A Covariant and Gauge-Invariant Approach

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COSMIC MICROWAVE BACKGROUND ANISOTROPIES IN THE CDM MODEL: A COVARIANT AND GAUGE-INVARIANT APPROACH

Anthony Challinor1 & Anthony Lasenby2 MRAO, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK.

ABSTRACT

We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the Cosmic Microwave Background (CMB) radiation. We use the physically appealing covariant approach to cosmological perturbations, which ensures that all variables are gauge-invariant and have a clear physical interpretation. We derive the complete set of frame-independent linearised equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker (FRW) Cold Dark Matter (CDM) universe. These equations include the contributions of scalar, vector and tensor modes in a unified manner. Frame-independent equations describing scalar perturbations, which are valid for any value of the background curvature, are obtained by placing appropriate covariant restrictions on the gauge-invariant variables. We derive the analytic solution of these equations in the early radiation dominated universe, and present the results of a numerical simulation of the standard CDM model. Our results confirm those obtained by other groups, who have worked carefully with non-covariant methods in specific gauges, but are derived here in a completely transparent fashion.

Subject headings: cosmic microwave background — cosmology: theory — gravitation — large-scale structure of universe

1. Introduction

The cosmic microwave background radiation (CMB) occupies a central role in modern cosmology. It provides us with a unique record of conditions along our past lightcone back

[email protected] [email protected] –2– to the epoch of decoupling (last scattering), when the optical depth to Thomson scattering rises suddenly due to Hydrogen recombination. Accurate observations of the CMB anisotropy should allow us to distinguish between models of structure formation and, in the case of non- seeded models, to infer the spectrum of initial perturbations in the early universe. Essential to this programme is the accurate and reliable calculation of the anisotropy predicted in viable cosmological models. Such calculations have a long history, beginning with Sachs & Wolfe (1967) who investi- ◦ gated the anisotropy on large scales (∼> 1 ) by calculating the redshift back to last scattering along null geodesics in a perturbed universe. On smaller angular scales one must address the detailed local processes occurring in the electron/baryon plasma prior to recombination, and the effects of non-instantaneous last scattering. These processes, which give rise to a wealth of structure in the CMB power spectrum on intermediate scales and damping on small scales (see, for example, Silk (1967, 1968)), are best addressed by following the photon distribution function directly from an early epoch in the history of the universe to the current point of observation. This requires a numerical integration of the Boltzmann equation, and has been carried out by many groups, of which Peebles & Yu (1970), Bond & Efstathiou (1984, 1987), Hu & Sugiyama (1995), Ma & Bertshinger (1995), Seljak & Zaldarriaga (1996) is a representative sample. The calculation of CMB anisotropies is simple in principle, but in reality is plagued with subtle gauge issues (Stoeger et al. 1995; Challinor & Lasenby 1998). These problems arise because of the gauge-freedom in specifying a map between the real universe and the unperturbed background model (Ellis & Bruni 1989), which is usually taken to be a Friedmann-Robertson-Walker (FRW) universe. The map identifies points in the real universe with points in the background model, thus defining the perturbation in any quantity of interest. (The choice of a map is often phrased in terms of the freedom to choose coordinates in the real universe.) Unless the map is constructed so that physically equivalent background models map points in one manifold to physically equivalent points in the other, the perturbation equations will admit unphysical gauge-mode solutions, which arise from mapping the background model to inequivalent points in itself. Furthermore, the perturbation in a quantity such as the density δρ will necessarily be unobservable unless the map is specified completely in terms of the results of physical measurements in the real universe. The problem of gauge-mode solutions can be eliminated by working exclusively with gauge-invariant variables: variables that are independent of the choice of map between the real universe and the background model. The gauge-invariant variables introduced by Bardeen (1980) provide a well-known example, and have been used in several calculations of CMB anisotropy (see, for example, Abbott & Schaefer (1986) and Panek (1986)). However, –3– even when gauge-invariant variables are employed, delicate gauge issues remain in Sachs- Wolfe type analyses which are related to the definition of the temperature perturbation and the placing of the last scattering surface (Panek 1986; Stoeger, Ellis, & Xu 1994; Stoeger et al. 1995). Although Bardeen’s gauge-invariant variables alleviate the problem of spuri- ous gauge-mode solutions to the perturbation equations, their physical interpretation is only transparent for certain gauge-choices. Further undesirable properties include the necessity to perform an initial non-local harmonic decomposition of the perturbations before the gauge- invariant variables may be defined and their equations of motion obtained, and the fact that, by construction, Bardeen’s variables are only gauge-invariant for those gauge-transformations which preserve the scalar, vector or tensor character of the metric perturbation. An alter- native scheme for the gauge-invariant treatment of cosmological perturbations was given by Ellis and coworkers (Ellis & Bruni 1989; Ellis, Hwang, & Bruni 1989) who built on the covariant approach to cosmology discussed, for example, by Hawking (1966). In this approach, covariantly-defined, gauge-invariant variables are constructed which are physically transparent and independent of any harmonic analysis or scalar, vector and tensor decomposition. In Dunsby (1997) and Challinor & Lasenby (1998) the covariant and gauge-invariant approach was applied to CMB anisotropies under the instantaneous recombination approximation. Simple expressions for the gauge-invariant temperature anisotropy were obtained, which are straightforward to interpret physically and are not obscured by the gauge ambiguities of some previous treatments. In this paper, we apply the covariant and gauge-invariant approach to the full kinetic theory calculation of CMB anisotropies on all angular scales. Our motivation for reconsid- ering this problem is two-fold. Firstly, it is our belief that the covariant and gauge-invariant description of cosmological perturbations provides a powerful set of tools for the formulation of the basic perturbation equations, and their subsequent interpretation, which are supe- rior to the techniques usually employed in such calculations (including other gauge-invariant approaches, such as that due to Bardeen (1980)). In applying covariant methods to the problem of CMB anisotropies, we can expect the same advantages of physical clarity and unification which have already been demonstrated in other areas (Ellis et al. 1989; Bruni, Ellis, & Dunsby 1992; Dunsby, Bruni, & Ellis 1992; Dunsby, Bassett, & Ellis 1996; Tsagas & Barrow 1997). Our second motivation is to respond to the criticism (Stoeger, Ellis, & Xu 1994) that “the meaning of many of the computer calculations of the Sachs-Wolfe effect re- mains obscure, because the way in which they handle these (gauge) issues is not made explicit in their discussions”. By adopting covariantly-defined gauge-invariant variables throughout, and paying careful attention to residual gauge effects such as placement of the last scattering surface and the specification of initial conditions, we ensure that the criticism by Stoeger et al. (1994) cannot be levelled at the results presented here. –4–

For definiteness we consider the cold dark matter (CDM) model, although the methods we describe are straightforward to extend to other models. We have endeavoured to make this paper reasonably self-contained, so we begin with a brief overview of the covariant approach to cosmology and define the key variables we use to characterise the perturbations in Section 2. We then go on to present a complete set of frame-independent equations describing the evolution of the matter components and radiation in Section 3 in a nearly FRW universe (with arbitrary spatial curvature). These equations, which employ only covariantly defined, gauge-invariant variables, are independent of any harmonic analysis; they describe scalar, vector and tensor perturbations in a unified manner. Equations pertinent to a particular type of perturbation are obtained by placing appropriate restrictions on the covariant variables. We do this in Section 5 for the case of scalar perturbations, obtaining, after a covariant harmonic decomposition, a set of scalar equations valid for all values of the spatial curvature. In the covariant approach to cosmology the harmonic decomposition of the perturbations does not play a central role. The decomposition is performed at a late stage in the calculation to aid solution of the linearised equations. In particular, the harmonic decomposition is made after the covariant angular decomposition of the kinetic (Boltzmann) equations, so that these decompositions are made in the reverse order here to that which occurs in most other calculational schemes. The advantage of performing the angular expansion before the harmonic decomposition is that the angular dependence of the harmonic modes, which is dictated by the Boltzmann equation, is taken care of naturally by expanding the angular moments in covariant tensors derived from the appropriate harmonic functions. This allows a more streamlined and unified treatment of the different perturbation types than conventional methods, where the angular dependence of the Fourier modes is inserted “by hand” to ensure consistency of the angular decomposition with the Boltzmann equation. In Section 5 we obtain analytic solutions to the scalar equations, valid on large scales in the early universe, which we use as initial conditions for the numerical solution of the scalar equations, the results of which we describe in Section 6. We end with our conclusions in Sec- tion 7. Ultimately, our results confirm those of other groups (for example, Ma & Bertshinger (1995)) who have performed similar calculations by working carefully in specific gauges, but are obtained here in a physically transparent, reliable, and unified manner.

We employ standard general relativity and use a (+ −−−) metric signature. Our c c d conventions for the Riemann and Ricci tensors are ﬁxed by [∇a, ∇b]u = −Rabd u ,and c Rab ≡Racb . Round brackets around indices denote symmetrisation on the indices enclosed, and square brackets denote antisymmetrisation. We use units with c = G = 1 throughout, and a unit of distance of Mpc for numerical work. –5–

2. The Covariant Approach to Cosmology

In this section, we summarise the covariant approach to cosmology (see, for example, Hawking (1966)). We begin by choosing a velocity field ua, which is defined physically in such a manner that if the universe is exactly Friedmann-Robertson-Walker (FRW) the velocity reduces to that of the fundamental observers. This property of ua is necessary to ensure gauge-invariance of the variables defined below. We refer to the choice of velocity as a frame choice. In this paper, we defer making an explicit frame choice until we discuss scalar perturbations in Section 5, when it is very convenient to choose ua to coincide with a the velocity of the CDM component. The velocity u defines a projection tensor hab which projects into the space perpendicular to ua (the instantaneous rest-space of observers moving with velocity ua):

hab ≡ gab − uaub, (2-1)

where gab is the spacetime metric. Since hab is a projection tensor it satisﬁes

c a a hab = h(ab),hahcb = hab,ha=3,uhab =0. (2-2)

We employ the projection tensor to deﬁne a spatial covariant derivative (3)∇a which b...c a acting on a tensor T d...e returns a tensor which is orthogonal to u on every index:

(3) a b...c a b c t u p r...s ∇ T d...e ≡ hphr ...hshd ...he∇ T t...u, (2-3)

where ∇a denotes the usual covariant derivative. If the velocity ﬁeld ua has vanishing vorticity (see later) (3)∇a reduces to the covariant derivative in the hypersurfaces orthogonal to ua. The covariant derivative of the velocity decomposes as

1 ∇aub = $ab + σab + 3 θhab + uawb, (2-4)

b a a where wa ≡ u ∇bua is the acceleration, which satisfies u wa = 0, the scalar θ ≡∇ ua =3H is the volume expansion rate (H is the local Hubble parameter), $ab ≡∇[aub]+w[aub] is the a (3) vorticity tensor, which satisfies $ab = $[ab] and u $ab =0,andσab ≡ ∇(aub) −θhab/3isthe a a shear tensor which satisfies σab = σ(ab), σa =0anduσab = 0. The non-trivial integrability condition

(3) (3) ˙ ∇[a ∇b]φ = −$abφ, (2-5)

a for any scalar ﬁeld φ, where an overdot denotes the action of the operator u ∇a, follows from the deﬁnition of the vorticity. Note in particular that in an evolving universe (φ˙ = 0), spatial –6– gradients are necessarily non-vanishing in the presence of vorticity. This behaviour, which is a consequence of there being no global hypersurfaces which are everywhere orthogonal to ua if the vorticity does not vanish, is central to the discussion of vector perturbations. For vanishing vorticity, the 3-Ricci scalar (or intrinsic-curvature scalar) (3)R in the hypersurfaces orthogonal to ua evaluates to

(3) 2 2 ab R =2κρ − 3 θ + σabσ , (2-6)

where ρ is the total energy density in the ua frame. In an exact FRW universe the vorticity, shear and acceleration vanish identically. We regard them as ﬁrst-order variables (denoted O(1)) in an almost FRW universe, so that products of such variables may be dropped from any expression in the linearised calculation that we consider here. Other ﬁrst-order variables may be obtained by taking the spatial gradient of scalar quantities. Such quantities are gauge-invariant by construction since they vanish identically in an exact FRW universe. We shall make use of the comoving fractional spatial gradient of the density ρ(i) of a species i, S X (i) ≡ (3)∇ ρ(i), (2-7) a ρ(i) a and the comoving spatial gradient of the expansion

(3) Za ≡ S ∇aθ. (2-8)

The scalar S is a local scale factor satisfying

a (3) a S˙ ≡ u ∇aS = HS, ∇ S = O(1), (2-9)

which removes the eﬀects of the expansion from the spatial gradients deﬁned above. The (i) vector Xa is a manifestly covariant and gauge-invariant characterisation of the density inhomogeneity.

a The matter stress-energy tensor Tab decomposes with respect to u as

Tab ≡ ρuaub +2u(aqb) −phab + πab, (2-10)

a b b c where ρ ≡Tabu u is the density of matter (measured by a comoving observer), qa ≡ haTbcu a ab is the energy (or heat) flux and is orthogonal to u , p ≡−habT /3 is the isotropic pressure, c d and the symmetric traceless tensor πab ≡ hahb Tcd + phab is the anisotropic stress, which is a also orthogonal to u . In an exact FRW universe, isotropy restricts Tab to perfect-fluid form, –7– so that in an almost FRW universe the heat flux and isotropic stress may be treated as first- order variables. The final first-order gauge-invariant variables we require derive from the

Weyl tensor Wabcd, which vanishes in an exact FRW universe due to isotropy. The electric and magnetic parts of the Weyl tensor, denoted by Eab and Bab respectively, are symmetric traceless tensors, orthogonal to ua, which we deﬁne by

c d Eab ≡ u u Wacbd (2-11) 1 c d ef Bab ≡−2uuηac Wefbd, (2-12) √ where ηabcd is the covariant permutation tensor with η0123 = − −g.

2.1. Linearised Perturbation Equations for the Total Matter Variables

Exact equations describing the propagation of the total matter variables (such as the total density ρ), the kinematic variables, and the electric and magnetic parts of the Weyl tensor, and the constraints between them, follow from the Ricci identity

c c d (∇a∇b −∇b∇a)u =−Rabd u , (2-13) and the Bianchi identity

b 1 −∇ Wabcd −∇[cRd]a − 6ga[c∇d]R=0. (2-14)

The Riemann tensor is expressed in terms of Eab, Bab and the Ricci tensor, Rab, and the Einstein equation is used to substitute for the Ricci tensor in terms of the matter stress- energy tensor. On linearising the equations that result from this procedure (Bruni, Dunsby, & Ellis 1992), one obtains ﬁve constraint equations:

(3) c (3) c d e Bab + ∇ $d(a + ∇ σd(a ηb)ce u =0 (2-15) (3) b 1 cd b b(3) c d ∇ Bab − 2 κ (ρ + p)ηab u $cd + ηabcdu ∇ q =0 (2-16) (3) b h 1 (3) (3) b i ∇ Eab − 6 κ 2 ∇aρ +2θqa +3 ∇ πab =0 (2-17) (3) b (3) b 2 (3) ∇ $ab + ∇ σab − 3 ∇aθ − κqa =0 (2-18) (3) c d ab ∇ ηabcdu $ =0, (2-19) –8– and seven propagation equations:

(3) c d e 1 E˙ab + θEab + ∇ Bd(aηb)ce u + κ 3(ρ + p)σab 6 (2-20) (3) 1 (3) hc +3 ∇(aqb) − 3 hab ∇ qc − 3˙πab − θπab =0 ˙ (3) c 1 (3) c d ie Bab + θBab − ∇ Ed(a + 2 κ ∇ πd(a ηb)ce u =0 (2-21) 2 (3) 1 (3) c 1 σ˙ab + 3 θσab − ∇(awb) − 3 hab ∇ wc + Eab + 2 κπab =0 (2-22) (3) 2 $˙ab − ∇[awb] + 3 θ$ab =0 (2-23) 4 (3) b (3) q˙a+3θqa +(ρ+p)wa + ∇ πab − ∇ap =0 (2-24) ˙ 1 2 (3) a 1 θ+3θ − ∇ wa + 2 κ(ρ +3p) = 0 (2-25) (3) a ρ˙ + θ(ρ + p)+ ∇ qa =0, (2-26) ˙ d where Tab...c ≡ u ∇dTab...c. The constraint equations do not involve time derivatives, and so they serve to constrain initial data for the problem. The propagation equations are consistent with the constraint equations in the sense that the constraints are preserved in time by the propagation equations if they are satisﬁed initially. The consistency of the exact equations follows from their derivation from the exact ﬁeld equations, and is preserved by the linearisation procedure. Including a cosmological constant Λ in the above equations is straightforward; one adds a contribution Λ/κ to the total density ρ, and subtracts the same term from the total pressure p. There is some redundancy in the full set of linear equations. For example, equa-

tion (2-15), which determines Bab in terms of the vorticity and the shear, along with equation (2-18) and the integrability condition given as equation (2-5) imply equation (2-16). Similarly, equation (2-21) follows from equation (2-15) and the propagation equations for

the shear (eq. [2-22]) and the vorticity (eq. [2-23]). It follows that Bab may be eliminated from the linearised equations in favour of the vorticity and the shear by making use of equation (2-15). This elimination is useful when discussing the propagation of vector and tensor modes. The usual Friedmann equations describing homogeneous and isotropic cosmological models are readily obtained from the full set of covariant equations, since in an exact FRW universe the only non-trivial propagation equations are the Raychaudhuri equation (eq. [2-25]) and the energy conservation equation (eq. [2-26]), which reduce to the Fried- mann equation ˙ 2 1 H + H = − 6 κ(ρ +3p), (2-27) and the usual equation for the density evolution ρ˙ = −3H(ρ + p). (2-28) –9–

The second Friedmann equation is obtained as a ﬁrst integral of these two equations:

2 K 1 H + S2 = 3 κρ, (2-29) where 6K/S2 is the intrinsic curvature scalar of the surfaces of constant cosmic time.

The fractional comoving spatial gradient of the density, Xa, and the comoving spatial gradient of the expansion rate, Za, are the key variables in the covariant discussion of the growth of inhomogeneity in the universe (Ellis et al. 1989). It is useful to have available the

propagation equations for these variables. For Xa, we take the spatial gradient of the density evolution equation (eq. [2-26]) and commute the space and time derivatives, to obtain

(3) (3) b (3) ρX˙a +(ρ+p)(Za −Sθwa)+S ∇a ∇ qb + Sθ ∇ap − θpXa =0. (2-30)

For Za , we take the spatial gradient of the Raychaudhuri equation (eq. [2-25]) which gives

˙ 2 1 2 1 1 (3) (3) (3) (3) b Za + 3 θZa − S 3 θ + 2 κ(ρ +3p) wa + 2κS ∇aρ +3 ∇ap − S ∇a ∇ wb =0, (2-31)

For an ideal ﬂuid (qa = πab =0whenwechooseua to be the ﬂuid velocity), with a barotropic equation of state p = p(ρ), the propagation equations for Xa and Za combine with the momentum conservation equation (eq. [2-19]) and the integrability condition, given as equa-

tion (2-5), to give an inhomogeneous second-order equation for Xa (Ellis, Bruni, & Hwang 1990). For a simple equation of state p =(γ−1)ρ,whereγis a constant, the second-order equation is

¨ 5 ˙ 1 1 2 3K Xa + 3 − γ θXa + 2 (γ − 2)(3γ − 2) 3 θ + S2 Xa (3) 2 2K (3) b +(γ−1) ∇ Xa + S2 Xa +2γ(γ−1)Sθ ∇ $ab =0. (2-32) From this equation, it is straightforward to recover the usual results for the growth of inhomogeneities in an almost FRW universe (Ellis & Bruni 1989). The inhomogeneous term describes the coupling between the vorticity and the spatial gradient of the density, which arises from the lack of global hypersurfaces orthogonal to ua in the presence of non-vanishing vorticity. In reality, the universe cannot be described by a barotropic perfect ﬂuid. A more careful analysis of the individual matter components is required, which we present in the next section.

3. Equations for Individual Matter Components

In this paper we concentrate on CDM models, so the matter components that we must consider are the photons and neutrinos, which are the only relativistic species, and the –10–

tightly-coupled baryon/electron system and the CDM, which are both non-relativistic over the epoch of interest. We consider the description of each of these components separately in this section.

3.1. Photons

In relativistic kinetic theory (see, for example, Misner, Thorne, & Wheeler (1973)), the photons are described by a scalar valued distribution function f (γ)(x, p). An observer sees f (γ)(x, p)d3xd3p photons at the spacetime point x in a proper volume d3x, with covariant momentum pa in a proper volume d3p of momentum space. The photon momentum pa decomposes with respect to the velocity ua as

pa = E(ua + ea), (3-1)

a where E = p ua is the energy of the photon, as measured by an observer moving with velocity a a u ,andea is a unit spacelike vector which is orthogonal to u :

a a e ea = −1,eua=0, (3-2)

which describes the propagation direction of the photon in the instantaneous rest space of the observer. With this decomposition of the momentum, we may write the photon distribution function in the form f (γ)(E,e) when convenient, where the dependence on spacetime position (γ) x has been left implicit. The stress-energy tensor Tab for the photons may then be written as

(γ) (γ) Tab = dEdΩ Ef (E,e)papb, (3-3) Z where the measure dΩ denotes an integral over solid angles. The photon energy density ρ(γ), (γ) (γ) the heat ﬂux qa , and the anisotropic stress πab are given by integrals of the three lowest moments of the photon distribution function:

ρ(γ) = dEdΩ E3f (γ)(E,e)(3-4) Z (γ)3 (γ) qa=dEdΩ E f (E,e)ea (3-5) Z (γ) 3 (γ) 1 (γ) πab = dEdΩ E f (E,e)eaeb + 3ρ hab. (3-6) Z In the absence of scattering, the photon distribution is conserved in phase space. De- noting the photon position by xa(λ) and the momentum by pa(λ), the path in phase space –11–

is described by the equations dxa = pa (3-7) dλ a b p ∇ap =0, (3-8) where λ is an aﬃne parameter along the null geodesic xa(λ). Denoting the Liouville operator by L,wehave d Lf(γ)(x, p)= f (γ)(xa(λ),pa(λ)) = 0, (3-9) dλ in the absence of collisions. Over the epoch of interest here, the photons are not collisionless, but instead are interacting with a thermal distribution of electrons and baryons. The dominant contribution to the scattering comes from Compton scattering oﬀ free electrons,

which have number density ne in the baryon/electron rest frame. Since the average energy of a CMB photon is small compared to the electron mass well after electron-positron annihi- lation, we may approximate the Compton scattering by Thomson scattering. Furthermore, since the kinetic temperature of the electrons (which equals the radiation temperature prior to recombination) is small compared to the electron mass, the electrons are non-relativistic and we may ignore the effects of thermal motion of the electrons (in the average rest frame of the baryon/electron system) on the scattering. Our final assumption is to ignore polarisation of the radiation. Thomson scattering of an unpolarised but anisotropic distribution of radiation leads to the generation of polarisation, which then affects the temperature anisotropy

because of the polarisation dependence of the Thomson cross section σT . In this manner, polarisation of the CMB is generated through recombination and its neglect leads to errors of a few percent (Hu et al. 1995) in the predicted temperature anisotropy. We hope to develop a covariant version of the radiative transfer equations including polarisation in the near future, which should simplify the physical interpretation of the transfer equations. In the presence of scattering, the photon distribution function evolves according to the collisional Boltzmann equation,

Lf (γ)(x, p)=C, (3-10) where the collision operator for Thomson scattering is

a (b) (γ) (γ) C = neσT p ua f+ (x, p) − f (x, p) , (3-11) (b) (γ) where ua is the covariant velocity of the baryon/electron system and f+ (x, p) describes scattering into the phase space element under consideration:

2 (γ) 3 (γ) 0 ab (b) 0(b) f (x, p)= f (x, p ) 1+ g e e dΩ 0(b) , (3-12) + 16π a b e Z h i –12–

(b) (b) where ea is the photon direction relative to ua

(b) (b) (b) (b) a(b) pa = E ua + ea ,E=pua, (3-13) 0(b) (b) 0 and ea is the initial direction (relative to ua ) of the photon whose initial momentum is pa and ﬁnal momentum is pa. We write the baryon velocity in the form

(b) (b) ua = ua + va , (3-14)

(b) a where va is the first-order, covariant and gauge-invariant relative velocity between u and a (b) the baryon frame, which satisfies u va = O(2). Using equations (3-1) and (3-13), we find

2 2 ab (b) 0(b) a (b) a 0(b) a 0(b) cd (b) 0(b) g ea eb =(1−2e va ) e ea −2e ea g vc ed + O(2) (3-15) 0(b) a (b) E = E 1 − e va + O(2), (3-16) 0(b) 0a (b) (b) wherewehaveusedthefactthatE ≡p ua is equal to E , since there is no energy (γ) transfer from Thomson scattering in the rest frame of the electron. The quantity f+ (x, p)− f (γ)(x, p) is first-order since it vanishes in an exact FRW universe. It follows that to first- a (b) order, we may replace p ua by E in equation (3-11). Multiplying the Boltzmann equation by E 2 and integrating over energies, we find

2 (γ) a (b) 0(b) 0(b)3 (γ) 3 (γ) dE E Lf (E,e)=neσT 1−4e va dE E f+ (x, p) − neσT dE E f (E,e), Z Z Z (3-17)

(γ) where we have made use of equation (3-16). The integral involving f+ (x, p) may be simpli- ﬁed by using equation (3-16) to obtain

16π 0(b) 0(b)3 (γ) 4 (γ) (γ) a b dE E f+ (x, p)= ρ +πab e e , (3-18) 3 Z 3 which is correct to ﬁrst-order. Using this result, the Boltzmann equation reduces to 3 4 dE E2Lf (γ)(E,e)= n σ 1 − eav(b) ρ(γ) + π(γ)eaeb − n σ dE E3f (γ)(E,e). 16π e T 3 a ab e T Z Z (3-19)

This covariant form of the Boltzmann equation is used in Challinor & Lasenby (1998) to discuss CMB anisotropies from scalar perturbations on angular scales above the damping scale. Note that the equation is fully covariant with all variables observable in the real universe, is valid for arbitrary type of perturbation (scalar, vector and tensor), employs no harmonic decomposition and is valid for any background FRW model. The numerical solution of the Boltzmann equation (eq. [3-19]) is greatly facilitated by decomposing the equation into covariantly-deﬁned angular moments. The majority of recent –13–

calculations (for example, Seljak & Zaldarriaga (1996)) perform an angular decomposition of the Boltzmann equation after specifying the perturbation type and performing the appropriate harmonic expansions. The procedure is straightforward for scalar perturbations in a K = 0 universe, where the Fourier mode of the perturbation in the distribution function may be assumed to be axisymmetric about the wavevector k (this assumption is consistent with the evolution implied by the Boltzmann equation), allowing an angular expansion in Legendre polynomials alone. However, for tensor perturbations the situation is not so straightforward (see, for example, Kosowsky (1996)), since the Boltzmann equation does not then support axisymmetric modes. Instead, the necessary azimuthal dependence of the Fourier components of the perturbation in the distribution function, which is diﬀerent for the two polarisations of the tensor modes, must be put in by hand prior to a Legendre expansion in the polar angle. This procedure may be eliminated by performing a covariant angular expansion of f (γ)(x, p) prior to specifying the perturbation type or background FRW model. The covariant (tensor) moment equations that result may then be solved for any type of perturbation (and any background curvature K) by expanding in covariant tensors derived from the appropriate harmonic functions (see Section 5 for the case of scalar perturbations). This procedure automatically takes care of the required angular dependencies of the harmonic components of the distribution function, allowing a streamlined and uniﬁed treatment of all perturbation types in background FRW models with arbitrary spatial curvature. The covariant angular expansion of the photon distribution function takes the form (El- lis, Matravers, & Treciokas 1983)

∞ f (γ)(E,e)= F (l) ea1 ea2 ...eal, (3-20) a1...al Xl=0 where the tensors F (l) have an implicit dependence on spacetime position x and energy E a1...al and are totally symmetric, traceless and orthogonal to ua:

F (l) = F (l) ,ga1a2F(l) =0,ua1F(l) =0. (3-21) a1...al (a1...al) a1a2...al a1...al

Employing the expansion given in equation (3-20), the action of the Liouville operator on the f (γ)(E,e) reduces to

∞ f (γ)(E,e)= ∂ F (l) ea1 ∂ E + pb F (l) ea1 + lF (l) pb ea1 ea2 ...eal. L E a1...al λ ∇b a1...al a1...al ∇b Xl=0 h i (3-22)

Using the geodesic equation, we ﬁnd that

c b b c b b b habp ∇ce = −E σbce e ea + σabe + w eaeb + wa − $abe , (3-23) –14–

which is first-order. In an exact FRW universe, isotropy restricts F (l) = 0 for l>0, so a1...al that in an almost FRW universe F (l) = (1) for l not equal to zero. It follows that the a1...al O last term in equation (3-22) makes only a second-order contribution and may be dropped in the linear calculation considered here. Inserting the expansion given in equation (3-20) into the Boltzmann equation (eq. [3-19]) and performing a covariant angular expansion of the resulting equation gives a set of moment equations which are equivalent to the original Boltzmann equation. The linearised calculation is straightforward, although a little care is needed for the first three moments since F (0) is a zero-order quantity. (The exact expansion of the left-hand side of the Boltzmann equation, equation (3-22), is given in Ellis et al. (1983).) For l = 0, 1 and 2, we find

(γ) 4 (γ) (3) a (γ) ρ˙ + 3 θρ + ∇ qa =0 (3-24) (γ) 4 (γ) (3) b (γ) 4 (γ) 1 (3) (γ) 4 (γ) (b) (γ) q˙a +3θqa + ∇ πab + 3 ρ wa − 3 ∇aρ = neσT 3 ρ va − qa (3-25) (γ) 4 (γ) (3) c (3) 2 (3) (γ) 1 (3) c (γ) 8 (γ) 9 (γ) π˙ ab + 3 θπab + ∇ Jabc − 5 ∇(aqb) − 3 hab ∇ qc − 15 ρ σab = − 10 neσT πab , (3-26)

and, for l ≥ 3,

J˙(l) + 4 θJ(l) + (3) bJ (l+1) l (3) J (l−1) a1...al 3 a1...al ∇ ba1...al − (2l+1) ∇(a1 a2...al) (l−1) (3) b (l−1) (l) − ∇ J ha a ) = −neσT J . (3-27) (2l−1) b(a1...al−2 l−1 l a1...al The tensors J (l) , which are traceless, totally symmetric and orthogonal to ua, are derived a1...al from the F (l) by integrating over energy: a1...al

4π(−2)l(l!)2 ∞ J (l) dE E3F (l) . (3-28) a1...al ≡ a1...al (2l + 1)(2l)! Z0 The constant factor is chosen to simplify algebraic factors in the moment equations. Using equations (3-4–3-6), the lowest three moments relate simply to the energy density, heat ﬂux and anisotropic stress:

(γ) (0) (γ) (1) (γ) (2) ρ = J ,qa =Ja ,πab = Jab . (3-29)

It is straightforward to show that the tensor

(3) (l−1) (l−1) (3) b (l−1) ∇ a J − ∇ J ha a , (3-30) ( 1 a2...al) (2l−1) b(a1...al−2 l−1 l) which appears in equation (3-27) is traceless, symmetric, and orthogonal to ua, as required. –15–

It will be observed that for l ≥ 3, the moment equations link the l − 1, l and l +1 angular moments of the (integrated) distribution function, while the l = 2 equation also involves the density ρ(γ) which is the l = 0 moment. The exact moment equations which arise from expanding the Liouville equation in covariant harmonics also couple the l +2 and l 2 moments to J (l) (Ellis et al. 1983), but these terms are second-order for l 3 − a1...al ≥ and so do not appear in the linearised equations presented here. In the exact expansion of the Liouville equation, the coefficient of the l + 2 angular moment in the exact propagation equation for J (l) is the shear σ , which leads to the result that the angular expansion of a1...al ab the distribution function for non-interacting radiation can only truncate (J (l) = 0 for all a1...al l greater than some L) if the shear vanishes (Ellis 1996). This exact result, which is lost in linearised theory which permits truncated distribution functions with non-vanishing shear, is an example of a linearisation instability (see Ellis & Dunsby (1997) for more examples). However, this is not problematic for the linearised calculation of CMB anisotropies since it is never claimed that the higher-order moments of the photon distribution vanish exactly. Instead, the series is truncated (with suitable care to avoid reflection of power back down the series) for numerical convenience. The truncation is performed with L large enough so that there is no significant effect on the J (l) for the range of l of interest. a1...al Finally, by taking the spatial gradient of equation (3-24), and commuting the space and time derivatives, we find the propagation equation for the comoving fractional spatial (γ) gradient of the photon density, Xa ,

˙ (γ) 4 S (3) a(3) b (γ) 4 Xa + 3 Za + ρ(γ) ∇ ∇ qb − 3 Sθwa =0, (3-31) where Za is the comoving spatial gradient of the volume expansion.

3.2. Neutrinos

We consider only massless neutrinos, and these are non-interacting over the epoch of interest. It follows that their distribution function f (ν)(x, p) satisﬁes the Liouville equation Lf (ν)(x, p) = 0. Expanding the neutrino distribution function in covariant angular harmonics, we arrive at the moment equations for the tensors G(l) , which are deﬁned in the same a1...al manner as the J (l) , but with the photon distribution function replaced by the neutrino a1...al distribution. These moment equations are the same as the photon equations, but with the –16–

scattering terms omitted:

(ν) 4 (ν) (3) a (ν) ρ˙ + 3 θρ + ∇ qa =0 (3-32) (ν) 4 (ν) (3) b (ν) 4 (ν) 1 (3) a (ν) q˙a +3θqa + ∇ πab + 3 ρ − 3 ∇ ρ =0 (3-33) (ν) 4 (ν) (3) c (3) 2 (3) (ν) 1 (3) c (ν) 8 (ν) π˙ab + 3 θπab + ∇ Gabc − 5 ∇(aqb) − 3 hab ∇ qc − 15 ρ σab =0, (3-34) and for l ≥ 3,

G˙ (l) + 4 θG(l) + (3) bG(l+1) l (3) G(l−1) a1...al 3 a1...al ∇ ba1...al − (2l+1) ∇(a1 a2...al) (l−1) (3) b (l−1) − ∇ G ha a =0. (3-35) (2l−1) b(a1...al−2 l−1 l) The propagation equation for the comoving fractional spatial gradient of the neutrino density, (ν) Xa , follows from equation (3-32):

˙ (ν) 4 S (3) a(3) b (ν) 4 Xa + 3 Za + ρ(ν) ∇ ∇ qb − 3 Sθwa =0. (3-36)

3.3. Baryons

Over the epoch of interest here, the electrons and baryons are non-relativistic, and may be approximated by a tightly-coupled ideal fluid (the coupling arising from Coulomb scattering). The energy density of the fluid is ρ(b), which includes contributions from both the baryonic species and the electrons, the fluid pressure is p(b), and the velocity of the fluid (b) (b) (b) a (b) is ua = ua + va , where the first-order relative velocity va satisfies u va = O(2). The linearised baryon stress-energy tensor evaluates to

(b) (b) (b) (b) (b) (b) Tab = ρ uaub − p hab +2(ρ +p )u(avb) , (3-37)

(b) (b) (b) which shows that there is a heat ﬂux (ρ + p )va due to the baryon motion relative to the a (b) (b) u frame. The equations of motion for ρ and va follow from the conservation of baryon plus photon stress-energy (the baryons and photons interact through non-gravitational eﬀects only with themselves):

(3) a (b) (3) a (γ) ∇ Tab + ∇ Tab =0. (3-38)

Using the l =0andl= 1 moment equations for the photon distribution, we ﬁnd the propagation equation for the baryon energy density:

(b) (b) (b) (b) (b) (3) a (b) ρ˙ +(ρ +p )θ+(ρ +p ) ∇ va =0, (3-39) –17–

(b) and a propagation equation for va :

(b) (b) (b) 1 (b) (b) (b) (b) (b) (3) (b) (ρ + p )(v ˙a + wa)+ 3(ρ +p )θva +˙p va − ∇ap 4 (γ) (b) (γ) + neσT 3 ρ va − qa =0, (3-40) which must be supplemented by an equation of state linking p(b) and ρ(b). The final term in equation (3-40) describes the exchange of momentum between the radiation and the baryon/electron fluid as a result of Thomson scattering. There is no such term in equation (3-39) since both the radiation drag force and the baryon velocity relative to the ua frame are first-order, which give only a second-order rate of energy transfer in the ua frame. Energy transfer due to thermal motion of the electrons in the baryon rest frame has negli- (b) (b) (b) gible effect on ρ since the electrons are non-relativistic; kBT me,whereT is the baryon kinetic temperature (assumed equal to the electron kinetic temperature), and me is the electron mass.

(b) Taking the spatial gradient of equation (3-39) gives the propagation equation for Xa , the fractional comoving spatial gradient of the baryon energy density:

(b) ˙ (b) (b) (b) (3) (3) b (b) (3) (b) (b) (b) ρ Xa + ρ + p Za +S ∇a ∇ vb − Sθwa + Sθ ∇ap − θp Xa =0. (3-41)

We have retained all terms involving the baryon pressure p(b) in the equations of this section. In practice, over epochs where the baryons are non-relativistic (p(b) ρ(b)), the only pressure term that need be retained is the term (3)∇ap(b) which appears in equation (3-40). This term appears as a small correction to the total sound speed in the tightly-coupled baryon/photon plasma, and is potentially signiﬁcant during the acoustic oscillations in the plasma.

3.4. Cold Dark Matter

We will only consider cold dark matter (CDM) here, which may be described as a pressureless ideal ﬂuid. Hot dark matter (HDM) would require a phase space description, which is more complicated than for photons or neutrinos since the energy dependence cannot be integrated out of the Boltzmann equation for massive particles. Both CDM and HDM are considered in Ma & Bertshinger (1995), where the calculations for scalar perturbations are performed in the synchronous and conformal Newtonian gauges. The CDM has energy (c) (c) (c) density ρ in its rest frame, which has velocity ua = ua + va , with the ﬁrst-order relative (c) a (c) velocity va satisfying u va = O(2). The linearised CDM stress-energy tensor evaluates to

which is conserved since the CDM interacts with other species only through gravity. The (c) (c) (c) conservation of Tab gives the propagation equations for ρ and va :

(c) Since the CDM moves on geodesics, the velocity ua provides a convenient frame choice. With this choice, the acceleration wa vanishes. We use the CDM frame to define the fundamental velocity ua in Section 5, where we discuss scalar perturbations in the CDM model. For the moment, however, we continue to leave the choice of frame unspecified for generality. The final equation that we require is the propagation equation for the fractional comoving (c) spatial gradient of the density, Xa . This follows from equation (3-43) on taking the spatial gradient:

˙ (c) (3) (3) b (c) Xa + Za + S ∇a ∇ vb − Sθwa =0. (3-45)

The equations for the matter components that we have described in this section combine with the covariant equations of Section 2 to give a complete description of the evolution of inhomogeneity and anisotropy in a fully covariant and gauge-invariant manner. The equations given in Section 2 make use of the total energy density and pressure, heat ﬂux and anisotropic stress. These quantities are related to the individual matter components in the CDM model by

ρ = ρ(γ) + ρ(ν) + ρ(b) + ρ(c) (3-46) 1 (γ) 1 (ν) (b) p = 3 ρ + 3 ρ + p (3-47) (γ) (ν) (b) (b) (b) (c) (c) qa = qa + qa + ρ + p va + ρ va (3-48) (γ) (ν) πab = πab + πab . (3-49)

The equations given here are both covariant and gauge-invariant. Employing gauge- invariant variables ensures that the problem of gauge-mode solutions does not arise, and that all quantities are independent of the choice of map between the real universe and a background FRW model. We have only considered the linearised equations here, but the linearisation procedure is not fundamental to the covariant and gauge-invariant approach. It is straightforward to derive the exact, non-linear equations for the total ﬂuid variables (Bruni et al. 1992), and, in principle, the exact collision term could be employed in the Boltzmann equation, allowing an exact covariant angular decomposition. Unlike in Bardeen’s gauge- invariant approach (Bardeen 1980), the deﬁnition of the variables employed here does not require that the perturbations be in the linear regime, and furthermore, the variables do –19–

not depend on the non-local decomposition of the perturbations into scalar, vector and tensor type and the associated harmonic analysis. The covariant approach describes scalar, vector and tensor modes in a unified manner, although decomposing the linear perturbations is useful to aid solution of the linearised equations late on in the calculation. A further advantage of the covariant and gauge-invariant approach over that introduced by Bardeen, is that only covariantly defined variables are employed, which are simple to interpret physically. In contrast, the Bardeen variables are constructed by taking linear combinations of (gauge- dependent) metric and matter perturbations in such a way that the resulting variable is gauge-invariant (for small gauge-transformations which preserve the scalar, vector or tensor structure of the metric perturbation). These variables have simple physical interpretations only for certain specific gauge choices. Finally, note that we have not yet had to specify whether the background FRW model is open, flat or closed. However, we have made the implicit assumption that the universe is almost FRW when specifying the zero and first-order variables in the linearisation procedure.

4. The CMB Temperature Anisotropy

The energy-integrated moments J (l) of the photon distribution function provide a a1...al fully covariant description of the CMB temperature anisotropy. In the ua frame, denote the

average bolometric temperature on the sky at the point x by T0(x), so that

4 1 3 (γ) 1 (0) T0 ∝ 4π dEdΩ E f (x, p)= 4πJ , (4-1) Z

which is just the Stefan-Boltzmann law. We use the fractional temperature variation δT (e) from the full-sky average T0 to characterise the temperature perturbation along the spatial direction e in a gauge-invariant and covariant manner (Maartens, Ellis, & Stoeger 1995; Dunsby 1997). It follows that

4 4π 1+δ (e) = dE E3f (γ)(x, p), (4-2) T J(0) Z so that to ﬁrst-order 1 ∞ (2l + 1)(2l)! (l) a1 al δT (e)= J e ...e . (4-3) 4ρ(γ) (−2)l(l!)2 a1...al Xl=1 The right-hand side of equation (4-3) is the covariant angular expansion of the temperature anisotropy. The tensors J (l) thus provide a natural covariant description of the CMB a1...al anisotropy. They may be related to the more familiar alm components in the spherical –20–

harmonic expansion of δT (e) by introducing an orthogonal triad in the instantaneous rest space at x,sothatea =(sinθcosφ, sinθsinφ, cosθ). Then the two expansions are related by

l (2l + 1)(2l)! (γ) (l) a1 al 4ρ almYlm(θ, φ)= J e ...e . (4-4) (−2)l(l!)2 a1...al mX=−l Squaring this expression and integrating over solid angles, we ﬁnd the important rotational invariant 1 l 4π (2l)! 2 a1b1 albl (l) (l) |alm| = h ...h J J . (4-5) 2l +1 (4ρ(γ))2 (−2)l(l!)2 a1...al b1...bl mX=−l

The quantity on the left is a quadratic estimator for the CMB power spectrum (the Cl), which we see is related to the covariant tensors J (l) in a very simple manner. a1...al

5. Scalar Perturbations

Up to this point, we have treated the scalar, vector and tensor modes of linear theory in a unified manner. However, to obtain solutions to the covariant equations it proves useful to consider scalar, vector and tensor modes separately. In this paper we shall consider only scalar modes; the extension to tensor modes is straightforward and will be dealt with in a future paper (vector modes decay in an expanding universe in the absence of defects, and so are not likely to have a significant effect on the CMB in inflationary models). In the covariant approach to cosmology, we characterise scalar perturbations by demanding that the magnetic part of the Weyl tensor and the vorticity be at most second-order. Setting

Bab = O(2) ensures that gravitational waves are excluded to first-order, and demanding that a $ab = O(2) ensures that the density gradients seen by an observer in the u frame arise from clumping of the density, (3)∇2ρ = O(1), and not from kinematic effects due to vorticity (the absence of flow-orthogonal hypersurfaces), which give (3)∇2ρ = O(2)inanalmostFRW universe. Note that we do not classify scalar perturbations as having Bab = $ab = 0 (to all orders), which is only a highly restricted subset of the full set of scalar solutions. For example, in an (exactly) irrotational dust-filled universe (a “silent” universe), it can be shown from

the exact non-linear equations that demanding Bab = 0 forces the solution into a very small class, which probably all have high symmetry (Ellis 1996), and so cannot represent a very

general perturbation. This arises because requiring that Bab = 0 be preserved along the ﬂow lines introduces a series of complex constraints which reduce greatly the size of the solution

set. However, requiring only that Bab and $ab be at most second-order gives a much larger class of solutions because only two new constraints are introduced, and these are necessarily preserved by the propagation equations. The solutions with Bab and $ab vanishing exactly –21–

comprise a very small subset of the larger class of exact solutions which we classify as scalar perturbations.

On setting Bab =0and$ab = 0, (equality to zero in the linearised theory should be taken to imply that the quantity is at most second-order) we see from equation (2-15) that

(3) c d e (3) (3) c ∇ σd(aηb)ce u =0 ⇒ ∇[a ∇ σb]c =0, (5-1)

where the antisymmetrisation is on the indices a and b in the right-hand equation. This

is a necessary condition for σab to be constructed from a scalar potential. It follows from equations (2-18) and (2-23) that

(3) (3) ∇[aqb] =0, ∇[awb] =0, (5-2)

so that the heat ﬂux and acceleration may be written as spatial gradients of scalar ﬁelds (3) (making use of the integrability condition given as equation (2-5)). Consistency of ∇[aqb] = 0 with equation (2-24) for qa then requires that

(3) (3) c (3) (3) c ∇[a ∇ πb]c =0 ⇒ ∇[a ∇ Eb]c =0, (5-3)

with the implication following from equation (2-17). It follows that all vector variables, such (3) b as qa and ∇ Eab, may be derived from scalar potentials. The new constraint, given as equation (5-1), is only consistent with the propagation equations if Eab and πab satisfy

(3) c d e 1 (3) c d e ∇ Ed(aηb)ce u = − 2 κ ∇ πd(aηb)ce u . (5-4)

In the absence of anisotropic stress, we see that the left-hand side of equation (5-4) is

constrained to be zero, which is consistent with the propagation equation for Eab,givenas equation (2-20), with πab = 0. If the anisotropic stress does not vanish, we include the constraint

(3) c d e (3) c d e ∇ Ed(aηb)ce u =0 ⇒ ∇ πd(aηb)ce u =0, (5-5)

in the deﬁnition of a scalar mode, which is easily shown to be consistent with the propagation

equation for Eab. Requiring consistency of equation (5-5) with the propagation equation for πab implied by the photon and neutrino Boltzmann hierarchy yields a series of constraints on the moments J (l) and G(l) , which are necessary conditions for them to be derived a1...al a1...al from scalar potentials. The new constraint equations that we have introduced, by restricting the solution to be a scalar mode, may be satisﬁed by constructing the covariant and gauge-invariant variables from tensors derived from scalar potentials by taking appropriate spatial covariant derivatives –22–

of the scalar functions. It proves convenient to separate the temporal and spatial aspects of the problem by expanding the scalar potentials in the (almost) eigenfunctions Q(k) of the generalised Helmholtz equation (Hawking 1966; Ellis et al. 1989)

(3) 2 (k) (3) a(3) (k) k2 (k) ∇ Q ≡ ∇ ∇aQ = S2 Q + O(1), (5-6)

which are constructed to satisfy

Q˙ (k) = O(1). (5-7)

(In general, we cannot impose eq. [5-6] and Q˙ (k) = 0 consistently, but we can restrict Q˙ (k) to be at most ﬁrst-order.) These equations determine Q(k) to zero-order, which is all that is required for linear theory. The allowed values of the eigenvalues k2/S2 are determined by the scalar curvature of the background model (since Q(k) are only needed to zero-order). In a ﬂat model, K =0,kis a comoving continuous wavenumber ≥ 0. In closed models, K>0, k takes only discrete values with k2 = γ(γ +2)where γ is a non-zero, positive integer. In open models, K<0, k again takes continuous values, but with the restriction k ≥ 1. More details may be found in Harrison (1967). The eigenfunctions Q(k) are labelled by the lumped index k. This index, which determines the eigenvalue k2/S2, should be understood to distinguish implicitly the distinct degenerate eigenfunctions which all have the same eigenvalue k2/S2. This multiple use of the symbol k should not cause any confusion, since the lumped index will always appear as a superscript or subscript. A function of the eigenvalue k will be denoted with the eigenvalue as an argument, for example A(k), to distinguish it from the (k) quantity Ak which depends on the mode label k and not just the eigenvalue. From the Q (k) we form a vector Qa ,

(k) S (3) (k) Qa ≡ k ∇aQ , (5-8)

which is orthogonal to ua and is parallel transported to ﬁrst-order along the ﬂow lines:

a (k) ˙(k) u Qa =0, Qa =O(1). (5-9)

We deﬁne totally symmetric tensors of rank l, Q(k) , by the recursion formula (for l>1) a1...al

(k) S (3) (k) l−1 (3) b (k) Q = ∇(a Q − ∇ Q ha a ) . (5-10) a1...al k 1 a2...al) 2l−1 b(a1...al−2 l−1 l These tensors satisfy the properties

ua1 Q(k) =0,ha1a2Q(k) =0, Q˙(k) = (1), (5-11) a1a2...al a1a2...al a1a2...al O which are readily proved by induction. –23–

The scalar functions Q(k) are the covariant generalisations of the scalar eigenfunctions of the Laplace-Beltrami operator on the homogeneous spatial sections of the background FRW model, which are usually employed in the harmonic decomposition of perturbed quantities (see, for example, Bardeen (1980)). In the covariant approach, attention is focused on a velocity field ua, rather than a spatial slicing of spacetime, so it is natural to employ harmonic functions defined by equation (5-6). Some of the differential properties of the (k) derived tensors Qa1...al (for l ≤ 2) are given in the appendix to Bruni et al. (1992). We add two more results to this list which will be useful later:

(3) a1 Q(k) = l k 1 (l2 1) K Q(k) (5-12) ∇ a1a2...al 2l−1 S − − k2 a2...al (3) 2Q(k) = k2 1 l(l +1)K Q(k) , (5-13) ∇ a1...al S2 − k2 a1...al which may be derived from the recursion relation given as equation (5-10) and the deﬁnition in equation (5-6). The additional constraints introduced by the conditions for a scalar mode are satisﬁed identically if we construct the gauge-invariant variables in the following manner:

(i) (i) (k) k2 (k) Xa = kXk Qa , Za = S ZkQa (5-14) Xk Xk (i) (i) (i) (k) (i) (i) (i) (k) qa = ρ qk Qa ,πab = ρ πk Qab (5-15) Xk Xk k2 (k) k (k) Eab = S2 ΦkQab ,σab = S σkQab (5-16) Xk Xk (i) (i) (k) (k) va = vk Qa ,wa=wkQa, (5-17) Xk Xk where i labels the particle species (and we omit the label when referring to total fluid variables). The symbolic summation in these expressions is a sum over eigenfunctions of equation (5-6). For closed background models, the sum is discrete, but in the flat and open cases the summation should be understood as an integral over the continuous label k,which (i) distinguishes distinct eigenfunctions. The scalar expansion coefficients, such as Xk ,are themselves first-order gauge-invariant variables, which satisfy

(3) a (i) ∇ Xk = O(2). (5-18)

They are labelled by the lumped index k. Finally, we assume that the higher-order angular moments of the photon and neutrino distribution functions may also be expanded in the Q(k) harmonics. That J (l) and G(l) may be derived from scalar potentials appears a1...al a1...al a1...al to be a necessary consequence of the constraints we have imposed in the deﬁning properties of a scalar mode. By considering the zero-order form of the scalar harmonics Q(k),and –24–

derived tensors, it is straightforward to show that this condition is equivalent to the usual assumption that the Fourier components of the distribution functions are axisymmetric about the wavevector k (see, for example, Seljak (1996)). With this condition, we have

J (l) = ρ(γ) J(l)Q(k) ,G(l)= ρ(ν) G(l)Q(k) , (5-19) a1...al k a1...al a1...al k a1...al Xk Xk for photons and neutrinos respectively.

5.1. The Scalar Equations

It is now a simple matter to substitute the harmonic expansions of the covariant variables into the constraint and propagation equations given in Sections 2 and 3, to obtain equations for the scalar expansion coeﬃcients which describe scalar perturbations in a covariant and gauge-invariant manner. To simplify matters, we assume that the variations in baryon pressure p(b) due to entropy variations are negligible compared to those arising from variations in ρ(b),sothatwemaywrite

a (b) 2 a (b) ∇p =cs∇ρ , (5-20)

where cs is the adiabatic sound speed in the baryon/electron fluid (this is different to the total sound speed in the tightly-coupled baryon/photon fluid). With this assumption, we obtain the following equations for scalar perturbations: for the spatial gradients of the densities, we find

˙ (γ) k 4 (γ) 4 S Xk = − S 3 Zk + qk + 3 k θwk (5-21) ˙ (ν) k 4 (ν) 4 S Xk = − S 3 Zk + qk + 3 k θwk (5-22) ˙ (c) k (c) S Xk = − S Zk + vk + k θwk (5-23) ˙ (b) p(b) k (b) S p(b) 2 (b) Xk = 1+ ρ(b) −S(Zk +vk )+ kθwk + ρ(b) − cs θXk , (5-24) h i for the spatial gradient of the expansion, we ﬁnd

˙ 1 1 S (γ) (γ) (ν) (ν) (c) (c) 2 (b) (b) Zk = − 3 θZk − 2 k κ 2 ρ Xk + ρ Xk + ρ Xk +(1+3cs)ρ Xk h S2 1 2 1 i + k2wk 3θ + 2κ(ρ+3p) , (5-25) h i the heat ﬂuxes satisfy

(γ) 2 k (γ) 1 k (γ) 4 4 (b) (γ) q˙k + 3 S πk − 3 S Xk + 3 wk = neσT 3 vk − qk (5-26) (ν) 2 k (ν) 1 k (ν) 4 q˙k + 3 S πk − 3 S Xk + 3 wk =0, (5-27) –25–

and for the baryon and CDM peculiar velocities

(c) 1 (c) v˙k + 3 θvk + wk =0 (5-28) p(b) (b) 1 2 (b) k 2 (b) ρ(γ) 4 (b) (γ) 1+ ρ(b) v˙k + 3(1 − 3cs)θvk + wk − S csXk = −neσT ρ(b) 3vk − qk . (5-29)

The propagation equations for the anisotropic stresses are

(γ) 3 k 8K (3) 2 k (γ) 8 k 9 (γ) π˙ k + 5 S 1 − k2 Jk − 5 S qk − 15 S σk = − 10 neσT πk (5-30) (ν) 3 k 8K (3) 2 k (ν) 8 k π˙ k + 5 S 1 − k2 Gk − 5 S qk − 15 S σk =0, (5-31) and the remaining moment equations, for l ≥ 3, are

˙(l) k l+1 K (l+1) l (l−1) (l) Jk + S 2l+1 1 − l(l +2)k2 Jk − 2l+1 Jk = −neσT Jk (5-32) ˙ (l) k hl+1 K (l+1) l (l−1)i Gk + S 2l+1 1 − l(l +2)k2 Gk − 2l+1 Gk =0. (5-33) h i The propagation equations for Eab and σab become

2 k ˙ 1 1 k 1 1 S (Φk + 3 θΦk)+ 2Sκρ(γσk + qk)+ 6κρθ(3γ − 1)πk − 2 κρπ˙ k =0 (5-34) 2 k 1 k 1 S(˙σk + 3θσk − wk)+ S Φk + 2κρπk =0, (5-35) where γ is deﬁned in terms of the total pressure p and density ρ by p =(γ−1)ρ. Finally, the remaining constraint equations become

3 k 3K k 3K 2 S 1 − k2 Φk − S κρ Xk + 1 − k2 πk − κρθqk =0 (5-36) 2 2 k h 3Ki 3 S Zk − 1− k2 σk +κρqk =0. (5-37) h i The variables Xk, qk and πk refer to the total matter, and are given in terms of the component variables by

(γ) (γ) (ν) (ν) (b) (b) (c) (c) ρXk = ρ Xk + ρ Xk + ρ Xk + ρ Xk (5-38) (γ) (γ) (ν) (ν) (b) (b) (b) (c) (c) ρqk = ρ qk + ρ qk +(ρ +p )vk +ρ vk (5-39) (γ) (γ) (ν) (ν) ρπk = ρ πk + ρ πk . (5-40)

These equations give a complete description of the evolution of homogeneity and anisotropy from scalar perturbations in an almost FRW universe with any spatial curvature. The system closes up once a choice for the velocity ua is made, and it is straightforward to check –26–

that the constraint equations are consistent with the propagation equations. At this point, it is convenient to make a choice of frame. In the CDM model, the rest frame of the CDM deﬁnes a geodesic frame, which provides a convenient choice for ua, since the acceleration then vanishes identically. We assume that this frame choice has been made in the rest of this paper.

(l) (l) The equations for Jk and Gk for l ≥ 3 are equivalent to those usually found in the literature (see, for example, Ma & Bertshinger (1995) and set K = 0). This is because the moments of the perturbed distribution function, used in such gauge-dependent calculations, are gauge-invariant for l ≥ 1. (The l = 1 moment does depend on the choice of coordinates in the real universe, but is independent of the mapping onto the background model, since the background distribution function has no angular dependence.) Gauge-invariant versions of the usual synchronous-gauge equations (Ma & Bertshinger 1995) are obtained by taking a (c) u to coincide with the CDM velocity, so that wa and va vanish.

5.2. Initial Conditions on Super-Horizon Scales

In this subsection, we analytically extract the solution of the scalar perturbation equations in the radiation dominated era. We shall only consider modes with |K|/k2 1sothat we may ignore terms involving K in the scalar equations. Associated with each mode there is a characteristic length scale, S/k. The condition |K|/k2 1isequivalenttorequiring that this length scale be small compared to the curvature radius of the universe. For such modes, k is eﬀectively a comoving wavenumber. We shall also require that the mode be well outside the horizon scale 1/H, so that we consider only those modes satisfying

2 H2S2 1 Hk |K| , (5-41) where Hk ≡ SH/k is the ratio of the characteristic length scale to the horizon scale, and H2S2/|K| is the (squared) ratio of the curvature radius to the horizon scale. If the universe may be approximated by a K = 0 universe to zero-order, equation (5-41) reduces to Hk 1. The approximate analytic solution may be used to provide initial conditions for a numerical integration of the scalar equations (see Section 6). Well before decoupling, the baryons and photons are tightly-coupled because of the high opacity to Thomson scattering. This scattering damps the photon moments for l ≥ 2, but adipole(l= 1) moment can survive if the baryon velocity does not coincide with the CDM (l) (b) (γ) velocity. To a good approximation, we may ignore the Jk for l ≥ 2, and set vk =3qk /4 so that the radiation is isotropic in the rest frame of the baryons. This is the lowest-order term in the tight-coupling approximation (see Section 6.2). Similarly, we expect that the –27–

higher-order neutrino moments will also be small in the early universe, since the neutrinos were in thermal equilibrium prior to their decoupling. Furthermore, the baryon and CDM densities, ρ(b) and ρ(c), are negligible compared to the radiation and neutrino densities, ρ(γ) and ρ(ν), in the radiation dominated era. A useful ﬁrst approximation to the full set of scalar equations is obtained by setting the (l) neutrino moments, Gk , to zero for l ≥ 2. It is convenient to take the dependent variable to −1 be x ≡Hk instead of the proper time t along the ﬂow lines, so that the scalar propagation equations of the previous section reduce to the following set:

2 0 (γ) (ν) x Zk + xZk +3 (1 − R)Xk + RXk =0 (5-42) 2 0 3 (γ) (ν) xΦk +xΦk +2σk + 2 (1 − R)qk + Rqk =0 (5-43) 0 xσk + σk + xΦk =0 (5-44) (γ)0 4 (γ) Xk +3Zk +qk =0 (5-45) (ν)0 4 (ν) Xk +3Zk +qk =0 (5-46) (γ)0 1 (γ) qk −3Xk =0 (5-47) (ν)0 1 (ν) qk −3Xk =0, (5-48) where a prime denotes diﬀerentiation with respect to x, and we have used the zero-order Friedmann equation in the form H2 = κρ/3 since the curvature term may be neglected by equation (5-41). We have followed Ma & Bertshinger (1995) by introducing the dimensionless quantity R deﬁned by

ρ(ν) R ≡ . (5-49) ρ(ν) + ρ(γ) After neutrino decoupling, R is a constant which depends only on the number of neutrino species. The remaining equations that we require are the two scalar constraints which reduce to

3 (γ) (ν) (γ) (ν) 2x Φk − 3x (1 − R)Xk + RXk − 9 (1 − R)qk + Rqk =0 (5-50) 2 (γ) (ν) 2x(Zk −σk)+9 (1 − R)qk + Rqk =0. (5-51)

This set of equations give a closed equation for Φk:

00 0 3xΦk + 12Φk + xΦk =0. (5-52)

This equation should be compared to the fourth-order equation for the metric perturbation variable in the synchronous gauge (see, for example, Ma & Bershinger (1995)). The –28–

fourth-order equation admits four linearly-independent solutions, but two of the solutions are gauge modes that arise from mapping an exact FRW universe to itself. The gauge- invariant approach adopted here ensures that such gauge modes do not arise. This is evident from equation (5-52) which is only a second-order equation. The two linearly-independent solutions of this equation both describe physical perturbations in the Weyl tensor, which vanishes for an exact FRW universe. It is now straightforward to ﬁnd the general solution of

equations (5-42)–(5-51). There are two solutions with non-vanishing Weyl tensor (Φk 6=0), which we write as

−3 Φk = −3y (Cy − D)cosy−(C+Dy)siny (5-53) √ −3 2 Zk =3 3y 2(C + Dy)cosy+2(Cy − D)siny−C(2 + y ) (5-54) √ −2 σk =3 3y Dcosy + C siny − Cy (5-55) √ (γ) −1 qk = −4 3y C cosy − D siny − C (5-56) √ (ν) 2 3 −1 qk = − R y (2RC + Dy)cosy+(Cy − 2RD)siny−2RC (5-57) (γ) −2 Xk =12y (C+Dy)cosy+(Cy − D)siny−C (5-58) (ν) 6 −2 2 2 Xk = R y (2RC − Cy +2RDy)cosy+(2RCy − 2RD + Dy )siny−2RC , (5-59) √ where y ≡ x/ 3, and C and D are constants. There are also three solutions with vanishing Weyl tensor (Φk = 0), which we write as √ 3 −3 2 Zk = 4 A3y 2+y (5-60) √ 3 −1 σk = 4 A3y (5-61) (γ) q = √1 A cosy A siny + A y−1 (5-62) k − 3 1 − 2 3 (ν) q = √R−1 ( A cosy + A siny) √1 A y−1 (5-63) k 3R − 1 2 − 3 3 (γ) −2 Xk = A1 siny + A2 cosy + A3y (5-64) (ν) R−1 −2 Xk = R (A1 siny + A2 cosy)+A3y , (5-65) where A1, A2 and A3 are further constants. The solution with only A3 non-zero describes a radiation dominated universe which is exactly FRW except that the CDM has a peculiar (c) (0) (0) velocity (relative to the velocity of the fundamental observers) va = va /S,whereva is a first-order vector, orthogonal to the fundamental velocity, which is parallel transported along (0) the fundamental flow lines:v ˙a = O(2). This can be seen most clearly by adopting the energy frame, defined by the condition qa = 0. This is arguably a better choice to make in the early universe, since ua is then defined in terms of the dominant matter components, rather than a –29–

minority component, such as the CDM, which has little eﬀect on the gravitational dynamics. Choosing the energy frame, and ignoring anisotropic stresses (which are frame-independent in linear theory), the CDM relative velocity evolves according to

(c) 1 (c) 1 (γ) (ν) v˙a + 3 θva + 4S (1 − R)Xa + RXa =0, (5-66) in the radiation dominated era. Since the CDM interacts with the other matter components through gravity, and since the gravitational inﬂuence of the CDM on the dominant matter components may be ignored during radiation domination, equation (5-66) is the only equation governing the evolution of the perturbations which makes reference to the CDM. It follows that any solution of equation (5-66), deﬁnes a valid solution to the linearised perturbation (c) (0) equations. The solution, which has va = va /S in an otherwise FRW universe corresponds to the solution labelled by A3 in equations (5-61)–(5-65). Note that this solution decays in an expanding universe. Following standard practice, we assume that this mode may be

ignored (A3 = 0) since it would require highly asymmetric initial conditions at the end of the inflationary epoch if the decaying mode was significant during the epoch of interest here. Similar comments apply to the mode labelled by D in equations (5-53)–(5-59). An important subclass of these solutions describe adiabatic modes. We assume that the appropriate covariant and gauge-invariant definition of adiabaticity is that

(3)∇ ρ(i) (3)∇ ρ(j) a = a , (5-67) ρ(i) + p(i) ρ(j) + p(j) for all species i and j (Bruni et al. 1992). This condition, which is frame-independent in linear theory, is the natural covariant generalisation of the (gauge-invariant) condition

δρ(i) δρ(j) = , (5-68) ρ¯(i) +¯p(i) ρ¯(j) +¯p(j) where δρ(i) = ρ(i) − ρ¯(i) is the usual gauge-dependent density perturbation, and overbars denote the background quantity. Demanding approximate adiabaticity between the photons and the neutrinos leaves only one free constant of integration, which we take to be C.The remaining constants are A1 = A3 = D =0andA2 =−6C. Note that the constants of integration will depend on the mode label k, in general, so we have C = Ck. This adiabatic solution may be developed further by including higher-moments of the neutrino distribution function, and ﬁnding a series expansion of the (enlarged) system in 3 (ν) (3) x. To obtain solutions correct to O(x ), it is necessary to retain πk and Gk .Theseries –30–

solution that results is

389R+700 2 (γ) (4R+15)C 2 Φk = C 1 − 168(2R+25) x , Xk = 6(R+5) x (4R+15)C 4R+5 3 (ν) (4R+15)C 2 Zk = − 4(R+5) x − 18(4R+15) x , Xk = 6(R+5) x 5C 112R2−16R−1050 3 (γ)(4R+15)C 3 σk = − 2(R+5) x + 2520(2R+25) x ,qk=54(R+5) x (ν) 2C 2 (ν)(4R+23)C 3 πk = − 3(R+5) x ,qk=54(R+5) x (3) 5C 3 (l)l Gk = − 63(R+5) x ,Gk=O(x) for l>3. (5-69)

Note that on large scales (x 1), the harmonic coefficient Φk of Eab is constant along the (0) 2 ˙(0) flow lines. It follows that on large scales, we may write Eab = Eab /S ,whereEab = O(2). The series solution given in equation (5-69) is adiabatic between the photons and neutrinos to O(x3), but the adiabaticity is broken by the higher-order terms. This difference in the dynamic behaviour of radiation and neutrinos is due to their different kinetic equations; the neutrinos are collisionless which allows higher-order angular moments in the distribution function to grow, but the radiation is tightly-coupled to the baryon fluid which prevents the growth of higher-order moments. The baryon relative velocity v(b) is determined by the condition that the radiation be nearly isotropic in the rest frame of the baryons:

(b) 3 (γ) vk = 4 qk , (5-70) and the spatial gradients of the baryon and CDM follow from the adiabaticity condition:

(b) (c) 3 (γ) 3 (ν) Xk = Xk = 4 Xk = 4 Xk , (5-71) where we have neglected the small eﬀect of baryon pressure. The series solution given as equation (5-69) was used to provide adiabatic initial conditions for the numerical solution of the perturbation equations, discussed in the next section.

6. Adiabatic Scalar Perturbations in a K =0Universe

In this section, we discuss the calculation of the CMB power spectrum from initially adiabatic scalar perturbations in an almost FRW universe with negligible spatial curvature. The evolution of anisotropy in the CMB and inhomogeneities in the density ﬁelds, resulting from scalar perturbations, may be found by solving numerically the equations presented in Section 5, with initial conditions determined from the analytic solutions of the previous section. For adiabatic perturbations, the speciﬁcation of initial conditions is particularly

simple; there is a single function Ck of the mode label k to set. This function gives the (constant) amplitude of the harmonic component of the electric part of the Weyl tensor on super-horizon scales. –31–

6.1. The CMB Power Spectrum

The gauge-invariant temperature perturbation from the mean, denoted by δT (e), is given by equation (4-3). Substituting for the harmonic expansion of the angular moments J (l) , a1...al we find 1 ∞ (2l + 1)(2l)! (l) (k) a1 al δT (e)= CkT (k)Qa ...a e ...e , (6-1) 4 (−2)l(l!)2 " 1 l # Xl=1 Xk where we have introduced the radiation transfer function T (l)(k), which is a function of the (l) eigenvalue k only. The transfer function is defined to be the value of Jk for the initial condition Ck = 1. Since the dynamics of a scalar mode labelled by the index k depends only on the eigenvalue of the eigenfunction Q(k), the transfer function is a function of the (l) (l) eigenvalue k only. For the linearised theory considered here, we have Jk = CkT (k). We have come as far as we can without making a specific choice for the scalar harmonic functions Q(k). To proceed, we introduce an almost FRW coordinate system (Ellis 1996) as follows. If the perturbations in the universe are only of scalar type, then the velocity ua is hypersurface orthogonal, so that we may label the orthogonal hypersurfaces with a time label t. Furthermore, since we have chosen ua to be the CDM velocity, which is geodesic, the flow orthogonal hypersurfaces may be labelled unambiguously with proper time along the flow lines, so that ua = ∇at. The orthogonal hypersurfaces depart from being spaces of constant curvature only at first-order, so we can introduce comoving spatial coordinates xi, in such a way that our (synchronous) coordinate system is almost FRW in form. (Latin indices, such as i, run from 1 to 3.) It is then straightforward to show that the functions eik·x,wherek·x=kixi and ki are constants, satisfy the defining equations for the scalar harmonic functions, equations (5-6) and (5-7), with k2 = kiki,inanalmostFRWuniverse with negligible spatial curvature. It follows that we may take

Q(k) = eik·x. (6-2)

For the open and flat cases, the appropriate generalisations of the eik·x (Harrison 1967) (k) (l) should be used for the Q . Note that the expansion coefficients, such as Jk , depend on the detailed choice of the scalar harmonics Q(k), but that covariant tensors, such as J (l)Q(k) , k k a1...al are independent of this choice. If vector perturbations are also significant, weP cannot use the velocity ua to define a time coordinate in the manner described above. Instead, an almost FRW coordinate system should be constructed using an irrotational and geodesic velocity fieldu â, which is close to our chosen fundamental velocity ua. Using this velocity field, almost FRW coordinates can be constructed by the above procedure (Ellis 1996). The resulting Q(k) will satisfy the defining (zero-order) properties of the scalar harmonics in the ua frame, since the relative velocity ofu â is first-order. –32–

a a a i Since u ea =0,andu =∇t, we can always choose the x so that at our observation point ea =(0,S−1ei), with eiei = 1 (for example, one can choose the xi so that Sxi are locally Cartesian coordinates in the constant time hypersurface). Then it follows that, to zero-order,

(2i)l(l!)2 (k) a1 al (k) Q e ...e = Pl(µ)Q , (6-3) a1...al (2l)!

i i where µ ≡ k e /k,andPl(µ) are the Legendre polynomials. This results demonstrates that expanding the angular moments of the distribution function in the covariant tensors Q(k) is a1...al equivalent to the usual Legendre expansion of the Fourier modes of the distribution function (which are axisymmetric about the wave vector k), in an almost FRW universe, where spatial curvature may be neglected. Following standard practice, we make the assumption that we inhabit one realisation of a

stochastic ensemble of universes, so that the Ck are random variables. (The physical basis, on which this assumption rests, is that initial ﬂuctuations were generated from causal quantum processes in the early universe, such as during a period of inﬂation; see for example, Kolb & Turner (1990). Given our chosen form for the Q(k), statistical isotropy of the ensemble

demands that the covariance matrix for the Ck takes the following form:

∗ 2 hCkCk0 i = C (k)δkk0, (6-4)

where C2(k) is the primordial power spectrum which is a function of the eigenvalue k.

The δkk0 appearing in equation (6-4) is defined by k δkk0Ak = Ak0,whereAk is an arbitrary 2 function of the mode label k. The CMB power spectrumP Cl is defined by Cl ≡h|alm| i,where alm are the coefficients in the spherical harmonic expansion of the temperature anisotropy (see Section 4). Substituting the harmonic expansion of the J (l) into equation (4-5), and a1...al using the zero-order result

(−2)l(l!)2 ha1b1 ...halblQ(k) Q(k)∗ = , (6-5) a1...al a1...al (2l)!

which follows from Q(k) = eik·x, we ﬁnd the familiar expression for the CMB power spectrum in terms of the transfer functions and the primordial power:

2 2 (l) 2 Cl = π d lnkC (k) T (k) . (6-6) Z We make the standard assumption that on large scales, the primordial power spectrum may be approximated by a power-law of the form C2(k) ∝ kns−1. Many inﬂationary models predict that the scalar index ns will be close to unity (Kolb & Turner 1990). The case ns =1 –33–

describes the scale-invariant spectrum. This term arises from considering the logarithmic power spectrum in Fourier space of the (gauge-dependent) fractional density perturbation

δρ evaluated at horizon crossing. An analogous result can be found in the covariant and gauge-invariant approach. We evaluate the logarithmic power spectrum of the dimensionless (3) (3) vector Da ≡ ∇aρ/ρH,where ∇aρ is evaluated in the energy frame (qa = 0), by making use of equation (2-17), with the contribution from anisotropic stress neglected, and the (3) a frame-invariance of ∇ Eab in linear theory, to ﬁnd that

a −6 2 ∂ lnkhD Dai∝Hk C (k). (6-7)

In deriving this relation, which is valid before the modes labelled by k reenter the Hubble radius, we have assumed that only the fastest growing scalar mode is signiﬁcant so that

Φk is constant before horizon crossing. For given k, the logarithmic power in Da evolves in time due to the presence of Hk on the right-hand side of equation (6-7). However, at horizon crossing Hk falls below some critical value of order unity which is independent of k. It follows that for the scalar index ns = 1, the logarithmic power in Da at horizon crossing is independent of scale.

6.2. The Tight-Coupling Approximation

At early times, when the baryons and photons are tightly-coupled, the radiation is nearly isotropic in the frame of the baryons. In this limit, it is convenient to replace the propagation equations for the J (l) with l 1, and for the baryon relative velocity v(b), a1...al ≥ a with approximate equations which may be developed by an expansion in the reciprocal of the

photon mean free path 1/neσT . The approximate equations are simpler to solve numerically than the exact equations since the former do not include the large Thomson scattering terms present in the latter. For scalar perturbations, it is simplest to work directly with the harmonic expansion (l) (b) coeﬃcients, Jk and vk . The relevant timescales in the problem are the photon mean −1 free time tc ≡ neσT , the expansion timescale tH ≡ H , and the light travel time across the wavelength of the mode under consideration, tk ≡ S/k. In the tight-coupling approximation, we expand in the small dimensionless numbers tc/tH and tc/tk, so that the procedure is valid for tc min(tH ,tk). While a mode is outside the horizon, min(tH ,tk)=tH,whereas min(tH ,tk)=tk during the acoustic oscillations. In the CDM frame (wa = 0), the procedure is similar to that usually employed (Peebles & Yu 1970; Ma & Bertshinger 1995) in the synchronous-gauge. We combine the propagation equations given in Section 5.1 for the (l) (b) photon moments Jk (l ≥ 1) and the baryon relative velocity vk , to get the exact (in linear –34–

theory) equations:

(b) 3 ˙ −1 (b) 1 −1 (γ) 2 −1 (b) 1 −1 (γ) (1 + R)˙vk =−4R∆k −tH vk + 4Rtk Xk + cstk Xk − 2 Rtk πk (6-8) ˙ 4 −1 (b) 1 −1 (γ) 4 2 −1 (b) 2 −1 (γ) (1 + R)∆k = −tc ∆k − 3 tH vk − 3 tk Xk + 3 cstk Xk + 3 tk πk (6-9) (γ) 10h (γ) 3 −1 (3) 2 −1 8 −1 (b) i πk = − 9 tc π˙ k + 5 tk Jk − 5 tk ∆k − 15 tk (σk + vk ) (6-10) (l) ˙h(l) −1 l+1 (l+1) l (l−1) i Jk = −tc Jk + tk 2l+1 Jk − 2l+1 Jk l ≥ 3, (6-11) h i (γ) (b) (γ) (b) where ∆ ≡ qk − 4vk /3, and for this section, R ≡ 4ρ /3ρ . Iterating these equations gives the tight-coupling expansions

(b) (b) (b) v˙k =˙v0 +˙v1 +..., ∆k =∆1 +∆2 +..., (6-12) (γ) (γ) (γ) (l) (l) (l) πk = π1 + π2 + ..., Jk =Jl−1 +Jl +..., (6-13) where the subscript on the variables on the right-hand side denotes the order in the expansion parameter =max(tc/tH ,tc/tk). To avoid cluttering of indices, we leave the mode label k implicit. We shall only require the results to ﬁrst-order in :

(b) 1 1 k (γ) k 2 (b) (b) v˙0 = 1+R 4 S RXk + S csXk − Hvk (6-14) (b) 3R2 R k (γ) v˙1 = − 2(1+R)2 H∆1 − 2(1+R) S π1

R tc ˙ 2 −1 (b) tc (γ) ˙(γ) 2 ˙(b) − 2 4 (H +2H )H v + 2HX +X −4c X 4(1+R) tH k tk k k s k (6-15) h i 1 tc (γ) 2 (b) tc (b) ∆1 = X − 4c X +4 v (6-16) 3(1+R) tk k s k tH k

(γ) 16 tc h (b) i π = σk + v . (6-17) 1 27 tk k (γ) The propagation equation for qk in the tight-coupling approximation may be obtained from ˙ (γ) (b) (b) (γ) the exact equation (6-8), with ∆k replaced byq ˙k − 4˙vk /3, andv ˙k and πk replaced by their tight-coupling expansions.

6.3. Numerical Results

We are now in a position to evolve an initial set of perturbations from early times to the present in an almost FRW universe with negligible spatial curvature, and to calculate the power spectrum of the CMB anisotropies that result. In this section we present the results of −1 −1 a numerical simulation in the standard CDM model, with parameters H0 = 50kms Mpc , baryon fraction Ωb =0.05, CDM fraction Ωc =0.95, Helium fraction 0.24, zero cosmological constant, and a scale-invariant spectrum of initially adiabatic conditions (ns =1). –35–

Our code to solve the covariant and gauge-invariant perturbation equations in the CDM frame, including the Boltzmann hierarchies for the photons and neutrinos, was based on the serial COSMICS code developed by Bertschinger and Bode, and described in Ma & Bertshinger (1995). The COSMICS package, including full documentation, is available at http://arcturus.mit.edu/cosmics. We modiﬁed the COSMICS code to solve the covariant equations given in this paper, for the matter variables and for the spatial gradient of the expansion, Za. The shear, which is required to solve the Boltzmann hierarchies for the photons and neutrinos, was determined from the equation (2-18). The electric part of the Weyl ten-

sor, Eab, could then be determined from equation (2-17). Our calculations of the zero-order ionisation history of the universe, which fully include the eﬀects of Helium and Hydrogen recombination, followed Ma and Bertschinger (1995), as did our truncation schemes for the photon and neutrino Boltzmann hierarchies. The ﬁrst-order tight-coupling approximation

was used at sufficiently early times that max(tc/tH ,tc/tk) 1. (i) In Figure 1 we show the variation of the harmonic coefficients Xk of the comoving fractional spatial gradients in the CDM frame, against redshift in the standard CDM model. Similar plots were given by Ma & Bertschinger (1995) for the Fourier components of the (gauge-dependent) density perturbations δ(i) ≡ (ρ(i) − ρ¯(i))/ρ¯(i), whereρ ¯(i) is the density of the species i in the background model. Our results, given in Figure 1, agree well with the synchronous-gauge results of Ma & Bertshinger (1995). This is because the constant time surfaces in this gauge are orthogonal to the CDM velocity, so that X (i) is a covariant measure of the density inhomogeneity in these surfaces. Although δ(i) is gauge-dependent in the synchronous gauge, the gauge-conditions restrict this gauge-dependence to transformations of the form δ(i) 7→ δ(i) − αρ/¯˙ ρ¯,whereαis a first-order constant. It follows that the Fourier coefficients of δ(i) are gauge-invariant away from k = 0 in Fourier space. The qualitative behaviour of the comoving density gradients can be seen directly from their propagation equations. For scalar perturbations, it is simplest to work directly with (i) the equations of motion (5-21)–(5-24) for the harmonic coefficients Xk in the CDM frame. Eliminating the spatial gradients of the expansion, Za, we find the following second-order –36–

Fig. 1.— The variation of the harmonic coeﬃcients of the fractional comoving density gradients in the CDM frame with redshift in the standard CDM model: Ωc =0.95, Ωb =0.05, −1 −1 −1 H0 = 50kms Mpc , with Helium fraction 0.24, for k =0.01, 0.1and1.0Mpc .The normalisation is chosen so that Φk = 1 for all k at early times. –37–

equations:

¨(γ) 2 ˙ (γ) 1 k2 (γ) 2 (i) (i) (i) 1 k (γ) Xk + 3 θXk + 3 S2 Xk = 3 κ ρ +3p Xk − 3Sθqk i X (6-18) 2 k2 (γ) k 4 (b) (γ) + 3 S2 πk − neσT S 3vk − qk ¨(ν) 2 ˙ (ν) 1 k2 (ν) 2 (i) (i) (i) 1 k (ν) 2 k2 (ν) Xk + 3 θXk + 3 S2 Xk = 3 κ ρ +3p Xk − 3Sθqk + 3 S2 πk (6-19) i X ¨(b) 2 ˙ (b) 2 k2 (b) 1 (i) (i) (i) ρ(γ) k 4 (b) (γ) Xk + 3 θXk + cs S2 Xk = 2 κ ρ +3p Xk +neσT ρ(b) S 3vk −qk i X (6-20) ¨(c) 2 ˙ (c) 1 (i) (i) (i) Xk + 3 θXk = 2 κ ρ +3p Xk , (6-21) i X 2 2 2 (b) where we have ignored baryon pressure except in the acoustic term cs(k /S )Xk ,which can be signiﬁcant on the small scales. In the limiting case that the mode is well outside the

Hubble radius (Hk 1), the equations of motion for each component reduce to the common form

¨(i) 2 ˙ (i) 1 p(i) (j) (j) (j) Xk + 3 θXk = κ 2 1+ ρ(i) ρ +3p Xk . (6-22) j X For adiabatic initial conditions, it is clear that the adiabatic condition, given as equation (5-67), is maintained while the mode is outside the Hubble radius. Solving equation (6-22) for adiabatic perturbations gives growing modes proportional to t and t2/3 during radiation and matter domination respectively. If a mode enters the Hubble radius prior to last scattering, the photon/baryon ﬂuid, which is still tightly-coupled, undergoes acoustic oscillations. To lowest-order in the tight-

coupling parameter, max(tc/tH ,tc/tk), the photon and baryon perturbations remain adiabatic, evolving according to

(γ) (γ) 2 2 (γ) (i) (b) ¨ 2 ˙ R+3cs k 2 (i) (i) 4R k Xk + 3 θXk + 3(1+R) S2 Xk = 3 κ ρ +3p Xk − 9(1+R) S θvk , i X (6-23)

where R ≡ 4ρ(γ)/3ρ(b). The solution of the homogeneous equation describes acoustic oscil- 2 lationsinaﬂuidwithsoundspeedsquared(R+3cs)/3(1 + R), which are damped by the expansion of the universe. However, the oscillations are driven gravitationally by the gradient (3)∇(ρ+3p), which gives an almost constant amplitude oscillation in the radiation dominated era. The Silk damping which is visible in Figure 1 for k =1.0Mpc−1 at z ' 10−3.5 arises from photon diﬀusion (which is not described by the lowest-order tight-coupling approximation) so is not described by equation (6-23). The neutrino perturbation also oscillates once inside the Hubble radius in the radiation dominated region, while the power-law growth of –38–

Fig. 2.— The power spectrum of scalar CMB anisotropies in the standard CDM model: −1 −1 Ωc =0.95, Ωb =0.05, H0 = 50kms Mpc , ns = 1, with Helium fraction 0.24. The normalisation of the vertical scale is arbitrary.

the CDM is impeded by the gravitational attraction of the oscillating dominant component (the inhomogeneous term in eq. [6-21]). In the matter dominated era, the CDM becomes the dominant component, so we again see power-law growth of the CDM perturbation on all scales. Before last scattering, the photons and baryons remain tightly-coupled, but the character of the (3)∇(ρ +3p) driving term in equation (6-23) changes from an oscillation to a power-law as the CDM becomes dominant. At last scattering, the photons and baryons decouple. The baryons no longer feel the pressure support provided by the photons; the Jeans’ length of the baryons is very small and the acoustic term in equation (6-20) is negligible. The (3)∇(ρ +3p) driving term attracts the baryons into the potential wells caused (b) (c) principally by inhomogeneity of the CDM, so that Xa relaxes to Xa as a power-law. After last scattering, the photons and neutrinos continue to undergo driven oscillations, which (γ) (ν) 2 (c) decay towards the particular integral Xk = Xk =6HkXk . In Figure 2 we show the CMB power spectrum calculated from a simulation in the standard CDM model. On large scales, the plateau arises from the usual potential fluctuations (k) ∆ k ΦkQ /3 on the last scattering surface (Sachs & Wolfe 1967). The oscillations in the CMBP power spectrum on smaller scales (the Doppler peaks) arise from the acoustic oscil- –39– lations in the baryon/photon fluid. These oscillations give rise to strongly scale-dependent gradients of the photon energy density in the energy-frame, which in the approximation of instantaneous recombination can be interpreted as temperature variations across the last scattering surface, and a local scale-dependent distortion of the last scattering surface relative to the energy frame. Since the last scattering surface is well approximated by a hypersurface of constant radiation temperature (so that recombination does occur there), it is more correct to interpret the Doppler peaks in terms of the local variations in redshift along null geodesics back to the last scattering surface, than in terms of temperature variations on the last scattering surface. (There is another significant contribution to the Doppler peaks, which is of dipole nature on the last scattering surface, and tends to fill in the power spectrum near the first Doppler peak; see Hu & Sugiyama (1995) and Challinor & Lasenby (1989) for more details.) On the smallest scales, the power spectrum is damped due to photon diffusion in the photon/baryon plasma prior to recombination.

7. Conclusion

We have shown how the full kinetic-theory calculation of the evolution of CMB anisotropies and density inhomogeneities can be performed in the covariant and gauge-invariant approach to cosmology (Ellis & Bruni 1989; Ellis et al. 1989). Adopting covariantly-defined, gauge- invariant variables throughout ensured that our discussion avoided the gauge ambiguities that appear in certain gauges, and that all variables had a clear, physical interpretation. We presented a unified set of equations describing the evolution of photon and neutrino anisotropies and cosmological perturbations in the CDM model, which were independent of a decomposition into scalar, vector or tensor modes and the associated harmonic analysis. We obtained equations describing scalar perturbations from the full frame-independent set by imposing covariant restrictions on the gauge-invariant variables, and provided the analytic solution of these equations at early times. Finally, we discussed the results of a numerical solution of these equations in an almost FRW universe with negligible spatial curvature in the standard CDM model. The equations for tensor perturbations may be obtained in a similar manner to the scalar equations. This work, which shows the real advantage of the covariant angular decomposition of the relativistic distribution functions, will be presented in a future publication. Our results confirm those of other groups (see, for example, Ma & Bertshinger (1995) and Seljak & Zaldarriaga (1996), who have obtained their results by employing non-covariant methods in specific gauges. Typically, these methods require one to keep careful track of all residual gauge-freedom, both to enable identification of any gauge-mode solutions, and to –40–

ensure that the final results quoted are gauge-invariant (and hence observable). Fortunately, the isotropy of the photon distribution function in an exact FRW universe ensures that the CMB power spectrum, as calculated from the gauge-dependent perturbation to the distribution function, is gauge-invariant for l ≥ 1. This, combined with the fact that it is not necessary to place the last scattering surface in kinetic theory calculations, saves the majority of the Boltzmann calculations in the literature from the gauge problems inherent in many Sachs-Wolfe type analyses (Stoeger et al. 1995), despite recent claims to the contrary (Ellis & Dunsby 1997). The need to keep track of residual gauge-freedom evaporates if one employs gauge-invariant variables, such as in the covariant approach, or that due to Bardeen (1980). However, by adopting variables which are also covariantly-defined, the covariant approach provides the additional advantages of unification (of the various perturbation types) and physical clarity of both methods and predictions.

The development of the COSMICS package was supported by the NSF under grant AST-9318185.

REFERENCES

Abbott, L. F., & Schaefer, R. K. 1986, ApJ, 308, 546

Bardeen, J. M. 1980, Phys. Rev. D, 22, 1882

Bond, J. R., & Efstathiou, G. 1984, ApJ, 285, L45

Bond, J. R., & Efstathiou, G. 1987, MNRAS, 226, 655

Bruni, M., Dunsby, P. K. S., & Ellis, G. F. R. 1992, ApJ, 395, 34

Bruni, M., Ellis, G. F. R., & Dunsby, P. K. S. 1992, Class. Quantum. Grav., 9, 921

Challinor, A. D., & Lasenby, A. N. 1998, A covariant and gauge-invariant analysis of CMB anisotropies from scalar perturbations, submitted

Dunsby, P. K. S. 1997, Class. Quantum Grav., 14, 3391

Dunsby, P. K. S., Bassett, B. A. C. C., & Ellis, G. F. R. 1997, Class. Quantum. Grav., 14, 1215

Dunsby, P. K. S., Bruni, M., & Ellis. G. F. R. 1992, ApJ, 395, 54 –41–

Ellis, G. F. R. 1996, in Current Topics in Astrofundamental Physics, ed. N. S´anchez & A. Zicichi (Singapore: World Scientiﬁc), 3

Ellis, G. F. R., & Bruni, M. 1989, Phys. Rev. D, 40, 1804

Ellis, G. F. R., Bruni, M., & Hwang, J. 1990, Phys. Rev. D, 42, 1035

Ellis, G. F. R, & Dunsby, P. K. S. 1997, in Current Topics in Astrofundamental Physics, ed. N. S´anchez (Dordrecht, Kluwer Academic), in press

Ellis, G. F. R., Hwang, J., & Bruni, M. 1989, Phys. Rev. D, 40, 1819

Ellis, G. F. R., Matravers, D. R., & Treciokas, R. 1983, Annals of Physics, 150, 455

Harrison, E. R. 1967, Rev. Mod. Phys., 39, 862

Hawking, S. W. 1966, ApJ, 145, 544

Hu, W., Scott, D., Sugiyama, N., & White, M. 1995, Phys. Rev. D, 52, 5498

Hu, W., & Sugiyama, N. 1995, ApJ, 444, 489

Kolb, E. W., & Turner, M. S. 1990, The Early Universe (California: Addison-Wesley)

Kosowsky, A. 1996, Annals of Physics, 246, 49

Ma, C. P., & Bertshinger, E. 1995, ApJ, 455, 7

Maartens, R., Ellis, G. F. R., & Stoeger, W. R. 1995, Phys. Rev. D, 51, 1525

Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation (San Francisco: W. H. Freeman and Company)

Pank, M. 1986, Phys. Rev. D, 34, 416

Peebles, P. J. E., & Yu, J. T. 1970, ApJ, 162, 815

Sachs, R. K., & Wolfe, A. M. 1967, ApJ, 147, 73

Seljak, U., & Zaldarriaga, M. 1996, ApJ, 469, 437

Silk, J. 1967, Nature, 215, 1155

Silk, J. 1968, ApJ, 151, 459

Stoeger, W. R., Ellis, G. F. R., & Xu, C. M. 1994, Phys. Rev. D, 49, 1845 –42–

Stoeger, W. R., Xu, C. M., Ellis, G. F. R., & Katz, M. 1995, ApJ, 445, 17

Tsagas, C. G., & Barrow, J. D. 1997, Class. Quantum. Grav., 14, 2539

This preprint was prepared with the AAS LATEX macros v4.0.