Nonlinear Effects in the Cosmic Microwave Background

General Relativity and Gravitation, Vol. 32, No. 6, 2000

Nonlinear Effects in the Cosmic MicrowaveBackground

Roy Maartens1

Received November 29, 1999

Major advances in the observation and theory of cosmic microwave back-ground anisotropies have opened up a new era in cosmology. This hasencouraged the hope that the fundamental parameters of cosmology willbe determined to high accuracy in the near future. However, this opti-mism should not obscure the ongoing need for theoretical developmentsthat go beyond the highly successful but simplified standard model.Such developments include improvements in observational modelling (e.g.foregrounds, non-Gaussian features), extensions and alternatives to thesimplest inflationary paradigm (e.g. non-adiabatic effects, defects), andinvestigation of nonlinear effects. In addition to well known nonlineareffects such as the Rees–Sciama and Ostriker-Vishniac effects, furthernonlinear effects have recently been identified. These include a Rees–Sciama-type tensor effect, time-delay effects of scalar and tensor lensing,nonlinear Thomson scattering effects and a nonlinear shear effect. Someof the nonlinear effects and their potential implications are discussed.

KEY WORDS : Cosmology ; microwave background radiation

1. INTRODUCTION

In 1948, Alpher and Herman predicted a cosmic microwave background(cmb) radiation, which remained a theoretical possibility until the seren-dipitous discovery by Penzias and Wilson in 1965. A potential crisis forcosmology, arising from the apparent perfect isotropy of the cmb, was

1 Relativity and Cosmology Group, Division of Mathematics and Statistics, PortsmouthUniversity, Portsmouth PO1 2EG, UK. E-mail: [email protected]

1075

0001-7701/00/0600-1075$18.00/0 c2000 Plenum Publishing Corporation

1076 Maartens

turned into a dramatic success when in 1992 the COBE satellite detectedsmall anisotropies at the level predicted by the standard inflationary colddark matter model. The confluence of theoretical and observational suc-cesses has opened up a new era of precision-tested cosmology, exemplifiedby the upcoming launch of the MAP and Planck satellites (see e.g. Ref. 1).Together with parallel advances in galactic, supernovae, lensing and otherobservations, this has promoted the hope that the fundamental parametersof the standard cosmological model (Ωm, ΩΛ, h, n, . . . ) will be determinedto high accuracy, and that in some sense, cosmology will be “solved” upto details of fine-tuning.

However, physics rarely turns out be as simple as this, and optimismneeds to be tempered by a realisation that future observations could entailnot only the resolution of old problems, but also the generation of new andunexpected ones (see also Ref. 2). The new “golden age” of cosmology isperhaps better seen as the end of the beginning, rather than the beginningof the end.

It is clearly necessary to refine and develop the predictions of thestandard model in readiness for the emerging observational data, and mucheffort has been put into this (see e.g. Refs. 3–7). But it is also necessaryto develop beyond the simplified standard model, in recognition of therich complexity of the universe, and in anticipation of new surprises fromfuture observations. This encompasses a range of theoretical advances:

• The modelling of observations needs significant development in rela-tion to foreground extraction, non-Gaussian features, data analysis,etc. (see e.g. Refs. 8,9).

• Extensions and alternatives to the simple inflationary paradigm needto be explored, including investigation of non-adiabatic effects (duringinflation and reheating), of string cosmology (see e.g. Ref. 10), andof defect theories (see e.g. Refs. 11,12). Such exploration can leadto unexpected results; for example, it has recently been shown [13]that resonant amplification of metric perturbations in preheating afterinflation could re-process the primordial power spectrum, and leavean imprint on cmb anistropies.• Nonlinear effects in cmb anisotropies require further study.

This article focuses on the latter aspect, in particular on recently iden-tified nonlinear effects. Well-known nonlinear effects include the Ostriker–Vishniac [14] and kinetic Sunyaev–Zeldovich [15] effects, which are localeffects that arise from the modulation of the Doppler effect with inho-mogeneities in the optical depth. These effects could restore small-scalepower via hot cluster gas or re-ionisation (see e.g. Ref. 16). The Rees–

Nonlinear Effects in the Cosmic Microwave Background 1077

Sciama effect [17] arises as a second-order contribution to the integratedSachs–Wolfe term. Nonlinear integrated effects can in principle be signif-icant because of the long photon paths after last scattering. In practicethe Rees–Sciama effect is typically very small. More recently, it has beenshown [18] that a larger effect follows from gravitational lensing of cmb

photons by density perturbations.Considerable effort has been put into calculating the imprint of these

particular nonlinear effects on cmb anisotropies, and this work is of greatimportance for isolating the secondary anisotropies and revealing the pri-mordial spectrum [19]. What has received less attention is a systematicapproach to nonlinear effects in general. This is important because it couldreveal new sources of secondary anisotropies that may be significant. Fur-thermore, a systematic analysis of second-order effects forms the basisfor estimating the theoretical errors associated with the linear analysis onwhich cmb science rests.

Mollerach and Matarrese [20] recently developed a systematic second-order Sachs–Wolfe analysis, building on previous work by Pyne and Car-roll [21]. They found the second-order metric perturbations to a matter-dominated flat universe in the Poisson gauge by transforming the knownsynchronous gauge solutions, using the methods of Bruni et al . [22]. Thenthese solutions were used to find the second-order Sachs–Wolfe effect. Themain relevant effects which they identified are:

• in addition to the Rees–Sciama effect, another nonlinear integratedSachs–Wolfe effect representing corrections to the linear gravitationalwave contribution;

• in addition to gravitational lensing by linear density perturbations,gravitational lensing by linear gravitational wave modes;

• time delay effects of scalar and tensor lensing;• a coupling of the velocity at last scattering with the perturbed photon

wave vector.

An alternative approach to nonlinear effects is developed in [23], andwill be briefly described here. Instead of starting from a background modeland perturbing, this approach starts from the fully nonlinear model of theinhomogeneous universe. The approach is thus more general, althoughthis generality comes at the cost of greater difficulty in quantifying theeffects on cmb anisotropies. More importantly, the alternative approachcovers nonlinearity not only in the metric, but also in Thomson scattering,and in relative motions of the particle species. (Second-order scatteringeffects have previously been investigated via a different approach by Huet al . [24].) It thus offers a foundation for a comprehensive second- and

1078 Maartens

higher-order analysis, taking into account all sources of nonlinearity. (Non-linear polarisation has not been treated, although key elements of such atreatment can be found in Challinor’s linear analysis [25].)

The key local effects identified in the new approach are:• at each multipole order , a coupling of baryonic bulk velocity to the

radiation brightness multipoles of order ± 1;• a coupling of the acceleration, vorticity and shear at order to the

brightness multipoles of order , ± 1, ± 2;• for 1, the shear coupling is dominant and could be important.

These local effects need to be integrated to evaluate their impact oncmb anisotropies, but a qualitative understanding has been achieved as afirst step.

2. NONLINEAR DYNAMICS

The covariant Lagrangian, or 1+3 covariant, approach is based on aphysical choice of a 4-velocity vector field ua. All variables are in principlephysically measurable by comoving observers. This approach is inherentlynonlinear. It starts from the inhomogeneous and anisotropic universe,without a priori restrictions on the degree of inhomogeneity and anisotropy,and then applies the Friedmann limit when required. The basic theoreticalingredients are: the covariant Lagrangian hydrodynamics of Ehlers andEllis [26,27], and the perturbation theory of Hawking [28] and Ellis andBruni [29] which is derived from it; the covariant Lagrangian approach tokinetic theory of Ellis et al . [30]; and the covariant analysis of temperatureanisotropies introduced by Maartens et al . [31] and developed by Challinorand Lasenby [32,25] and Gebbie et al . [33].

The projected symmetric tracefree (pstf) parts of vectors and rank-2tensors are

V〈a〉 = habVb , S〈ab〉 = h(a

chb)d − 1

3hcdhabScd ,

where hab = gab + uaub is the projector, with gab the spacetime metric.The skew part of a projected rank-2 tensor is spatially dual to the pro-jected vector Sa = 1

2εabcS[bc], where εabc = ηabcdu

d is the projection ofthe spacetime alternating tensor. Any projected rank-2 tensor has theirreducible decomposition

Sab = 13Shab + εabcS

c + S〈ab〉 ,

where S = Scdhcd is the spatial trace. A covariant vector product and its

generalization to pstf rank-2 tensors are

[V,W ]a = εabcVbW c , [S,Q]a = εabcS

bdQ

cd.


The covariant derivative ∇a produces time and spatial derivatives

Ja······b = uc∇cJa······b , DcJa······b = hc

dhae · · ·hbf∇dJe······f .

The projected derivative Da further splits irreducibly into a spatial diver-gence and curl [34]

divV = DaVa , (divS)a = DbSab ,

curlVa = εabcDbV c , curlSab = εcd(aDcSb)

d.

Relative motion of comoving observers is encoded in the kinematicquantities: the expansion Θ = Daua, the 4-acceleration Aa ≡ ua = A〈a〉,the vorticity ωa = − 1

2curlua, and the shear σab = D〈aub〉. The dynamicquantities describe the sources of the gravitational field: the (total) energydensity ρ = Tabu

aub, isotropic pressure p = 13habT

ab, energy flux qa =−T〈a〉bub, and anisotropic stress πab = T〈ab〉, where Tab is the total energy-momentum tensor. The locally free gravitational field, i.e. the part of thespacetime curvature not directly determined locally by dynamic sources,is given by the Weyl tensor Cabcd. This splits irreducibly into the gravito-electric and gravito-magnetic fields

Eab = Cacbducud = E〈ab〉 , Hab = 1

2εacdCcdbeu

e = H〈ab〉 ,

which provide a covariant Lagrangian description of tidal forces and grav-itational radiation.

The Ricci identity for ua and the Bianchi identities ∇dCabcd =∇[a(−Rb]c + 1

6Rgb]c) produce the fundamental evolution and constraintequations governing the above covariant quantities [26,27]. Einstein’sequations are incorporated via the algebraic replacement of the Ricci ten-sor Rab by Tab− 1

2Tccgab. These equations, in fully nonlinear form and for

a general source of the gravitational field, areEvolution:

ρ + (ρ + p)Θ + div q = −2Aaqa − σabπab , (1)Θ + 1

3Θ2 + 12 (ρ + 3p)− divA

= −σabσab + 2ωaωa + AaAa (2)

q〈a〉 + 43Θqa + (ρ + p)Aa + Dap + (divπ)a

= −σabqb + [ω, q]a −Abπab , (3)ω〈a〉 + 2

3Θωa + 12curlAa = σabω

b , (4)

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σ〈ab〉 + 23Θσab + Eab − 1

2πab −D〈aAb〉= −σc〈aσb〉c − ω〈aωb〉 + A〈aAb〉 , (5)

E〈ab〉 + ΘEab − curlHab + 12 (ρ + p)σab

+ 12 π〈ab〉 +

16Θπab + 1

2D〈aqb〉= −A〈aqb〉 + 2Acεcd(aHb)

d + 3σc〈aEb〉c

− ωcεcd(aEb)d − 1

2σc〈aπb〉c − 1

2ωcεcd(aπb)

d, (6)

H〈ab〉 + ΘHab + curlEab − 12curlπab

= 3σc〈aHb〉c − ωcεcd(aHb)

d − 2Acεcd(aEb)d

− 32ω〈aqb〉 +

12σ

c(aεb)cdq

d. (7)

Constraint :

divω = Aaωa , (8)(divσ)a − curlωa − 2

3DaΘ + qa = −2[ω,A]a , (9)curlσab + D〈aωb〉 −Hab = −2A〈aωb〉 , (10)

(divE)a + 12 (divπ)a − 1

3Daρ + 13Θqa

= [σ,H]a − 3Habωb + 1

2σabqb − 3

2 [ω, q]a , (11)(divH)a + 1

2curl qa − (ρ + p)ωa= −[σ,E]a − 1

2 [σ, π]a + 3Eabωb − 12πabω

b . (12)

If the universe is almost Friedmann, then quantities that vanish inthe Friedmann limit are O(ε), where ε is a dimensionless smallness para-meter, and the quantities are suitably normalised (e.g.

√σabσab/Θ < ε,

etc.). Linearisation reduces all the right hand sides of the evolution andconstraint equations to zero.

For a given choice of fundamental frame ua, each of the species Iwhich source the gravitational field has relative velocity vI and 4-velocity

uaI

= γI(ua + va

I), va

Iua = 0 . (13)

If the 4-velocities are close, i.e. if the frames are in non-relativistic relativemotion, then O(v2) terms may be dropped from the equations, except ifwe include nonlinear kinematic, dynamic and gravito-electric/magnetic ef-fects, in which case, for consistency, we must retain O(ε0v2) terms such asρv2, which are of the same order of magnitude in general as O(ε2) terms.This is the physically relevant nonlinear regime, i.e. the case where only


nonrelativistic bulk velocities are considered, but no restrictions are im-posed on non-velocity terms, and we neglect only terms O(εv2, v3). Second-order effects involve neglecting also terms O(ε3), while linearisation meansneglecting terms O(ε2, εv, v2).

The dynamic quantities in the evolution and constraint equations (1)–(12) are the total quantities, with contributions from all dynamically sig-nificant particle species. Thus

T ab =∑I

T abI

= ρuaub + phab + 2q(aub) + πab, (14)

T abI

= ρIuaIuaI

+ pIhabI

+ 2q(aIub)I

+ πabI

. (15)

The dynamic quantities ρI, · · · in eq. (15) are as measured in the I-frame.

For cold dark matter (cdm), baryons, photons and neutrinos

pC

= 0 = qaC

= πabC

, qaB

= 0 = πabB

, pR

= 13ρR , p

N= 1

3ρN ,

where we have chosen the unique 4-velocity in the cdm and baryonic caseswhich follows from modelling these fluids as perfect. The cosmologicalconstant is characterized by

pV

= −ρV

= −Λ , qaV

= 0 = πabV

, vaV

= 0 .

The conservation equations for the species are best given in the over-all ua-frame, as are the evolution and constraint equations above. Thisrequires the expressions for the partial dynamic quantities as measured inthe overall frame. The inverse velocity relation is

ua = γI(uaI

+ v∗aI

), v∗aI

= −γI(vaI

+ v2Iua),

where v∗aI

uIa = 0, and v∗a

Iv∗Ia = va

IvIa. Then the dynamic quantities of

species I as measured in the overall ua-frame are

ρ∗I

= ρI

+ γ2Iv2I(ρI+p

I) + 2γ

IqaIvIa + πab

IvIavIb, (16)

p∗I

= pI

+ 13γ2

Iv2I(ρI+p

I) + 2γ

IqaIvIa + πab

IvIavIb, (17)

q∗aI

= qaI

+ (ρI+p

I)vaI

+ (γI−1)qa

I− γ

IqbIvIbu

a

+ γ2Iv2I(ρI+p

I)vaI

+ πabI

vIb − πbc

IvIbvIcu

a, (18)

π∗abI

= πabI

+ −2u(aπb)cI

vIc + πbc

IvIbvIcu

aub+ − 1

3πcdI

vIcvIdh

ab + γ2I(ρI

+ pI)v〈aI

vb〉I

+ 2γIv〈aI

qb〉I. (19)

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These are the nonlinear generalisations of well-known linearised results,which correspond to removing all terms in braces, dramatically simplifyingthe expressions. To linear order, there is no difference in the dynamicquantities when measured in the I-frame or the fundamental frame, apartfrom a simple velocity correction to the energy flux. But in the generalnonlinear case, this is no longer true.

The total dynamic quantities are simply given by

ρ =∑

ρ∗I, p =

∑p∗I, qa =

∑q∗aI

, πab =∑

π∗abI

.

A convenient choice for each partial 4-velocity uaI

is the energy frame, i.e.qaI

= 0 for each I. In the fundamental frame, the partial energy fluxes donot vanish, i.e. q∗a

I= 0, and the total energy flux is given by

qa =∑

[ (ρI

+ pI)vaI

+ πabI

vIb + O(εv2

I, v3I) ].

Then the dynamic quantities of cdm as measured in the fundamental frameare

ρ∗C

= γ2CρC, p∗

C= 1

3γ2Cv2CρC, (20)

q∗aC

= γ2CρCvaC, π∗ab

C= γ2

CρCv〈aC

vb〉C

, (21)

while for baryonic matter

ρ∗B

= γ2B(1 + w

Bv2B)ρB, p∗

B= [w

B+ 1

3γ2Bv2B(1 + w

B) ]ρ

B, (22)

q∗aC

= γ2B(1 + w

B)ρBvaB, π∗ab

B= γ2

B(1 + w

B)ρBv〈aB

vb〉B

, (23)

where wB≡ p

B/ρB. For radiation and neutrinos, the dynamic quantities

relative to the ua-frame are found directly via kinetic theory below.The total energy-momentum tensor is conserved, i.e. ∇bT ab = 0,

which is equivalent to the evolution equations (1) and (3). The partialenergy-momentum tensors obey

∇bT abI = JaI

= U∗Iua + M∗a

I, (24)

where U∗I

is the rate of energy density transfer to species I as measured inthe ua-frame, and M∗a

I= M∗〈a〉

Iis the rate of momentum density transfer.

ThusJaC

= 0 = JaN, Ja

R= −Ja

B= U

Tua + Ma

T,


where the Thomson rates are, to O(εv2B, v3B),

UT

= nEσT( 43ρ∗Rv2B− q∗a

RvBa),

MaT

= nEσT( 43ρ∗RvaB− q∗a

R+ π∗ab

RvBb),

as shown below. Here nE

is the free electron number density, and σT

isthe Thomson cross-section. Note that beyond linear order there is energytransfer, i.e. U

T= 0.

Using eqs. (20)–(23) in (24), we find that, to O(εv2, v3), for cdm

ρC

+ ΘρC

+ ρCdiv v

C

= −(ρCv2C)· − 4

3v2CΘρ

C− va

CDaρC − 2ρ

CAav

aC, (25)

vaC

+ 13Θva

C+ Aa

= AbvbCua − σabv

bC

+ [ω, vC]a − vb

CDbv

aC, (26)

and for baryonic matter

ρB

+ Θ(1 + wB)ρB

+ (1 + wB)ρBdiv v

B

= −[(1 + wB)ρBv2B]· − 4

3v2BΘ(1 + w

B)ρB− va

BDa[(1 + w

B)ρB]

− 2(1 + wB)ρBAav

aB− n

EσT( 43ρ∗Rv2B− q∗a

RvBa) , (27)

(1 + wB)vaB

+ ( 13 − c2

B)Θva

B+ (1 + w

B)Aa

+ ρ−1B

DapB

+ ρ−1B

nEσT(ρ∗RvaB− q∗a

R)

= (1 + wB)AbvbBu

a − (1 + wB)σabvbB + (1 + w

B) [ω, v

B]a

− (1 + wB)vbBDbv

aB

+ c2B(1 + w

B)(div v

B)vaB− ρ−1

BnEσTπ∗abR

vBb , (28)

where c2B≡ p

B/ρB

(this equals the adiabatic sound speed only to linearorder). These are the nonlinear energy conservation and relative velocityequations for matter . Linearization reduces the right hand sides to zero,dramatically simplifying the equations. The conservation equations for themassless species (radiation and neutrinos) are found below.

3. NONLINEAR THOMSON SCATTERING

In covariant Lagrangian kinetic theory [30] the photon 4-momentumis split as

pa = E(ua + ea) , eaea = 1 , eaua = 0 ,

1084 Maartens

where E = −uapa is the energy and ea = p〈a〉/E is the direction, asmeasured by a comoving (fundamental) observer. The photon distributionfunction is expanded in covariant harmonics

f(x, p) = f(x,E, e) = F + Faea + Fabe

aeb + · · · =∑≥0

FAl(x,E)e〈Al〉,

where eA ≡ ea1ea2 · · · ea and

Fa···b = F〈a···b〉 ⇔ Fa···b = F(a···b) , Fa···bub = 0 = Fa···bchbc .

The first 3 multipoles arise from the radiation energy-momentum tensor,

T abR

(x) =∫

papbf(x, p)d3p = ρ∗Ruaub + 1

3ρ∗Rhab + 2q∗(a

Rub) + π∗ab

R.

The radiation brightness multipoles are

Πa1···a =∫

E3Fa1···adE ,

so that (dropping asterisks) Π = ρR/4π, Πa = 3qa

R/4π and Πab = 15πab

R/8π.

These multipoles define the temperature fluctuations [31].The Boltzmann equation is

dfdv≡ pa

∂f

∂xa− Γabcpbpc

∂f

∂pa= C[f ] = b + bae

a + babeaeb + · · · ,

where the collision term C[f ] determines the rate of change of f due toemission, absorption and scattering processes. For simplicity, the effectsof polarization are neglected (see Ref. 25 for a linear treatment), and

C[f ] = σTnEEB[f(x, p)− f(x, p)],

where EB

= −pauaB is the photon energy relative to the baryonic frame uaB,

and [32]

f(x, p) =3

16π

∫f(x, p′) [1 + (ea

Be′Ba)

2]dΩ′B. (29)

Here e′Ba is the initial and ea

Bis the final direction, so that

p′a = EB(uaB

+ e′aB

), pa = EB(uaB

+ eaB),


where we have used E′B

= EB, which follows since the scattering is elastic.

The exact forms of the photon energy and direction in the baryonic frameare

EB

= EγB(1− va

Bea),

eaB

=1

γB(1− vc

Bec)

[ea + γ2B(vbBeb − v2

B)ua + γ2

B(vbBeb − 1)va

B].

Since the baryonic frame will move non-relativistically relative to thefundamental frame in all cases of physical interest, it is sufficient to lin-earize only in v

B, and not in the other quantities. Thus we drop terms in

O(εv2B, v3B) but do not neglect terms that are O(ε0v2

B, εv

B) or O(ε2) relative

to the background. It follows from eq. (29) that

4π∫

fE3BdE

B= (ρ

R)B

+ 34 (πab

R)BeBaeBb ,

where the dynamic radiation quantities are evaluated in the baryonicframe. Transforming back to the fundamental frame, we find, toO(εv2

B, v3B),

(ρR)B

= ρR[1 + 4

3v2B]− 2qa

RvBa ,

(πabR

)B

= πabR

+ 2vBcπ

c(aR

ub) − 2q〈aR

vb〉B

+ 43ρRv

〈aB

vb〉B

.

Using the above equations and various identities [23], and defining theenergy-integrated scattering multipoles

KA =∫

E2bAdE ,

we find that to O(εv2B, v3B)

K = nEσT[ 43Πv2

B− 1

3ΠavBa], (30)

Ka = −nEσT[Πa − 4Πva

B− 2

5ΠabvBb], (31)

Kab = −nEσT[ 910Πab − 1

2Π〈avb〉B− 3

7ΠabcvBc − 3Πv〈a

Bvb〉B

], (32)

Kabc = −nEσT[Πabc − 3

2Π〈abvc〉B− 4

9ΠabcdvBd], (33)

and, for > 3:

KA = −nEσT

[ΠA −Π〈A−1va〉

B−

( + 12 + 3

)ΠAav

Ba

]. (34)

1086 Maartens

Equations (30)–(34) are a nonlinear generalisation of the linearised Thom-son scattering results [32]. They show clearly the coupling of baryonic bulkvelocity to the radiation multipoles, arising from local nonlinear effects inThomson scattering.

The multipoles of E−1df/dv are derived in [30] and [23], using differ-ent methods. The result is

F〈A〉 −13

ΘEF ′A + D〈aFA−1〉 +( + 1)(2 + 3)

DaFaA

− ( + 1)(2 + 3)

E−(+1) [E+2FaA ]′Aa − E[E1−F〈A−1 ]

′Aa〉

− ωbεbc(aFA−1)c − ( + 1)( + 2)

(2 + 3)(2 + 5)E−(+2)[E+3FabA ]

′σab

− 2(2 + 3)

E−1/2[E3/2Fb〈A−1 ]′σa〉

b − E−1[E2−F〈A−2 ]′σa−1a〉 ,

where a prime denotes ∂/∂E. This result is exact and holds for any photonor (massless) neutrino distribution in any spacetime. Integrating, it leadsto

KA = Π〈A〉 +43

ΘΠA + D〈aΠA−1〉 +( + 1)(2 + 3)

DbΠbA

− ( + 1)(− 2)(2 + 3)

AbΠbA + ( + 3)A〈aΠA−1〉

− ωbεbc(aΠA−1)c − (− 1)( + 1)( + 2)

(2 + 3)(2 + 5)σbcΠbcA

+5

(2 + 3)σb〈aΠA−1〉b − ( + 2)σ〈aa−1ΠA−2〉 . (35)

For decoupled neutrinos, KAN

= 0, while for (unpolarised) photons un-dergoing Thomson scattering, the left hand side of eq. (35) is given byeqs. (30)–(34), which are exact in the kinematic and dynamic quantities,but first order in the baryonic bulk velocity. These equations show the cou-pling of acceleration, vorticity and shear to the radiation multipoles thatarises at nonlinear level.

The monopole and dipole of eq. (35) imply photon conservation ofenergy and momentum density (to O(εv2

B, v3B)):

ρR

+ 43Θρ

R+ Daq

aR

+ 2AaqaR + σabπabR

= nEσT( 43ρRv

2B− qa

RvBa), (36)

q〈a〉R

+ 43Θqa

R+ 4

3ρRAa + 1

3DaρR

+ DbπabR

+ σabqbR− [ω, q

R]a + Abπ

abR

= nEσT( 43ρRv

aB− qa

R+ πab

RvBb) . (37)


The quadrupole evolution equation is

π〈ab〉R

+43

ΘπabR

+815

ρRσab +

25

D〈aqb〉R

+8π35

DcΠabc

+ 2A〈aqb〉R− 2ωcεcd(aπb)dR

+27σc〈aπb〉c

R− 32π

315σcdΠabcd

= −nEσT

[910

πabR− 1

5q〈aR

vb〉B− 8π

35Πabcv

Bc −25ρRv〈aB

vb〉B

], (38)

and the higher multipoles ( > 3) evolve according to

Π〈A〉 +43

ΘΠA + D〈aΠA−1〉 +( + 1)(2 + 3)

DbΠbA

− ( + 1)(− 2)(2 + 3)

AbΠbA + ( + 3)A〈aΠA−1〉

− ωbεbc(aΠA−1)c − (− 1)( + 1)( + 2)

(2 + 3)(2 + 5)σbcΠbcA

+5

(2 + 3)σb〈aΠA−1〉b − ( + 2)σ〈aa−1ΠA−2〉

= −nEσT

[ΠA −Π〈A−1va〉

B−

( + 12 + 3

)ΠAav

Ba

]. (39)

For = 3, the second term in square brackets on the right of eq. (39) mustbe multiplied by 3

2 . The temperature fluctuation multipoles are determinedby the radiation brightness multipoles ΠA [23].

These equations show in a transparent and covariant form preciselywhich physical effects are directly responsible for the evolution of cmb

anisotropies in an inhomogeneous universe. They apply at second andhigher-order, and are readily specialised to the linear case. They showhow the matter generates anisotropies: directly through interaction withthe radiation, as encoded in the Thomson scattering terms on the right ofeqs. (36)–(39); and indirectly through the generation of inhomogeneitiesin the gravitational field via the field equations (1)–(12) and the evolutionequation (28) for the baryonic velocity va

B. This in turn feeds back into

the multipole equations via the kinematic quantities, the baryonic veloc-ity va

B, and the spatial gradient DaρR in the dipole equation (37). The

coupling of the multipole equations themselves provides an up and downcascade of effects, shown in general by eq. (39). Power is transmittedto the -multipole by lower multipoles through the dominant (linear) dis-tortion term D〈aΠA−1〉, as well as through nonlinear terms coupled to

1088 Maartens

the 4-acceleration (A〈aΠA−1〉), baryonic velocity (v〈aB

ΠA−1〉), and shear(σ〈aa−1ΠA−2〉). Simultaneously, power cascades down from higher mul-tipoles through the linear divergence term (div Π)A , and the nonlinearterms coupled to Aa, va

Band σab. The vorticity coupling does not trans-

mit across multipole levels.

4. CONCLUSION

A range of nonlinear effects is identified via a systematic covariantanalysis. These include the following.• Nonlinear relative velocity corrections, as exemplified in the dynamic

quantities in eqs. (16)–(19) and in the bulk velocity equations (26)and (28).

• Nonlinear Thomson scattering corrections affect the baryonic and ra-diation conservation equations, entailing a coupling of the baryonicbulk velocity va

Bto the radiation energy density, momentum density

and anisotropic stress. The evolution of the radiation quadrupole πabR

also acquires nonlinear Thomson corrections, which couple vaB

to theradiation dipole qa

Rand octopole Πabc. Linearization, by removing

these terms, has the effect of removing the nonlinear contribution ofthe radiation multipoles ΠA±1 to the collision multipole KA .

• Nonlinear kinematic corrections introduce additional acceleration andshear terms. Vorticity corrections are purely nonlinear, i.e. a linearapproach could give the false impression that vorticity has no directeffect at all on the evolution of cmb anisotropies. However, for veryhigh , i.e. on very small angular scales, the nonlinear vorticity termcould in principle be non-negligible. The general evolution equation(39) for the radiation brightness multipoles ΠA shows that five suc-cessive multipoles, i.e. for − 2, · · ·, + 2, are linked together in thenonlinear case. The 4-acceleration Aa couples to the ± 1 multipoles,the vorticity ωa couples to the multipole, and the shear σab couples tothe ±2 and multipoles. All of these couplings are nonlinear, exceptfor = 1 in the case of Aa, and = 2 in the case of σab. These lattercouplings that survive linearisation are shown in the dipole equation(37) (i.e. ρ

RAa) and the quadrupole equation (38) (i.e. ρ

Rσab). The

latter term drives Silk damping during the decoupling process. Thedisappearance of most of the kinematic terms upon linearisation isfurther reflected in the fact that the linearised equations link onlythree successive moments, i.e. , ± 1.

• A crucial feature of the nonlinear kinematic terms is that some of themscale like for large , as already noted in the case of vorticity. There


are no purely linear terms with this property, which has an importantconsequence, i.e. that for very high multipoles (corresponding to verysmall angular scales in cmb observations), certain nonlinear termsmay reach the same order of magnitude as the linear contributions.(Note that the same effect applies to the neutrino background.) Therelevant nonlinear terms in eq. (39) are (for 1):

− [ 14σbcΠbcA + σ〈aa−1ΠA−2〉

+ A〈aΠA−1〉 + 12AbΠ

bA + ωbεbc〈aΠA−1〉c].

Any observable imprint of this effect will be made after last scattering.In the free-streaming era, it is reasonable to neglect the vorticity rel-ative to the shear. We can remove the acceleration term by choosingua as the dynamically dominant cold dark matter frame (i.e. choosingvaC

= 0). It follows that the nonlinear correction to the rate of changeof the linearised fluctuation multipoles is

δ(ΠA) ∼ [ 14σbcΠbcA + σ〈aa−1ΠA−2〉] for 1 . (40)

The linear solutions for ΠA and σab can be used in eq. (40) to estimatethe correction to second order. Its effect on observed anisotropies willbe estimated by integrating δ(ΠA) from last scattering to now.• Further quantitative analysis of the new nonlinear effects in [20,23] is

needed to identify more clearly which effects could be observationallysignificant. This would also clarify the relationship between variousnonlinear effects that have been derived under different assumptionsand using different approaches [14,15,17,20,23,24].

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge the inspiration provided by the energy,enthusiasm and ideas of George Ellis, whose influence is widely felt inrelativistic cosmology.

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Nonlinear Effects in the Cosmic Microwave Background