The influence of structure formation on the cosmic expansion

# The influence of structure formation on the cosmic expansion

Chris Clarkson, Kishore Ananda and Julien Larena Cosmology & Gravity Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa
###### Abstract

We investigate the effect that the average backreaction of structure formation has on the dynamics of the cosmological expansion, within the concordance model. Our approach in the Poisson gauge is fully consistent up to second-order in a perturbative expansion about a flat Friedmann background, including a cosmological constant. We discuss the key length scales which are inherent in any averaging procedure of this kind. We identify an intrinsic homogeneity scale that arises from the averaging procedure, beyond which a residual offset remains in the expansion rate and deceleration parameter. In the case of the deceleration parameter, this can lead to a quite large increase in the value, and may therefore have important ramifications for dark energy measurements, even if the underlying nature of dark energy is a cosmological constant. We give the intrinsic variance that affects the value of the effective Hubble rate and deceleration parameter. These considerations serve to add extra intrinsic errors to our determination of the cosmological parameters, and, in particular, may render attempts to measure the Hubble constant to percent precision overly optimistic.

###### pacs:
04.20.-q, 98.80.-k, 98.80.Jk, 91.30.Cd

## I Introduction

The universe appears to be close to homogeneous and isotropic, on average, on large scales, but it exhibits a very clumpy distribution of matter on small scales. To account for this structure, the standard cosmological model relies on the separation of the geometry of spacetime into a perfectly homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) background describing the large scale properties of the universe, such as the expansion rate, and small fluctuations around this background solution. This provides a straightforward perturbative treatment of the growth of structure under the influence of gravitation. The explicit construction of the background by a smoothing or averaging procedure applied to the clumpy Universe is often ignored, and the background appears as an artificial mathematical object used to perform the calculations of gauge invariant quantities characterising the physical properties of the clumpy universe.

The essence of this ‘averaging problem’ comes when we try to match the late time universe today, which is full of structure, to the early time universe, which isn’t. At the end of inflation we are left with a universe with curvature characterised by some constant ( in some units), and cosmological constant, , which are fixed for all time (and might be zero), and perturbations which are of tiny amplitude and well outside the Hubble radius; there is no averaging problem at this time, and the idea of background plus perturbations is very natural and simple to define. Fast-forward to today, where structures are non-linear, are inside the Hubble radius, and many have broken away from the cosmic expansion altogether. We may still apparently describe the universe as FLRW plus perturbations to high accuracy; that is, it is natural and seemingly correct to define a FLRW background, but it is implicitly assumed that this background is the same one that we are left with at the end of inflation, in terms of and . Mathematically we can follow a model from inflation to today, but when we try to fit our models to observations to describe our local universe we are implicitly smoothing over structure, and this can contaminate what we think our inflationary background FLRW model should be. Indeed, it is not clear that the background smoothed model should actually obey the field equations at all. Within the standard paradigm, then, the averaging problem also becomes a fitting problem; are the background parameters we are fitting with the CMB actually the same as those when fitting SNIa? (See KMM () for a discussion of these issues.)

Because of the non-linearity of the Einstein field equations, the explicit construction of a homogeneous background is far from trivial and it has been know for a long time that the local fluctuations may influence the way the Universe behaves on average Ellis84 (); this effect is usually dubbed backreaction and has started to be investigated in detail (see, e.g., backreactionbiblio1 (); backreactionbiblio2 (); Rasanen (); KMNR (); backreactionbiblio3 (); backreactionbiblio4 (); backreactionbiblio5 (); wilt0 (); wilt1 (); wilt2 (); wilt3 (); backreactionbiblio6 (); backreactionbiblio7 (); backreactionbiblio8 (); lischwarz (); lischwarz1 (); lischwarz2 (); Iain1 (); Iain2 (); Iain3 (); Kasai (); coley1 (); coley2 (); AverageArbCoord () and references therein), often as a possible solution to the dark energy problem itself. The problem is mathematical: if we have an inhomogeneous matter distribution in some spacetime, and we try to calculate a homogeneous ‘background’ by smoothing the matter content and calculating the new smoothed metric, we get a different answer than if we smooth the metric directly; this difference is usually termed backreaction, in this context.

In this work, we explore the changes to the background due to the presence of perturbations by explicitly calculating the effects of backreaction up to second-order in perturbation theory. Similar investigations have been done in the past in the synchronous gauge lischwarz (); lischwarz1 (); lischwarz2 (), and in the Poisson gauge (see e.g. KMNR (); Rasanen (); Kasai (); Iain1 (); Iain2 ()). The authors of these previous works have mainly considered only terms which are quadratic in linear quantities (ignoring the second-order Bardeen potentials), and/or a pure Einstein de-Sitter Universe (ignoring the cosmological constant).

Here, we present an analysis which is consistent up to second order, both for Einstein de-Sitter and the concordance model – flat CDM – as underlying background solutions. To perform our averages we argue that the rest frame of the gravitational field is a natural choice of spatial hypersurface, and that this may be unambiguously defined by the vanishing of the magnetic part of the Weyl tensor, where this is possible. We then define the effective expansion rate through the average of the matter fluid expansion projected onto these spatial hypersurfaces, rather than through the expansion 4-scalar for matter (as in Rasanen (); KMNR ()) or through the expansion of the ‘observer’ (defined below) flow lines (as in Kasai (); Iain1 (); Iain2 (); Iain3 () – though as we show below, this is in fact the rest frame of the gravitational potential). Our choice can be justified by the fact that the averaging process used in this work is frame (observer) dependent, so that, for a matter of consistency, one wants to retain the observer dependence on the quantities that are averaged also. On the one hand, the usual expansion 4-scalar for matter is the observed expansion only for observers comoving with the fluid so that it is only appropriate for averaging with respect to a set of comoving observers. On the other hand, the expansion of the observer flow lines is clearly frame dependent, but it does not encompass any information about the matter flow, apart from its background part; the averaged homogeneity being a characteristic of the total matter flow, it seems then more natural to retain a quantity that takes into account the fluctuations in the matter distribution and its peculiar velocity with respect to the observer’s rest-frame.

Moreover, we include an important characteristic effect of backreaction, viz. the existence of an intrinsic variance on quantities such that the Hubble rate and the deceleration parameter (as noted by lischwarz1 (); KMNR ()). This is important because it gives us our ‘ error’ which is an intrinsic unknown when identifying a background based on Gaussian perturbations around FLRW.

The paper is organized as follows. In Section 2 we describe the averaging formalism used in the rest of the paper and based on a recent generalization AverageArbCoord () of the standard averaging procedure (defined in a comoving coordinate system) to arbitrary coordinate systems. In Section 3, we apply this formalism to the averaging of the cosmological model in the Poisson gauge up to second order. In Section 4, we present the effects on the Hubble rate and deceleration parameter, with a particular emphasis on the importance of the different scales involved (smoothing and averaging) in the process. We find that the effect on the quantities themselves is small (as expected in perturbation theory) but that it can be quite large in terms of the variance affecting these quantities. Our results are quantitatively in agreement with the results obtained in the synchronous gauge lischwarz (); lischwarz1 (); lischwarz2 (). Finally, Section 5 summarizes the results and addresses quickly possible future developements of this work.

## Ii Averaging formalism in an arbitrary coordinate system

In this paper, we are concerned with estimating the backreaction effect in the Poisson gauge of perturbation theory. In this gauge, the 4-velocity of the matter fluid is tilted with respect to the timelike normal vector of the coordinate system, and this makes it difficult to use the standard averaging procedure (see backreactionbiblio1 ()), that has been developed with respect to observers comoving with the cosmic matter fluid. A recent work AverageArbCoord () has generalized the averaging procedure to an arbitrary coordinate system, which allows us to estimate consistently the backreaction effect in the Poisson gauge. This section briefly presents this formalism and discusses the assumptions at the root of it. Essentially, we introduce two velocity fields, one in the rest-frame of the matter content, and another arbitrary one on which we fix an arbitrary family of ‘observers’. The role of the observers is to introduce spacelike hypersurfaces on which we perform our average; these hypersurfaces are tilted with respect to the spacelike hypersurfaces defined by the fluid (tilted in the sense that their normal vectors are tilted). We fix the coordinates with respect to the observers, and so loosely refer to this as the coordinate frame.

Throughout the paper, we will suppose that gravitation is well described by general relativity on all scales and that the cosmic matter fluid can be considered as a perfect fluid. Moreover, Latin letters of the beginning of the alphabet will denote spacetime indices, and Latin letters in the middle the alphabet will denote spatial indices.

### ii.1 1+3 threading of spacetime

We consider a set of observers defined at each point of the spacetime manifold , and characterized by a unit 4-velocity field that is everywhere timelike and future directed, i.e. , with zero vorticity. This 4-velocity field induces a natural foliation of spacetime by a continuous set of space-like hypersurfaces locally orthogonal to . Then we can define the projection tensor field onto these hypersurfaces as that is a well-defined Riemannian metric for these hypersurfaces. The line element can then be written, with respect to this foliation:

 ds2=−(N2−NiNi)dt2+2Nidtdxi+hijdxidxj , (1)

where we have introduced respectively the lapse function and the shift 3-vector such that the components of the 4-velocity read:

 na=1N(1,−Ni) ,    na=N(−1,0,0,0) . (2)

For the purposes of this paper, the hypersurfaces orthogonal to are characterized by two quantities:

• the intrinsic curvature of the hypersurfaces: , where is the 3-Ricci curvature of the hypersurfaces;

• the extrinsic curvature (or second fundamental form): that encodes the way the hypersurfaces are embedded in the manifold .

In the following, we will assume that the matter content can be well described by a perfect fluid (not necessarily irrotational) of energy density , pressure and 4-velocity (with ), so that its stress-energy tensor reads:

 Tab=(ρ+p)uaub+pgab . (3)

Note that in this work, the 4-velocity of matter is not necessarily aligned with the 4-velocity of the observers , so that there exists a vector field corresponding to the relative velocity of the matter fluid with respect to the fundamental observers. is space-like and orthogonal to () and one has:

 ua=γ(na+va)   with   γ=1√1−v2 , (4)

where is the usual Lorentz factor and . Thanks to the 1+3 foliation, the Einstein field equations can be separated in two different sets: the constraint equations that have to be satisfied on every hypersurface, and the evolution equations, that prescribe how the fields evolve from one hypersurface to another infinitesimally close. The Hamiltonian constraint reads:

 R−KijKji+K2=16πGϵ+2Λ ,~{}~{}~{}% where     ϵ=Tabnanb=γ2ρ+(γ2−1)p , (5)

where . The momentum constraint is:

 ~∇iKij−~∇jK=8πGJj ,~{}~{}~{}% where~{}~{}~{} Jj=−Tabnahbj=γ2(ρ+p)vj , (6)

where we have defined the projected covariant 3-derivative on the spatial hypersurfaces of any tensor field : . The evolution equation for the first fundamental form reads:

 1N∂thij=−2Kij+2N~∇(jNi) , (7)

and for the second fundamental form, one has:

 1N∂tKij=Rij+KKij−Λδij−1N~∇j~∇iN+1N(Kik~∇jNk−Kkj~∇kNi+Nk~∇kKij)−8πG(Sij+12(ϵ−Skk)δij) , (8)

with . These equations have to be supplemented by the energy-momentum conservation for the matter fluid: . We will now introduce the standard decomposition for the covariant spatial derivatives of the 4-vectors in terms of their trace, symmetric trace-free and antisymmetric parts. Writing for any quantity , one has:

 ∇anb = −na˙nb+13ξhab+Σab with    ξ≡hcahda∇cnd~{}~{}~{} and ~{}~{}~{}Σab≡hcahdb∇(cnd)−13ξhab ; ∇aub = −na(2˙nb+v˙vnb)−γ~∇avnb+13θhab+σab+ωab with    θ≡hcahda∇cud ,~{}~{}~{} σab≡hcahdb∇(cud)−13θhab~{}~{}~{} and ~{}~{}~{}ωab≡hcahdb∇[cud] ; ∇avb = −na(˙vb+Xb)+Yanb+13κhab+βab+Wab with    κ≡hcahda∇cvd ,~{}~{}~{} βab≡hcahdb∇(cvd)−13κhab and    Xa≡nbhca∇cvb , ~{}~{}~{}Yb≡nchab∇cva , ~{}~{}~{}% Wab≡hcahdb∇[cvd] .

In these relations, , and denote the isotropic expansion rates of the 4-velocities , and of the peculiar velocity respectively, and , and their shears, with respect to the threading of spacetime induced by the vector . and are, respectively, the vorticities111Since it defines the foliation, is vorticity free by definition. of and in this same foliation. These quantities are those measured by the observers with 4-velocities in their instantaneous rest-frame. In particular , and differ from the usual expansion, shear and vorticity of the matter fluid as measured by observers comoving with this matter fluid (by acceleration terms essentially), that are defined by the decomposition of . For example, the expansions are linked by the relation:

 Θ≡∇aua=θ+γ(γ2va˙va−na˙va) . (12)

Using (4), one can relate these quantities as follows:

 ξ = γ−1θ−κ−γ2B (13) Σab = γ−1σab−βab−γ2(B(ab)−13Bhab) (14) Wab = γ−1ωab−γ2B[ab] , (15)

where we have introduced the tensor:

 Bab≡13κ(vanb+vavb)+βcavcnb+βcavcvb+Wcavcnb+Wcavcvb , (16)

whose trace is given by . In our notation, angular and round brackets denote the antisymmetric and symmetric parts, respectively, of a tensor projected with . Let’s finally introduce the following notation for convenience:

 θB ≡ −γκ−γ3B (17) σBij ≡ −γβij−γ3(B(ij)−13Bhij) , (18)

so that:

 ξ=γ−1(θ+θB) , (19) Σij=γ−1(σij+σBij) . (20)

The non-local, free gravitational field is described by the Weyl tensor. Given a timelike vector this is split into electric and magnetic parts. For example, with respect to these are

 E(n)ab=Cacbdncnd    and    H(n)ab=∗Cacbdncnd, (21)

where is the Weyl tensor and is its dual. Analogous definitions exist for the vector field . This means that observers in the frame of the fluid and observers in the coordinate frame observe this electric-magnetc split differently (see mge () for the transformation relations between the two), analogously to boosted observers measuring different electric and magnetic parts of the electromagnetic field. In particular, in certain gravitational fields there may exist a special frame whereby one of these two components vanishes. For example, in so-called silent universes which are not conformally flat, there exists a preferred frame in which the magnetic part of the Weyl tensor is zero – such a frame may be considered the rest-frame of the gravitational field. In spacetimes where this is possible, it is unique as follows from the transformation laws in mge (), and there exist (at least) two physical, well motivated, frames: the rest-frame of the fluid, and the rest-frame of the non-local gravitational field. We return to this below.

### ii.2 General averaging procedure

The non-locality of the spatial average that is usually calculated manifests itself problematically when interpreting averaged quantities. What do they mean? In which spacetime do they exist – the rough or the smooth? – representing physical things (i.e., where are they tensorial objects)? It is tempting to average the fluid expansion of a lumpy spacetime, say, and to think of this as actually being in some sense the ‘averaged fluid expansion rate’. Normally, of course, the expansion rate of a fluid is a covariantly defined local object, and so unambiguous when defined in the local rest-frame of the fluid. But it is not understood how to define the rest-frame of the non-local smoothed fluid in a covariant way, and the ‘average expansion rate’ picks up this ambiguity.

Furthermore, any definition of an averaged expansion rate in the spacetime in which we started is not covariant from a 4-dimensional perspective because the average is with respect to spatial hypersurfaces defined by in the un-smoothed spacetime, and so implicitly rely on some coordinates, and, hence, a mapping between the unsmoothed and smoothed spacetimes. How do we choose these coordinates? Scalar averaging approaches give us access to some averaged quantities and backreaction terms, but not the spacetime in which they exist as the objects they are supposed to represent. Analogously to gauge freedom, this ambiguity leaves implicit choices to make about what objects we consider important, as well as what they mean. Previous analyses have fixed these freedoms in one way or another.

Ideally, we would like to construct a smooth FLRW ‘spacetime’ from an lumpy inhomogeneous spacetime by averaging over structure. This would, in principle, have a metric

 ds2eff=−dτ2+a2Dγijdyidyj , (22)

where is the cosmic time and a scale factor, the subscript indicating that it has been obtained at a certain spatial scale characteristic of a compact spatial domain , which is large enough so that a homogeneity scale has been reached; in this case will be a metric of constant curvature. This is not to imply that this would be a spacetime in the usual sense because the normal observational relations may not follow directly from this FLRW metric – these may have to be calculated separately crap (); rass (). However, in the context of perturbation theory the effect of renormalisation of the background appears naturally as a first step in this procedure.

In the context of averaging a perturbed spacetime we consider below, we can imagine choosing coordinates which straddle both the rough and smooth spacetimes. In particular, we can choose our coordinates in the rough spacetime such that they become the ones we want in the smooth spacetime. We will choose our coordinates such that the time coordinate in the rough spacetime becomes the proper time coordinate in the smoothed one: that is, we will set as well as . (We will also set .) This effectively de-synchronises the clocks between the rough (with proper time when ) and smoothed (proper time ) spacetimes. The averaging operator we define is simply the Riemannian average over the domain in the surfaces orthogonal to (i.e., const.):

 ⟨ψ⟩D≡1VD∫DJd3xψ(t,xi) , (23)

and is well defined for any scalar function . This choice has the property that the commutator between partial time derivatives and spatial averages which reads, when ,

 [∂t⋅,⟨⋅⟩D]ψ(t,xi)=⟨Nξψ⟩D−⟨Nξ⟩D⟨ψ⟩D , (24)

is zero when we consider perturbed Einstein-de Sitter models below at second-order (that is, the expansion of the spatial hypersurfaces when scaled by is, for the perturbed model we consider below, ; with as it is for pure dust, the commutator vanishes at second-order).

Of the three expansion rates we have introduced, tells us the expansion of the coordinate grid, and so is not physically attached to the fluid. measures the fluid expansion rate in its own rest frame, and is the sensible choice of expansion rate with which to characterise the rough spacetime. However, as we have mentioned, after smoothing, the rest-frame of the fluid will change in a way which is not yet known, and will depend on the domain. Any expansion rate we try to investigate must take this into account and so allow for a tilt between the fluid and the normal to the hypersurfaces we use to average. The expansion rate , which is the trace of the expansion tensor of the fluid projected into the coordinate rest-frame in which the averaging takes place is the most natural choice when accounting for peculiar velocities in this way. We shall define our observers, which define the spatial hypersurfaces on which the averaging takes place, by the rest frame of the gravitational field; that is, the frame in which . At second-order in a perturbation expansion, this is different from the rest-frame of the matter.

Thus, in the rough spacetime, when we consider the length-scale associated with , we have

 13θ=na∇alnℓ=1ℓdℓdtprop=1Nℓ∂ℓ∂t.

Hence, if represents the proper time in the smoothed spacetime, we can define a pre-synchronised, smoothed, Hubble parameter using as backreactionbiblio2 ()

 HD≡13⟨Nθ⟩D=13VD∫DJd3xNθ . (25)

We may think of this as the average Hubble parameter which preserves the length-scale after smoothing, according to the pre-chosen proper time in the smoothed spacetime. This is referred to as the scaled t-Hubble parameter in backreactionbiblio2 (), where it is introduced to preserve the structure of the averaged equations. This is not a unique choice and we refer to wilt0 () for a detailed discussion. In particular, this Hubble parameter changes under a re-scaling of ; in the context of perturbations of FLRW, we use to be the proper time in the background – if we were to use conformal time, say, would be an averaged conformal Hubble parameter. We can use this to then define the effective scale factor for the averaged model as the function obeying:

Using the commutation relation, one can average the scalar part of the Einstein field equations to obtain a set of two equations giving the behavior of the effective scale factor (see AverageArbCoord () with ):

where we have defined the standard kinematical backreaction:

 (29)

 LD ≡ (30) PD ≡ (31) KD ≡ ⟨N2γ−1θBθ⟩D−3⟨Nγ−1θB⟩DHD (32) FD = (33)

## Iii Averaging perturbed FLRW models

We consider the backreaction effect in a perturbed FLRW model, with a flat background and a cosmological constant. We are interested in the backreaction effect at late times, and so assume the matter in our model is comoving cold dark matter plus baryons. We shall consider scalar modes up to second-order, and ignore vectors and tensors throughout, as the first-order contributions are small and it has been shown that the vectors and tensors induced by first-order scalars are also sub-dominant, although they could provide a slight correction to the results presented here MHM (); LAC (); LACM (); ACW (); BSTI (). In the Poisson gauge222In this work, we call a gauge the choice of both the frame, i.e. , and the coordinate set on hypersurfaces orthogonal to . the metric reads BMR ()

 ds2=−(1+2Φ+Φ(2))dt2+a2(1−2Ψ−Ψ(2))δijdxidxj. (34)

The first-order scalar perturbations are given by , and the second-order by . In this form we have the metric in its Newtonian form, which we may think of as the local rest-frame of the gravitational field. Comparing with the general metric given above, we are in a gauge with zero shift vector, and . Moreover, the matter fluid has a peculiar velocity where , where is the first order part and the second order part of the velocity potential. The spatial metric is obviously . The Poisson gauge is particularly elegant for scalar perturbations because with defined orthogonal to the spatial metric , the Weyl tensor becomes

 E(n)ij = 12(h aih bj−13hijhab){~∇a~∇b[Φ+Ψ−Φ2−Ψ2+12(Φ(2)+Ψ(2))]+~∇aΦ~∇bΦ−~∇aΨ~∇bΨ} (35) H(n)ij = 0. (36)

In the rest frame , then, the gravitational field is silent, and, with is a pure potential field. Hence, may be considered as the rest-frame of the gravitational field, and so defines natural hypersurfaces with which to perform our averages. By contrast, in the frame the Weyl tensor has non-zero  mge ().

As time coordinates we shall use conformal time, , proper time in the background, , and background redshift , related by , interchangeably, where the background Hubble rate is given by

 H(z)2=H20[Ω0(1+z)3+1−Ω0] (37)

and is the present-day matter density parameter. Note that in the perturbed spacetime the parameter is just a time coordinate, even though we refer to it as redshift. The Raychaudhuri equation in the background, , gives the deceleration parameter

 qnormal(z)=−1H2¨aa=−1+1+zH(z)dHdz=−1+32Ωm(z), (38)

where

 Ωm(z)=Ω0(1+z)3[Ω0(1+z)3+1−Ω0]1/2, (39)

gives the evolution of the density parameter; the density parameter associated with the cosmological constant is .

For a single fluid with zero pressure and no anisotropic stress , and obeys the Bardeen equation

 Φ′′+3HΦ′+a2ΛΦ=0=¨Φ+4H˙Φ+ΛΦ. (40)

and , and is the conformal Hubble rate. All first-order quantities can be derived from ; for example,

 v(1)=−23aH2Ωm(˙Φ+HΦ), (41)

and is the source of the second-order scalars. The solution to the growing mode of the Bardeen equation may be written as

 Φ(η,x)=g(η)Φ0(x) (42)

where is the Bardeen potential today () and is the growth suppression factor, which may be approximated, in terms of redshift, as lahav (); Carroll ()

 g(z)=52g∞Ωm(z){Ωm(z)4/7−ΩΛ(z)+[1+12Ωm(z)][1+170ΩΛ(z)]}−1. (43)

and is chosen so that .

We define our Fourier transform as (suppressing any temporal quantities)

 Φ(x)=1(2π)3/2∫d3kΦ(k)eik⋅x, (44)

where . The power spectrum of is defined by

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Φ(k)Φ(k′)=2π2k3PΦ(k)δ(k+k′), (45)

where an overbar denotes an ensemble average. Assuming scale-invariant initial conditions from inflation, this is given by

 PΦ(z,k)=(3ΔR5g∞)2g(z)2T(k)2 (46)

where is the normalised transfer function, is the primordial power of the curvature perturbation, with WMAP5 () at a scale .

The second-order solutions for and are given by BMR (). We quote their results directly:

 Ψ(2)(η,x) = (B1(η)−2g(η)gm−103(anl−1)g(η)gm)Φ20 (47) +(B2(η)−43g(η)gm)[∇−2(∂iΦ0∂iΦ0)−3∇−4∂i∂j(∂iΦ0∂jΦ0)] +B3(η)∇−2∂i∂j(∂iΦ0∂jΦ0)+B4(η)∂iΦ0∂iΦ0, Φ(2)(η,x) = (B1(η)+4g2(η)−2g(η)gm−103(anl−1)g(η)gm)Φ20 (48) +B3(η)∇−2∂i∂j(∂iΦ0∂jΦ0)+B4(η)∂iΦ0∂iΦ0,

where with the following definitions

 ~B1(η) = ∫ηηmd~ηH2(~η)(f(~η)−1)2C(η,~η),~B2(η)=2∫ηηmd~ηH2(~η)[2(f(~η)−1)2−3+3Ωm(~η)]C(η,~η), (49) ~B3(η) = 43∫ηηmd~η(e(~η)+32)C(η,~η),~B4(η)=−∫ηηmd~ηC(η,~η), (50)

and

 (51)

with and

 f(η)=1+g′(η)Hg(η). (52)

denotes the value of , deep in the matter era before the cosmological constant was important. We also have which denotes any primordial non-Gaussianity present. We shall set this to unity, representing a single field slow-roll inflationary model, in what follows.

We shall also require the perturbed energy density up to second-order

 κ2δ2ρ = 2a2∂2Ψ(2)−6H˙Ψ(2)−6H2Φ(2)+24H2Φ2+6˙Φ2+16a2Φ∂2Φ (53) −83a2H2Ωm[H2(1−94Ωm)∂kΦ∂kΦ+2H∂kΦ∂k˙Φ+∂k˙Φ∂k˙Φ]

(where ), and the Laplacian of the perturbed velocity:

 3aH2Ω2m ∂2υ(2) = −2Ωm(∂2˙Ψ(2)+H∂2Φ(2)) (54) +4H(Ωm−2)Φ∂2Φ−4(Ωm+2)˙Φ∂2Φ−4(3Ωm+2)Φ∂2˙Φ−8H˙Φ∂2˙Φ +4H(Ωm−2)∂kΦ∂kΦ−16(Ωm+1)∂kΦ∂k˙Φ−8H∂k˙Φ∂k˙Φ +83a2H2[H∂2Φ ∂2Φ+∂2Φ ∂2˙Φ+H∂kΦ ∂2∂kΦ+∂k˙Φ ∂2∂kΦ].

### iii.1 The averaged perturbed EFE

By making use of the expansion at second order of the three dimensional volume element

 J=a3[1−3Ψ+32(Ψ2−Ψ(2))], (55)

for any scalar function , the Riemannian average can be expanded in terms of the Euclidean average over the domain

 ⟨Υ⟩=∫Dd3xΥ∫Dd3x (56)

on the background space slices as:

 ⟨Υ⟩D=Υ(0)+⟨Υ(1)⟩+⟨Υ(2)⟩+3[⟨Υ(1)⟩⟨Ψ⟩−⟨Υ(1)Ψ⟩] , (57)

where , and denote respectively the background, first order and second order parts of the scalar function . In the rest of the paper, every spatial average will be a Euclidean one.

The averaged Hubble rate as defined by equation (25) is given by:

 HD = H−⟨˙Φ⟩−2(1+z)29H2Ωm(H⟨∂2Φ⟩+⟨∂2˙Φ⟩)+⟨Φ ˙Φ⟩ (58) +2(1+z)29H3Ω2m{2HΩm[H⟨Φ ∂2Φ⟩+⟨Φ ∂2˙Φ⟩]+(1+3Ωm)H2⟨∂kΦ ∂kΦ⟩+(2+3Ωm)H⟨∂kΦ ∂k˙Φ⟩+⟨∂k˙Φ ∂k˙Φ⟩} −3⟨Φ⟩⟨˙Φ⟩−2(1+z)23H2Ωm[H⟨Φ⟩⟨∂2Φ⟩+⟨Φ⟩⟨∂2˙Φ⟩] −12⟨˙Ψ(2)⟩+16(1+z)⟨∂2υ(2)⟩.

The averaged Friedmann equation (27) reads:

 H2D = H2−16(QD−LD+RD)−2H⟨˙Φ⟩+23(1+z)2⟨∂2Φ⟩−4H2⟨Φ2⟩+2H⟨Φ ˙Φ⟩−23(1+z)2⟨Φ ∂2Φ⟩ (59) +4(1+z)29H2Ω2m(1+Ωm)[H2⟨∂kΦ ∂kΦ⟩+2H⟨∂kΦ ∂k˙Φ⟩+⟨∂k˙Φ ∂k˙Φ⟩]−6H⟨Φ⟩⟨˙Φ⟩+2(1+z)2⟨Φ⟩⟨∂2Φ⟩ +H2⟨Φ(2)⟩+κ26⟨δ2ρ⟩,

Finally, the averaged acceleration equation (28), which gives an effective Raychaudhuri equation, reads:

 3∂2taDaD = (60) +9H2(1−Ωm)⟨Φ⟩+3H⟨˙Φ⟩−(1+z)2⟨∂2Φ⟩ +3H2(9Ωm−7)⟨Φ2⟩−3H⟨Φ ˙Φ⟩+(1+z)2⟨Φ ∂2Φ⟩ +(1+z)23H2Ω2m(4−9Ωm)[H2⟨∂kΦ ∂kΦ⟩+2H⟨∂kΦ ∂k˙Φ⟩+⟨∂k˙Φ ∂k˙Φ⟩] +9H⟨Φ⟩⟨˙Φ⟩+27H2(1−Ωm)⟨Φ⟩2−3(1+z)2⟨Φ⟩⟨∂2Φ⟩ +3H2(1−32Ωm)⟨Φ(2)⟩−κ24⟨δ2ρ⟩,

The averaged curvature term is:

 RD = 2(1+z)2[2⟨∂2Φ⟩+6⟨Φ ∂2Φ⟩+3⟨∂kΦ ∂kΦ⟩+6⟨Φ⟩⟨∂2Φ⟩+⟨∂2Ψ(2)⟩], (61)

and the additional backreaction terms are:

 FD = 4(1+z)23H2Ω2m[H2⟨∂kΦ ∂kΦ⟩+2H⟨∂k