Perturbation theory for cosmologies with non-linear structure

Transcription

1 Perturbaton theory for cosmologes wth non-lnear structure Goldberg, S; Gallagher, C; Clfton, T 2017 Amercan Physcal Socety Ths s a pre-copyedted, author-produced verson of an artcle accepted for publcaton n Physcal evew D followng peer revew. The verson of record s avalable For addtonal nformaton about ths publcaton clck ths lnk. Informaton about ths research object was correct at the tme of download; we occasonally make correctons to records, please therefore check the publshed record when ctng. For more nformaton contact scholarlycommuncatons@qmul.ac.uk

2 Perturbaton theory for cosmologes wth non-lnear structure Sopha. Goldberg, Chrstopher S. Gallagher, and Tmothy Clfton School of Physcs and Astronomy, Queen ary Unversty of London, le End oad, London E1 4S, UK. arxv: v6 [gr-qc] 27 Oct 2017 The next generaton of cosmologcal surveys wll operate over unprecedented scales, and wll therefore provde exctng new opportuntes for testng general relatvty. The standard method for modellng the structures that these surveys wll observe s to use cosmologcal perturbaton theory for lnear structures on horzon-szed scales, and ewtonan gravty for non-lnear structures on much smaller scales. We propose a two-parameter formalsm that generalzes ths approach, thereby allowng nteractons between large and small scales to be studed n a self-consstent and well-defned way. Ths uses both post-ewtonan gravty and cosmologcal perturbaton theory, and can be used to model realstc cosmologcal scenaros ncludng matter, radaton and a cosmologcal constant. We fnd that the resultng feld equatons can be wrtten as a herarchcal set of perturbaton equatons. At leadng-order, these equatons allow us to recover a standard set of Fredmann equatons, as well as a ewton-posson equaton for the nhomogeneous part of the ewtonan energy densty n an expandng background. For the perturbatons n the large-scale cosmology, however, we fnd that the feld equatons are sourced by both non-lnear and mode-mxng terms, due to the exstence of small-scale structures. These extra terms should be expected to gve rse to new gravtatonal effects, through the mxng of gravtatonal modes on small and large scales effects that are beyond the scope of standard lnear cosmologcal perturbaton theory. We expect our formalsm to be useful for accurately modellng gravtatonal physcs n unverses that contan non-lnear structures, and for nvestgatng the effects of non-lnear gravty n the era of ultra-large-scale surveys. I. ITODUCTIO There s hope that the next generaton of astronomcal surveys[1 3], whch wll collect data on scales comparable to the cosmologcal horzon, wll have suffcent precson to provde a new testng ground for non-lnear relatvstc gravty [4 7]. Ths s a partcularly exctng prospect as, to date, non-lnear gravtatonal effects have only been observed n the solar system [8 11], bnary pulsar systems [12, 13], and the newly dscovered bnary black hole mergers observed usng LIGO [14 16]. The observaton of correspondng effects n cosmology would allow general relatvty to be nvestgated on entrely new length and tme scales, as well as n a totally dfferent physcal envronment. There s no tellng what ths mght reveal about Ensten s theory. To develop a mathematcal formalsm for nvestgatng the non-lnear propertes of gravty n cosmology s, however, a hghly non-trval task. There s now a substantal lterature dedcated to developng dfferent approaches to modellng non-lnear gravtatonal physcs n cosmology. The most common approach s a drect mplementaton of second-order cosmologcal perturbaton theory [17 19], whch allows relatvstc gravtatonal perturbatons around a homogeneous and sotropc background to be modelled n the presence of lnear densty contrasts. Other approaches, however, have started to mport technques from post-ewtonan gravty [20 22], where gravtatonal felds are assumed to be slowly varyng and where non-lnear densty contrasts can be consstently modelled. These two dfferent approaches are the obvous non-lnear extensons of the standard method that has been so successful when consderng lnear gravty n cosmology; usng cosmologcal perturbaton theory to model lnear structures on scales above the homogenety scale 100pc, and ewtonan gravty to model non-lnear structures on smaller scales. Ths looks very natural at lnear order n the gravtatonal felds, partly because the equatons of ewtonan gravty can be recovered as the quas-statc lmt of cosmologcal perturbaton theory when the gravtatonal felds slowly vary n tme. Indeed, ths pleasng feature extends to non-lnear gravtatonal felds n cosmology [20]. However, f one wants to consder non-lnear gravty n a unverse that smultaneously contans lnear structures on large scales and non-lnear structures on small scales, then one must adopt a more sophstcated approach. Ths s because, when consderng the quas-statc lmt, the terms that have been relegated to hgher-order can no longer be entrely forgotten; they can and should be expected to appear n the next-to-leadng-order gravtatonal feld equatons. Ths could be at second-order on small scales, but could n prncple be at what s usually thought of as Electronc address: s.r.goldberg@qmul.ac.uk Electronc address: c.s.gallagher@qmul.ac.uk Electronc address: t.clfton@qmul.ac.uk

3 frst-order on large scales. The queston of how to construct a perturbatve expanson that can systematcally perform the requred re-orderng, and produce a self-consstent and well-motvated set of feld equatons on all scales, s the purpose of ths paper. Ths work s a development of the two-parameter perturbatve expanson proposed and nvestgated n a prevous paper [23]. We extend and apply ths work to the case of realstc cosmologcal models that contan relatvstc fluds wth barotropc equatons of state, as well as a cosmologcal constant, Λ, and non-relatvstc dust-lke matter that can be used to model dark matter and baryons. The result s a set of equatons that can be used to calculate the effect of small-scale structure on the leadng-order perturbatons on large scales. These equatons contan terms that are quadratc n short-scale potentals and can be wrtten as an effectve flud [24], as well as mode-mxng terms 1 that couple scalar, vector and tensor perturbatons n the large-scale cosmology. Both of these two types of terms offer exctng possbltes for testng non-lnear gravty wth upcomng surveys. In Secton II we gve the fnal feld equatons for the two-parameter perturbatve expanson constructed durng ths paper, wrtten n ther effectve flud form. In Secton III we dscuss the constructon of the two-parameter perturbaton expanson. Secton IV contans a study of nfntesmal gauge transformatons n the context of twoparameter expansons, as well as the constructon of gauge-nvarant varables. The applcaton of these varables to the feld equatons s then dscussed n Secton V. We use Greek ndces to represent spacetme coordnates, and Latn ndces for spatal coordnates. Dots refer to dfferentaton wth respect to cosmc tme ϕ ϕ t, and dashes refer to dfferentaton wth respect to conformal tme ϕ ϕ τ. Sngle spatal dervatves are gven by ϕ, ϕ ϕ, and 2 refers to the Laplacan assocated wth spatal partal dervatves wth respect to the comovng coordnates. Throughout, we use unts n whch G = c = 1. 2 II. PETUBED FIELD EQUATIOS In ths secton we wll present the perturbed feld equatons that result from smultaneously consderng lnear structures on large scales, and non-lnear sngle-stream structures on small scales. The quanttes that appear n these equatons wll then be explaned further n the sectons that follow, wth techncal detals reserved for the appendces. We hope ths wll allow the busy reader to see the most physcally nterestng aspects of ths work wthout frst havng to undergo the lengthy mathematcal consderatons that went nto ther dervaton. It wll, however, requre the reader to temporarly suspend ther satsfacton of knowng exactly what all the quanttes nvolved represent, as we then move on to more carefully lay out ther specfc techncal defntons and propertes. The mult-stream generalzaton should follow straghtforwardly. The feld equatons we present wll be expressed n terms of the followng set of gauge-nvarant gravtatonal felds: as well as a correspondng set of gauge-nvarant matter perturbatons: {U,φ,ψ,S,h }, 1 {δρ,δρ,δp,v,v }, 2 where ρ, p and v correspond to the energy-densty, pressure and pecular velocty, respectvely. These quanttes all represent fluctuatons about a spatally-flat FLW geometry, whch n a partcular choce of coordnates can be wrtten as [ ds 2 = a 2 τ 1+2U+2φdτ U 2ψδ +2h dx dx j 2S dτdx ], 3 where a s the scale factor and τ s a conformal tme coordnate. In what follows we wll also requre the Hubble rate n conformal tme, whch we wrte as H a /a. There are a couple of thngs that requre some explanaton before we can proceed further. Frst, U has been removed from the scalar potentals φ and ψ because we ntend t to correspond to the ewtonan gravtatonal potental. A precse explanaton of what we mean by ths wll come n the followng sectons. For now t suffcent to understand U as the leadng-order part of the gravtatonal feld produced by non-lnear densty contrasts. The potentals φ and ψ, as well as h and S, contan nformaton about both the large-scale cosmologcal potentals and the small-scale post-ewtonan potentals. Agan, the precse meanng of ths wll become clearer n subsequent sectons. Lkewse, 1 ote that we use mode-mxng to descrbe the couplng of scalar, vector and tensor perturbatons. We wll use scale-mxng to refer to the couplng of large-scale and short-scale perturbatons due to quadratc terms.

4 the ewtonan densty contrast s δρ, and the cosmologcal and post-ewtonan densty contrast s gven by δρ. The former of these s allowed to be arbtrarly large, whle the latter s requred to be small agan, to be made precse later on. Smlar comments apply to v and v. After smultaneously expandng the feld equatons n post-ewtonan and cosmologcal perturbaton theores we fnd the leadng-order parts are gven by the effectve Fredmann equatons H 2 = 8πa2 3 ρ+ 1 3 Λa2 +Oη 4 and H = 4πa2 3 ρ+3 p+ 1 3 Λa2 +Oη 4, 4 where ρ = ρ + ρ and p = p are the leadng order parts of the spatal averages of the energy densty and pressure, respectvely. They have both radaton ρ and p, and dark and baryonc matter ρ contrbutons. The ewtonan gravtatonal feld equaton occurs at the same order n our expanson, and s gven by 2 U = 4πa 2 δρ +Oη 4, 5 where η s the expanson parameter for the post-ewtonan expanson. In ths case t s used to characterse the sze of structure on scales of order the homogenety scale; the largest-scale at whch the post-ewtonan expanson can sensbly be performed. The reader may note that only dark matter and baryonc matter contrbute to δρ, and not radaton. Subsequent orders of the perturbaton expanson n the feld equatons yeld the followng two equatons for the scalar part of the gravtatonal feld: φ+hφ +Hψ +ψ +2H φ = 4πa2 2 δρ+δρeff +3δp+3δp eff D Uh 8πa2 3 δρ ψ φ+oη 5 6 and ψ Hψ H 2 φ = 4πa2 1 δρ+δρeff D Uh 16πa2 δρ ψ+oη 5, 7 3 where D ϕ ϕ, 1 3 δ 2 ϕ s the trace-free second dervatve operator on any feld ϕ, and where perturbatons n radaton, and dark and baryonc matter contrbute to both δρ and δp. The reader wll note that these equatons contan extra terms when compared to standard cosmologcal perturbaton theory. Frst, there are effectve energy densty and pressure terms, δρ eff and δp eff. These are solely due to the presence of non-lnear structures on small scales, and are gven explctly n Eqs. 10 and 11, below. In other words, by wrtng the feld equatons n an effectve flud descrpton, one can clearly dentfy that small-scale non-lneartes lead to, amongst other thngs, an effectve pressure on large-scales. Second, n the above equatons, the potental U couples to h and there are extra source terms on the rght-hand-sdeof these equatons that are lnear n φ and ψ. These nteracton terms do not exst n standard cosmologcal perturbaton theory and vansh n the lmt n whch non-lnear small-scale structures vansh. In general, the nteracton terms should be expected to produce mode-mxng between scalar, vector and tensor parts of the gravtatonal feld on cosmologcal scales and couplng between dfferent Fourer modes n Fourer-space. The remanng parts of the gravtatonal feld are the vector and tensor modes. For the vectors we fnd that we can wrte the followng sngle equaton to descrbe S, accurate up to order Oη 5 : 2 S +4 ψ +Hφ +16πa 2 ρ+ p+δρ v S = 16πa 2 Q eff 8πa 2 δρ S 2 j US j +Oη 5. 8 We can take the leadng-order part of ths equaton, at Oη 3, and wrte t as the followng smple Posson equaton 2 S +4 U +HU+16πa 2 ρ+ p v = 16πa 2 δρ v +Oη 4. 9 The leadng-order part of the vector gravtatonal feld, gven by the soluton to Eq. 9, s only sourced by small-scale quanttes, and s a hundred tmes greater than mght navely be expected from cosmologcal perturbaton theory. Ths s the equaton that was dentfed n the post-fredmann approach of [20], and solved for numercally n [25]. For the full vector equaton 8, accurate up to Oη 5, t can be seen that there exsts sources on both small and large scales and mode-mxng whch s mssng from [27, 28]. Ths equaton has an effectve energy flux, Q eff, whch s due to small scale potentals. It also has extra source terms on the rght-hand-sde that are lnear n S. Both of these vansh when small-scale structures are absent. The explct expresson for Q eff s gven n Eq. 12, below, along wth the other effectve flud quanttes. The fnal feld equatons we requre, n order to complete our set to the desred accuracy, s gven as follows: 2 h h 2Hh +D φ ψ 2H j S j S = 8πa2 Π eff 8πa2 δρ h +4 k Uh j k +2D Uφ+ψ+Oη 5, 3

5 where angle brackets around ndces ndcate a symmetrc and trace-free operaton has been used, so that T T 1 3 δ T kk for any feld T. Ths equaton can be used to determne the tensor part of the gravtatonal feld, h. It also has an effectve flud source, Π eff, whch ths tme acts as an effectve ansotropc stress and s formed from the quadratc contractons of the lower-order small-scale potentals, see Eq. 13. Agan, the non-lnear structure on small scales couples the large-scale scalar and tensor parts of the cosmologcal gravtatonal felds, and agan we have addtonal terms on the rght-hand-sde that are lnear n h, resultng n mode-mxng. As promsed, the effectve flud quanttes n the perturbaton equatons above are gven as follows: δρ eff = ρ+ p+δρ v 2 1 πa 2U 2 U+ 3 4πa 2 H 2 U+HU 1 2 U2 10 δp eff = 1 3 ρ+ p+δρ v 2 1 4πa 2 U +3HU 7 6 U2 +a 2 UΛ 8π p + 1 3πa 2U 2 U 11 Q eff = ρ+ p+δρ v + 1 4πa 2 U +HU 12 Π eff = ρ+ p+δρ v v j 1 4πa 2 U j U 1 2πa 2UD U. 13 It can be seen that each of these quanttes was constructed only from varables that correspond to small-scale gravtatonal felds, or background quanttes whch wll be shown later to be calculated from the average of smallscale quanttes. We therefore have a herarchy of equatons that can be solved order-by-order: frst, the Fredmann and ewtonan equatons 4 and 5, and then the equatons that contan large-scale perturbatons The former of these sets are already calculated routnely n modern cosmologcal -body smulatons. The latter are modfed versons of the usual cosmologcal perturbaton equatons on large scales, and can be used to fnd post- ewtonan equatons on small scales as recently solved for numercally n [22, 26 28]. Fnally, note that the above effectve quanttes, n Eqs , contan terms that would normally only be ncluded n second or thrd order n cosmologcal perturbaton theory. In partcular, the term δρ v v j n Eq. 13 would appear at thrd order n standard perturbaton theory, but here should be expected to source a gravtatonal slp n the leadng-order part of the large-scale physcs. Our approach can be compared to the effectve flud approach studed prevously n [24, 29], as well as the large and small wavelength splt used n [30, 31]. As a fnal comment, before movng on to explan the orgn of these equatons and gve detaled explanatons of the gauge nvarant quanttes nvolved, we note that the usual trck of separatng equatons lke 8 and 10 nto scalar, vector and tensor parts s much more dffcult to apply here. Ths s due to the fact terms lke D Uφ + ψ do not have scalar, vector and tensor parts that are easy to dentfy. Ths term, for example, s a scalar multpled by a tensor, and n general should be expected to contan scalar, vector and tensor parts. Ths does not mean that such a separaton s mpossble ndeed we very much expect t to be possble. It just means that the resultng equatons are very messy to wrte down, whch s the reason why we have chosen to present these equatons wthout such a decomposton. The reader should also be warned that manpulaton of these equatons s consderably more dffcult than n ether cosmologcal perturbaton theory or standard post-ewtonan theory. Ths s due to dfferent dervatve operators changng the order to the terms they operate on n dfferent ways. Ths wll be made clearer n the sectons that follow, and expanded upon n more detal n a subsequent publcaton [32]. 4 III. PETUBATIVE FAEWOKS In order to explan the orgn of the equatons presented n Secton II we frst need to outlne the perturbatve expansons used n ther dervaton. These are the post-ewtonan expanson and cosmologcal perturbaton theory, whch wll be used to descrbe perturbatons on small and large scales, respectvely. For the former of these expansons we use the expanson parameterη 1, whle for the latter we use ǫ 1. We assume that any feld Q can be expanded n both ǫ and η as follows: Q = n,m 1 n!m! Qn,m, 14 where Q n,m s a quantty of order Oǫ n η m. The characterstclength scales on whch post-ewtonan and cosmologcal perturbatons exst and vary on wll be labelled L and L C, respectvely. Let us now brefly outlne the essental features of these two dfferent expanson schemes, before consderng how they can be performed smultaneously.

6 5 A. Post-ewtonan theory Post-ewtonan gravty s a slow-moton and weak-feld expanson of the feld equatons, usually appled to systems that are much larger than ther Schwarzschld radus but also much smaller than the Hubble radus[33]. Characterstc dmensonless veloctes on such scales are usually small, from whch t follows that tme dervatves of felds are also small n comparson to spatal dervatves: ϕ ϕ ϕ L, 15 where L s the spatal length scale over whch these perturbatons are expected to vary, and ϕ represents metrc potentals. The perturbatve order-of-smallness of metrc potentals s gven by notng that the magntude of pecular veloctes are small, v 0,1 η, and assgnng terms wth tme dervatves an extra order-of-smallness n η, compared to spatal dervatves. The magntude of the gravtatonal potentals, n terms of η, can then be determned through the geodesc equatons for freely fallng partcles and the feld equatons. Ths process can also be used to assgn orders of smallness to the sources of stress-energy [33, 34]. At ths stage orders-of-smallness n η can be assgned to all matter felds and gravtatonal potentals. The perturbed energy densty and pressure for a perfect flud are gven by the expansons ρ = ρ 0, ρ0, and p = p 0,4 +..., respectvely, where ellpses denote hgher-order terms. The quantty ρ 0,2 s the ewtonan energy densty, whle ρ 0,4 and p 0,4 are the post-ewtonan contrbutons to energy densty and pressure. The reader may note that there s no pressure term at Oη 2. Ths s no accdent f such a term were to be ncluded then the stress-energy conservaton equatons would requre t to be spatally homogeneous. Ths means that barotropc fluds wth p = wρ and w 0 do not ft nto post-ewtonan gravty n a natural way, unless they are dffuse enough to be consdered post-ewtonan.e. to occur only at Oη 4 or above. Smlarly, order-of-smallness can be assgned to the metrc components. The mnmal set of metrc perturbatons requred to consstently expand a tme-dependent background metrc, g µν 0,0 t, usng the post-ewtonan expanson, s therefore gven by δg 00 = δg 0,2 00 t,x+ 1 2 δg0,4 00 t,x..., δg = δg 0,2 t,x+..., and δg 0 = δg 0,3 0 t,x+.... It s mportant to note that the post-ewtonan formalsm outlned above was formulated to nclude matter, but not radaton or a cosmologcal constant. Ths wll be dscussed further n Secton IIID below, n the context of ncorporatng post-ewtonan expansons nto cosmology. For further detals about post-ewtonan expansons, the reader s referred to the textbooks by Wll [33], and Posson and Wll [34]. B. Cosmologcal perturbaton theory Cosmologcal perturbaton theory s a weak-feld expanson about the spatally homogeneous and sotropc class of Fredmann-Lemaître-obertson-Walker FLW geometres: dr ds 2 = dt 2 +a 2 2 t 1 kr 2 +r2 dθ 2 +sn 2 θdφ 2, 16 where at s the scale factor and k s the spatal curvature we take k = 0 for the rest of ths paper. All perturbatons to all matter and gravtatonal felds n ths perturbatve expanson are taken to occur at the same order: ǫ v 1,0 δρ 1,0 δp 1,0 δg 1,0 µν, 17 where L C s the characterstc length scale gravtatonal felds vary on, and s necessary above to compare the dmensonless expanson parameter, pecular velocty and gravtatonal potentals to dmensonful quanttes lke the perturbed energy densty and pressure. Cosmologcal perturbaton theory also assumes that tme and space dervatves do not change the order-of-smallness of a quantty, so we have ϕ ϕ ϕ L C. 18 Ths result occurs because the perturbatons can be as large as, or larger than, the horzon, and s n stark contrast wth the post-ewtonan expanson outlned prevously. It means that orders at whch terms appear n the two

7 expansons cannot n general be expected to be comparable, a dfference that changes the character of the resultng feld equatons. For our present purpose t s mportant to note that standard cosmologcal perturbaton theory s desgned n such a way that t can be appled to many dfferent epochs n the Unverse. Specfcally, gven a flud wth equaton of state p = wρ then both the background and perturbed equatons can be wrtten down straghtforwardly. Ths allows the theory to be appled to both the radaton-domnated and matter-domnated stages of the Unverse s evoluton, as well as to the current cosmologcal-constant-domnated epoch. Ths s a versatlty that s absent from the standard approach n post-ewtonan gravty, as radaton and Λ are completely neglgble for the study of gravty n the Solar System, bnary pulsars, and other such very-small-scale astrophyscal envronments. If one wants to apply such expansons to super-clusters n a cosmologcal context, however, then more care may be requred. For further explanaton of cosmologcal perturbaton theory, the reader s referred to the revew by alk and Wands [35]. 6 C. Two-parameter perturbaton theory In realty, both post-ewtonan and cosmologcal perturbatons should be expected to exst n any realstc model of the Unverse [36]. In ths secton we revew and extend the two-parameter framework developed n [23] that smultaneously performs a perturbatve expanson n both sectors. In the next secton we wll extend ths formalsm to nclude radaton and a cosmologcal constant. Our two-parameter expanson n both ǫ and η wll be constructed around an FLW geometry, correspondng to the lne-element n Eq. 16. Ths s the standard background for cosmologcal perturbaton theory, but s so far lttle used for post-ewtonan gravty see however [20, 21]. evertheless, t can be shown that both expansons can be performed n such a background n an entrely self-consstent and well-posed way [23, 37, 38]. The frst step n dong ths s to expand the total energy densty and pressure n both ǫ and η: ρ = ρ 0,0 +ρ 0,2 +ρ 1,0 +ρ 1,1 +ρ 1, ρ0, p = p 0,0 +p 1,0 +p 1, p0, The terms ρ 0,0 and p 0,0 can be consdered as the backgroundenergy densty and pressure, as they arenot perturbed n ether ǫ or η. All other terms correspond to perturbatons at the order ndcated by the superscrpt, but we have neglected to nclude δ symbols before them to keep the notaton as compact as possble. To be even more precse, the orders-of-magntude of these perturbed quanttes are gven by ρ 0,0 1 L 2, ρ n,0 ǫn C L 2, ρ 0,m ηm C L 2 and ρ n,m ǫn η m L 2, 21 where {m,n} +, and agan L C and L are the characterstclength scales of the cosmologcaland post-ewtonan sytems, respectvely. A smlar expresson holds for the expanson of p. The length scales are necessary n the denomnators of these expressons, as ρ s a quantty wth dmenson L 2, and because t only makes sense to compare the magntude of quanttes wth the same dmensons. The frst thng to notce about Eq. 19 s that the mxed-order terms do not always appear at the same order as the product of post-ewtonan and cosmologcal terms.e. we have ncluded ρ 1,1, even though there s no Oη term n the post-ewtonan expanson. The reason for ths s that such terms are necessarly generated by arbtrary gauge transformatons, and so cannot be assumed to vansh n general, even f they are assumed to do so n one partcular coordnate system. Secondly, we have ncluded a background energy densty, whch was absent n ef. [23]. Ths wll be useful for ncludng radaton. We also need to expand the metrc n both ǫ and η, whch we do as follows: g 00 = g 0,0 00 +g 0,2 00 +g 1,0 00 +g 1,1 00 +g 1, g0, = 1+h 0,2 00 +h 1,0 00 +h 1,1 00 +h 1, h0, g = g 0,0 +g 0,2 +g 1,0 +g 1,1 +g 1, g0, = a 2 δ +h 0,2 +h 1,0 +h 1,1 +h 1, h0, g 0 = g 1, = a h 1,0 0 +h 0,3 0 +h 1, , 0 +g 0,3 0 +g 1,2

8 where n the second lne of each of these equatons we have chosen our background metrc g 0,0 µν to be the flat FLW metrc from Eq. 16, and smultaneously defned the perturbatons h µν. The orders of magntude of each of the perturbatons to each of the components of ths metrc are the mnmal set requred to self-consstently account for the gravtatonal felds of the two-parameter perturbed perfect flud dscussed above, n any arbtrary coordnate system. We fnd that the ncluson of radaton and a cosmologcal constant does not requre the ntroducton of any new metrc potentals at any new order, so the form of Eqs s the same as n ef. [23]. The fnal ngredent of the feld equatons that must be perturbed s the pecular velocty, v. Ths s splt nto post-ewtonan and cosmologcal parts such that v = v 0,1 +v 1,0 +..., 25 whch leads to the followng components of the reference four-velocty u µ : u 0 = 1+ 1 h 0,2 00 +h 1, v0,1 v 0, u = 1 a v 0,1 +v 1,0 +..., 27 whch are derved usng the normalzaton condton u µ u µ = 1, and Eqs The components of the two-parameter perturbed energy-momentum tensor that arse from these equatons are gven n Appendx A. The components of the cc tensor are unchanged from ef. [23], and can be found n the appendx of that paper. The reader should note that wthn the context of the two-parameter formalsm, tme dervatves are taken to add an extra order-of-smallness, η, compared to spatal dervatves whenever they act on an object that contans any non-zero perturbaton n ts post-ewtonan sector. So, for example, we take 7 ρ 0,2 η ρ 0,2 η3 L 3 and ρ 1,1 η ρ 1,1 ǫη2 L 3 whlst ρ 1,0 ρ 1,0 ǫ L 3. C As n Eq. 21, the purpose of ths s to reflect the expectaton that quanttes perturbed n the post-ewtonan sector should be slowly varyng n tme and change over spatal length scales L, whle quanttes that are perturbed only n the cosmologcal sector should vary equally over both tme and length scales L C. Ths s explaned further n [23]. D. Includng radaton and Λ Let us now consder how to add radaton and Λ to our two-parameterexpanson. For radaton ths can be acheved by wrtng ρ = ρ +ρ, p = p +p and v = v +v, 28 where ρ and p are the energy densty and pressure of non-relatvstc matter, ρ and p are the energy densty and pressure of radaton, and v and v are the pecular veloctes of the matter and radaton fluds. We then want to expand each of these new quanttes n ǫ and η, whch we do accordng to and ρ = ρ 0,2 p = p 1,0 +ρ1,0 +p1,2 +ρ1, p0,4 +ρ1, ρ0, ρ = ρ 0,0 +ρ 1,0 +ρ 1, ρ0, p = p 0,0 +p 1,0 +p 1, p0,4 +..., 32 v = v 0,1 +v1,0 +..., v = v 0,1 +v 1, These equatons can, of course, be compared to Eqs. 19, 20 and 25 to read off values for the perturbatons to the total energy densty, pressure and v. They can also be seen to generalse those from ef. [23] by the ncluson of ρ and p, as well as by the ncluson of an extra mxed-order term p 1,2 and a factor of 1/2 n front of p0,4.

9 The reader wll note that the expansons of the matter and radaton fluds have not been performed n an dentcal way: We have omtted a tme-dependent background-level contrbuton to the matter energy densty and pressure, and a ewtonan-level contrbuton to the radaton energy densty and pressure, so that ρ 0,0 = p0,0 = 0 and ρ0,2 = p 0,2 = 0. The former of these s neglected because t corresponds to a term that could otherwse be as large as the ewtonan rest mass energy densty ρ 0,2, whch we consder to be hghly unphyscal. In the real unverse there s no tme-dependent background matter component to the energy densty, ρ 0,0 t. Ths s because the leadng-order contrbuton to ρ s n fact domnated by the nhomogeneous rest mass of galaxes, dust etc., whch s exactly what ρ 0,2 xµ corresponds to. Furthermore, ρ 0,0 would necessarly have to be a functon of tme only and because there s no dscernble homogeneous flud of non-relatvstc matter wth ths magntude n the real Unverse 2. The term p 0,0 could be neglected on smlar grounds, but must also vansh because of the requrement p ρ n non-relatvstc matter. Let us now consder the expanson of ρ and p gven n Eqs. 31 and 32. For ths purpose t s useful to consder the stress-energy conservaton equaton for the total stress-energy tensor T µν : µ T µν = µ T µν +T µν = 0, 34 where T µν and T µν are the matter and radaton contrbutons to the total stress-energy tensor, respectvely. Ths mples µ T µν = Q ν and µ T µν = Q ν, where Q ν 0 for nteractng fluds and Q ν = 0 for non-nteractng fluds. In ether case, the lowest-order part of Eq. 34 s gven by p 0,0 = 0, 35 whch mples p 0,0 = p 0,0 t s a functon of tme only. If we now take p = 1 3 ρ, then ths result mples that the leadng-order part of the energy densty n radaton must also be spatally homogeneous, such that ρ 0,0 = ρ 0,0 t. Ths s, n fact, exactly what s requred for a background-level contrbuton to the energy densty n an FLW model. A smlar argument can now be used to understand why t would be napproprate to nclude a term ρ 0,2 n Eq. 31. Such a term would mply the exstence of p 0,2 whch, agan through the conservaton equatons, can be shown to be necessarly spatally homogeneous. Such a term would therefore be functonally degenerate wth ρ 0,0, as they are both functons of tme only, and would therefore show up n every concevable set of equatons n exactly the same way. We can therefore neglect both ρ 0,2 and p 0,2 wthout any loss of generalty. oreover, the term ρ 0,2 t would be ewtonan n sze, and such a term would be hghly unusual n normal post-ewtonan gravty. We therefore fnd that the lowest order at whch nhomogeneous perturbatons n radaton ft nto our two-parameter expanson s at order Op 1,0 OǫL 2 C, whch corresponds to a cosmologcal-scale perturbaton. The reader may also note that there s no term ρ 1,1 n Eq. 31, whereas there s a term ρ 1,1 n Eq. 29. The ρ 1,1 s necessary because a term of the form ρ0,2, ξ1,0 s always generated under a general nfntesmal gauge transformaton [23] where ξ 1,0 s a part of the gauge generator see Secton IV. Ths mples there must n general exst a term ρ 1,1 n the expanson of ρ, because even f we artfcally exclude t n one coordnate system, t wll be generated n another. However, a smlar argument does not apply to ρ 1,1, because the gauge transformaton ρ0,0 does not generate any terms of the same order as ρ 1,1. Ths can be seen to be true because ρ0,0 s a functon of tme only, such that ρ 0,0, ξ 1,0 = 0. Of course, the same argument would apply to a term of the form ρ 0,2, f t had been ncluded, as ths term s also tme dependent. Ths means that we can set ρ 1,1 = p 1,1 = 0 n any coordnate system, and the same result wll hold n any other coordnate system related by an nfntesmal gauge transformaton. Fnally, let us consder the cosmologcal constant Λ. We assgn an order of magntude and dmensons to the cosmologcal constant n the followng way: 8 Λ = Λ 0,0 1 L C 2 In fact, the exstence of such a component corresponds to a breakdown of standard perturbaton theory [39].

10 Ths choce s motvated by the fact that the cosmologcal constant n the standard model of cosmology must be of background order, n order for t to be nfluental n the Fredmann equatons at late tmes. There s also no pont n perturbng t n ether ǫ or η, as t s a constant, and the Taylor expanson s trval. The cosmologcal constant therefore fts naturally nto our two-parameter expanson at lowest-order, as a cosmologcal background quantty wth correspondng scale L 2 C. The full feld equatons, n arbtrary coordnates, are gven n terms of the perturbed quanttes ntroduced n ths secton n Appendx B. In the sectons that follow we wll perform gauge transformatons n order to determne how these quanttes transform between dfferent coordnate systems. We wll then construct a set of gauge-nvarant quanttes that obey the same equatons n any coordnate system, before ntroducng combnatons of potentals that can be taken n order to wrte the gauge-nvarant feld equatons n terms of an effectve flud. 9 IV. COSTUCTIG GAUGE-IVAIAT VAIABLES A general nfntesmal gauge transformaton between coordnate systems can be wrtten as x µ x µ = e ξα α x µ, 37 where ξ µ s the gauge generator. All tensors, T, are taken to transform under the gauge transformaton n Eq. 37 as T = e L ξ T = T +L ξ T L2 ξt +..., 38 where L ξ denotes the Le dervatve operator wth respect to ξ µ. Ths exponental map results n an nvertble transformaton, and can be appled to both the metrc and the stress-energy tensor. We must now expand the components of the gauge generator n terms of ǫ and η, whch we do as follows: ξ 0 = ξ 1,00 +ξ 0,30 +ξ 1,20 + ǫl C +η 3 L +ǫη 3 L ξ = ξ 1,0 +ξ 0,2 +ξ 1,1 +ξ 1, ξ0,4 ǫl C +η 2 L +ǫη 2 L +η 4 L These non-vanshng components of the gauge generator have been chosen so that no new components of the metrc or the stress-energy tensor are generated by ths transformaton, whch s an mportant condton to ensure the problem s beng treated n a self-consstent manner. A. Infntesmal coordnate transformatons In order to perform nfntesmal coordnate transformatons t s useful to decompose the perturbed gauge generator nto a scalar and a dvergence-free vector. Omttng superscrpts, these can be wrtten as ξ 0 δt and ξ δx, +δx, 41 where δx, = 0. The transformaton of the metrc perturbatons due to a gauge transformaton of ths type are unchanged from the dust-only case, and are gven n Secton V of ef. [23]. In the remander of ths secton we wll outlne how the presence of radaton affects the transformaton propertes of the matter felds {ρ,p,v,λ}. Ths s done usng the result from Eq. 38, and by solvng for the decomposed matter varables. In order to present these results n a form that can be used for cosmology we choose to take L /L C η. Ths means that we are restrctng the post-ewtonan sector of our expanson to apply on scales below about 100pc, whch s realstc also about the sze of the homogenety scale. Ths s deal for consderng the nfluence of galaxes, clusters and super-clusters on large-scale lnear cosmologcal perturbatons. We also choose, wthout loss of generalty,

11 10 to express our results n terms of L. Gven ths, the total energy densty transforms as follows: ρ 0,0 + ρ 0,2 = ρ 0,0 +ρ 0,2 η2 L 2 ρ 1,1 = ρ 1,1 +ρ 0,2, δx 1,0, +δx1,0 ǫη L whle the total pressure transforms as ρ 1,0 + ρ 1,2 = ρ 1,0 +ρ 1,2 + ρ 0,0 +ρ 0,2 δt 1,0 ǫη2 L 2 44 ρ 0,4 = ρ 0,4 +2ρ 0,2, δx 0,2, +δx0,2 η4 L 2, 45 p 0,0 = p 0,0 η2 L 2 p 1,0 + p 1,2 = p 1,0 +p 1,2 +ṗ 0,0 δt 1,00 2ȧ a p0,0 δt 1,00 ǫη2 L p 0,4 = p 0,4 η4 L The transformatons n Eqs. 43, 45 and 48 reman exactly the same as the dust-only case studed n ef. [23], whle all other transformatons are affected by the presence of the radaton. The term ρ 0,0 can be seen to transform n the same was as the ewtonan energy densty, ρ 0,2. Ths s not unexpected, as both quanttes have magntude L 2 C η2 L 2. Smlarly, ρ0,0 appears alongsde ρ 0,2 n the transformaton gven n Eq. 44. Wth the ncluson ofradaton, we fnd that p 0,0 s automatcallygaugenvarant. Furthermore, ascan be seen n Eq. 47, the ncluson of radaton means that the transformaton of the cosmologcal and mxed-order perturbatons to the pressure are no longer gauge nvarant as they were n the dust-only case. The reader may note that these results dffer from the quas-statc lmt of cosmologcal perturbaton theory, as space and tme dervatves are treated on a dfferent footng, and because veloctes come n at dfferent orders [41]. eanwhle, the pecular veloctes transform n the followng way: ṽ 1,0 = v 1,0 a δx 1,0, +δx 1,0 0,1 +v,j δx 1,0j, +δx 1,0j ǫ 49 ṽ 0,1 = v 0,1 η. 50 These transformatons are the same as n the dust-only case studed n ef. [23]. ote partcularly that n Eq. 49 the small-scale ewtonan velocty contrbutes to the transformaton of the large-scale velocty ths s a by-product of our two-parameter expanson, and s an effect that would otherwse only appear at second order n standard cosmologcal perturbaton theory. Fnally, we fnd that the cosmologcal constant Λ 0,0 does not transform under the gauge transformaton n Eq. 37, as t s a constant n space and tme: Λ 0,0 = Λ 0,0. 51 The transformatons above wll now be used to construct gauge-nvarant quanttes. B. Gauge nvarant quanttes Let us now create gauge-nvarant quanttes for the matter degrees of freedom n the presence of radaton and Λ. Such varables solate and remove superfluous degrees of freedom, as well as allowng the feld equatons to be wrtten n a greatly smplfed way. To do ths t s useful to perform an rreducble decomposton on the metrc. Omttng superscrpts for smplcty, and wthout loss of generalty, we can do ths as follows: h 00 φ, h 0 B, +B and h ψδ +E, +F,j + 1 2ĥ, 52

12 where B,F,ĥ are dvergenceless and ĥ s trace-free. Applyng the gauge transformaton 38 to the metrc components22-24 ths gves the transformaton rules for the rreducbly decomposed components, and allows gaugenvarant gravtatonal perturbatons to be constructed see Sectons V and VI of ef. [23]. The presence of radaton does not affect the constructon of gauge-nvarant gravtatonal perturbatons, but does affect the constructon of gauge-nvarant quanttes for the matter varables, whch s what we wll elaborate upon here. The method we use to calculate gauge-nvarant quanttes s as follows: we choose gauge generators δx,δx and δt such that the gauge transformed metrc potentals Ẽ = B = F = 0. We then substtute these gauge generators, now wrtten n terms of E,B and F, back nto the expressons for all of the transformed perturbatons presented n Secton IV. Because the orgnal gauge transformatons were wrtten down n a completely arbtrary coordnate system, these new results are automatcally gauge nvarant [35]. All such quanttes also reduce to metrc perturbatons n longtudnal gauge when E = B = F = 0, and have been explctly checked to be truly gauge nvarant. Toconstructgauge-nvarantmatterperturbatonswerequrethetransformatonlawsforE 1,0,B 1,0,F 1,0,E 0,2 and F 0,2 under Eq. 37. These are gven n [23] and are B 1,0 = B 1,0 +a δx 1,0 1 a δt1,0 ǫη 1 L 53 Ẽ 1,0 = E 1,0 +2δx 1,0 ǫη 2 L 2 54 F 1,0 = F 1,0 +2δx 1,0 ǫη 1 L 55 Ẽ 0,2 = E 0,2 +2δx 0,2 η 2 L 2 56 F 0,2 = F 0,2 +2δx 0,2 η 2 L. 57 For the total energy densty perturbatons t can then be seen that the followng quanttes are gauge nvarant: ρ 0,0 +ρ 0,2 = ρ 0,0 +ρ 0,2 58 ρ 1,1 = ρ 1,1 1 2 ρ0,2, E 1,0, +F 1,0 59 ρ 1,0 +ρ 1,2 = ρ 1,0 +ρ 1,2 + ρ 0,0 +ρ 0,2 ab 1,0 a2 2 Ė1,0 60 ρ 0,4 = ρ 0,4 ρ 0,2, E 0,2, +F 0,2. 61 Correspondngly, for the pressure perturbatons we fnd the followng gauge-nvarant quanttes: p 0,0 = p 0,0 62 p 1,0 +p 1,2 = p 1,0 +p 1,2 + ṗ 0,0 2ȧ ab a p0,0 a2 2 Ė1,0 63 and for the pecular velocty we construct p 0,4 = p 0,4, 64 v 0,1 = v 0,1 v 1,0 = v 1,0 + a Ė1,0, + 2 F 1,0 1 2 v0,1,j E 1,0,j +F 1,0j. 66 These last two quanttes can be separated nto scalar and dvergenceless vector parts n a straghtforward way. Fnally, the gauge-nvarant cosmologcal constant s trval to construct: Λ = Λ 0,0. 67 There are no further quanttes to consder n the stress-energy tensor, so, when combned wth the set of gaugenvarant metrc potentals {Φ,Ψ,B,h }, constructed n Secton VI of ef. [23], ths gves us a full set of gaugenvarant quanttes n our two-parameter perturbatve expanson. The feld equaton n terms of these gauge-nvarant varables are gven n Appendx C.

13 12 V. COSTUCTIG THE FIELD EQUATIOS The two-parameter expanson descrbed n the prevous sectons could n prncple be appled to numerous dfferent physcal systems. Whle the perturbed metrc and stress-energy tensor can be wrtten down wthout specfyng any specfc relatonshp between ether ǫ and η or L C and L, we must choose how to express these quanttes n terms of one another f we want to be able to solve a herarchcal set of feld equatons. In order to model a realstc unverse that has non-lnear structure on scales up to 100pc, as well as lnear structure on large scales, we choose L /L C η. On the other hand, to model a realstc unverse, gravtatonal potentals must have smlar magntude on both small and large scales, so we choose ǫ η 2, see Secton III of ef. [23] for justfcaton of ths. Both of these requrements are therefore satsfed by the choce ǫ η 2 L2 10 5, 68 where 10 5 s the typcal depth of a potental on both cosmologcal and post-ewtonan scales. Wth these relatons we can translate our two-parameter expanson nto effectvely a sngle-parameter expanson n η, and wrte the feld equatons order-by-order n η. We further choose to express the feld equatons n unts of L 2. Ths last choce has no partcular physcal sgnfcance, and s purely for expedency. We use Eq. 68 n the work that follows, as well n Secton II and Appendces B and C. A. Background and ewtonan cosmologcal equatons Wthn the formalsm outlned above, the Fredmann-lke equatons that govern the evoluton of the scale factor at, and hence the large-scale expanson of the Unverse, are not ndependent of the perturbatons. Ths can be seen explctly n Eqs. C3 and C4 from Appendx C, where the ewtonan mass densty and gravtatonal potental act as sources for the cosmologcal expanson. Ths s n some sense a very pleasng result; the large-scale expanson of space s drven by the same ewtonan mass that governs the leadng-order part of the gravtatonal feld on small scales. On the other hand, t means that our background s not by tself an exact soluton of Ensten s equatons. Ths stretches the meanng of what s usually mpled by the phrase perturbaton theory n Ensten s theory 3. evertheless, both the fundamental objects beng perturbed and the feld equatons themselves are beng consstently expanded n the perturbatve parameters ǫ and η, and we see no reason to expect ths expanson should not converge. Indeed the present expanson seems to have much better convergence propertes than the standard approach to cosmologcal perturbaton theory, n the presence of non-lnear structures [39]. Furthermore, a change of coordnates on a sub-horzon-szed regon of space can be shown to be sometrc to perturbed nkowsk space, wth the cosmologcal expanson arsng from boundary condtons at the edge of the regon [37]. In ths sense, the cosmologcal expanson can be consdered an emergent property, and the background on small-scales could equally well be consdered to be ether a Fredmann model or nkowsk space whch defntely s a soluton when ǫ = η = 0. In any case, we can now proceed n a smlar manner to Secton VIIA of ef. [23] to fnd the smplest way n whch to express the equatons that govern the large-scale expanson of space. In order to do ths, t us useful to calculate the averagemass densty and radaton densty on dstances above the homogenety scale, L hom 100pc [42]. These are gven by ρ V hom ρ 0,2 dv V hom dv and ρ V hom ρ 0,0 dv V hom dv = ρ 0,0, 69 where V hom ndcates the spatal volume assocated wth the homogenety scale. Of course, we know from Eq. 35 that there can be no small-scale nhomogenetes n the radaton flud. For the matter flud, on the other hand, small-scale fluctuatons most defntely do exst and are of order unty. To accommodate these fluctuatons we defne δρ 0,2 ρ 0,2 ρ. 70 Ths equaton mples that the leadng-order nhomogeneous part of the matter energy densty, δρ 0,2, s formally of the same order as the background component of the matter felds, ρ 0,2, both beng Oη 2 L 2. These quanttes can now be used to wrte Eqs. C3 and C4 nto a more useful form. 3 We are grateful to arco Brun for a number of stmulatng dscussons on ths pont.

14 To derve a set of effectve Fredmann equatons we frst ntegrate Eq. C4 over the volume correspondng to the homogenety scale: 3H 2 1a V 2 2 Φ 0,2 dv = 8π ρ 0,0 +ρ 0,2 +Λ dv, 71 hom V hom where H ȧ/a. Usng Gauss theorem ths can be wrtten as 3H 2 V hom 1 a 2 S hom Φ 0,2 ds = 8πρ +ρ V hom +ΛV hom If we now assume that on the homogenety scale there s no net flux of Φ 0,2 nto or out of the surface S hom, then the second term n Eq. 72 vanshes. Ths leaves us wth H 2 = 8π 3 ρ +ρ + Λ 3, 73 whch s exactly the same form as the standard Fredmann equaton n the presence of matter, radaton and a cosmologcal constant. What s more, the lowest-order parts of the stress-energy conservaton equatons yelds the results [40] ρ a 3 and ρ a 4, 74 whch are agan exactly as expected from Fredmann cosmology. Substtutng these results back nto Eq. C4 gves 2 Φ 0,2 = 8πa 2 δρ 0,2, 75 whch s dentcal to the standard equaton used n ewtonan -body smulatons for cosmology. In summary, we fnd that the leadng-order parts of the feld equatons, n the context of our two-parameter expanson, reproduce exactly the same results as standard Fredmann cosmology wth dust, radaton and a cosmologcal constant although the meanng of the equatons s slghtly dfferent. In the followng secton we wll fnd that ths s not the case when the non-lnear aspects of Ensten s equatons become mportant, on large scales. Fnally, we note that the Fredmann equaton s recovered from our expanson when δρ 0,2 = 0.e. when the leadng-order contrbuton to the energy densty s homogeneous. Ths s not the same condton as settng ǫ = η = 0, whch would correspond to an empty space wthn our framework. B. Leadng-order cosmologcal perturbaton equatons The equatons presented n Appendx C consttute a herarchy of feld equatons, where the equatons from Secton V A are the leadng-order parts. Once the Fredmann equaton73 and the ewtonan equaton75 have been solved, then ther solutons can be substtuted nto the remanng hgher-order equatons to gan a set of solutons for the leadng-order cosmologcal perturbatons. Ths latter set of solutons, at Oη 4 L 2, contan lnear-order cosmologcal large-scale potentals and post-ewtonan potentals from small scales. Wth ths n mnd, we therefore seek to recast the Oη 4 L 2 equatons n the form of the equatons of standard frst-order cosmologcal perturbaton theory, modfed by the addton of terms related to the exstence of nhomogenety on the length scale L. These terms wll be then be treated as formng the components of an effectve flud on large scales, whose characterstcs and behavour s determned by the small-scale gravtatonal physcs. Such an approach has smlartes to the effectve flud approaches n for example [24, 29], but n our case s also requred to reduce the number of gravtatonal degrees of freedom to be no more than the avalable number of feld equatons. In the end, we want to reduce to a set of sx perturbed feld equatons for sx degrees of freedom.e. the 10 degrees of freedom n the metrc mnus the four coordnate freedoms. At present, Eqs. C5-C14 from Appendx C contan a total of sxteen degrees of freedom: sx scalars Φ 1,0, Φ 1,2, Φ 0,4, Ψ 1,0, Ψ 1,2 and Ψ 0,4, sx n the tensors h 1,0, h 1,2 and h 0,4 and four n the vectors B 1,0 and B 1,2. Takng nto account the four degrees of freedom removed by gauge fxng mples that we need to remove sx degrees of freedom. Ths s acheved by defnng new sets

15 14 of varables as follows: U 1 2 Φ 0,2 +Φ 1,1 76 φ 1 2 Φ 1,0 +Φ 1, Φ0,4 77 ψ 1 2 Ψ 1,0 +Ψ 1, Ψ0,4 78 S j B 1,0 j +B 0,3 j +B 1,2 j 79 h 1 4 h 1,0 +h 1, h0,4, 80 and δρ δρ 0,2 +ρ 1,1 81 δρ ρ 1,0 +ρ 1, ρ0,4 82 δp p 1,0 +p 1, p0,4 83 v v 0,1 84 v v 1,0, 85 whch are exactly the sets of varables used n Secton II. A number of these new varables could be consdered to be composte quanttes, as they contan a number of dfferent perturbatve orders n the same varable. For example, the varable ψ s domnated by Oǫ terms on cosmologcal length scales L C, but contans smaller terms at Oη 4 on small-scales L. Ths s qute atypcal n perturbaton theory. However, the way n whch these quanttes arse together n the feld equatons suggest that they should be solved for together. The equatons dsplayed n Secton II can now be seen to be gven by those from Appendx C, but wrtten n terms of the varables n Eqs and n terms of conformal tme dτ a 1 dt. To solve ths system, one must solve the ewton-posson equaton and Fredmann equatons for U and at, respectvely. Ths should not be too challengng, as they are dentcal to the standard equatons used n ewtonan -body smulatons for Fredmann cosmology. From these results one can then solve for the effectve flud parameters, and then solve the cosmologcal perturbaton equatons. Ths wll nevtably be complcated by the addtonal mode-mxng terms n the cosmologcal perturbaton equatons, whch wll requre more sophstcated technques than at leadng-order n standard cosmologcal perturbaton theory. These terms suggest that t may n fact be possble to generate vector and tensor modes from scalar fluctuatons, whch s already well known n second-order cosmologcal perturbaton theory [17, 19], but s not usually seen at frst order. One should also note that these terms, for example 8πa2 3 δρ ψ φ, also mean that Fourer modes no longer decouple n a trval way as they do n standard frst-order perturbaton theory, even f no mode-mxng occurs. Ths s because the Fourer transforms of such terms are expressble only n terms of a convoluton ntegral over all Fourer modes. VI. COCLUSIOS We construct a two-parameter perturbaton expanson around an FLW background that smultaneously descrbes non-lnear structures on small-scales and lnear structures on large scales. oreover, t ncludes matter, radaton and a cosmologcal constant. In dong so we use both cosmologcal perturbaton theory and the post-ewtonan expanson. As ths expanson s able to model large densty contrasts and dfferent matter components t therefore both contans the essental features of the real Unverse and has a number of potental advantages over standard cosmologcal perturbaton theory. We derved the two-parameter perturbed feld equatons vald for structure on the order of a fracton on the horzon sze, the two-parameter gauge transformatons of the matter sector, and construct gauge-nvarant quanttes n ths sector. The consstency of the gauge transformatons requres not only gravtatonal potentals and matter perturbatons at the orders expected from post-ewtonan gravty and cosmologcal perturbaton theory alone, but also a number of others at orders n perturbaton whch may not navely have been expected. We have therefore dentfed a mnmal set of perturbatons that are requred for mathematcal consstency of the problem, and wrtten down gauge-nvarant versons of the feld equatons that contan them all. We fnd that the small-scale ewton-posson equaton for the scalar gravtatonal potental occurs at the same order n perturbatons as the Fredmann equaton, but that they can be separated after the ntroducton of a sutable