General primordial cosmic perturbation

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PHYSICAL REVIEW D, VOLUME 62, 083508

General primordial cosmic perturbation

Martin Bucher,* Kavilan Moodley,† and Neil Turok‡

DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingd~Received 22 April 1999; published 26 September 2000!

We consider the most general primordial cosmological perturbation in a universe filled with photons, bary-ons, neutrinos, and a hypothetical cold dark matter~CDM! component within the framework of linearizedperturbation theory. We present a careful discussion of the different allowed modes, distinguishing modeswhich are regular at early times, singular at early times, or pure gauge. As well as the familiar growing anddecaying adiabatic modes and the baryonic and CDM isocurvature modes, we identify twoneutrino isocurva-ture modes. In the first, the ratio of neutrinos to photons varies spatially but the net density perturbationvanishes. In the second the photon-baryon plasma and the neutrino fluid have a spatially varying relative bulkvelocity balanced so that the net momentum density vanishes. Possible mechanisms which could generate thetwo neutrino isocurvature modes are discussed. If one allows the most general regular primordial perturbation,all quadratic correlators of observables such as the microwave background anisotropy and matter perturbationsare completely determined by a 535, real, symmetric matrix-valued function of comoving wave number. In acompanion paper we examine prospects for detecting or constraining the amplitudes of the most generalallowed regular perturbations using present and future CMB data.

PACS number~s!: 98.80.2k

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I. INTRODUCTION

A key challenge of modern cosmology is understandthe nature of the primordial fluctuations that eventually ledthe formation of large scale structure in our universe. Opossibility is that the fluctuations were generated durinperiod of inflation prior to the radiation dominated era of thot big bang. As inflation ended the fluctuations would thhave been imprinted as initial conditions for the radiationon scales far beyond the Hubble radius. The second possity is that the structure was generated through some camechanism operating within the standard big bang radiaand matter eras. In this paper we focus on the first optthat the fluctuations were imprinted early in the radiationas linear fluctuations in the metric and in the matter aradiation content.

For several good reasons the possibility that the primdial perturbations were adiabatic has been the focus of minterest to date. If the relative abundances of different pticle species were determined directly from the Lagrangdescribing local physics, one would expect those abundaratios to be spatially constant because all regions of theverse would share an identical early history, independenthe long wavelength perturbations. The stress-energy prein the universe would then be characterized on large scby a single, spatially uniform equation of state. Such flucttions are termed adiabatic and are the simplest possibilityperturbing the matter content and the geometry of the uverse. They are also naturally predicted by the simplestflationary models@1#, although there also exist more complex models giving other types of perturbations@2#. For a

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]

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recent discussion see Ref.@3#.Nevertheless, there is noa priori reason why the situation

could not be more complicated with abundance ratios vaing from place to place. Perturbations of this sort are knoas isocurvature,or sometimesentropy,perturbations. Moststudies of isocurvature perturbations have examined thesibility that the primordial perturbations were entirely isocuvature with a vanishing primordial adiabatic component ahave sought to explore whether such pure isocurvature mels could explain the observed structure in the unive@4–7#.

In this paper we adopt a more phenomenologicalproach, which we believe is now warranted by the prospof upcoming precision measurements of the cosmic micwave background~CMB! anisotropy on small scales. Grounbased and balloon borne telescopes and the MicrowAnisotropy Probe~MAP! and Planck satellites will providevery detailed measurements of the primordial fluctuatio@8,9#. Most work on how to interpret this data has focusedparameter estimation starting from the assumption thatinitial perturbations were generated by an inflationary mospecified by a small number of undetermined free parame@10#. Adiabatic perturbations, characterized by an amplituand spectral index, as well as tensor fluctuations also cacterized by two parameters are usually assumed, and bon these assumptions a host of cosmological paramesuch asV total , Vb , VL , h, Nn , are to be inferred from theobserved CMB multipole moments. While the prospectssuch measurements are beguiling, they rely heavily onsumptions regarding the form of the primordial perturbtions. We feel that those assumptions are worth checkagainst the data using an approach that assumes neitheflation nor any other favorite theoretical model.

For learning about the fundamental physics responsfor structure formation, determining whether the primordfluctuations were in fact Gaussian and adiabatic is at leas

©2000 The American Physical Society08-1

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MARTIN BUCHER, KAVILAN MOODLEY, AND NEIL TUROK PHYSICAL REVIEW D 62 083508

important and interesting as measuring the values of coslogical parameters. For this purpose the relevant questionot whether primordial isocurvature perturbations offer aable alternative to adiabatic perturbations, which has bthe focus of prior work, but rather how and to what extecan observations constrain the presence of isocurvamodes. Rather than considering a particular isocurvamodel, it seems appropriate to consider the most generalmordial perturbation possible without adding new physicsthe hot big bang era. In other words, we assume a univfilled with neutrinos, photons, leptons and baryons, ancold dark matter~CDM! component and then try to placconstraints on the amplitudes of all possible perturbatmodes.

As mentioned above, we shall need to make a simplifyassumption to limit the parameter space of possible perbations to reasonable size. We shall assume that the fluctions were indeed primordial—that is, that the perturbatmodes were excited by physics operating at a very eepoch preceding the hot radiation dominated era and thanew dynamics influenced the perturbations at late timHere ‘‘late’’ means well before decoupling, so that whatevdecaying modes may have been excited earlier had a chto decay before influencing the observed structure in our cmic microwave sky. This assumption excludes cosmic demodels~i.e., strings, global textures, etc.! @11# of structureformation where a detailed understanding of the dynamicthe order parameter field is required to determine the unetime correlations at late times@12,13#.

How should the most general perturbation be characized? This question has historically generated some cosion in the literature. The standard terminology refers‘‘growing,’’ ‘‘decaying,’’ and ‘‘gauge’’ modes, the latter re-ferring to modes affected by general coordinate transfortions preserving the chosen gauge condition. In this papein most work on cosmological perturbations we shall usynchronous gauge. The variables in other common gau~e.g., Newtonian gauge! are merely linear combinations othe synchronous gauge variables and their time derivatiIn the synchronous gauge, the two gauge modes for scperturbations are easily identified. The first correspondstime-independent spatial reparametrizations of the constime hypersurfaces. In this mode the metric perturbationconstant and the matter is unperturbed. The second cosponds to deformations of these hypersurfaces through atially dependent shift in the time direction. For the lattgauge mode, which diverges at early times, both the mavariables and the metric are perturbed. Indeed this modebe regarded as a shift in the time of the big bang singularThe remaining modes are then defined modulo the two gamodes.

We characterize the remaining modes as either ‘‘regulor ‘‘singular,’’ according to their behavior as the time sinthe big bang tends to zero. This terminology is preferablethe standard one because it includes constant modes suthe neutrino velocity mode we shall discuss. In this papershall treat only the ‘‘regular’’ modes~i.e., those regular up tothe gauge modes!. There are several reasons for this. Singlar modes are necessarily decaying as one proceeds for

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in time away from the big bang, so if they are presentsome very early time with small amplitude so that a pertbative treatment is valid, they quickly become irrelevant.second reason is that even if the perturbation amplitudsmall, the perfect fluid approximation breaks down at eatimes for the singular modes. Higher moments in the Bomann hierarchy become progressively more important asgoes back in time, and specifying the initial conditions ivolves specifying an infinite number of constants. The pfect fluid approximation seems essential to a simple specation of primordial initial perturbations. Of coursedecaying modes might be produced by some late time phics ~such as cosmic defects! operating well after the bigbang, but our point here is that it would be very difficultcharacterize them without introducing explicit source termIn contrast, the ‘‘regular’’ modes are completely charactized by specifying the leading terms in a power series expsion in conformal time of the low moments of the phaspace density—that is, the density and velocity of the flui

Having characterized the regular perturbations byleading terms in the power series for the metric and fldensities and velocities, we note that any quadratic obsable ~e.g., the matter power spectrum or the cosmic micwave anisotropy power spectrum! is then completely determined by a primordial power spectral matrix, which raththan a single function ofk as it would be for growing modeadiabatic perturbations is a 535, real, symmetric matrixfunction of k. The off-diagonal elements establish corretions between the modes. As long as only quadratic obsables are considered, no assumption of Gaussianity isquired.

The counting arises as follows. Each cosmological fluiddescribed by two first order equations, so each fluid intduces two new perturbation modes. However, in the schronous gauge, as mentioned, there is one gauge modfecting the fluid perturbations. For a single fluid theretherefore just one regular, growing mode perturbation, ano physical~i.e., non-gauge! decaying mode. A convenienway to deal with the gauge mode is to identify it with thvelocity of the cold dark matter. By a coordinate choice thmay be chosen initially to be zero. If it is assumed thatcold dark matter couples to the other fluids only via gravithere is no scattering term to consider, and with coordinachosen so that the velocity is initially zero, it remains soall times. If we now introduce photons and baryons, fonew modes arise. The first is an adiabatic decaying moThere are also the baryon isocurvature and cold dark maisocurvature modes, where the initial conditions contequal and opposite perturbations of the photon densitythe baryon or cold dark matter densities, respectively. Tfourth new mode is that in which the photon and baryfluids have a relative velocity that diverges at early times athe fluid approximation breaks down. So far we have thregular modes. Let us now introduce neutrinos into the pture. We shall imagine we are setting up the perturbatiafter neutrino decoupling (T;MeV, t; seconds!. For ourpurposes there is no distinction between the different ntrino species or between neutrinos and antineutrinos, swe are only interested in how perturbations in the neutr

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GENERAL PRIMORDIAL COSMIC PERTURBATION PHYSICAL REVIEW D62 083508

fluid affect cosmological observations of the density andcrowave background today. Two new perturbation modesintroduced, the first a neutrino isocurvature density perturtion and the second a neutrino isocurvature velocity perbation. In the latter, we can arrange the neutrino and phobaryon fluids to have equal and opposite momentum denIn the approximation that we ignore the collision term copling neutrinos to the photon-baryon fluid, which is valafter neutrino decoupling, we shall show there is no divgence in this mode at early times.

The neutrino isocurvature velocity perturbation mayconsidered as primordial as long as one takes primordiamean ‘‘generated after one second’’ but well before photbaryon decoupling. Of course similar remarks apply tocold dark matter, baryon isocurvature or neutrino isocurture perturbations. Namely, if we go back far enough in cmic history, where the various conservation laws for baryor lepton number break down or where the cold dark mawas initially generated or reached thermal equilibrium, tha description of the form used here would also break doSince one second is still quite early, and certainly well befphoton-baryon decoupling, we regard it admissible to csider the possibility of ‘‘primordial’’ neutrino velocity perturbations. Possible mechanisms for generating shall becussed in a separate paper@28#.

A primeval baryon isocurvature~PBI! model was intro-duced by Peebles@4# in which a universe with just baryonsradiation, and neutrinos is assumed and primordial pertutions in the baryon-to-photon ratio are assumed. Sinceearly times the baryons contribute negligibly to the total dsity, such perturbations lack an adiabatic, or curvature, cponent at early times. Compared to anV51 CDM model,the PBI model gives~1! lower small-scale relative peculiavelocities,~2! greater large-scale flow velocities,~3! earlierre-ionization, and~4! earlier galaxy and star formation@6#.The consequences of the PBI model and comparisonobservations have been studied by a number of authors@15–20,22#.

Bond and Efstathiou@7,5# have considered a CDM isocuvature model in which the CDM-to-photon ratio varies sptially. A possible mechanism for generating such perturtions arises in models with axion dark matter in which scinvariant fluctuations of the axion fieldA(x) are convertedinto density fluctuations after the axion field acquires a min the QCD phase transition@21#. Under the assumption thaquantum fluctuations during inflation impart a scale frspectrum of fluctuations to the axion field, so thatdA(k);k23/2, and dA(k)!A, the mean axion field displacemefrom the vacuum, a scale-free spectrum of Gaussian fluctions in the axion-to-photon ratio is generated. The resultspectrum of density fluctuations today has the same polaw on large scales as for adiabatic fluctuations with a scfree spectrumPr(k);k1, but compared to adiabatic, scalfree perturbations the turnover toPr(k);k23 power lawbehavior on small scales occurs on a larger scale forisocurvature case. For the amplitude of density perturbatnormalized on small scales~for example, usings8), thisgives about 30 times more power in the matter perturbation large scales and entails an excessive CMB anisotrop

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large angles. This work has been extended to consider mtures of CDM isocurvature and adiabatic fluctuations, lodensity universes, and the predictions compared with mrecent CMB data@23,24#.

The neutrino isocurvature modes discussed here allowa spatially varying relative density of photons and neutrinand a relative velocity between the photon and neutrino coponents as well. For the neutrino isocurvature density mothe total density perturbation vanishes but the relative dsity of neutrinos and photons varies spatially. On superhzon scales, the neutrinos and photons evolve similarlyupon entering the horizon the neutrinos free stream, deoping nonvanishing higher moments of the neutrino phspace densityFn l(k), while because of Thomson scatterinthe photons continue to behave much as a perfect flThese distinct behaviors subsequently generate perturbain the total density. For the neutrino isocurvature velocmode, the rest frames of the neutrino and the photon fludo not coincide. The relative velocities are such that initiathe perturbation in the total momentum density vanishesthis last condition were not satisfied, the metric perturbatgenerated by this mode would diverge at early times, rending the mode a singular mode, which according to the dcussion above should be excluded. But the lack of divgence owing to this cancellation makes this an admissregular perturbation mode. The neutrino isocurvature msolutions are implicit in the work of Rebhan and Schwa@24# and of Challinor and Lasenby@25#, but their implica-tions were not explored.

A possible obstacle to exciting neutrino isocurvatumodes arises at early times from processes in which nenos are generated or scattered. If the neutrino chemicaltentials vanish so that there are in each generation precas many neutrinos as antineutrinos, a spatially varying rtive density can only be established at a temperature sciently low that the processes turning photons into neutriantineutrino pairs and vice versa had already been froout—that is, below a few MeV. Nonvanishing chemical ptentials for the various neutrino species can protect variatiin the ratio (rn /rg) from erasure by processes involvingnnannihilation, and observational constraints that would rout neutrino chemical potentials of the required magnituare lacking.L-violating processes mediated through spharons, unsuppressed at temperatures above the electrophase transition, can readjust the neutrino chemical potials mne

, mnm, and mnt

and, moreover, can convert leptoasymmetries in the neutrino sector into baryon asymmebut to the extent that the neutrino chemical potentials satmne

1mnm1mnt

50, no tendency favoring sphalerons ovanti-sphalerons is introduced and the net effect of these etroweak processes vanishes. Recall that the neutrino odensity is proportional tomne

21mnm

21mnt

2. Similarly, forthe neutrino isocurvature velocity mode, scattering of neunos with other components dampens this mode at very etimes, and for this mode to be relevant there must exisprocess capable of exciting it after this dampening effectfrozen out.

There exist many candidate mechanisms that might g

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erate these neutrino isocurvature modes. The neutrinosity isocurvature mode could be generated during inflatiothe theory included a light scalar field carrying lepton nuber, with mass much smaller than the Hubble constant duinflation @26#. During inflation this field would be excited buwould contribute negligibly to the density of the universAfter inflation, when the Hubble constant fell below thmass of the scalar field, it would oscillate and decay, proding a lepton asymmetry. We would expect the neutrchemical potential to be proportional to the scalar field valso the most natural possibility would be Gaussian, scalevariant perturbations in the neutrino chemical potential, wthe density perturbation in the neutrinos being proportioto the square of the chemical potential.

The neutrino velocity mode could have been excitedthe decay of relics such as cosmic strings, walls, or sustrings, surviving until after the neutrinos had decoupled athen decaying into neutrinos. The neutrino fluid producedsuch processes would be perturbed both in its densityvelocity, and these perturbations would be isocurvaturecharacter@28#. Another possibility for exciting the modearises from magnetic fields frozen into the plasma whstress gradients impart a velocity to the photon-leptbaryon plasma but not to the neutrino component.

II. IDENTIFYING THE MODES

We now turn to identifying the possible perturbatiomodes using synchronous gauge with the line element

ds25a2~t!@2dt21~d i j 1hi j !dxidxj #. ~1!

For spatial dependence of a given wave numberk, we define

hi j ~k,t!5eik•xF ki k jh~k,t!1S ki k j21

3d i j D6h~k,t!G .

~2!

Our discussion generally follows the notation of Ref.@27#.The linearized Einstein equations are

k2h21

2Hh5~4pGa2!dT0

0 ,

k2h5~4pGa2!~ r1 p!u,

h12Hh22k2h52~8pGa2!dTii ,

h16h12H~ h16h !22k2h52~24pGa2!~ r1 p!s~3!

where we define (r1 p)u5 ik jT0j and (r1 p)s52( ki k j

2 13 d i j )Ti

j . We defineH5a/a. With only a single fluid,u issimply the divergence (¹•v).

AssumingN fluids, labeled (J51, . . . ,N), one may re-write the above as

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JVJdJ ,

k2h53

2H 2(

JVJ~11wJ!uJ ,

h12Hh22k2h529H 2(J

VJcsJ2 dJ ,

h16h12H~ h16h !22k2h5212H 2Vnsn , ~4!

where wJ5pJ /rJ and cs25]pJ /]rJ . In the last equation

only the neutrino contribution to the anisotropic stress iscluded. At early times only the neutrino contribution is reevant, but later when the photons and baryons begin tocouple the photon quadrupole moment must alsoincluded.

For the photons, the equations of motion at early timbefore the baryonic component of the fluid was significaare

dg14

3ug1

2

3h50,

~5!

ug21

4k2dg50.

Similarly, for the neutrinos, after neutrino decoupling at teperatures of;1 MeV, we have

dn14

3un1

2

3h50,

un21

4k2dn1k2sn50,

sn52

15@2un1h16h#2

3

10kFn3 , ~6!

wheresn5Fn2/2 is the quadrupole moment of the neutrinphase space density andFn l is the l th multipole, in the no-tation used below.

The photon and neutrino evolution equations above dionly in the presence of an anisotropic stress termsn . Pho-tons, because of their frequent scattering by charged lepand baryons, at early times are unable to develop a quapole moment in their velocity distribution. Neutrinos, on thother hand, develop significant anisotropic stresses ucrossing the horizon. The addition of an extra degree of frdom that would result from the equation forsn is avoided bysettingsn50 at t50.1

1If we were to consider the physical decaying mode withh}t21, we would find this condition cannot be satisfied. As dcussed above, decaying modes are inevitably associated withbreakdown of the fluid approximation at early times.

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For a CDM component, the equations of motion are

dc1uc11

2h50,

uc1Huc50. ~7!

At later times, whenVb becomes comparable toVg , thephoton equations of motion are modified as follows:

dg14

3ug1

2

3h50,

ug2k2

4dg1anesT~ug2ub!50. ~8!

Herea is the scale factor,ne is the electron density, andsTis the Thomson scattering cross section. The baryon etions of motion are

db1ub11

2h50,

ub1Hub14

3

Vg

VbanesT~ub2ug!50. ~9!

At early times, the characteristic time for the synchronizatof the photon and baryon velocitiestbg'1/(nesT) is smallcompared to the expansion timetexp'at and to the oscilla-tion periodtosc'(at/k). ~We use units where the speedlight c51.! Any deviation of (ug2ub) from zero rapidlydecays away. This may be seen by subtracting the seconEqs. ~9! from the second of Eqs.~8! and regardingHub1 1

3 k2dg as a forcing term. In the limitsT→`, one obtainsub5ug—in other words, a tight coupling between the barons and the photons. Therefore, at early times we mayub5ug5ugb .

In the tight coupling approximation the evolution eqution for ugb is obtained by adding the second equationEqs. ~8! multiplied by 4

3 Vg to the second equation of Eq~9! multiplied by Vb , so that the scattering terms cancgiving

S 4

3Vg1VbD ugb52VbHugb1

1

3Vgk2dg ~10!

and the baryon and photon density contrasts evolve accing to

db52ugb21

2h,

dg524

3ugb2

2

3h. ~11!

In the absence of baryon isocurvature perturbations,dg5 4

3 db , making one of these equations redundant, but bequations are required for the most general type of pertution.

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Although an excellent approximation early on, the tigcoupling assumption breaks down at later times as the ptons and baryons decouple, and for a more accurate destion the two equations in Eqs.~8! for the time derivatives ofthe monopole and dipole moments of the velocity of tphoton phase space distribution must be replaced by thelowing infinite hierarchy of equations for the time derivtives of higher order moments of the photon distributifunction as well:

dg524

3ug2

2

3h,

ug5k2S dg

42

Fg2

2 D1anesT~ub2ug!,

Fg258

15ug2

3

5kFg31

4

15h1

8

5h2

9

10anesTFg2 ,

Fg l5k

2l 11@ lF g( l 21)2~ l 11!Fg( l 11)#2anesTFg l

~12!

where l>3 for the last equation. Initially, ast→0, Fg l50for l>2. If this condition were relaxed, an infinite numberdecaying modes would appear. The fluid approximationthe photons, Eqs.~8!, is obtained using only the first two othe above equations combined with the approximationFg250.

To describe the neutrinos during and after horizon croing requires a Boltzman hierarchy fordn , un , Fn l identicalto the one above except with the Thomson scattering teomitted. This assumes that neutrino masses are irreleva

Before solving the Einstein equations~4!, we first identifythe two gauge modes resulting from the residual gauge fdom remaining within synchronous gauge. A general infitesimal gauge transformation considered to linear ordercoordinate transformationxm→x8m(x) where xm5x8m

1em(x). From the transformation rule for tensors

gmn8 ~x8!5]xj

]x8m

]xh

]x8ngjh~x!, ~13!

it follows that

dgmn5ej]gmn

(0)

]xj1ej

,mgjn(0)1gmj

(0)ej,n ~14!

wheregmn(0) is the unperturbed, zeroth-order metric. Applyin

Eq. ~14! to the metric ds25a2(t)@2(12h00)dt2

12h0idtdxi1(d i j 1hi j )dxidxj #, where

t5t81T~x,t!, x5x81S~x,t! ~15!

with S andT regarded as infinitesimal, one obtains

dh00522a

aT~x,t!22T~x,t!,

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dh0i52T,i~x,t!1Si~x,t!,

dhi j 5Si , j1Sj ,i12a

ad i j T. ~16!

For the density contrast for a perfect fluid component labeby a, one obtainsd(da)523(11wa)HT. After a gaugetransformation the velocity is shifted bydv52S(x,t) for allcomponents.

The synchronous gauge conditiondh0050 implies thatT(x,t)5A(x)/a(t), anddh0i50 implies that

S~x,t!5B~x!1¹A~x!E dt

a~t!. ~17!

Therefore

dhi j 52a

a2A~x!d i j 12A,i j E dt

a~t!1Bi , j1Bj ,i . ~18!

For a radiation-dominated universe@with a(t)5t#, con-sidering for the moment only a single wave number, we fithat2

dhi j 5F S 2

t2 d i j 22kikj ln~t! DA~k!12Bkikj Geik•x ~19!

or

h5AF 6

t2 22k2 ln~t!G12B, h5AF21

t2 G . ~20!

To obtain a more accurate power series expansion closthe time of matter-radiation equality and to considerbaryon and CDM isocurvature modes, we assume a sfactor evolution for a universe filled with matter and radtion: a(t)5t1t2. At matter-radiation equalityaeq51/4 andteq5(A221)/2. SinceH5(2t11)/(t21t) has a pole att521, the resulting power series solutions are expectediverge beyond the unit circle.

For the matter-radiation universe, witha(t)5t1t2, Eq.~20! is modified to become

h5AF 6~112t!

t2~11t!222k2 lnS t

11t D G12B,

h52AF ~112t!

t2~11t!2G . ~21!

We now present the power series solutions.Rg and Rn

represent the fractional contribution of photons and neutrito the total density at early times deep within the radiatdominated epoch.~We ignore possible effects due to nonvnishing neutrino masses.! We also assume that we are co

2Note that the gauge modeA disagrees with Eqs.~94! and~95! ofRef. @27#, which are incorrect.

08350

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sidering the perturbations after the annihilation of electroand positrons, so that the latter have dumped their eneinto the photon background. ForNn species of massless neu

trinos we define R5 78 Nn( 4

11 )4/3 and we haveRg5(11R)21 andRn5R(11R)21.

Adiabatic growing mode:

h51

2k2t2,

h512514Rn

12~1514Rn!k2t2,

dc521

4k2t2,

db521

4k2t2,

dg521

3k2t2,

dn521

3k2t2,

uc50,

ugb521

36k4t3,

un521

36F2314Rn

1514RnGk4t3,

sn52

3~121Rn!k2t2. ~22!

Baryon isocurvature mode:

h54Vb,0t26Vb,0t2,

h522

3Vb,0t1Vb,0t

2,

dc522Vb,0t13Vb,0t2,

db5122Vb,0t13Vb,0t2,

dg528

3Vb,0t14Vb,0t

2,

dn528

3Vb,0t14Vb,0t

2,

uc50,

ugb521

3Vb,0k

2t2,

8-6

th

geM

ieocrly

hehenrfectity, in


un521

3Vb,0k

2t2,

sn522Vb,0

3~2Rn115!k2t3. ~23!

There is no regular baryon velocity mode because oftight coupling of the baryons to the photons.

CDM isocurvature mode:

h54Vc,0t26Vc,0t2,

h522

3Vc,0t1Vc,0t

2,

dc5122Vc,0t13Vc,0t2,

db522Vc,0t13Vc,0t2,

dg528

3Vc,0t14Vc,0t

2,

dn528

3Vc,0t14Vc,0t

2,

uc50,

ugb521

3Vc,0k

2t2,

un521

3Vc,0k

2t2,

sn522Vc,0

3~2Rn115!k2t3. ~24!

The CDM velocity mode may be identified with the gaumode. This can be seen from the equation for the CDvelocity, which is decoupled from the other equations.uCDMbehaves ast21 at early times. If there were several specof CDM, however, a new physical, non-gauge relative velity mode would arise, which would be divergent at eatimes.

Neutrino isocurvature density mode:

h5Vb,0Rn

10Rgk2t3,

h52Rn

6~1514Rn!k2t2,

dc52Vb,0Rn

20Rgk2t3,

db51

8

Rn

Rgk2t2,

08350

e

s-

dg52Rn

Rg1

1

6

Rn

Rgk2t2,

dn5121

6k2t2,

uc50,

ugb521

4

Rn

Rgk2t1

3Vb,0Rn

4Rg2

k2t2,

un51

4k2t,

sn51

2~1514Rn!k2t2. ~25!

Physically, one starts with a uniform energy density, with tsum of the neutrino and photon densities unperturbed. Wa mode enters the horizon, the photons behave as a pefluid while the neutrinos free stream, creating nonuniformin the energy density, pressure, and momentum densityturn generating metric perturbations.

Neutrino isocurvature velocity mode:

h53

2Vb,0

Rn

Rgkt2,

h524Rn

3~514Rn!kt

1S 2Vb,0Rn

4Rg1

20Rn

~514Rn!~1514Rn! D kt2,

dc523Vb,0

4

Rn

Rgkt2,

db5Rn

Rgkt2

3Vb,0Rn~Rg12!

4Rg2

kt2,

dg54

3

Rn

Rgkt2

Vb,0Rn~Rg12!

Rg2

kt2,

dn524

3kt2

Vb,0Rn

Rgkt2,

uc50,

ugb52Rn

Rgk1

3Vb,0Rn

Rg2

kt

1Rn

RgS 3Vb,0

Rg2

9Vb,02

Rg2 D kt21

Rn

6Rgk3t2,

8-7

ensnedr

f t

la

he

-

te

en-

the

ityy

to

arthec

ity

to

c-


un5k2~914Rn!

6~514Rn!k3t2,

sn54

3~514Rn!kt1

16Rn

~514Rn!~1514Rn!kt2,

Fn354

7~514Rn!k2t2. ~26!

Here the neutrinos and photons start with uniform total dsity and uniform density ratio but with relative velocitiematched so that initially the total momentum density vaishes. If the relative momenta were not perfectly matchthe metric perturbation generated would diverge at eatimes, as in the adiabatic decaying mode. But because operfect match, a divergence at early times is avoided.

It is also possible, at least in principle, to consider regumodes with higher moments ofFn l with l>3 excited ini-tially, as was considered in Ref.@24#; however, it is difficultto envision plausible mechanisms for exciting these higmoment modes.

Newtonian potentials. In this paper we have used synchronous gauge, but for completeness we give the formthe Newtonian potentials for the regular modes presenabove. The Newtonian potentialsf andc are related to thesynchronous variables as follows:

FIG. 1. CMB anisotropy for the neutrino isocurvature densmode. We plotl ( l 11)cl /2p for the neutrino isocurvature densitmode ~dashed curves! for initial power spectraPdn

;ka wherea523.0, . . . ,22.4, increasing in increments of 0.1 from bottomtop at largel. The adiabatic growing mode~solid curve! with ascale-invariant spectrum is included for comparison. All curvesnormalized to COBE. For the lowest curve the variations inphoton-to-neutrino ratio obey a scale-invariant initial power sptrum.

08350

-

-,

lyhe

r

r

ofd

f51

2k2 @ h16h1H~ h16h !#,

~27!

c5h2H

2k2@ h16h#.

We define the Newtonian potentials according to the convtion ds25a2(t)@2dt2(112f)1dxidxjd i j (122c)#. Wenow give the Newtonian potentials to leading order. Forgrowing adiabatic mode,

f510

~1514Rn!,

c510

~1514Rn!. ~28!

For the neutrino isocurvature density mode,

f522Rn

~1514Rn!,

c5Rn

~1514Rn!. ~29!

For the neutrino isocurvature velocity mode,

ee-

FIG. 2. CMB anisotropy for the neutrino isocurvature velocmode. We plotl ( l 11)cl /2p for the neutrino isocurvature velocitymode ~dashed curves! for initial power spectraPvn

;ka with a5

23.0, . . . ,22.0, increasing in increments of 0.2 from bottomtop at largel. Herea523.0 is the ‘‘scale invariant’’ distribution forvn, anda522.0 corresponds to a white noise initial power spetrum in the divergence of the velocity field.

8-8

rlinthtis

ge

thate

n-tapiis

deesi-

nnerbe

ay-va-sityureba-wo-theet-

-re

entshis

eres

s-ere

ldo-p-yn-


f524Rn

~1514Rn!k21t21,

c54Rn

~1514Rn!k21t21. ~30!

The potentials for the baryon isocurvature mode are

f5~4Rn215!Vb,0

2~1512Rn!t,

c52~4Rn215!Vb,0

6~1512Rn!t, ~31!

and for the CDM isocurvature mode are

f5~4Rn215!Vc,0

2~1512Rn!t,

c52~4Rn215!Vc,0

6~1512Rn!t. ~32!

It is curious that the Newtonian potential diverges at eatimes for the neutrino isocurvature velocity mode whilethe synchronous gauge there is no singularity. It appearsthe synchronous gauge is a more physical gauge and thaNewtonian gauge is inadequate for modes based on antropic stresses. The dimensionless Ricci curvaturea2(t)t2R~i.e., the Ricci curvature scaled to the horizon size! is non-singular at early times. In any case, the synchronous gaumore physical in the sense that its evolution islocal whereasin the Newtonian gauge the evolution of the shape ofconstant cosmic time hypersurfaces depends on how mbehaves infinitely far away~because the gauge choice is dfined in terms of thescalar-vector-tensordecomposition,which is nonlocal!. The divergence of the Newtonian potetials arises from the need to warp the surfaces of conscosmic time so as to put the metric in a spatially isotroform; however, there is nothing at all physical about th

..

s.-,

D

08350

y

attheo-

is

eter-

ntc

particular form. The neutrino isocurvature velocity mowas excluded in Ref.@25# because of the behavior of thNewtonian potentials at early times; however, when a phycal phenomemon can be described in a nonsingular main a certain gauge, its singularity in another gauge mustregarded as a coordinate singularity.

III. DISCUSSION

For each wave number, we have identified five nondecing modes: an adiabatic growing mode, a baryon isocurture mode, a CDM isocurvature mode, a neutrino denisocurvature mode, and a neutrino velocity isocurvatmode. Under the assumption that the primordial perturtions are small enough so that the linear theory suffices, tpoint derived observables are completely determined bygeneralization of the power spectrum given by the symmric, positive definite correlation matrix

^Am~k!An~2k!& ~33!

where the indices (m,n51, . . . ,5) label the modes, independently of whether or not the primordial fluctuations weGaussian. In Figs. 1 and 2 the shapes of the CMB momfor the neutrino isocurvature modes are indicated. In tcomputation the cosmological parametersH0550 km s21,Vb50.05, Vc50.95 were assumed. In a companion pap@14#, we examine prospects for constraining the amplitudof these modes using upcoming MAP and Planck data.

ACKNOWLEDGMENTS

We would like to thank David Spergel for useful discusions at an early stage of this project. The CMB spectra wcomputed using a modified version of the codeCMBFAST

written by Uros Seljak and Matias Zaldarriaga. We woulike to thank Anthony Challinor, Anthony Lasenby, and Dminik Schwarz for useful comments. This work was suported in part by the UK Particle Physics and AstronomResearch Council. K.M. was supported by the UK Commowealth Scholarship Commission.

rk,

J.

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re,

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e,’’

tt.

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08350

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Documents

General primordial cosmic perturbation