0Þig�; (54c)

Sð2Þ ¼ � 1

a2½h@A00ð@A’�H@A

0Þi�; (54d)

where is defined in Eq. (51b).Since we are working with a flat FLRW background, it

becomes convenient to transform our expressions in termsof Fourier space variables. This will also highlight theproblemwith large scale fluctuations which was mentionedin Sec. III. We will use the following Fourier transformconventions: For any scalar function fð�; ~xÞ, its Fouriertransform f ~kð�Þ satisfies

fð�; ~xÞ ¼Z d3k

ð2�Þ3 ei ~k� ~xf ~kð�Þ;

f ~kð�Þ ¼Zd3xe�i ~k� ~xfð�; ~xÞ:

(55)

Consider an average of a generic quadratic product of two

scalars fð1Þð ~xÞ and fð2Þð ~xÞ where we have suppressed thetime dependence since it simply goes along for a ride.Using the definition (41), and keeping in mind that thescalars are real, it is easy to show that we have

hfð1Þfð2Þið ~xÞ ¼Z d3k1d

3k2ð2�Þ6 W�

Lð ~k1 � ~k2; ~xÞfð1Þ~k1 fð2Þ�~k2; (56)

where WLð ~k; ~xÞ is the Fourier transform of the windowfunction WLð ~x; ~yÞ on the variable ~y, and the asterisk de-notes a complex conjugate.

In the present context, the functions fð1Þ and fð2Þ willtypically be derived in terms of the initial random fluctua-tions in the metric’~ki which are assumed to be drawn from

a statistically homogeneous and isotropic Gaussian distri-bution with some given power spectrum. In order to ulti-mately make contact with observations, it seems necessaryto perform a formal ensemble average over all possible

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realizations of this initial distribution of fluctuations. Thestatistical homogeneity and isotropy of the initial distribu-

tion implies that the functions fð1Þ and fð2Þ will satisfy arelation of the type

½fð1Þ~k1 fð2Þ�~k2

�ens ¼ ð2�Þ3�ð3Þð ~k1 � ~k2ÞPf1f2ðj ~k1jÞ; (57)

for some function Pf1f2ðk; �Þwhich is derivable in terms of

the initial power spectrum of metric fluctuations, and

where ½. . .�ens denotes an ensemble average and �ð3Þð ~kÞ isthe Dirac delta distribution.

Applying an ensemble average to Eq. (56) introduces a

Dirac delta which forces ~k1 ¼ ~k2. Further, the normaliza-tion condition on the window function in Eq. (41) impliesthat we have

WLð ~k ¼ 0; ~xÞ ¼ 1; (58)

which means that all dependence on the averaging scaleand domain drops out, and we are left with

½hfð1Þfð2Þi�ens ¼Z d3k

ð2�Þ3 Pf1f2ðkÞ: (59)

Note, however, that the right-hand side of Eq. (59) isprecisely what we would have obtained, had we treatedthe spatial average h. . .i to be the ensemble average ½. . .�ensto begin with. Therefore for all practical purposes, we arejustified in replacing all the spatial averages in the expres-sions for the correlation scalars (54), by ensembleaverages.

It is convenient to define the transfer function�kð�Þ viathe relation

’~kð�Þ ¼ ’~ki�kð�Þ: (60)

For the calculations in this paper, we shall assume that thecN gauge scalars ’ð�; ~xÞ and ð�; ~xÞ are equal

’ð�; ~xÞ ¼ ð�; ~xÞ; (61)

a choice which is valid in first order PT when anisotropicstresses are negligible (see Ref. [30]). This simplifies manyof the expressions we are dealing with. The Fourier trans-form of can be written, using Eqs. (51b) and (61), as

~kð�Þ ¼2

k2’~kð�Þ: (62)

Finally, in terms of the transfer function �kð�Þ and theinitial power spectrum of metric fluctuations defined by

½’~k1i’�~k2i�ens ¼ ð2�Þ3�ð3Þð ~k1 � ~k2ÞP’iðk1Þ; (63)

the correlation scalars (54) can be written as [compareEqs. (58)–(61) of Ref. [10] ]

P ð1Þ ¼ � 2

a2

Z dk

2�2k2P’iðkÞð�0

kÞ2; (64a)

Sð1Þ ¼ � 2

a2

Z dk

2�2k2P’iðkÞðk2�2

kÞ; (64b)

P ð1Þ þ P ð2Þ ¼ � 8Ha2

Z dk

2�2k2P’iðkÞð�k�

0kÞ; (64c)

Sð2Þ ¼ � 2

a2

Z dk

2�2k2P’iðkÞ�00

k

��k � 2H

k2�0k

�:

(64d)

These expressions highlight the problem of having a finiteamplitude for fluctuations at arbitrarily large length scales(k! 0), which was mentioned in Sec. III. As a concreteexample, consider the frequently discussed Harrison-Zel’dovich scale invariant spectrum [33] which satisfiesthe condition

k3P’iðkÞ ¼ constant: (65)

Equations (64) now show that, if the transfer function�kð�Þ has a finite time derivative at large scales (as itdoes in the standard scenarios—see the next section),

then the correlation objects P ð1Þ, P ð2Þ, and Sð2Þ all divergedue to contributions from the k! 0 regime. This demon-strates the importance of having an initial power spectrumin which the amplitude dies down sufficiently rapidly onlarge length scales (which is a known issue, see Ref. [32]).Perturbation theory cannot adequately describe the behav-ior of inhomogeneities with arbitrarily large length scales[34]. Keeping this in mind, we shall concentrate on initialpower spectra which display a long wavelength cutoff [35].Models of inflation leading to such power spectra havebeen discussed in the literature [36], and more encourag-ingly, analyses of WMAP data seem to indicate that such acutoff in the initial power spectrum is in fact realized in theuniverse [37].A final comment before proceeding to explicit calcula-

tions: In addition to picking up nontrivial correlation cor-rections in the cosmological equations, the averagingformalism also requires that the ‘‘cross-correlation’’ con-straints in Eq. (16) be satisfied. In the absence of vector andtensor modes, it is straightforward to show that the statis-tical homogeneity and isotropy of the scalar metric fluctu-ations implies that these constraints are identicallysatisfied. It will be an interesting exercise to analyze theconditions imposed by these constraints in the presence oftensor modes; we leave this for future work.

V. WORKED OUT EXAMPLES

We will now turn to some explicit calculations of thecorrelation integrals (backreaction), which will show thatthe magnitude of the effect remains negligibly small formost of the evolution duration in which linear PT is valid.At early times, linear PT is valid at practically all scalesincluding the smallest scales at which we wish to apply

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general relativity. As matter fluctuations grow, the smalllength scales progressively approach nonlinearity, and lin-ear PT breaks down at these scales. As we will see, how-ever, by the time a particular length scale becomesnonlinear, its contribution to the amplitude of the metricfluctuations correspondingly becomes negligible. In prac-tice therefore, one can extend the linear calculation wellinto the matter dominated era, with the expectation that theorder of magnitude of the various integrals will not changesignificantly due to nonlinear effects (see also the discus-sion in the last section).

The model we will use is the standard cold dark matter(sCDM) model consisting of radiation and CDM [30]. Wewill neglect the contribution of baryons, and at the end weshall discuss the effects this may have on the final results.We shall also discuss, without explicit calculation, theeffects which the introduction of a cosmological constantis likely to have. In the following, �r and �m denote thedensity parameters of radiation and CDM, respectively, atthe present epoch �0, with � denoting cosmic time. �r isassumed to contain contributions from photons and 3species of massless, out-of-equilibrium neutrinos. At the‘‘zeroth iteration’’ (see Sec. II C) we have�

1

a

da

d�

�2 ¼ H2ðaÞ ¼ H2

0

��m

a3þ�r

a4

�; (66)

where H0 is the standard Hubble constant, the scale factoris normalized so that að�0Þ ¼ 1, and H and H are relatedby

H ðaÞ ¼ aHðaÞ: (67)

The comoving wave number corresponding to the scalewhich enters at the matter radiation equality epoch is givenby

keq ¼ aeqHðaeqÞ ¼ H0

�2�2

m

�r

�1=2 �H0 � 105=2; (68)

where we have set (see Refs. [30,38] for details)

�r ¼ �photon þ 3�neutrino ¼ �photon

�1þ 3 � 7

8

�4

11

�4=3

�¼ 4:15� 10�5h�2; (69)

where h is the dimensionless Hubble parameter defined byH0 ¼ 100h km=s=Mpc. For all calculations we shall seth ¼ 0:72 [39].

A. EdS background and nonevolving potentials

Before dealing with the full model (which requires anumerical evolution) let us consider the simpler situation,described by an Einstein-de Sitter (EdS) background, withnegligible radiation and a nonevolving potential ’ ¼ ’ð ~xÞ(which is a consistent solution of the Einstein equations inthe sCDM model at least at subhorizon scales at late times[30]). Although not fully accurate, this example requires

some very simple integrals and will help to give us a feelfor the structure and magnitude of the backreaction.With a constant potential, the only correlation object

which survives is Sð1Þ, which evolves like�a�2, where thescale factor refers to the ‘‘zeroth iteration.’’ The constant ofproportionality can be written in terms of the BBKS trans-fer function TBBKSðk=keqÞ (after Bardeen, Bond, Kaiser,

and Szalay [40]), to give

S ð1Þ ¼ � 2

a2

Z dk

2�2k4P’iðkÞT2

BBKSðk=keqÞ; (70)

where we have [40]

TBBKSðxÞ ¼ ln½1þ 0:171x�ð0:171xÞ ½1þ 0:284xþ ð1:18xÞ2

þ ð0:399xÞ3 þ ð0:490xÞ4��0:25; (71)

where x � ðk=keqÞ.The integral in Eq. (70) is well behaved even in the

presence of power at arbitrarily large scales, for a (nearly)scale invariant spectrum. Since we are only looking for anestimate, we shall evaluate the integral in the absence of alarge scale cutoff, and leave a more accurate calculation forthe next subsection. For the initial spectrum given by

k3P’iðkÞ2�2

¼ Aðk=H0Þns�1; (72)

where the scalar spectral index ns is close to unity, theintegral in Eq. (70) can be easily performed numericallyand has the order of magnitude

Z dk

2�2k4P’iðkÞT2

BBKSðk=keqÞ � AðkeqÞ2 � AH20 � 105;

(73)

up to a numerical prefactor of order 1. Since the amplitudeof the power spectrum is A� 10�9 [41], the overall con-tribution of the backreaction is

Sð1Þ

H20

�� 1

a2ð10�4Þ: (74)

Now, as long as the correlation objects give a negligiblebackreaction to the usual background quantities, when weproceed with the next iteration, the effect of the backreac-tion on the evolution of the perturbations will also remainnegligible (at least at the leading order). Hence in practicethere will be essentially no difference between the zerothiteration and first iteration perturbation functions. Thisamounts to saying that, when the backreaction is negli-gible, convergence to the true solution for the scale factorat the leading order, is essentially achieved in a singlecalculation.

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B. Radiation and CDM without baryons

Let us now turn to the full sCDM model (withoutbaryons). An analytical discussion of this model in variousregions of ðk; �Þ space can be found e.g. in Ref. [30]. Sincewe are interested in integrals over k across a range ofepochs �, it is most convenient to solve this model nu-merically. It is further convenient to use ðlnaÞ in place of �,as the variable with which to advance the solution. Also, itis useful to introduce transfer functions like �kð�Þ for allthe relevant perturbation functions in exactly the samemanner [see Eq. (60)], namely, by pulling out a factor of’~ki, since the initial conditions are completely specified by

the initial metric perturbation. For a generic perturbationfunction s ~kð�Þ (other than the metric fluctuation ’~k), the

transfer function corresponding to s will be denoted by acaret, so that

s ~kð�Þ ¼ ’~kiskð�Þ: (75)

The relevant Einstein equations can be brought to thefollowing closed set of first order ordinary differentialequations [adapted from Eqs. (7.11)–(7.15) of Ref. [30] ],

@�k

@ðlnaÞ ¼ ���

1þ K2

3E2

��k þ 1

2E2a

��m�k þ 4

a�r�0k

��;

(76a)

@�k@ðlnaÞ ¼ �K

EVk þ 3

@�k

@ðlnaÞ ; (76b)

@�0k

@ðlnaÞ ¼ �K

E�1k þ @�k

@ðlnaÞ ; (76c)

@�1k

@ðlnaÞ ¼K

3Eð�0k þ�kÞ; (76d)

@Vk@ðlnaÞ ¼ �Vk þ K

E�k: (76e)

Here we have introduced the dimensionless variables

K � k

H0

; EðaÞ � H ðaÞH 0

¼ H ðaÞH0

; (77)

and the various perturbation functions are defined as fol-lows: �k is the k-space density contrast of CDM, �0k and�1k are the monopole and dipole moments, respectively, ofthe k-space temperature fluctuation of radiation, andð�iVkÞ is the k-space peculiar velocity scalar potential ofCDM [i.e., the real space peculiar velocity is vA ¼ @Avwhere v is the Fourier transform of ð�iVkÞ].

Assuming adiabatic perturbations, the initial conditionssatisfied by the transfer functions at a ¼ ai are (adaptedfrom Chapter 6 of Ref. [30])

�kðaiÞ ¼ 1; �kðaiÞ ¼ � 3

2; �0kðaiÞ ¼ � 1

2;

VkðaiÞ ¼ 3�1kðaiÞ ¼ 1

2

K

EðaiÞ : (78)

We choose ai ¼ 10�16, which corresponds to an initialbackground radiation temperature of T � 103 GeV.While this is not as far back in the past as the energy scaleof inflation (which is closer to �1015 GeV), it is on theedge of the energy scale where known physics begins [32].This makes Eq. (69) unrealistic since we have ignored allof the big bang nucleosynthesis and also the fact thatneutrinos were in equilibrium with other species at tem-peratures higher than about 1 MeV. However, the modifi-cations due to these additional details are not expected todrastically change the final results, and these assumptionslead to some simplifications in the code used. The goal hereis only to demonstrate an application of the formalism;more realistic calculations accounting for the effects ofbaryons can also be performed (see, e.g. Behrend et al.[10] who incorporate these effects for the postrecombina-tion era, albeit in the Buchert formalism).In order to partially account for the fact that inflationary

initial conditions are actually set much earlier than a ¼10�16, we impose an absolute small wavelength cutoff atthe scale which enters the horizon at the initial epochwhich we have chosen. In the above notation this corre-sponds to setting Kmax ¼ EðaiÞ � 1013. This makes sensesince scales satisfying K � Kmax have already entered thehorizon and decayed considerably by the epoch a ¼ 10�16.There is a source of error due to ignoring scales K * Kmax

which have not yet decayed significantly, but this errorrapidly decreases with time as progressively larger lengthscales enter the horizon and decay. [In fact, in practice tocompute the integrals at any given epoch a ¼ a�, one onlyneeds to have followed the evolution of modes with K <�5000Eða�Þ: more on this in the next subsection.] Moreimportant is the cutoff at longwavelengths, which we set atKmin ¼ 1 (corresponding to kmin ¼ H0), which is first anatural choice given that H�1

0 is the only large scale in the

system, and is second also guided by analyses of CMB datawhich have detected such a cutoff [37]. We will see thatreducing Kmin even by a few orders of magnitude does notaffect the final qualitative results significantly.

Numerical results

Equation (76) with initial conditions (78) were solvedwith a standard 4th order Runge-Kutta integrator withadaptive stepsize control (based on the algorithm given inRef. [42]). For the integrals in Eq. (64), only the function

�kðaÞ needs to be tracked accurately. Hence, although �0k

and �1k are difficult to follow accurately beyond the matterradiation equality aeq ¼ ð�r=�mÞ ’ 8� 10�5 due to

rapid oscillations, the integrals can still be reliably com-

puted since �0k and �1k do not significantly affect theevolution of �k in the matter dominated era [as seen inEq. (76a)].To see that known results are being reproduced by the

code, consider Figs. 1 and 2 as examples. Figure 1shows the evolution of two scales corresponding to

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K ¼ 1 (k ¼ H0 Mpc�1) and K ¼ 0:01. The first enters thehorizon at the present epoch, while the second remainssuperhorizon for the entire evolution, satisfying k� 1.In this limit an analytical solution exists in the sCDMmodel, due to Kodama and Sasaki [30,43], given by

�kðyÞ ¼ 1

10y3½16 ffiffiffiffiffiffiffiffiffiffiffiffi

1þ yp þ 9y3 þ 2y2 � 8y� 16�; (79)

where y � a=aeq, and this function is also shown. Clearly

all the curves in Fig. 1 are practically identical.Figure 2 shows the function �k normalized by its (con-

stant) value at large scales, at the epoch a ¼ 500aeq ’ 0:04

(which is well into the matter dominated era). The dottedline is the BBKS fitting form given in Eq. (71) with keqgiven by Eq. (68).

To numerically estimate the integrals in Eq. (64), thevalues of �k and its first and second derivatives withrespect to ðlnaÞ are needed across a range of K values.For reference, note that the following relations hold for ageneric function of time wð�Þ,

dw

d�¼ aH

dw

da¼ H

dw

dðlnaÞ : (80)

Based on the earlier discussion, the initial power spectrumP’iðkÞ is taken to satisfy

k3P’iðkÞ2�2

¼ A; for H0 < k< kmax ¼ H ðaiÞ; (81)

and zero otherwise, and we set

A ¼ 1:0� 10�9; (82)

which, for the sCDM model follows from the convention[see Eq. (6.100) of Ref. [30] ] A ¼ ð5�H=3Þ2 with �H 2� 10�5 (see, e.g. Ref. [41]).Consider Figs. 3 and 4, which highlight two issues

discussed earlier. Figure 3 shows the integrand of Sð1Þ atthree sample epochs, and we see that the integrand diesdown rapidly at increasingly smaller k values for progres-sively later epochs. (The other integrands, not displayedhere, also show this rapid decline for large k.) [We have notshown the integrand at the later two epochs for all values ofk since this was computationally expensive, but the declin-ing trend of the curves can be extrapolated to large k, whichis well understood analytically [30].] This justifies thestatement in the beginning of this section, that at any epocha� it is sufficient to have followed the evolution of scalessatisfying K < 5000Eða�Þ for computing the integrals.

Second, Fig. 4 shows the behavior of k3=2j�kj ¼ A1=2j�kjat the same three epochs, and comparing with Fig. 3 we seethat at any epoch, the region of k space where linear PT hasbroken down, does not contribute significantly to the in-tegrals. This is in line with the conjecture in Ref. [44] thatthe effects of the backreaction should remain small sincethe mass contained in the nonlinear scales is subdominant.Because of the structure of the integrals and the chosen

initial power spectrum, it is convenient to compute theintegrands in Eq. (64) equally spaced in ðlnKÞ, and thenperform the integrals using the extended Simpson’s rule[42]. If 2N þ 1 points are used to evaluate a given integral,resulting in a value IN say, then the error can be estimatedby computing the integral with 2N�1 þ 1 points to getIN�1, and estimating the relative error as jIN�1=INj �1. With N ¼ 10, the estimated errors in all the integrals atall epochs were typically less than 0.1%. A bigger error isincurred in computing the integrand itself at any given

epoch, leading to estimated errors of order �1% in Sð1Þ,P ð1Þ, and P ð1Þ þ P ð2Þ, with a larger error in Sð2Þ as ex-plained below.The second derivative @2�k=@ðlnaÞ2 proves to be diffi-

cult to track numerically. At early times, when most scales

0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1 10

Tra

nsfe

r fu

nctio

n

k (h Mpc-1)

Φk(a=500aeq)

TBBKS(k)

FIG. 2. The transfer function �k normalized by its constantvalue at large scales, at the epoch a ¼ 500aeq. The dotted line is

the BBKS transfer function (71).

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1

Φk

a

k = (0.01 H0) Mpc-1

k = H0 Mpc-1

Φk,analytic

FIG. 1. The evolution of two large scale modes. Also shown isthe Kodama-Sasaki analytical solution in the large scale limitk� 1, given by Eq. (79).

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are superhorizon, the Kodama-Sasaki analytical solution(79) is a good approximation for most values of k. Usingthis, one can see that at early times the value of thederivative is numerically very small, and is difficult toreliably estimate due to roundoff errors. For this reason

the integral Sð2Þ could not be accurately estimated at early

times. However, the structure of the integrand of Sð2Þ (64d)shows that the largest contribution comes from large(superhorizon) scales (the small scales being subdominantdue to the presence of�k and 1=k

2). An analysis using theKodama-Sasaki solution then shows in a fairly straightfor-ward manner that the behavior of the backreaction term is

jSð2Þ=H2j � 10�6ða=aeqÞðH0=kminÞ2 for our choices of pa-rameters, where ða=aeqÞ 1. At intermediate times

around a� aeq and later, although it becomes computa-

tionally expensive to obtain convergent values for thesecond derivative at all relevant scales [45], moderatelygood accuracy (1%–5%) can be achieved.

The results are shown in Fig. 5, in which the magnitudesof the correlation integrals of Eq. (64), normalized by theHubble parameter squaredH2ðaÞ ¼ ðH =aÞ2 are plotted asa function of the scale factor in a log-log plot. The values

for Sð2Þ are shown only for epochs later than a ’ 0:01aeq �10�6. We see that at all epochs, the correlation termsremain negligible compared to the chosen zeroth iterationbackground. Also, in the radiation dominated epoch all the

correlation scalars (except Sð2Þ whose evolution could notbe accurately obtained) track the �a�4 behavior of thebackground radiation density (see also the second paper inRef. [4]). The discussion above shows however that the

magnitude of Sð2Þ is far smaller than the other backreactionfunctions at early times, for a cutoff at kmin ¼ H0. On the

other hand, in the matter dominated epoch Sð1Þ dominates

the backreaction and settles into a curvaturelike �a�2

behavior (note that in the matter dominated epoch wehave H2 � a�3). This can also be compared with theresults of Ref. [3]. As for the signs of the correlations,

Sð1Þ, Sð2Þ, and P ð1Þ are negative throughout the evolution

while P ð1Þ þ P ð2Þ is positive throughout.Finally, a few comments regarding the effects of ignor-

ing baryons, nonlinear corrections, etc. Including baryonsin the problem (with a background density parameter of�b ’ 0:05) will lead to a significant suppression of smallscale power (by introducing pressure terms which will tendto wipe out inhomogeneities) and also a small suppressionof large scale power. This effect causes a (downward)change in the late time transfer function of roughly 15%–20% [30], and therefore cannot increase the contribution ofthe backreaction. Quasilinear corrections can lead to sig-

1e-05

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1 1 10 100 1000 10000

k3/

2 | δ k

|

k (h Mpc-1)

a = 0.0125aeq

a = aeq

a = 200aeq

FIG. 4. The dimensionless CDM density contrast. Togetherwith Fig. 3 this shows that nonlinear scales do not impact thebackreaction integrals significantly.

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1

Bac

krea

ctio

n

a

a=aeq

S(1)/H2

P(1)/H2

( P(1) + P(2) )/H2

S(2)/H2

FIG. 5. The correlation scalars (backreaction) for the sCDMmodel, normalized by H2ðaÞ. Sð1Þ, P ð1Þ, and Sð2Þ are negativedefinite and their magnitudes have been plotted. The vertical linemarks the epoch of matter radiation equality a ¼ aeq.

0.01

0.1

1

10

100

1000

1e+04

1e+05

1e+06

1e+07

1e+08

1e+09

0.0001 0.001 0.01 0.1 1 10 100 1000 10000

( k/

H0

)2Φ

k2

k (h Mpc-1)

a = 0.0125aeq

a = aeq

a = 200aeq

FIG. 3. The dimensionless integrand of Sð1Þ, namely, the func-tion ðk=H0Þ2�2

k, at three sample values of the scale factor. The

function dies down rapidly for large k, with the value at some kbeing progressively smaller with increasing scale factor. Thedeclining behavior of the curves for a ¼ aeq and a ¼ 200aeqextrapolates to large k.

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nificant changes in the transfer function, but do not causeshifts by several orders of magnitude (see Ref. [46] andreferences therein). Hence, accounting for changes due toquasilinear behavior will also not increase the magnitudeof the backreaction by a large amount (see also Ref. [10]).As for effects from fully nonlinear scales, we have seenthat these can be expected to remain small, or at least notorders of magnitude larger than those from linear scales(see also the discussion in the last section, and Ref. [3]).

Adding a cosmological constant (and retaining a flatbackground geometry) will change the qualitative featuresof the correlation functions by shifting the scale keq (due to

a reduced�m, which will also increase the power spectrumamplitude [30], but again not by orders of magnitude).Also, the late time behavior of the correlation scalars willbe affected since the potential �k will decay at late timesinstead of remaining constant. Regardless, the backreac-tion is expected to remain small even in this case (which isalso indicated by the calculations of Behrend et al. [10] inthe Buchert framework [5]).

VI. DISCUSSION

This paper has presented an analysis of cosmologicalperturbation theory (PT) in the fully covariant averagingframework of Zalaletdinov’s macroscopic gravity (MG)[6,7] and its restriction to spatial averaging [21]. Whilethis framework is generally covariant, the issue of gaugedependence in perturbation theory introduces certainsubtleties in the problem. We have shown that, providedone takes seriously the idea that the cosmological back-ground must be defined by an averaging procedure [23], itis possible to attach a gauge invariant meaning to theaveraging condition and the corresponding correlation ob-jects which appear as corrections to the cosmologicalequations. While there remains considerable freedom inan explicit choice of the averaging operator (through achoice of the volume preserving gauge used in its defini-tion), this freedom can be fixed by some additional require-ments based on knowledge of cosmological PT in thestandard implementations. In particular, we have seenthat properties of the conformal Newtonian or Poissongauge can be used to motivate a fully specified choice ofthe averaging operation adapted to first order PT.

One prerequisite to the formulation of a consistent aver-aging framework in the context of perturbation theory isthe absence of perturbative fluctuations with arbitrarilylarge wavelengths, since such fluctuations would rendermeaningless the notion of recovering a homogeneousbackground on averaging. This problem also manifesteditself in the correlation integrals (64), which diverge in thepresence of a finite amplitude of fluctuations as the wavenumber k! 0. Accordingly, all the calculations of thispaper have assumed that the initial power spectrum ofmetric fluctuations has a sharp cutoff at the scale corre-

sponding to k ¼ H0, a hypothesis which is in fact sup-ported by analysis of CMB data [37].The main purpose of this paper was to lay down the

formalism of MG in a language most convenient from thepoint of view of cosmological PT. This was accomplishedby writing Eqs. (47) and (54), and the Fourier space version(64) (with certain simplifying assumptions regarding vec-tor and tensor perturbations which can if needed be relaxedin a completely straightforward manner). This was supple-mented by calculations in the sCDM model [30] (which isthe flat FLRW model with radiation and cold dark matterbut no dark energy) with the additional simplification ofignoring the baryons. The analytical results of Sec. VA aswell as the more detailed numerical results of Sec. VBshow that the correlation objects or backreaction remainnegligibly small up to epochs corresponding to a scalefactor of a� 0:01. While the calculations ignored correc-tions from quasilinear and nonlinear scales, these are notexpected to contribute dramatically to the correlationsobtained here, an expectation which is justified by thework of Behrend et al. [10] and further by the calculationin Ref. [47] (see below).We have seen that, by using the framework of MG, we

have completely bypassed the problem mentioned in theIntroduction, which one faces when applying the Buchertframework to cosmological PT, namely, of having to dealwith two scale factors. Here, one has a single well-definedscale factor associated with the background metric, and itsevolution can be obtained in an iterative fashion as de-scribed in Sec. II C. In practice, we saw that, since thebackreaction is small, convergence can be achieved byessentially a single calculation, at least in the context offirst order PT.This brings us to a final, and very important issue: What

is the magnitude and behavior of the backreaction in thefully nonlinear regime of structure formation? There are (atleast) two possible avenues to approach this question. Thefirst is to set up the problem in a manner which is suitablefor N-body simulations. The iterative approach suggestedearlier can presumably be adapted to full-fledged N-bodycodes as well. While this is a possibility worth pursuing,there is also a less involved (but correspondingly lessrealistic) way of determining the effect of nonlinearitieson the backreaction, which is to study toy models ofstructure formation. As mentioned earlier, such a toymodel of spherical collapse was recently presented inRef. [20], and it was shown by an explicit coordinatetransformation, that the metric can be brought to theNewtonian form (1). Now, it is worth noting that, whileall the calculations of this paper assumed that first order PTis valid, the actual expressions for the correlation integralsin Eq. (54) only assume that the potentials ’ and satisfyj’j, j j 1. The dynamics governing the potentials isirrelevant at this stage. This means that, as long as one isinterested in leading order effects only, the expressions in

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Eq. (54) can be directly applied to any model of structureformation where the metric can be brought to the confor-mal Newtonian form [48]: in particular, they can be appliedto the model of Ref. [20]. This has been done in Ref. [47],and one finds that, even in the fully nonlinear regime, theeffect of the backreaction remains negligible. In this con-text see also Ref. [3] for a third approach.

To conclude, all our calculations and arguments seem toindicate that the averaging of perturbative inhomogeneitiesin a consistent manner appears to lead only to very smalleffects. This does not, however, mean that the effects donot exist. It remains to be seen whether any observableconsequences of the backreaction may be detectable byfuture experiments.

ACKNOWLEDGMENTS

It is a pleasure to thank Professor T. P. Singh for hispatient guidance and for many insightful conversations,and Professor T. Padmanabhan for his encouragementand support, and for fruitful discussions. I am grateful toProfessor Roustam Zalaletdinov for his patience during anearlier correspondence which laid the foundations of thiswork. I thank Jasjeet Bagla and the astrophysics group atHRI, Allahabad, for their hospitality during a visit, duringwhich this work was begun. Finally, I am grateful to AlokMaharana, Subhabrata Majumdar, and Rakesh Tibrewalafor useful discussions.

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