Cosmic microwave background anisotropies: Nonlinear dynamics

Transcription

1 Cosmic microwve bckground nisotropies: onliner dynmics oy Mrtens, Tim Gebbie nd George F.. Ellis School of Computer Science nd Mthemtics, Portsmouth University, Portsmouth PO1 2EG, Englnd Deprtment of Mthemtics nd Applied Mthemtics, University of Cpe Town, Cpe Town 7701, South Afric We develop new pproch to locl nonliner effects in cosmic microwve bckground nisotropies, nd discuss the qulittive fetures of these effects. ew couplings of the bryonic velocity to rdition multipoles re found, rising from nonliner Thomson scttering effects. We lso find new nonliner sher effect on smll ngulr scles. The full set of evolution nd constrint equtions is derived, including the nonliner generliztions of the rdition multipole hierrchy, nd of the dynmics of multi-fluids. These equtions govern rdition nisotropies in ny inhomogeneous spcetime, but their min ppliction is to second-order effects in universe tht is close to the Friedmnn models. Qulittive nlysis is given here, nd quntittive clcultions re tken up in further ppers. to pper Phys. ev. D 59 (1999). TODUCTO ecent nd upcoming dvnces in observtions of the cosmic microwve bckground (CM) rdition re fuelling the construction of incresingly sophisticted nd detiled models to predict the nisotropy on smll ngulr scles. Such models require highly specific input in order to produce numericl results, nd they involve intricte problems of computtion. As complement to such specific predictive models, it is lso useful to pursue more qulittive nd nlyticl investigtion of CM nisotropies. A generl qulittive nlysis does not rely on detiled ssumptions bout the origin of primordil fluctutions, the density prmeters of the bckground, reioniztion nd structure formtion history, etc. nsted, the im is to better understnd the underlying physicl nd geometric fctors in the dynmics of rdition nisotropies, nd hopefully to uncover new results nd insights. n this pper, we follow such n pproch, nd develop new nlysis of locl nonliner effects in CM nisotropies. We re ble to give physiclly trnsprent qulittive nlysis of how inhomogeneities nd reltive motions produce nonliner effects in CM nisotropies. We derive the nonliner generliztion of Thomson scttering, nd we find new nonliner sher effect on smll scles. We use 1+3 covrint pproch (i.e., covrint Lgrngin pproch) to CM nisotropies, bsed on choice of physiclly determined 4-velocity vector field u. This llows us to derive the exct nonliner equtions for physicl quntities s mesured by observers moving with tht 4-velocity. Then the nonliner equtions provide covrint bsis for investigting second-order effects, s well s for linerizing bout Friedmnn-Lemitre-obertson-Wlker (FLW) bckground. The bsic theoreticl ingredients re: () the covrint Lgrngin dynmics of Ehlers nd Ellis 1,2], nd the perturbtion theory of Hwking 3] nd Ellis nd runi 4] which is derived from it; (b) the 1+3 covrint kinetic theory formlism of Ellis, Trecioks nd Mtrvers 5,6] (which builds on work by Ehlers, Geren nd Schs 7], Trecioks nd Ellis 8] nd Thorne 9]); nd (c) the 1+3 covrint nlysis of temperture nisotropies due to Mrtens, Ellis nd Stoeger 10]. The well-developed study of CM nisotropies is bsed on the pioneering results in CM physics (Schs nd Wolfe 11], ees nd Scim 12], Peebles nd Yu 13], Sunyev nd Zeldovich 14], Grishchuk nd Zeldovich 15], nd others), nd on the development of guge-invrint perturbtion theory, prticulrly by rdeen 16] nd Kodm nd Sski 17] (building of the work of Lifshitz 18]). There re comprehensive nd detiled models see e.g. Hu nd Sugiym 19 21], M nd ertschinger 22], Seljk et l ], Durrer nd Khnishvili 27]. These provide the roy.mrtens@port.c.uk 1

2 bsis for sophisticted predictions nd comprisons with the observtions of recent, current nd future stellite nd ground-bsed experiments. The hope is tht this inter-ply between theory nd observtion (including the lrge-scle glctic distribution nd other observtions), in the context of infltionry cosmology, will produce ccurte vlues for the vrious prmeters tht chrcterize the stndrd models, thus llowing theorists to discriminte between competing models (see for exmple 28,29]). While these ppers hve provided ner-exhustive tretment of mny of the issues involved in CM physics, there re number of resons for pursuing complementry 1+3 covrint pproch, s developed in 10,30 38]. Firstly, the covrint pproch by its very nture incorportes nonliner effects. This pproch strts from the inhomogeneous nd nisotropic universe, without priori restrictions on the degree of inhomogeneity nd nisotropy, nd then pplies the lineriztion limit when required. The 1+3 covrint equtions governing CM nisotropies re thus pplicble in fully nonliner generlity. These equtions cn then be specilized in vrious wys in ddition to stndrd FLW-lineriztion. Second-order effects in n lmost-flw universe probbly form the most importnt possibility, given the incresing ccurcy nd refinement of observtions. The study of CM nisotropies in homogeneous inchi universes with lrge nisotropy is nother possibility tht flows directly from the generl nonliner equtions. Such pplictions will be the subject of future ppers in the progrmme. The current pper is concerned with setting up the generl dynmicl equtions nd identifying the qulittive nture of nonliner effects. (The generl lgebric equtions re derived in 37].) Secondly, the 1+3 covrint pproch is bsed entirely on quntities with direct nd trnsprent physicl nd geometric interprettion, nd the fundmentl quntities describing nisotropy nd inhomogeneity re ll utomticlly guge-invrint when suitble covrint choice of fundmentl 4-velocity hs been mde. As consequence the pproch leds to results with unmbiguous physicl mening (provided the fundmentl 4-velocity field is chosen in physiclly unique nd pproprite wy; we discuss the vrious options below). This pproch hs been developed in the context of density perturbtions 4,39 54] nd grvittionl wve perturbtions 3,55 59]. (See lso 60] for recent review.) n reltion to CM nisotropies, the covrint Lgrngin pproch ws initited 1 by Stoeger, Mrtens nd Ellis 30], who proved the following result: if ll comoving observers in n expnding universe region mesure the nisotropy of the CM fter lst scttering to be smll, then the universe is lmost FLW in tht region. 2 o priori ssumptions re mde on the spcetime geometry, or on the source nd nture of CM nisotropies, so tht this result provides generl theoreticl underpinning for CM nlysis in perturbed FLW universes. t effectively constitutes proof of the stbility of the corresponding exct-isotropy result of Ehlers, Geren nd Schs 7]. The wek Copernicn principle implicit in the ssumption tht ll fundmentl observers see smll nisotropy is in principle prtilly testble vi the Sunyev-Zeldovich effect (see 32] nd references therein). The qulittive result ws extended into quntittive set of limits on the nisotropy nd inhomogeneity of the universe imposed by the observed degree of CM nisotropy, independently of ny ssumptions on cosmic dynmics or perturbtions before recombintion 10,31 33]. More recently, this pproch to CM nisotropies in n lmost FLW universe hs been extended by Dunsby 34], who derived 1+3 covrint version of the Schs-Wolfe formul, nd by Chllinor nd Lsenby 35,36], who performed comprehensive 1+3 covrint nlysis of the imprint of sclr perturbtions on the CM, confirming the results of other pproches from this viewpoint nd bringing new insights nd clrifictions vi the covrint pproch. n 35], they lso discuss qulittively the imprint of tensor perturbtions on the CM, in the covrint pproch (see 38] for quntittive results). This pper is closely relted to, nd prtly dependent upon, ll of these previous 1+3 covrint nlyses. t extends nd generlizes spects of these ppers, using nd developing the covrint nonliner Einstein-oltzmnnhydrodynmic formlism. We nlyze the nonliner dynmics of rdition nisotropies, with the min ppliction being second-order effects in n lmost FLW universe. We identify nd describe the qulittive fetures of such effects. This lys the bsis for generliztion of results on well known second-order effects such s the ees- Scim nd Vishnic effects (see e.g. 19]), nd on recent second-order corrections of the Schs-Wolfe effect 63,64]. Developing quntittive nlysis on the bsis of the equtions nd qulittive nlysis given here is the subject of further work. Ultimtely this involves the solution of prtil differentil equtions, which requires in prticulr choice of coordintes, breking covrince. However, the 1+3 covrint pproch mens tht ll the equtions nd vribles hve direct nd trnsprent physicl mening. 1 A 1+3 covrint pproch to CM nisotropy ws independently outlined by onnno nd omno 61] in generl terms, using flux-limited diffusion theory, but the detiled implictions of smll CM nisotropy were not pursued. 2 ote the importnce of expnsion: sttic isotropic cosmology with rbitrrily lrge inhomogeneity cn be constructed in which ll observers see isotropic CM 62]. 2

3 n Section, the covrint Lgrngin formlism for reltivistic cosmology is briefly summrized. Section develops n exct 1+3 covrint tretment of multi-fluids nd their reltive velocities, building on 52]. n Section V, the covrint Lgrngin pproch to kinetic theory is outlined. Section V develops nonliner tretment of Thomson scttering, which identifies new couplings of the bryonic reltive velocity to the rdition multipoles. We derive the hierrchy of exct covrint multipole equtions which rise from the oltzmnn eqution. This section uses nd generlizes combintion of the results of Ellis et l. 5] on the multipoles of the oltzmnn eqution in generl, Mrtens et l. 10] on covrint description of temperture fluctutions, nd Chllinor nd Lsenby 36] on Thomson scttering. The equtions constitute covrint nd nonliner generliztion of previous linerized tretments. n Section V, we consider qulittive implictions of the nonliner equtions. We identify the role of the kinemtic quntities in the nonliner terms, nd comment on the implictions for second-order effects, which include new nonliner sher correction to CM nisotropies on smll ngulr scles. We lso give the multipole equtions for the cse where the rdition nisotropy is smll, but spcetime nisotropy nd inhomogeneity re unrestricted. 3 Finlly, we give the linerized form of the multipole equtions, regining the equtions of Chllinor nd Lsenby 36]. This provides covrint Lgrngin version of the more usul metric-bsed formlism of guge-invrint perturbtions (see e.g. 65,20,22,23]). n further pper 66], the linerized equtions derived here re expnded in covrint sclr modes, nd this is used to determine nlytic properties of CM liner nisotropy formtion. We follow the nottion nd conventions of 2,5,10], with the improvements nd developments introduced by 67,48]. n prticulr: the units re such tht c, 8πG nd k re equl 1; the signture is ( + ++); spcetime indices re, b, =0,1,2,3; the curvture tensors re bcd = d Γ bc +, b = c cb nd =, nd the icci identity is b] u c = 1 2 bcdu d ; A l denotes the index string 1 2 l,nde A l denotes the tensor product e 1 e 2 e l ; (squre) round brckets enclosing indices denote the (nti-)symmetric prt, nd ngled brckets denote the projected symmetric nd trcefree (PSTF) prt (defined below). The sptilly projected prt of the covrint derivtive is denoted by D, following 67]. 4 The pproximte equlity symbol, s in J 0, indictes equlity up to first (liner) order in n lmost-flw spcetime.. COVAAT LAGAGA FOMALSM ELATVSTC COSMOLOGY The Ehlers-Ellis 1+3 formlism 1,2,68] is covrint Lgrngin pproch, i.e. every quntity hs nturl interprettion in terms of observers comoving with the fundmentl 4-velocity u (where u u = 1). Provided this is defined uniquely in n invrint mnner, ll relted quntities hve direct physicl or geometric mening, nd my in principle be mesured in the instntneous rest spce of the comoving fundmentl observers. Any coordinte system or tetrd cn be used when specific clcultions re mde. These fetures re crucil prt of the strengths of the formlism nd of the perturbtion theory tht is derived from it. We will follow the stremlining nd development of the formlism given by Mrtens 67], the essence of which is to mke explicit use of irreducible quntities nd derivtives, nd to develop the identities which these quntities nd derivtives obey (see lso 48,57,58,51,52]). The bsic lgebric tensors re: () the projector h b = g b +u u b,whereg b is the spcetime metric, which projects into the instntneous rest spce of comoving observers; nd (b) the projected lternting tensor ε bc = η bcd u d,where η bcd = g δ 0 δ 1 bδ 2 cδ 3 d] is the spcetime lternting tensor. Thus η bcd =2u ε b]cd 2ε bc u d], ε bc ε def =3!h d h b e h c] f. The projected symmetric trcefree (PSTF) prts of vectors nd rnk-2 tensors re V = h b V b, S b = { h ( c h b) d 1 3 hcd h b } Scd, with higher rnk formuls given in 37]. The skew prt of projected rnk-2 tensor is sptilly dul to the projected vector S = 1 2 ε bcs bc], nd then ny projected rnk-2 tensor hs the irreducible covrint decomposition S b = 1 3 Sh b + ε bc S c + S b, where S = S cd h cd is the sptil trce. n the 1+3 covrint formlism, ll quntities re either sclrs, projected vectors or PSTF tensors. The equtions governing these quntities involve covrint vector product nd its generliztion to PSTF rnk-2 tensors: 3 This cse will pply before decoupling, in order to be consistent with the lmost-flw result quoted bove. 4 n 4,43,45,35,36] it is denoted (3), while in 10,34] it is. 3

4 V,W] = ε bc V b W c, S, Q] = ε bc S b dq cd. The covrint derivtive defines 1+3 covrint time nd sptil derivtives J b = u c c J b, D c J b = h c d h e h b f d J e f. ote tht D c h b =0=D d ε bc, while ḣb =2u ( u b) nd ε bc =3u ε bc]d u d. The projected derivtive D further splits irreducibly into 1+3 covrint sptil divergence nd curl 67] nd 1+3 covrint sptil distortion 57] div V =D V, (div S) =D b S b, curl V = ε bc D b V c, curl S b = ε cd( D c S d b), D V b =D ( V b) 1 3 (div V ) h b, D S bc =D ( S bc) 2 5 h (b (div S) c). ote tht div curl is not in generl zero, for vectors or rnk-2 tensors (see 67,48,51,58] for the relevnt formuls). The covrint irreducible decompositions of the derivtives of sclrs, vectors nd rnk-2 tensors re given in exct (nonliner) form by 57] ψ = ψu +D ψ, (1) { } b V = u b V + A c V c { u + u 1 3 ΘV b + σ bc V c } +ω, V ] b (div V ) h b 1 2 ε bccurl V c +D V b, (2) c S b = u c {Ṡ b +2u ( S b)d A d} { +2u 1 ( 3 ΘS b)c+s d b) (σ cd ε cde ω e ) } (div S) h b c 2 3 ε dc(curl S b) d +D S bc. (3) The lgebric correction terms in equtions (2) nd (3) rise from the reltive motion of comoving observers, s encoded in the kinemtic quntities: the expnsion Θ = D u, the 4-ccelertion A u = A, the vorticity 5 ω = 1 2 curl u, nd the sher σ b =D u b. Thus, by Eq. (2) b u = A u b Θh b + ε bc ω c + σ b. The irreducible prts of the icci identities produce commuttion identities for the irreducible derivtive opertors. n the simplest cse of sclrs: curl D ψ ε bc D b D c] ψ = 2 ψω, (4) D ψ h b (D b ψ) = ψa ΘD ψ + σ b D b ψ +ω, Dψ]. (5) dentity (4) reflects the crucil reltion of vorticity to non-integrbility; non-zero ω implies tht there re no constnttime 3-surfces everywhere orthogonl to u, since the instntneous rest spces cnnot be ptched together smoothly. 6 dentity (5) is the key to deriving evolution equtions for sptil grdients, which covrintly chrcterize inhomogeneity 4]. Further identities re given in 40,67,69,57,48]. The kinemtic quntities govern the reltive motion of neighboring fundmentl world-lines, nd describe the universl expnsion nd its locl nisotropies. The dynmic quntities describe the sources of the grvittionl field, nd directly determine the icci curvture loclly vi Einstein s field equtions. They re the (totl) energy density ρ = T b u u b, isotropic pressure p = 1 3 h bt b, energy flux q = T b u b, nd nisotropic stress π b = T b,wheret b is the totl energy-momentum tensor. The loclly free grvittionl field, i.e. the prt of the spcetime curvture not 5 The vorticity tensor ω b = ε bc ω c is often used, but we prefer to use the irreducible vector ω. The sign conventions, following 1,2], re such tht in the ewtonin limit, ω = 1 v.otethtd b ω 2 b =curlω. 6 n this cse, which hs no ewtonin counterprt, the D opertor is not intrinsic to 3-surfce, but it is still well-defined sptil projection of in ech instntneous rest spce. 4

5 directly determined loclly by dynmic sources, is given by the Weyl tensor C bcd. This splits irreducibly into the grvito-electric nd grvito-mgnetic fields E b = C cbd u c u d = E b, H b = 1 2 ε cdc cd beu e = H b, which provide covrint Lgrngin description of tidl forces nd grvittionl rdition. An FLW (bckground) universe, with its unique preferred 4-velocity u, is covrintly chrcterized s follows: dynmics: D ρ =0=D p,q =0,π b =0; kinemtics: D Θ=0,A =0=ω,σ b =0; grvito-electric/mgnetic field: E b =0=H b. The Hubble rte is H = 1 3Θ=ȧ/, where(t) is the scle fctor nd t is cosmic proper time. n sptilly homogeneous but nisotropic universes (inchi nd Kntowski-Schs models), the quntities q, π b, σ b, E b nd H b in the preceding list my be non-zero. The icci identity for u nd the inchi identities d C bcd = ( b]c g b]c) produce the fundmentl evolution nd constrint equtions governing the bove covrint quntities 1,2]. Einstein s equtions re incorported 7 vi the lgebric replcement of the icci tensor b by T b 1 2 T c c g b. These equtions, in exct (nonliner) form nd for generl source of the grvittionl field, re 57]: Evolution: Constrint: ρ +(ρ+p)θ+divq= 2A q σ b π b, (6) Θ+ 1 3 Θ (ρ+3p) div A = σ bσ b +2ω ω +A A, (7) q Θq +(ρ+p)a +D p+(divπ) = σ b q b +ω, q] A b π b, (8) ω Θω curl A = σ b ω b, (9) σ b Θσ b + E b 1 2 π b D A b = σ c σ c b ω ω b + A A b, (10) Ė b +ΘE b curl H b (ρ + p)σ b π b D q b Θπ b = A q b +2A c ε cd( H b) d +3σ c E b c Ḣ b +ΘH b +curle b 1 2 curl π b =3σ c H b c ω c ε cd( H b) d ω c ε cd( E b) d 1 2 σc π b c 1 2 ωc ε cd( π b) d, (11) 2A c ε cd( E b) d 3 2 ω q b σc (ε b)cd q d. (12) div ω = A ω, (13) (div σ) curl ω 2 3 D Θ+q = 2ω, A], (14) curl σ b +D ω b H b = 2A ω b, (15) (div E) (div π) 1 3 D ρ Θq =σ, H] 3H b ω b σ bq b 3 2 ω, q], (16) (div H) curl q (ρ + p)ω = σ, E] 1 2 σ, π] +3E b ω b 1 2 π bω b. (17) f the universe is close to n FLW model, then quntities tht vnish in the FLW limit re O(ɛ), where ɛ is dimensionless smllness prmeter, nd the quntities re suitbly normlized (e.g. σb σ b /H < ɛ, etc.). The bove equtions re covrintly nd guge-invrintly linerized 4] by dropping ll terms O(ɛ 2 ), nd by replcing sclr coefficients of O(ɛ) terms by their bckground vlues. This lineriztion reduces ll the right hnd sides of the evolution nd constrint equtions to zero COVAAT OLEA AALYSS OF MULT-FLUDS The formlism described bove pplies for ny covrint choice of u. f the physics picks out only one u,then tht becomes the nturl nd obvious 4-velocity to use. n complex multi-fluid sitution, however, there re vrious 7 ote tht one constrint Einstein eqution is not explicitly contined in this set see 2,70]. 5

6 possible choices. The different prticle species in cosmology will ech hve distinct 4-velocities; we could choose ny of these s the fundmentl frme, nd other choices such s the centre of mss frme re lso possible. This llows vriety of covrint choices of 4-velocities, ech leding to slightly different 1+3 covrint description. One cn regrd choice between these different possibilities s prtil guge-fixing (but determined in covrint nd physicl wy). Any differences between such 4-velocities will be O(ɛ) in the lmost-flw cse nd will dispper in the FLW limit, 8 s is required in consistent 1+3 covrint nd guge-invrint lineriztion bout n FLW model (see 4,35] for further discussion). n ddition to the issue of lineriztion, one cn lso sk more generlly wht the impct of chnge of fundmentl frme is on the kinemtic, dynmic nd grvito-electric/mgnetic quntities. f n initil choice u is replced by new choice ũ,then ũ =γ(u +v ) where v u =0,γ=(1 v v ) 1/2, (18) where v is the (covrint) velocity of the new frme reltive to the originl frme. The exct trnsformtions of ll relevnt quntities re given in the ppendix, nd re tken from 52]. To liner order, the trnsformtions tke the form: Θ Θ+divv, Ã A + v +Hv, ω ω 1 2 curl v, σ b σ b +D v b, ρ ρ, p p, q q (ρ+p)v, π b π b, Ẽ b E b, Hb H b. Suppose now tht choice of fundmentl frme hs been mde. (For the purposes of this pper, we will not need to specify such choice.) Then we need to consider the velocities of ech species which source the grvittionl field, reltive to the fundmentl frme. f the 4-velocities re close, i.e. if the frmes re in non-reltivistic reltive motion, then O(v 2 ) terms my be dropped from the equtions, except if we include nonliner kinemtic, dynmic nd grvito-electric/mgnetic effects, in which cse, for consistency, we must retin O(ɛ 0 v 2 )termssuchsρv 2, which re of the sme order of mgnitude in generl s O(ɛ 2 ) terms. (See 72].) f the universe is close to FLW, then O(ɛ 0 v 2 ) terms my be neglected, together with O(ɛv) ndo(ɛ 2 )terms. n summry, there re two different lineriztions: () linerizing in reltive velocities (i.e. ssuming ll species hve nonreltivistic bulk motion reltive to the fundmentl frme), without linerizing in the kinemtic, dynmic nd grvito-electric/mgnetic quntities tht covrintly chrcterize the spcetime; (b) FLW-lineriztion, which implies the specil cse of () obtined by lso linerizing in the kinemtic, dynmic nd grvito-electric/mgnetic quntities. Clerly () is more generl, nd we cn tke it to be the physiclly relevnt nonliner regime, i.e. the cse where only nonreltivistic verge velocities 9 re considered, but no other ssumptions re mde on the physicl or geometric quntities. n cse (), no restrictions re imposed on non-velocity terms, nd we neglect only terms O(ɛv 2,v 3 ). n cse (b), we neglect terms O(ɛ 2,ɛv,v 2 ). Covrint second-order effects ginst n FLW bckground re included within (), when we neglect terms O(ɛ 3 ). (ote tht guge-invrince is fr more subtle problem t second order thn t first order: see runi et l. 73].) The dynmic quntities in the evolution nd constrint equtions (6) (17) re the totl quntities, with contributions from ll dynmiclly significnt prticle species. Thus T b = T b = ρu u b + ph b +2q ( u b) +π b, (19) T b = ρ u u + p hb +2q ( u b) +π b, (20) where lbels the species. We include rdition photons ( = ), bryonic mtter ( = ) modelled s perfect fluid, cold drk mtter ( = C) modelled s dust over the er of interest for CM nisotropies, neutrinos ( = ) 8 A similr sitution occurs in reltivistic thermodynmics, where suitble 4-velocities re close to the equilibrium 4-velocity, nd hence to ech other 71]. 9 Of course, this implies no restrictions on the velocities of individul prticles within ny species. 6

7 (ssumed to be mssless), nd cosmologicl constnt ( = V ). 10 ote tht the dynmic quntities ρ, in eqution (20) re s mesured in the -frme, whose 4-velocity is given by Thus we hve u = γ ( u + v ),v u =0. (21) p C =0=q C =πb C, q =0=πb, (22) p = 1 3 ρ, p = 1 3 ρ, (23) where we hve chosen the unique 4-velocity in the cold drk mtter nd bryonic cses which follows from modelling these fluids s perfect. The cosmologicl constnt is chrcterized by p V = ρ V = Λ, q =0=πb, v =0. V V V The conservtion equtions for the species re best given in the overll u -frme, in terms of the velocities v of species reltive to this frme. Furthermore, the evolution nd constrint equtions of Section re ll given in terms of the u -frme. Thus we need the expressions for the prtil dynmic quntities s mesured in the overll frme. The velocity formul inverse to eqution (21) is ), v u = γ ( u + v = γ ( v +v2 u), (24) where v u = 0, nd v v = vv. Using this reltion together with the generl trnsformtion equtions (A8) (A11), or directly from the bove equtions, we find the following exct (nonliner) equtions for the dynmic quntities of species s mesured in the overll u -frme: ρ = ρ + { γ 2 v2 p = p + { 1 3 γ 2 q = q π b v2 +(ρ +p )v + { (γ 1)q = π b + { (ρ + p )+2γ q v +π b v v b}, (25) (ρ + p )+2γ q v +π b 2u ( π b)c γ qb v bu + γ 2 v2 v c + π bc v bv cu u b} v v b}, (26) (ρ + p ) v + πb v b π bc v bv cu }, (27) + { 1 3 πcd v cv dh b + γ 2 (ρ + p ) } v v b +2γ v q b. (28) These expressions re the nonliner generliztion of well-known linerized results (see e.g. 45,71]). FLW lineriztion implies tht v 1forech, nd we neglect ll terms which re O(v 2)orO(ɛv ). This removes ll terms in brces, drmticlly simplifying the expressions: ρ ρ, p p, q q +(ρ +p )v, π b π b. To liner order, there is no difference in the dynmic quntities when mesured in the -frme or the fundmentl frme, prt from simple velocity correction to the energy flux. ut in the generl nonliner cse, this is no longer true. The totl dynmic quntities re simply given by ρ = ρ, p = p, q = q, π b = π b. ote tht the equtions (25) (28) hve been written to mke cler the liner prts, so tht the irreducible nture is not explicit. rreducibility (in the u -frme) is reveled on using the reltions 10 A more generl tretment, incorporting ll the sources which re currently believed to be potentilly significnt, would lso include dynmic sclr field tht survives fter infltion ( quintessence ), nd hot drk mtter in the form of mssive neutrinos (see 28] for survey with further references). Our min im is not detiled nd comprehensive model with numericl predictions, but qulittive discussion focusing on the underlying dynmic nd geometric effects t nonliner nd liner level tht re brought out clerly by 1+3 covrint pproch. n principle our pproch is redily generlized to include other sources of the grvittionl field. 7

8 q h bq b = q qbv bu, π b h ch b dπ cd = π b 2u ( π b)c v c + π cd v cv du u b. The exct equtions show in detil the specific couplings nd contributions of ll prtil dynmic quntities in the totl quntities. For exmple, it is cler tht in sptilly homogeneous but nisotropic models, the prtil energy fluxes q contribute to the totl energy density, pressure nd nisotropic stress t first order in the velocities v, while the prtil nisotropic stresses π b contribute to the totl energy flux t first order in v. The totl nd prtil 4-velocities define corresponding number 4-currents: = nu + j =, =n u +j, (29) where n nd n re the number densities, j nd j re the number fluxes, nd j u =0=j u. t follows tht n = j = n j = = n + ( j + n v { (γ 1)n + j v }, (30) ) + { (γ 1)n v vb j bu }, (31) where the strred quntities re s mesured in the u -frme. Lineriztion removes the terms in brces, regining the expressions in 45,71]. Four-velocities my be chosen in number of covrint nd physicl wys. The min choices re 75,71]: () the energy (Lndu-Lifshitz) frme, defined by vnishing energy flux, nd (b) the prticle (Eckrt) frme, defined by vnishing prticle number flux. For given single fluid, these frmes coincide in equilibrium, but in generl they re different. For ech prtil u, ny chnge in choice u ũ leds to trnsformtions in the prtil dynmic quntities, tht re given by equtions (A8) (A11) in the ppendix. For the fundmentl u, chnge in choice leds in ddition to trnsformtions of the kinemtic quntities, given by equtions (A4) (A7), nd of the grvito-electric/mgnetic field, given by equtions (A12) (A13). A convenient choice for ech prtil four-velocity u is the energy frme, i.e. q =0forech(this is the obvious choice in the cses = C, ). As mesured in the fundmentl frme, the prtil energy fluxes do not vnish, i.e. 0, nd the totl energy flux is given by q q = (ρ + p ) v + πb v b + O(ɛv 2,v3 )]. (32) With this choice, using the bove equtions, we find the following expressions for the dynmic quntities of mtter s mesured in the fundmentl frme. For cold drk mtter: For bryonic mtter: ρ C = γ2 C ρ C, p C = 1 3 γ2 C v2 C ρ C, (33) q = γ2ρ C C C v, π b C C = γ 2 ρ C C v vb. (34) C C ρ = ( γ2 1+w v 2 ) ρ, p = w γ2v2(1 + w )] ρ, (35) q = γ2(1 + w )ρ v, π b = γ 2 (1 + w )ρ v vb, (36) where w p /ρ. n the cse of rdition nd neutrinos, we will evlute the dynmic quntities reltive to the u -frme directly vi kinetic theory, in the next section. The totl energy-momentum tensor is conserved, i.e. b T b = 0, which is equivlent to the evolution equtions (6) nd (8). The prtil energy-momentum tensors obey b T b = J = U u + M, (37) where U is the rte of energy density trnsfer to species s mesured in the u -frme, nd M = M is the rte of momentum density trnsfer to species, smesuredintheu -frme. Cold drk mtter nd neutrinos re decoupled during the period of relevnce for CM nisotropies, while rdition nd bryons re coupled through Thomson scttering. Thus 8

9 J C =0=J, J = J = U T u +M T, (38) where the Thomson rtes re U T M T ( = n E σ 4 T 3 ρ v2 q = n σ ( 4 E T 3 ρ v q v ) + O(ɛv 2,v3 ), (39) ) + O(ɛv 2,v3 ), (40) + π bv b s given by Eq. (63), derived in Section V. Here n E is the free electron number density, nd is the Thomson cross-section. ote tht to liner order, there is no energy trnsfer, i.e. U T 0. Using equtions (33) (36) in (37), we find tht for cold drk mtter nd for bryonic mtter ρ C +Θρ C +ρ C div v C = ( ρ C v 2 ) C 4 3 v2θρ C C v D C ρ C 2ρ C A v + O(ɛv2,v3 ), (41) C C C v + 1 C 3 Θv + A = A C b v b u σ C bv b C +ω, v C ] v b D C bv + O(ɛv2,v3 ), (42) C C C ρ +Θ(1+w )ρ +(1+w )ρ div v = (1 + w )ρ v] v2θ(1 + w )ρ v D (1 + w )ρ ] 2(1 + w )ρ A v n ( 4 E 3 ρ v2 q v ) + O(ɛv 2,v3 ), (43) (1 + w ) v +( ) 1 3 c2 Θv +(1+w )A +ρ 1 D p +ρ 1 n ( ) E ρ v q =(1+w )A b v b u (1 + w )σ bv b +(1+w )ω, v ] (1 + w )v b D bv + c2(1 + w )(div v )v ρ 1 n σ E T π bv b + O(ɛv 2,v3 ), (44) where c 2 ṗ / ρ (this equls the dibtic sound speed only to liner order). These conservtion equtions generlize those given in 36] to the nonliner cse. FLW lineriztion reduces the right hnd sides of these equtions to zero, drmticlly simplifying the equtions. The conservtion equtions for the mssless species (rdition nd neutrinos) re given below. ote from Eq. (42) tht if the cold drk mtter frme is chosen s the fundmentl frme, then the 4-ccelertion vnishes, i.e. v = 0 implies A C = 0. This is the choice of fundmentl frme dvocted in 36]. V. COVAAT LAGAGA KETC THEOY eltivistic kinetic theory (see e.g ]) provides self-consistent microscopiclly bsed tretment where there is nturl unifying frmework in which to del with gs of prticles in circumstnces rnging from hydrodynmic to free-streming behvior. The photon gs undergoes trnsition from hydrodynmic tight coupling with mtter, through the process of decoupling from mtter, to non-hydrodynmic free streming. This trnsition is chrcterized by the evolution of the photon men free pth from effectively zero to effectively infinity. The rnge of behvior cn ppropritely be described by kinetic theory with Thomson scttering 79,80], nd the bryonic mtter with which rdition intercts cn resonbly be described hydrodynmiclly during these times. (The bsic physics of rdition nd mtter nd density perturbtions in cosmology ws developed in the works of Schs nd Wolfe 11], Silk 81], Peebles nd Yu 13], Weinberg 82], nd others.) n the covrint Lgrngin pproch of 5] (see lso 7,8]), the photon 4-momentum p (where p p = 0) is split s p = E(u + e ), e e =1,e u =0, (45) where E = u p is the energy nd e = p /E is the direction, s mesured by comoving (fundmentl) observer. Then the photon distribution function is decomposed into covrint hrmonics vi the expnsion 5,9] f(x, p) =f(x, E, e) =F+F e +F b e e b + = l 0F Al (x, E)e A l, (46) 9

10 where e A l e 1 e 2 e l,nde Al provides representtion of the rottion group 37]. The covrint multipoles re irreducible since they re PSTF, i.e. F b = F b F b = F ( b), F b u b =0=F bc h bc. They encode the nisotropy structure of the distribution in the sme wy s the usul sphericl hrmonic expnsion f = l 0 +l m= l f m l (x, E)Y m l ( e ), but here () the F Al re covrint, nd thus independent of ny choice of coordintes in momentum spce, unlike the fl m;(b)f A l is rnk-l tensor field on spcetime for ech fixed E, nd directly determines the l-multipole of rdition nisotropy fter integrtion over E. The multipoles cn be recovered from the distribution function vi 5,37] F Al = 1 l f(x, E, e)e Al dω, with l =4π (l!)2 2 l (2l +1)!, (47) where dω = d 2 e is solid ngle in momentum spce. A further useful identity is 5] e A l dω = 4π 0 l odd, l +1 h (12 h 34 h l 1 l ) l even. (48) The first 3 multipoles rise from the rdition energy-momentum tensor, which is T b (x) = p p b f(x, p)d 3 p = ρ u u b ρ hb +2q ( ub) +π b, where d 3 p = EdEdΩ is the covrint volume element on the future null cone t event x. t follows tht the dynmic quntities of the rdition (in the u -frme) re: ρ =4π E 3 FdE, q = 4π E 3 F de, π b = 8π E 3 F b de. (49) From now on, we drop the sterisks from the rdition dynmic quntities reltive to the fundmentl frme, since we do not need to relte them to their vlues in the rdition frme. We extend these dynmic quntities to ll multipole orders by defining 11 5] Π 1 l = E 3 F 1 l de, (50) so tht Π = ρ /4π, Π =3q /4πnd Πb =15π b/8π. The oltzmnn eqution is df f p dv x Γ bcp b p c f = Cf], (51) p where p =dx /dvnd Cf] is the collision term, which determines the rte of chnge of f due to emission, bsorption nd scttering processes. This term is lso decomposed into covrint hrmonics: Cf] = l 0b Al (x, E)e A l = b + b e + b b e e b +, (52) where the multipoles b Al = b Al encode covrint irreducible properties of the prticle interctions. Then the oltzmnn eqution is equivlent to n infinite hierrchy of covrint multipole equtions 11 ecuse photons re mssless, we do not need the complexity of the moment definitions used in 5]. n 36], J (l) A l where J (l) A l = l Π Al. From now on, ll energy integrls will be understood to be over the rnge 0 E. is used, 10

11 L Al (x, E) =b Al F Am ](x, E), where L Al re the multipoles of df/dv, nd will be given in the next section. These multipole equtions re tensor field equtions on spcetime for ech vlue of the photon energy E (but note tht energy chnges long ech photon pth). Given the solutions F Al (x, E) of the equtions, the reltion (46) then determines the full photon distribution f(x, E, e) s sclr field over phse spce. Over the period of importnce for CM nisotropies, i.e. considerbly fter electron-positron nnihiltion, the verge photon energy is much less thn the electron rest mss nd the electron therml energy my be neglected, so tht the Compton interction between photons nd electrons (the dominnt interction between rdition nd mtter) my resonbly be described in the Thomson limit. (See 72] for refinements.) We will lso neglect the effects of polriztion (see e.g. 24]). For Thomson scttering Cf] = n E E f(x, p) f(x, p) ], (53) where E = p u is the photon energy reltive to the bryonic (i.e. bryon-electron) frme u,nd f(x, p) determines the number of photons scttered into the phse spce volume element t (x, p). The differentil Thomson cross-section is proportionl to 1 + cos 2 α,whereαis the ngle between initil nd finl photon directions in the bryonic frme. Thus cos α = e e where e is the initil nd e is the finl direction, so tht p ( = E u + ) e, p ( = E u +e ), wherewehveusede =E, which follows since the scttering is elstic. Here u is given by Eq. (21), where v is the velocity of the bryonic frme reltive to the fundmentl frme u, with v u =0. Then f is given by 36,72] f(x, p) = 3 16π f(x, p ) 1+ ( e ) ] 2 e dω. (54) The exct forms of the photon energy nd direction in the bryonic frme follow on using equtions (21) nd (A2): E ( = Eγ 1 v e ), (55) e = 1 e γ (1 v c e + γ 2 ( v b c) e b v) 2 u + γ 2 ( v b e b 1 ) v ]. (56) Anisotropic scttering will source polriztion, nd smll errors re introduced by ssuming tht the rdition remins unpolrized 83]. A fully consistent nd generl tretment requires the incorportion of polriztion. However, for simplicity, nd in line with mny previous tretments, we will neglect polriztion effects. V. THE OLEA MULTPOLE HEACHY The full oltzmnn eqution in photon phse spce contins more informtion thn necessry to nlyze rdition nisotropies in n inhomogeneous universe. For tht purpose, when the rdition is close to blck-body we do not require the full spectrl behviour of the distribution multipoles, but only the energy-integrted multipoles. The monopole leds to the verge temperture, while the higher order multipoles determine the temperture fluctutions. The 1+3 covrint nd guge-invrint definition of the verge temperture T is given by 10] ρ (x) =4π E 3 F(x, E)dE = rt (x) 4, (57) where r is the rdition constnt. f f is close to Plnck distribution, then T is the therml blck-body verge temperture. ut note tht no notion of bckground temperture is involved in this definition. There is n ll-sky verge implied in (57). Fluctutions cross the sky re mesured by integrting the higher multipoles ( precise definition is given below), i.e. the fluctutions re determined by the Π 1 l (l 1) defined in Eq. (50). The form of Cf] shows tht covrint equtions for the temperture fluctutions rise from decomposing the energy-integrted oltzmnn eqution E 2 df dv de = E 2 Cf]dE (58) 11

12 into 1+3 covrint multipoles. We begin with the right hnd side, which requires the covrint form of the Thomson scttering term (54). Since the bryonic frme will move nonreltivisticlly reltive to the fundmentl frme in ll cses of physicl interest, it is sufficient to linerize only in v, nd not in the other quntities. Thus we drop terms in O(ɛv 2,v3 ) but do not neglect terms tht re O(ɛ0 v 2,ɛv )oro(ɛ2 ) reltive to the FLW limiting bckground. n other words, we mke no restrictions on the geometric nd physicl quntities tht covrintly chrcterize the spcetime, prt from ssuming nonreltivistic reltive verge velocity for mtter. The resulting expression will in prticulr be pplicble for covrint second-order effects in FLW bckgrounds (recognising tht polriztion effects should be included for complete tretment), or for first-order effects in inchi bckgrounds. For brevity, we will use the nottion O3] O(ɛv 2,v3 ), noting tht this does not imply ny second-order restriction on the dynmic, kinemtic nd grvito-electric/mgnetic quntities. t follows from equtions (48) nd (54) tht 4π fe 3 de =(ρ ) +3 4 (πb) e e b, (59) where the dynmic rdition quntities re evluted in the bryonic frme. This pproch relies on the frmetrnsformtions given in the ppendix, nd llows us to evlute the Thomson scttering integrl more directly nd clerly thn other pproches. n the process, we re lso generlizing to include nonliner effects. We use equtions (A8) nd (A11) to trnsform bck to the fundmentl frme: 12 ] (ρ ) = ρ v2 2q v +O3], (π b ) = πb +2v cπ c( ub) 2q vb ρ v vb +O3]. ow E 2 Cf]dE = n E 1+3v c e c+ ( ] v c 2 e 3 c) 2 v2 n ddition, we need the following identity, vlid for ny projected vector v : E 3 fde n E 1 v c e c+ 1 2 v2 ] fe 3 de +O3]. (60) v e f = 1 3 F v + Fv F bv b] e + F v b F bcv c] e e b + = ( ] l+1 F Al 1 v l + )F Alv e Al. (61) 2l+3 l 0 (Here nd subsequently, we use the convention tht F Al =0forl<0.) This identity my be proved using Eq. (48) nd the identity (see 5], p. 470): ( ) l V b S Al = V (b S Al ) V c S c(al 1 h l b) 2l +1 where S Al = S Al. (62) Using the bove equtions, we find tht 13 4π E 2 Cf]dE = n E σ 4 T 3 ρ v2 qv ] n E 3q 4ρ v 3πb v b] e 27 n E 4 πb 3 2 q vb 12 n E 4πΠ bc 45 4 π b vc ] 7 ππbc v c 3ρ v vb e e b ] 16 9 ππbcd v d e e b e c + +O3]. (63) 12 As noted in Section, we retin the O(v 2 )termin(ρ ) since ρ 13 A. Chllinor hs independently derived the sme result 38]. is zero-order. 12

13 ow it is cler from equtions (59) nd (60) tht the first four multipoles re ffected by Thomson scttering differently thn the higher multipoles. This is confirmed by the form of eqution (63). Defining the energy-integrted scttering multipoles K Al = E 2 b Al de, we find from Eq. (63) tht K = n E σ 4 T 3 Πv2 1 3 Π v ] + O3], (64) K = n E Π 4Πv 2 5 Πb v b] + O3], (65) ] K b 9 = n E 10 Πb 1 2 Π v b 3 7 Πbc v c 3Πv vb + O3], (66) ] K bc = n E Π bc 3 2 Π b v c 4 9 Πbcd v d + O3], (67) nd, for l>3: ( ) ] K A l = n E Π A l Π A l 1 v l l +1 Π Al v 2l+3 +O3]. (68) Equtions (64) (68) re nonliner generliztion of the results given by Chllinor nd Lsenby 36]. They show the new coupling of bryonic bulk velocity to the rdition multipoles, rising from locl nonliner effects in Thomson scttering. f we linerize fully, i.e. neglect ll terms contining v except the ρ v term in the dipole K, which is first-order, then our equtions reduce to those in 36]. The generlized nonliner equtions pply to the nlysis of second-order effects on n FLW bckground, to first-order effects on sptilly homogeneous but nisotropic bckground, nd more generlly, to ny sitution where the bryonic frme is non-reltivistic reltive to the fundmentl u -frme. ext we require the multipoles of df/dv. These cn be red directly from the generl expressions first derived in 5], which re exct, 1+3 covrint nd lso include the cse of mssive prticles. For clrity nd completeness, we outline n lterntive, 1+3 covrint derivtion (the derivtion in 5] uses tetrds). We require the identity 7,8] de dv = E2 1 3 Θ+A e +σ b e e b], (69) which follows directly from E = p u, p b b p =0nd b u = A u b +D b u.then d dv F 1 l (x, E)e 1 e l ]= d E l F 1 dv l (x, E)p 1 p l] =E { 1 3 Θ+A be b +σ bc e b e c]( lf 1 l EF ) 1 l e 1 e l ]} +(u 1 +e 1 ) (u l +e l ) F 1 l + e b b F 1 l, where prime denotes / E. The first term is redily put into irreducible PSTF form using the identity (62) with V = A, nd its extension to the cse when V is replced by rnk-2 PSTF tensor W b (see 5], p. 470), with W b = σ b. n the second term, when the round brckets re expnded, only those terms with t most one u r survive, nd u F = A F, u b F b = ( 1 3 Θhb + σ b ε bc ) ω c Fb. Thus the covrint multipoles b Al of df/dv re E 1 b Al = F Al 1 3 ΘEF A l +D l F Al 1 + (l+1) (2l +3) D F Al (l +1) (2l +3) E (l+1) E l+2 ] F Al A E l E 1 l ] F Al 1 Al lω b ε bc(l F Al 1 ) c (l +1)(l+2) (2l + 3)(2l +5) E (l+2) E l+3 ] F bal σ b 2l ] (2l +3) E 1/2 E 3/2 σl F b Al 1 b E l 1 E 2 l ] σl 1 F Al 2 l. (70) 13

14 This regins the result of 5] eqution (4.12)] in the mssless cse, with minor corrections. The form given here benefits from the stremlined version of the 1+3 covrint formlism. We reiterte tht this result is exct nd holds for ny photon or (mssless) neutrino distribution in ny spcetime. We now multiply Eq. (70) by E 3 nd integrte over ll energies, using integrtion by prts nd the fct tht E n F 0sE for ny positive n. Weobtin the multipole equtions tht determine the brightness multipoles Π Al : K Al = Π Al ΘΠ A l +D l Π Al 1 + (l+1) (2l +3) Db Π bal (l +1)(l 2) A b Π bal +(l+3)a l Π Al 1 lω b ε bc(l Π Al 1 ) c (2l +3) (l 1)(l +1)(l+2) σ bc 5l Π bcal + (2l + 3)(2l +5) (2l +3) σb l Π Al 1 b (l+2)σ l l 1 Π Al 2. (71) Once gin, this is n exct result, nd it holds lso for ny collision term, i.e. ny K Al. For decoupled neutrinos, we hve K A l = 0 in this eqution. For photons undergoing Thomson scttering, the left hnd side of Eq. (71) is given by Eq. (68), which is exct in the kinemtic nd dynmic quntities, but first order in the reltive bryonic velocity. The equtions (68) nd (71) thus constitute nonliner generliztion of the FLW-linerized cse given by Chllinor nd Lsenby 36]. These equtions describe evolution long the timelike world-lines of fundmentl observers, not long the lightlike geodesics of photon motion. The timelike integrtion is relted to light cone integrtions by mking homogeneity ssumptions bout the distribution of mtter in (spcelike) surfces of constnt time, s is discussed in 84]. The monopole nd dipole of eqution (71) give the evolution equtions of energy nd momentum density: K = Π+ 4 3 ΘΠ D Π A Π σb Π b, (72) K = Π ΘΠ +D Π+ 2 5 D bπ b A bπ b +4ΠA ω, Π] + σ b Π b. (73) n the cse of neutrinos, K =0=K, these express the conservtion of energy nd momentum:14 ρ Θρ +D q = 2A q σ bπ b, (74) q Θq ρ A D ρ +D b π b = ω, q ] σ bq b A bπ b. (75) FLW-lineriztion reduces the right hnd sides to zero. For photons, K nd K re given by equtions (64) nd (65), nd determine the Thomson rtes of trnsfer in equtions (39) nd (40): U T =4πK, M T = 4π 3 K. (76) Finlly, we return to the definition of temperture nisotropies. As noted bove, these re determined by the Π Al. Generlizing the linerized 1+3 covrint pproch in 10], we define the temperture fluctution τ(x, e) vi the directionl bolometric brightness: 1/4 4π T (x)1+τ(x, e)] = E 3 f(x, E, e)de]. (77) r This is 1+3 covrint nd guge-invrint definition which is lso exct. We cn rewrite it explicitly in terms of the Π Al : ( ) 4π τ(x, e) = 1+ Π Al e A l ρ l 1 1/4 1=τ e +τ b e e b +. (78) 14 As in the photon cse, we omit the sterisks on the neutrino dynmic quntities, since we do not require their vlues in the neutrino frme. 14

15 n principle, we cn extrct the irreducible PSTF temperture fluctution multipoles by using the inversion in Eq. (47): τ Al (x) = 1 l τ(x, e)e Al dω. (79) n the lmost-flw cse, when τ is O(ɛ), we regin from Eq. (78) the linerized definition given in 10]: ( ) π Π Al, (80) τ Al where l 1. n prticulr, the dipole nd qudrupole re ρ τ 3q 4ρ nd τ b 15πb. (81) 2ρ V. QUALTATVE MPLCATOS OF THE OLEA DYAMCAL EFFECTS n Section, we gve the nonliner evolution nd constrint equtions governing the kinemtic, totl dynmic nd grvito-electric/mgnetic quntities see equtions (6) (17). n these equtions, the totl dynmic quntities re, using the results of Section : ρ = ρ + ρ + ( 1+vC) 2 ρc + 1+(1+w )v] 2 ρ +Λ+O3], (82) p = 1 3 ρ ρ v2ρ + w C C (1 + w ] )v2 ρ Λ+O3], (83) q = q + q + ρv +(1+w )ρ C C v +O3], (84) π b = π b C v vb C C v vb +O3]. (85) The conservtion equtions for mtter were given in Section see equtions (41) (44). For neutrinos, the equtions were given in Section V see equtions (74) nd (75). For photons, the equtions follow from the results of Section Vs: q ρ Θρ +D q +2A q +σ bπ b = n E ( 4 3 ρ v2 qv ) + O3], (86) Θq ρ A D ρ +D b π b +σ bq b ω, q ] + A b π b = n ( 4 E 3 ρ v q + πbv b) + O3]. (87) The nonliner dynmicl equtions re completed by the integrted oltzmnn multipole equtions given in Section V see Eq. (71). For neutrinos (l 2): 0= Π A l ΘΠA l +D l Π A l 1 + (l+1) (2l +3) D bπ ba l (l +1)(l 2) A b Π ba l +(l+3)a l Π A l 1 (2l +3) lω b ε bc( l Π A l 1) c (l 1)(l +1)(l+2) σ bc Π bca 5l l + (2l + 3)(2l +5) (2l +3) σ b l Π A l 1 b (l+2)σ l l 1 Π A l 2. (88) For photons, the qudrupole evolution eqution is π b Θπb ρ σ b D q b + 8π 35 D cπ bc +2A q b = n E 2ωc ε cd ( π b)d 9 10 πb σ c π b c 1 5 q vb 8π 32π 315 σ cdπ bcd 35 Πbc v c 2 5 ρ v vb ] + O3]. (89) 15

16 n the free-streming cse n E = 0, eqution (89) reduces to the result first given in 30]. This qudrupole evolution eqution is centrl to the proof tht lmost-isotropy of the CM fter lst scttering implies lmost-homogeneity of the universe 30]. The higher multipoles (l >3) evolve ccording to Π Al ΘΠA l +D l Π Al 1 + (l+1) (2l +3) D bπ ba l (l +1)(l 2) A b Π ba l +(l+3)a l Π Al 1 lω b ( ε l bc Π A l 1)c (2l +3) (l 1)(l +1)(l+2) σ bc Π bca l 5l + (2l + 3)(2l +5) (2l +3) σ b l Π Al 1 b (l+2)σ l l 1 Π A l 2 ( ) ] = n E Π A l Π A l 1 v l l+1 Π Al v 2l+3 +O3]. (90) For l = 3, the second term in squre brckets on the right of Eq. (90) must be multiplied by 3 2. The temperture fluctution multipoles τ Al re determined in principle from the rdition dynmic multipoles Π Al vi equtions (78) nd (79). These equtions show in trnsprent nd explicitly 1+3 covrint nd guge-invrint form precisely which physicl effects re directly responsible for the evolution of CM nisotropies in n inhomogeneous universe. They show how the mtter content of the universe genertes nisotropies. This hppens directly through direct interction of mtter with the rdition, s encoded in the Thomson scttering terms on the right of equtions (86), (87), (89) nd (90). And it hppens indirectly, s mtter genertes inhomogeneities in the grvittionl field vi the field equtions (6) (17) nd the evolution eqution (44) for the bryonic velocity v. This in turn feeds bck into the multipole equtions vi the kinemtic quntities, the bryonic velocity v, nd the sptil grdient D ρ in the dipole eqution (87). The coupling of the multipole equtions themselves provides n up nd down cscde of effects, shown in generl by eqution (90). Power is trnsmitted to the l-multipole by lower multipoles through the dominnt (liner) distortion term D l Π Al 1, s well s through nonliner terms coupled to the 4-ccelertion (A l Π Al 1 ), bryonic velocity (v l ΠAl 1 ), nd sher (σ l l 1 Π Al 2 ). Simultneously, power cscdes down from higher multipoles through the liner divergence term (div Π) A l, nd the nonliner terms coupled to A, v nd σb. (ote tht the vorticity coupling does not trnsmit cross multipole levels.) The equtions for the rdition (nd neutrino) multipoles generlize the equtions given by Chllinor nd Lsenby 36], to which they reduce when we remove ll terms O(ɛv )ndo(ɛ 2 ). n this cse, i.e. FLW-lineriztion, there is mjor simplifiction of the equtions: nd for l 3 π b ρ Θρ +divq 0, (91) q +4Hq ρ A D ρ +(divπ ) ( n E σ 4 T 3 ρ v q ), (92) +4Hπb ρ σb D q b + 8π 35 (div Π)b 9 10 n σ E T πb, (93) Π A l +4HΠ A l +D l Π A l 1 + (l+1) (2l +3) (div Π)A l n E Π A l. (94) These linerized equtions, together with the linerized equtions governing the kinemtic nd free grvittionl quntities, given by equtions (6) (17) with zero right hnd sides, my be covrintly split into sclr, vector nd tensor modes, s described in 43,35,36]. The modes cn then be expnded in covrint eigentensors of the comoving Lplcin, nd the Fourier coefficients obey ordinry differentil equtions, fcilitting numericl integrtion. Such integrtions re performed for sclr modes by Chllinor nd Lsenby 36], with further nlyticl results given in 35,36,66,38]. However, in the nonliner cse, it is no longer possible to split into sclr, vector nd tensor modes 63,64,73]. A simple illustrtion of this rises in dust spcetimes, which my be considered s simplified model fter lst scttering if we neglect the dynmicl effects of bryons, rdition nd neutrinos. f one ttempts to crry over the linerized sclr-mode conditions 43,36] ω =0=H b, 16