Article

Cosmic microwave background anisotropies from scaling seeds : global defect models

DURRER, Ruth, KUNZ, Martin, MELCHIORRI, Alessandro

Abstract

We investigate the global texture model of structure formation in cosmogonies with a nonzero cosmological constant for different values of the Hubble parameter. We find that the absence of significant acoustic peaks and little power on large scales are robust predictions of these models. However, from a careful comparison with data we conclude that at present we cannot safely reject the model on the grounds of present CMB data. Exclusion by means of galaxy correlation data requires assumptions on biasing and statistics. New, very stringent constraints come from peculiar velocities. Investigating the large-N limit, we argue that our main conclusions apply to all global O(N) models of structure formation.

Reference

DURRER, Ruth, KUNZ, Martin, MELCHIORRI, Alessandro. Cosmic microwave background anisotropies from scaling seeds : global defect models. Physical Review. D, 1999, vol. 59, no. 123005, p. 26

DOI : 10.1103/PhysRevD.59.123005 arxiv : astro-ph/9811174

Available at: http://archive-ouverte.unige.ch/unige:976

Disclaimer: layout of this document may differ from the published version.

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Cosmic Microwave Background Anisotropies from Scaling Seeds

Global Defect Mo dels

R Durrer M Kunz and A Melchiorri

Departement de Physique Theorique Universite de Geneve quai Ernest Ansermet CH Geneve Switzerland

cosmic uid only gravitationally and which represents al

ways a small fraction of the total energy of the universe

They induce geometrical p erturbations but their inu

Abstract

ence on the evolution of the background universe can b e

We investigate the global texture mo del of structure forma

neglected Furthermore in rst order p erturbation the

tion in cosmogonies with nonzero cosmological constant for

ory seeds evolve according to the unp erturb ed spacetime

dierent values of the Hubble parameter We nd that the

geometry

absence of signicant acoustic p eaks and little p ower on large

Here we mainly investigate the mo dels of structure for

scales are robust predictions of these mo dels However from a

mation with global texture This mo dels for

matter

careful comparison with data we conclude that at present we

show discrepancies with the observed intermediate

cannot safely reject the mo del on the grounds of present CMB

scale CMB anisotropies and with the galaxy p ower sp ec

data Exclusion by means of galaxy correlation data requires

trum on large scales Recently it has b een argued

assumptions on biasing and statistics New very stringent

that the addition of a cosmological constant leads to b et

constraints come from p eculiar velo cities

ter agreement with data for the cosmic string mo del of

Investigating the largeN limit we argue that our main

structure formation We analyze this question for the

conclusions apply to all global O N mo dels of structure for

texture mo del by using ab initio simulation of cosmic

mation

texture as describ ed in Ref We determine the CMB

anisotropies the dark matter p ower sp ectrum and the

bulk velo cities for these mo dels We also compare our

results with the largeN limit of global O N mo dels

and we discuss briey which typ e of parameter changes

I INTRODUCTION

in the p oint functions of the seeds may lead to b etter

agreement with data

Recently a lot of eort has gone into the determina

We nd that the absence of signicant acoustic p eaks

tion of cosmological parameters from measurements of

in the CMB anisotropy sp ectrum is a robust result for

cosmic microwave background CMB anisotropies es

global texture as well as for the largeN limit for all

p ecially in view of the two planned satellite exp eriments

choices of cosmological parameters investigated Further

MAP and PLANCK However we b elieve it is im

more the dark matter p ower sp ectrum on large scales

p ortant to b e aware of the heavy mo deling which enters

h Mp c is substantially lower than the measured

these results In general simple p ower law initial sp ectra

galaxy p ower sp ectrum

for scalar and tensor p erturbations and vanishing vector

However comparing our CMB anisotropy sp ectra with

p erturbations are assumed as predicted from ination

present data we cannot safely reject the mo del On large

To repro duce observational data the comp osition of the

angular scales the CMB sp ectrum is in quite go o d agree

dark matter and the cosmological parameters as well as

ment with the COBE data set while on smaller scales we

the input sp ectrum and the scalar to tensor ratio are

nd a signicant disagreement only with the Saskato on

varied

exp eriment Furthermore for nonsatellite exp eriments

We want to take a dierent approach We mo dify the

foreground contamination remains a serious problem due

mo del for structure formation We assume that cosmic

to the limited sky and frequency coverage

structure was induced by scaling seeds Using a simplied

The dark matter p ower sp ectra are clearly to o low on

and not very accurate treatment for the photon prop

large scales but in view of the unresolved biasing prob

agation we have already shown that some key observa

lem we feel reluctant to rule out the mo dels on these

tions can b e repro duced within a very restricted family of

grounds A much clearer rejection may come from the

scaling seed mo dels Here we want to outline in detail

bulk velo city on large scales Our prediction is by a fac

a more accurate computation with a fully gaugeinvariant

tor to lower than the POTENT result on large scales

Boltzmann co de esp ecially adapted to treat mo dels with

Since global texture and the largeN limit lead to very

sources In this pap er we follow the philosophy of a gen

similar results we conclude that all global O N mo dels

eral analysis of scaling seed mo dels motivated in Ref

of structure formation for the cosmogonies investigated

Seeds are an inhomogeneously distributed form of mat

in this work are ruled out if the bulk velo city on scales of

ter like eg top ological defects which interacts with the

h Mp c is around kms or if the CMB primordial

In the next subsection we parameterize in a completely anisotropies p ower sp ectrum really shows a structure of

general way the degrees of freedom of the seed energy mo p eaks on subdegree angular scales

mentum tensor Subsection B is devoted to the p ertur This pap er is the rst of a series of analyses of mo d

bation of Einsteins equations and the uid equations of els with scaling seeds We therefore fully present the

motion Next we treat the Boltzmann p erturbation equa formalism used for our calculations in the next section

tion In Subsection D we explain how we determine the There we also explain in detail the eigenvector expan

p ower sp ectra of CMB anisotropies density uctuation sion which allows to calculate the CMB anisotropies and

and p eculiar velo cities by means of the derived p ertur matter p ower sp ectra in mo dels with seeds from the two

bation equations and the unequal time correlators of the p oint functions of the seeds alone This section can b e

seed energy momentum tensor which are obtained by skipp ed if the reader is mainly interested in the results

numerical simulations In Subsection E we give the ini Section is devoted to a brief description of the numeri

tial conditions and a brief description of our Boltzmann cal simulations In Section we analyze our results and

co de in Section we draw some conclusions Two app endices

are devoted to detailed denitions of the p erturbation

variables and to some technical derivations

A The seed energy momentum tensor

Notation We always work in a spatially at Fried

mann universe The metric is given by

Since the energy momentum tensor of the seeds

i j

ds at dt dx dx

ij

has no homogeneous background contribution it is

gauge invariant by itself according to the StewartWalker

where t denotes conformal time

Lemma

Greek indices denote spacetime co ordinates to

can b e calculated by solving the matter equa

whereas Latin ones run from to Three dimensional

tions for the seeds in the Friedmann background geometry

vectors are denoted by b old face characters

Since has no background comp onent it satises the

unp erturb ed conservation equations We decomp ose

into scalar vector and tensor contributions They

I I THE FORMALISM

decouple within linear p erturbation theory and it is thus

p ossible to write the equations for each of these contribu

Anisotropies in the CMB are small and can thus b e

tions separately As always unless noted otherwise we

describ ed by rst order cosmological p erturbation theory

work in Fourier space We parameterize the scalar S

which we apply throughout We neglect the nonlinear

vector V and tensor T contributions to in the

evolution of density uctuations on smaller scales Since

form

mo dels with seeds are genuinely nonGaussian the usual

numerical Nb o dy simulations which start from Gaussian

S

M f

initial conditions cannot b e used to describ e the evolution

S

on smaller scales

iM k f

j v

j

Gaugeinvariant p erturbation equations for cosmolog

S

ical mo dels with seeds have b een derived in Refs

f M k f k k f

p j l j l

j l

Here we follow the notation and use the results presented

v V

in Ref Denitions of all the gaugeinvariant p ertur

M w

j j

bation variables used here in terms of p erturbations of

V

iM k w k w

the metric the energy momentum tensor and the bright

j l

j

j l l

ness are given in App endix A for completeness

T

M

ij

We consider a background universe with density pa j l

rameter consisting of photons cold

m

Here M denotes a typical mass scale of the seeds In the

dark matter CDM baryons and neutrinos At very

case of top ological defects we set M where is the

early times z z photons and baryons form

dec

v

symmetry breaking scale The vectors w and w

a p erfectly coupled ideal uid As time evolves and as

is a transverse traceless tensor are transverse and

the electron density drops due to recombination of pri

ij

mordial helium and hydrogen Compton scattering b e

j

v i

comes less frequent and higher moments in the photon

k w k w k

j ij

distribution develop This ep o ch has to b e describ ed by

From the full energy momentum tensor which may

a Boltzmann equation Long after recombination free

contain scalar vector and tensor contributions the scalar

electrons are so sparse that the collision term can b e ne

parts f and f of a given Fourier mo de are determined

v

glected and photons evolve according to the collisionless

by

Boltzmann or Liouville equation During the ep o ch of

interest here neutrinos are always collisionless and thus

j

ik k M f

j v

ob ey the Liouville equation

i j k l

B Einsteins equations and the uid equations

k k k M f

ij ij k l

scalar perturbations

On the other hand f and f are also determined in terms

v

of f and f by energy and momentum conservation

p

Scalar p erturbations of the geometry have two degrees

a

of freedom which can b e cast in terms of the gauge

f k f f f

v p

a

invariant Bardeen p otentials and For New

tonian forms of matter is nothing else than the

Newtonian gravitational p otential For matter with sig

a

f f k f f

nicant anisotropic stresses and dier In geo

v p v

a

metrical terms the former represents the lapse function

v

of the zeroshear hyp ersurfaces while the latter is a mea

Once f is known it is easy to extract M w

j v

j

sure of their curvature In the presence of seeds the

we use ik M f For w

j v

i

Bardeen p otentials are given by

S

j

ik k M w

lj

s m

lj l

s m

v

Again w can also b e obtained in terms of w by

l l

where the indices refer to contributions from a source

means of momentum conservation

sm

the seed and the cosmic uid resp ectively The seed

a

v v

Bardeen p otentials are given in Eqs and

w w k w

l l l

a

To describ e the scalar p erturbations of the energy mo

mentum tensor of a given matter comp onent we use the

The geometry p erturbations induced by the seeds are

variables D a gaugeinvariant variable for density uc

g

characterized by the Bardeen p otentials and for

s s

tuations V the p otential of p eculiar velo city uctua

scalar p erturbations the p otential for the shear of the ex

tions and a p otential for anisotropic stresses which

s

trinsic curvature for vector p erturbations and the

vanishes for CDM and baryons A denition of these

s

for tensor p erturba gravitational wave amplitude H

ij

variables in terms of the comp onents of the energy mo

tions Detailed denitions of these variables and their

mentum tensor of the uids and the metric p erturbations

geometrical interpretation are given in Ref see also

can b e found in Refs or and in App endix A

App endix A Einsteins equations link the seed p ertur

Subscripts and sup erscripts or denote the ra

c b

bations of the geometry to the energy momentum tensor

diation CDM baryon or neutrino uids resp ectively

of the seeds Dening the dimensionless small parameter

Einsteins equations yield the following relation for the

matter part of the Bardeen p otentials

GM

Ga

c b

we obtain

D D D

m c b

g g g

k

a

D f g

c b

g

k f f

s v

a

a

k f V V f

c c s s

a

v s

w k

i i

V g V

b b

a

s s s

H k H H

ij ij ij ij

Ga

a

p p

m m

k

Eqs to would determine the geometric p ertur

bations if the cosmic uid were p erfectly unp erturb ed

Note the app earance of on the rhs

s m

In a realistic situation however we have to add the uid

of Eq Using the decomp ositions we can

p erturbations in the geometry which are dened in the

solve for and in terms of the uid variables and the

next subsection Only the total geometrical p erturba

seeds With the help of Friedmanns equation Eqs

tions are determined via Einsteins equations In this

and can then b e written in the form

sense Eqs to should b e regarded as deni

s

s

tions for and H

s s

ij

a

k x x x x

A description of the numerical calculation of the energy

c b

a

momentum tensor of the seeds for global texture is given

c b

x x D xD x

c

g g

in Section I I I

a

k x V x V x V

c c b b

a

a

k V D

g

x V k

s

a

f

V k D k

g

a

b

k x x

D D

g g

a

V V

b

Here we have normalized the scale factor such that

a to day The density parameters always repre



The conservation equations for neutrinos are not very

sent the values of the corresp onding density parameter

useful since they involve anisotropic stresses and thus

to day Here stands for or To avoid any

 c b

do not close At the temp eratures of interest to us

confusion we have intro duced the variables x for the



T MeV neutrinos have to b e evolved by means of

time dep endent density parameters

the Liouville equation which we discuss in the next sub

section

Once the baryon contribution to the baryonphoton

x

a a a

c b

uid b ecomes nonnegligible and the imp erfect coupling

a

cb

of photons and baryons has to b e taken into account for

x

cb

a a a

a accuracy of the results the redshift corresp onding

c b

to this ep o ch is around z we evolve also the

The uid variables of photons and neutrinos are ob

photons with a Boltzmann equation The equation of

tained by integrating the scalar brightness p erturbations

motion for the baryons is then

which we denote by M t k n and N t k n resp ec

S S

b

tively over directions n

D k V

b

g

Z

a n

T e

V V k V V

b b b

D M d

S

g

a

b

Z

i

The last term in Eq represents the photon drag

V k nM d

S

k

force induced by nonrelativistic Compton scattering

T

S

is the Thomson cross section and n denotes the num

e

b er density of free electrons At very early times when

Z

n t the Thomson drag just forces V V

T e b

k M d k n

S

which together with Eqs and implies

k

S An interesting phenomenon often called comp ensa

tion can b e imp ortant on sup er horizon scales k t

Z

If we neglect anisotropic stresses of photons and neutri

D N d

S

g

nos and take into account that O D O k tV and

g

Z

i

O V O k t for k t Eqs and lead to

V k nN d

S

k

O O k t f

s

S

Z

Hence if anisotropic stresses are relatively small f

the resulting gravitational p otential on sup er horizon

k N d k n

s

S

k

scales is much smaller than the one induced by the seeds

S

alone One must b e very careful not to over interpret this

comp ensation which is by no means related to causality

A systematic denition of the mo des and is given

but is due to the initial condition D V A

j j

g t!

in the next subsection

thorough discussion of this issue is found in Refs

The equation of motion for CDM is given by energy

As we shall see in the next section for textures and

s

and momentum conservation

f are actually of the same order Therefore Eq

do es not lead to comp ensation but it indicates that CMB

c

k V D

c

g

anisotropies on very large scales SachsWolfe eect are

dominated by the amplitude of seed anisotropic stresses

a

V V k

c c

a

During the very tight coupling regime z z we may

The quantities which we want to calculate and compare

dec

neglect the baryon contribution in the energy momentum

with observations are the CDM density p ower sp ectrum

conservation of the baryonphoton plasma We then have

and the p eculiar velo city p ower sp ectrum to day

c

P k hjD k t j i see next section and transfer it onto the baryons via

g

the photon drag force

and

P k hjV k t j i

v c

n a

T e

b b b

a

b

Here h i denotes an ensemble average over mo dels

Note that even though D and V are gauge invariant

g

The photon vorticity is given by an integral over the vec

quantities which do not agree with eg the corresp ond

tor typ e photon brightness p erturbation M

V

ing quantities in synchronous gauge this dierence is

Z

very small on subhorizon scales of order k t and can

nM d

V

thus b e ignored

On subhorizon scales the seeds decay and CDM p er

where the integral is over photon directions n In terms

turbations evolve freely We then have like in ination

of the development presented in the next section for k

ary mo dels

p ointing in z direction we obtain

P k H P k k

v

m

V V V V

Equivalently we have for neutrinos

vector perturbations

Z

nN d

V

Vector p erturbations of the geometry have two degrees

of freedom which can b e cast in a divergence free vector

V V V V

eld A gaugeinvariant quantity describing vector p er

turbations of the geometry is a vector p otential for the

The vector equations of motion for photons and neutrinos

shear tensor of the ft constg hyp ersurfaces Like for

are discussed in the next section

scalar p erturbations we split the contribution to into a

source term coming from the seeds given in the previous

subsection and a part due to the vector p erturbations in

tensor perturbations

the uid

Metric p erturbations also have two tensorial degrees

s m

of freedom gravity waves which are represented by the

The p erturbation of Einsteins equation for is

m

two helicity states of a transverse traceless tensor see

App endix A As b efore we split the geometry p erturba

a

tion into a part induced by the seeds and a part due to

k x x

m c c

a

the matter uids

m s

x x

b b

H H H

ij

ij ij

Here is the uid vorticity which generates the vec

The only matter p erturbations which generate gravity



tor typ e shear of the equal time hyp ersurfaces see Ap

waves are tensor typ e anisotropic stresses which are

p endix A By denition vector p erturbations are trans

present in the photon and neutrino uids The p ertur

verse

bation of Einsteins equation yields

k k k k a

m s 

m m m

H k H H

ij ij ij

a

It is interesting to note that vector p erturbations in

a

the geometry do not induce any vector p erturbations in

x x

ij ij

a

the CDM up to unphysical gauge mo des since no geo

metric terms enter the momentum conservation for CDM

The relation b etween the tensor brightness p erturba

vorticity

tions M N and the tensor anisotropic stresses

T T

ij

a

is given by and

c c

ij

a

Z

hence we may simply set This is also the case

c

M d n n

ij T i j

ij

for the tightly coupled baryon radiation plasma But as

Z

so on as higher moments in the photon distribution build

N d n n

ij T i j

ij

up they feel the vector p erturbations in the geometry

In terms of the development presented in the next section scalar perturbations

for k p ointing in z direction we have

We expand the brightness M k n t in the form

S

1

X

S

M n k t i t kP

S

  

where P denotes the Legendre p olynomial of order

S

and is the asso ciated multip ole moment An analo

  

gous decomp osition also applies to the amplitude of p o

Q

larization anisotropy M n k t and we denote the as

S

We nd that the eect of anisotropic stresses of photons

S

so ciated multip ole moment by q

and neutrinos is less than in the nal result and

The Boltzmann equation for scalar p erturbations in

hence we have neglected them

the photon brightness and p olarization is

M ik M ik

S S

C The Boltzmann equation

P Q a n D M iV

T e S b

g

When particle interactions are less frequent the uid

approximation is not sucient and we have to describ e

Q Q

the given particle sp ecies by a Boltzmann equation in or

M ik M

S S

der to take into account phenomena like collisional and

Q

directional disp ersion In the case of massless particles

a n M P Q

T e

S

like massless neutrini or photons the Boltzmann equa

tion can b e integrated over energy and we obtain an

where

equation for the brightness p erturbation which dep ends

S S S

only on momentum directions As b efore we split the

Q q q

brightness p erturbation into a scalar vector and tensor

The rst term on the right hand side of Eq repre

comp onent and we discuss the p erturbation equation of

sents the gravitational interaction photons without col

each of them separately

lisions move along lightlike geo desics of the p erturb ed

M M M M

S V T

geometry while the term in square brackets is the colli

sion integral for nonrelativistic Compton scattering

and

Inserting expansion into Eqs and using

N N N N

S V T

the standard recursion relations for Legendre p olynomi

als we obtain the following series of coupled equations

The functions M and N dep end on the wave vector k

the photon neutrino direction n and conformal time t

S S

k

Linear p olarization of photons induced by Compton scat

Q

tering is describ ed by the variable M the Stokes pa

rameter Q dep ending on the same variables We cho ose

k

S S S

for each kmo de a reference system with z axis parallel

to k For scalar p erturbations we achieve in this way

S

azimuthal symmetry the left hand side of the Boltz

k a n V

T e b

mann equation and therefore also the brightness M de

p end only on k n and can b e develop ed in Legendre

p olynomials

k

S S S

The left hand side of the Boltzmann equation for vector

and tensor p erturbations also determines the azimuthal

S

Q a n

dep endence of M for vector and tensor p erturbations as

T e

we shall see in detail

h i

S S S

k

S

1

a n for

We could in principle add higher spin comp onents to the

T e

distribution functions But they are not seeded by gravity and

and

since photons and neutrinos interact at high enough temp er

atures they are also absent in the initial conditions

h i

k

S S S

n n n and

q q q

ij i j ij

Z

S

M n M d

Q a n q

ij ij V

T e

We use co ordinates for which k is parallel to the z axis

For the neutrinos we obtain the same equations just

Then

without collision integral

1

X

S

N n k t i t kP

S

and

p p

and

cos sin n

h i

S S S

k

With the ansatz

p

V

M k n t k t cos M

V

k

V

M k t sin

We are interested in the p ower sp ectrum of CMB

the equations for M decouple and the right hand side

anisotropies which is dened by

of Eq dep ends only on Like for scalar p erturba

X

T T

tions we expand M in Legendre p olynomials

0

n n C P cos

0

n
n cos

T T

1

X

V

V

M k t i t kP

Here h i denotes the ensemble average over mo dels

where

We assume that an ergo dic hyp othesis is satised and

Eq then leads to

we can interchange spatial and ensemble averages The

V V

problem that actual observations can average at b est over

M ik M ik

one horizon volume is known under the name cosmic

V V

b V

a n M i

variance It severely restricts the accuracy with which

T e

for example low multip oles of CMB anisotropies ob

served in our horizon volume can b e predicted for a given With Eq this can b e expressed as the following set

V

mo del

of coupled equations for the variables

Using the addition theorem of spherical harmonics one

V V V

b

obtains with the Fourier transform conventions adopted

k a n

T e

here for details see App endix B

Z

S S

k

V V V

C k dk hj t k j i

V V

S

k a n

T e

where the sup erscript indicates that Eq gives

the contribution from scalar p erturbations

and

i h

k

V V V

vector perturbations

V

a n for

T e

Vector p erturbations are very small on angular scales

For neutrino p erturbations we obtain the same equa

corresp onding to where Compton scattering and

tions up to the collision term We rep eat them here for

thus p olarization b ecome relevant We therefore neglect

completeness

p olarization in this case The Boltzmann equation for

h i

k

V V V

vector p erturbations then reads

M ik nM in kn

V V

k

a n n M n M

T e b V ij ij

As for scalar p erturbations the CMB anisotropy p ower

sp ectrum is obtained by integration over k space One

where

nds see App endix B

Z

V V

D Eigenvector expansion of the source correlators

hj t k t k j i

V

C k dk

In the previous subsections we have derived a closed

system of linear dierential equations with source terms

The source terms are linear combinations of the seed en

Here the fact that there are two equal contributions from

ergy momentum tensor which is determined by numerical

b oth p olarization states statistical isotropy is

simulations A given realization of our mo del has random

taken care of

initial conditions the seed energy momentum tensor is a

random variable In principle we could calculate the in

 c

tensor perturbations

k t V k t k t duced random variables D

g

c

etc for to realizations of our mo del and de

termine the exp ectation values P k P k and C by

For tensor p erturbations and a wave vector k p ointing

v

averaging This pro cedure has b een adapted in Ref

into the direction the only non vanishing comp onents

for a seed energy momentum tensor mo deled by a few

of the p erturb ed metric tensor are H H H

random parameters

and H H H Neglecting p olarization the



In the case of a seed energy momentum tensor coming

Boltzmann equation for tensor p erturbations is

entirely from numerical simulations this pro cedure is not

i j

H M ik M n n

ij T T

feasible The rst and most imp ortant b ottleneck is the

dynamical range of the simulations which is ab out

a n M n M

T e T ij ij

in our largest simulation taking around hours

CPU time on a NEC SX sup ercomputer To determine

With the ansatz

the C s for we need a dynamical range of

T

0

M k n t M k t cos

T

ab out in k space this means k k

max min

T

where k and k are the maximum and minimum

max min

k t sin M



wave numb ers which contribute to the C s within our

T

accuracy

the two mo des M decouple completely and the right



With brute force this problem is thus not tractable

hand side of Eq dep ends only on We can then

with present or near future computing capabilities But

expand the mo des in terms of Legendre p olynomials

there are a series of theoretical observations which reduce

1

X

T

T

the problem to a feasible one

M k t i t kP

For each wave vector k given we have to solve a system

of linear p erturbation equations with random sources

where Eq now b ecomes

D X S

T T

M ik M H

Here D is a time dep endent linear dierential op erator

T T T

T

a n M

T e

X is the vector of our matter p erturbation variables sp ec

ied in the previous subsections photons CDM baryons

leading to the series of coupled equations for the co e

and neutrini total length up to and S is the ran

T

cients

dom source term consisting of linear combinations of the

seed energy momentum tensor

T T

H k

For given initial conditions this equation can b e solved

0

T T T

by means of a Greens function kernel G t t in the

a n

T e

form

h i

Z

k

t

T T T

dtG t t kS t k X t k

j l l j

t

in

T

a n for

T e

We want to compute p ower sp ectra or more generally

As b efore the CMB anisotropy p ower sp ectrum is ob

quadratic exp ectation values of the form

tained by integration over k space see App endix B



hX t kX t ki

j

l

Z

T

hj j i

T

C k dk

which according to Eq are given by



hX t kX t ki

j

l

where

Z Z

t t

T T T

0  0

dt G t t k dtG t t k

j m T

ln

t t

in in

 0

hS t kS t ki

m

n



The only information ab out the source random variable hf f i have to b e analytic and from dimensional consid

v

v

which we really need in order to compute p ower sp ectra erations see Ref

are therefore the unequal time two p oint correlators The functions C are shown in Figs to Panels a

ij

are obtained from numerical simulations Panels b rep

 0

hS t kS t ki

m

resent the same correlators for the largeN limit of global

n

O N mo dels see

This nearly trivial fact has b een exploited by many work

in the eld for the rst time probably in Ref

ers (a)

where the decoherence of mo dels with seeds has b een

discovered and later in Refs and others

To solve the enormous problem of dynamical range we

e use of scaling statistical isotropy and causality

mak 10

We call seeds scaling if their correlation functions

C dened by

1

k t M k t

0.1

0  0

C k t t h k t k t i

0.01

0.01

scale free ie the only dimensional parameters are 0.1 1

0 z

C are the variables t t and k themselves Up

in 1

r 10

to a certain numb er of dimensionless functions F of

n

p

0

0

tt z k and r tt the correlation functions are then

determined by the requirement of statistical isotropy

and by their dimension Causality requires

symmetries (b)

the functions F to b e analytic in z A more detailed

n

investigation of these arguments and their consequences

is presented in Ref There we show that statistical

isotropy and energy momentum conservation reduce the

correlators to ve such functions F to F

10

In cosmic string simulations energy and momentum

1

are not conserved Strings lo ose their energy by radiation

gravitational waves andor massive particles In this

of 0.1

functions of z and r are needed to describ e the

case 0.01

time correlators unequal 0.01

0.1

analytic functions generically are constant for Since 1z

1

small arguments z F r actually determines F n n r

p 10

0

for all values of k with z k tt Furthermore

FIG The two p oint

p

4  0

0

the correlation functions decay inside the horizon and we

tt correlation function C z r k h k t k t i is

11 s

s

shown Panel a represents the result from numerical sim

can safely set them to zero for z where they have

ulations of the texture mo del panel b shows the largeN

decayed by ab out two orders of magnitude see Figs

limit For xed r the correlator is constant for z and

to Making use of these generic prop erties of the

then decays Note also the symmetry under r r

correlators we have reduced the dynamical range needed

for our computation to ab out which can b e attained

In Fig we show C z r and in Fig the

ij

with the to simulations feasible on present

constant of the Taylor expansion for C is given as a

ij

sup ercomputers

function of r ie C r

ij

For the scalar part we need the correlators

s

Vector p erturbations are induced by which is

v

0 

seeded by w Transversality and dimensional argu

p

C z r k t i h k t

s

s

0

tt ments require the correlation function to b e of the form k

p

0 

v  v

p

k t i h k t C z r

0

s

0

s

k t i k tw hw tt k k k W z r

0

ij i j

j i

k tt

0 

p

k t i h k t C z r

s

s

0

k tt

Again as a consequence of causality the function W is



as well as C z r C z r The functions C are

analytic in z see The function W z r is plotted ij

p

0

tt comes from the in Fig In Figs and we graph W z and W r analytic in z The prefactor k

 

fact that the correlation functions hf f i k hf f i and

(a) (a)

10 10

1 1

0.1 0.1 0.01 0.01 0.01 0.01 0.1 0.1 1 1 z 1 z r 1 10 r 10

(b) (b)

10 10

1 1

0.1 0.1 0.01 0.01 0.01 0.01 0.1 0.1 1 1z 1z r 1

10 r 10

FIG The same as Fig but for

p

FIG The unequal time correla

 0 4

p

0

tt h k t k t i C z r k

4  0

s 22

s

0

tor jC z r j k tt jh k t k t ij is shown Note

12 s

s

that the r r symmetry is lost in this case

Symmetry transversality and tracelessness together

X

0 T T T  0

with statistical isotropy require the tensor correlator to

T x x v xv x

n n n

b e of the form see

n

V T S



0

v and v are orthonor where the series v

t t i h

n n

in

ij

lm

mal series of eigenvectors ordered according to the am

p

T z r k k k

il j m im j l ij lm ij l m

plitude of the corresp onding eigenvalue of the op erators

0

tt

C W and T resp ectively for a given weight function w

k k k k k k k k k k

lm i j il j m im l j j l i m j m l i

We then have

k k k k k

Z

i j l m

S S

0 0 0 S 0

x x w x dx v C x x v

ij

n

in j n

The functions T z r as well as T z and T r are

Z

shown in Figs to

0 V 0 0 0 V V

W x x v x w x dx v x

n n n

Clearly all correlations b etween scalar and vector

Z

scalar and tensor as well as vector and tensor p ertur

T 0 0 0 T 0 T

x v x w x dx T x x v

bations have to vanish

n n n

The scalar source correlation matrix C and the func

The eigenvectors and eigenvalues dep end on the weight

tions W and T can b e considered as kernels of p ositive

function w which can b e chosen to optimize the sp eed of

hermitian op erators in the variables x k t z r and

0 0

x k t z r which can b e diagonalized

X

S  S

0 0 S

x C x x xv v

ij

n

j n in

2

n

Here the assumption that the op erators C W and T are

X

0 V V V  0

traceclass enters This hyp othesis is veried numerically by

W x x v xv x

n n n

the fast convergence of the sums to

n

FIG The correlators C z are shown The solid FIG The correlators C r are shown in the same

ij ij

0

dashed and dotted lines represent C C and jC j resp ec line styles as in Fig but for z as function of r t t

22 11 12

tively Panel a is obtained from numerical simulations of the The stronger decoherence of the texture mo del is even more

texture mo del and panel b shows the largeN limit A strik evident here

ing dierence is that the largeN value for jC j is relatively

12

p

jC well approximated by the p erfectly coherent result C j

11 22

Instead of averaging over random solutions of Eq

while the texture curve for jC j lies nearly a factor lower

12

we can thus integrate Eq with the deterministic

n

source term v and sum up the resulting p ower sp ectra

convergence of the sums to In our mo dels we

The computational requirement for the determination of

found that scalar p erturbations typically need eigen

the p ower sp ectra of one seed mo del with given source

vectors whereas vector and tensor p erturbations need ve

term is thus on the order of n inationary mo dels This

S

to ten eigenvectors for an accuracy of a few p ercent see

eigenvector metho d has rst b een applied in Ref

Fig

A source is called totally coherent if the unequal

Inserting Eqs to in Eq leads to

time correlation functions can b e factorized This means

X

that only one eigenvector is relevant A simple totally

n n



hX k t X k t i k t k t X X

i n

j

j i

coherent approximation which however misses some im

n

p ortant characteristics of defect mo dels can b e obtained

by replacing the correlation matrix by the square ro ot of

the pro duct of equal time correlators

n

t is the solution of Eq with determin where X

i

q

n

 0

istic source term v

0

i

hS tS t i hjS tj ihjS t j i

i i j

j

Z

t

n n

dtG t t k v x k t k X

j l

This approximation is exact if the source evolution is lin

j

l

t

in

ear Then the dierent k mo des do not mix and the value

For the CMB anisotropy sp ectrum this gives

of the source term at xed k at a later time is given by its

n n n

value at initial time multiplied by some transfer function

S V T

X X X

S n V n T n

S V T

S k t S k t T k t t In this situation b e

C C C C

in in

n n n

comes an equality and the mo del is p erfectly coherent

n n n

Decoherence is due to the nonlinearity of the source evo

lution which induces a sweeping of p ower from one scale

n

into another Dierent wave numb ers k do not evolve in

C is the CMB anisotropy induced by the determinis

dep endently

tic source v and n is the numb er of eigenvalues which

n 

It is interesting to note that the p erfectly coherent ap

have to b e considered to achieve go o d accuracy (a)

−1 10

0.01 −2 10 0.1 1 1 z

r 10

FIG The vector correlator W z is plotted The solid

line represents the texture simulations and the dashed line is

the largeN result Up to a slight dierence in amplitude the

(b)

two results are very similar

−1 10

0.01 −2 10 0.1 1 1 z

r 10

FIG The vector correlator W z r is shown The tex

ture simulations panel a and the largeN limit panel b

give very similar results

FIG The vector correlator W r is shown The solid

line represents the texture simulations and the dashed line is

the largeN result Also here the two results are very similar

proximation leaves op en a choice of sign which has

The wings visible in the texture curve are probably not due

to b e p ositive if i j but which is undetermined oth

to a resolution problem but the b eginning of oscillations

erwise According to Schwarz inequality the correlator

 0

hS tS t i is b ounded by

i

j

q

absence of p ower on this angular scale however is not

 0

0

hjS tj ihjS t j i hS tS t i

a consequence of decoherence but is mainly due to the

i j i

j

q

anisotropic stresses of the source which lead to p ertur

0

hjS tj ihjS t j i

bations in the geometry inducing large scale C s Sachs i j

Wolfe but not to density uctuations Large anisotropic

Therefore for scalesvariables for which the Greens func

stresses are also at the origin of vector and tensor uctu

tion is not oscillating eg Sachs Wolfe scales the full

ations Our results are in agreement with Refs and

result always lies b etween the anticoherent minus sign

but we disagree with Ref which has found acous

and the coherent result We have veried this b ehavior

tic p eaks with an amplitude of ab out six in the coherent

numerically

approximation

The rst evidence that Doppler p eaks are suppressed

In the real universe p erfect scaling of the seed corre

in defect mo dels has b een obtained in the p erfectly co

lation functions is broken by the radiationmatter tran

herent approximation in Ref In Fig we show the

sition which takes place at the time of equal matter and

contributions to the C s from more and more eigenvec

radiation t h Mp c The time t is an

m

eq eq

tors A p erfectly coherent mo del has only one nonzero

additional scale which enters the problem and inuences

eigenvalue

the seed correlators Only in a purely radiation or mat

A comparison of the full result with the totally coher

ter dominated universe are the correlators strictly scale

ent approximation is presented in Fig There one

invariant This means actually that the k dep endence of

sees that decoherence do es smear out the oscillations

the correlators C W and T cannot really b e cast into

0

present in the fully coherent approximation and do es

a dep endence on x and x but that these functions de

0

somewhat damp the amplitude Decoherence thus pre

p end on t t and k in a more complicated way We have

vents the app earance of a series of acoustic p eaks The (a)

1

0.1

0.01 0.01 0.1

1

As Fig but for the tensor source function

1 z FIG

z r 10 T

(b)

1

0.1

0.01

As Fig but for the tensor source function 0.01 FIG

0.1

T r 1 1 z

r 10

and from interp olated source terms

FIG As Fig but for the tensor source function T z r

Clearly the eect of the radiation dominated early

state of the universe is relatively unimp ortant for the

to calculate and diagonalize the seed correlators for each

scales considered here The dierence b etween the pure

wave numb er k separately and the huge gain of dynami

matter era result and the interp olation is barely visible

cal range is lost as so on as scaling is lost

and thus not shown on the plot This seems to b e quite

In the actual case at hand however the deviation from

dierent for cosmic strings where the uctuations in the

scaling is weak and most of the scales of interest to us

radiation era are ab out twice as large as those in the mat

enter the horizon only in the matter dominated regime

ter era The radiation dominated era has very little

The b ehavior of the correlators in the radiation domi

eect on the key results which we are rep orting here

nated era is of minor imp ortance To solve the problem

namely the absence of acoustic p eaks and the missing

we calculate the correlator eigenvalues and eigenfunctions

p ower on very large scales

twice in a pure radiation and in a pure matter universe

In mo dels with cosmological constant there is actually

and we interp olate the source term from the radiation to

a second break of scale invariance at the matter tran

the matter ep o ch Denoting by v and v a given

m m r r

sition There we pro ceed in the same way as outlined

pair of eigenvalue and eigenvector in a matter and radia

ab ove Since defects cease to scale and disapp ear rapidly

tion universe resp ectively we cho ose as our deterministic

in an exp onentially expanding universe the eigenvalues

source function

for the dominated universe all vanish

p p

v t y t v k t y t v k t

r r m m

with eg

E Initial conditions and numerical implementation

t

eq

y t or y t exptt

eq

t t

eq

We numerically integrate our system of equations

from redshift z up to the present with the goal

or some other suitable interp olation function In Fig

to have one p ercent accuracy up to for a

we show the results for scalar vector and tensor p ertur

given source term We use the integration metho d de

bations resp ectively using purely radiation dominated era 10 100 1000 101 101

0 0 V l

10 10 Vector +1)C

l -1 -1 (

l 10 10

10-2 10-2

0 10 0

10 Scalar S l

-1 +1)C

l 10 -1 ( 10 l

-2 10 10-2 T l 100 100 Tensor +1)C l ( l 10-1 10-1

10-2 10-2 10 100 1000

l

FIG The scalar vector and tensor contributions for the

texture mo del of structure formation are shown The dashed

lines show the contributions from single eigenfunctions while

the solid line represents the sum Note that the single con

tributions to the scalar and tensor sp ectrum do show oscil

lations which are however washed out in the sum Vector

p erturbations do not ob ey a wave equation and thus do not

FIG The sum of the rst few eigenfunctions of T x x

show oscillations

is shown for two dierent weight functions a logarithmic

w x and b linear w The rst long dashed rst

and second short dashed rst ten dotted and rst thirty

lowing a standard recombination scheme for H and

solid eigenfunctions are summed up The op en circles repre

H e for a helium abundance by mass of At high

sent the full correlation function Clearly the eigenfunctions

redshift z the Thomson opacity is very large

obtained by linear weighting converge much faster Here we

and photons and baryons are tightly coupled Due to

only show the equal time diagonal of the correlation matrix

the large Thomson drag term Eqs and b ecome

but the same b ehavior is also found in the C p ower sp ectrum

sti and dicult to solve numerically Therefore in this

which is sensitive to the full correlation matrix

limit we follow the metho d of Ref which is accurate

to second order in n see also Assuming

T e

a standard inationary mo del we obtain a single scalar

scrib ed in Refs and We sample the interval

p ower sp ectrum in few minutes seconds for the

log k h Mp c with minimum step size

tensor case on a PC class workstation which diers by

log for the scalar case and use a smo oth

less than one p ercent from the C s computed with other

ing algorithm to suppress high frequency sampling noise

co des

In order to save computing time we start the inte

Summing the scalar C s from the largest eigenvec

gration of the k s with harmonics adding new

tors in the tensor case for vector p erturbations

harmonics in the course of the integration We nd

typically repro duces the total sum to b etter than see

that typically harmonics are sucient for small

Fig

k values log k h Mp c while for higher k

log k h Mp c up to harmonics for

the scalar case for the tensor case are needed to

I I I THE NUMERICAL SIMULATIONS

achieve the desired accuracy Including more than har

monics for neutrinos corrects our results by less than one

As in previous work we consider a sp ontaneously

p ercent We obtain algebraically using Eq With

broken scalar eld with ON symmetry We use the

this choice of variables we avoid the numerical diculties

mo del approximation ie the equation of motion

present in conformal gauge where is determined

by numerical integration

2 2

The abundance of free electrons n is calculated fol

e 10 100 1000 1.0x104 Scalar Coherent 8.0x103 Tensor Vector 3 l 6.0x10 Total

4.0x103 l(l+1)C 2.0x103

0.0

Including decoherence 8.0x103

3 l 6.0x10

4.0x103 l(l+1)C 2.0x103

0.0 10 100 1000

l

FIG The C p ower sp ectrum for the texture scenario

is shown in the p erfectly coherent approximation top panel

and in the full eigenfunction expansion Even in the coher

ent approximation the acoustic p eaks are not higher than

the Sachs Wolfe plateau Decoherence just washes out the

structure but do es not signicantly damp the p eaks

1x104 1x104 Including radiation to matter transition 1x104 Radiation dominated 3

] 9x10 2 3

K 8x10

µ 3 [

l 7x10 6x103 5x103 l(l+1)C 4x103 3x103 2x103

1x103 2

FIG The ETCs C z hjj ik t panel a and

0 11

2

C z hjj ik t panel b are shown for dierent times

10 100 1000 22

grid units the times are t dashed t dotted

l In

t long dashed t dash dotted long dash

FIG The scalar vector tensor and total C p ower

dotted and t solid Clearly C scales much so oner

22

sp ectrum is shown from pure radiation sources and for an in

than C To safely arrive in the scaling regime one has to

11 terp olated source While the vector p erturbations are some

wait until t and C k t is b est determined at t

ij

what higher in the radiation era scalar and tensor p ertur

but k t

bations are higher in the matter era and the sum is nearly

unchanged

comoving time t x Dierent grid p oints are uncor

related at all earlier times

where is the rescaled eld

The use of nite dierences in the discretized action

We do not solve the equation of motion directly but

as well as in the calculation of the energy momentum

use a discretized version of the action

tensor intro duce immediately strong correlations b etween

Z

neighb oring grid p oints This problem manifests itself in

S d x a t

an initial phase of nonscaling b ehaviour the length of

which varies b etween x and x dep ending on the

where is a Lagrange multiplier which xes the eld

variable considered It is very imp ortant to use results

to the vacuum manifold this corresp onds to an innite

from the scaling regime only cf Fig

Higgs mass Tests have shown that this formalism agrees

In order to reduce the time necessary to reach scaling

well with the complementary approach of using the equa

and to improve the overall accuracy we try to cho ose the

tion of motion of a scalar eld with Mexican hat p otential

nite dierences in an optimal way Our current co de

and setting the inverse mass of the particle to the smallest

calculates all values in the center of each cubic cell dened

scale that can b e resolved in the simulation typically of

by the lattice The additional smo othing intro duced by

the order of GeV but tends to give b etter energy

this improves energymomentum conservation by several

momentum conservation

As we cannot trace the eld evolution from the unbro

ken phase through the phase transition due to the limited

dynamical range we cho ose initially a random eld at a

1

p ercent

To calculate the unequal time correlator UTC the

value of the observable under consideration is saved once

scaling is reached at time t we checked this by using dif c

0.1

ferent correlation times and then correlated at all follow

ing time steps While there is some danger of contaminat

ing the equal time correlator ETC which contributes

most strongly to the C s with nonscaling sources this

0.01

metho d ensures that the constant for k t is deter

mined with maximal precision for the ETCs This is

very imp ortant as the constants C W and

ij

T x the relative size of scalar vector and tensor

0.001

contributions of the SachsWolfe part and severely inu

0 0.1 0.2 0.3 0.4 0.5

ence the resulting C s In contrast the CMB sp ectrum

scale [fraction of total size]

FIG Energy momentum conservation of our numerical

seems quite stable under small variations of the shap e of

simulations is shown The lines represent the sum of the terms

the UTCs

which has to vanish if energy solid resp ectively momentum

The resulting UTCs are obtained numerically as func

dashed is conserved divided by the sum of the absolute

tions of the variables k t and t with t t and t xed

c c c

value of these terms The abscissa indicates the wavelength

They are then linearly interp olated to the required range

0

of the p erturbation as fraction of the size of the entire grid

We construct a hermitian matrix in k t and k t

with the values of k t chosen on a linear scale to maximize

the information content k t x The choice

max

mentum tensor b eing quadratic in the elds are much

of a linear scale ensures go o d convergence of the sum

weaker than in the texture mo del where the eld evolu

of the eigenvectors after diagonalization see Fig

tion itself is nonlinear Therefore decoherence which is

but still retains enough data p oints in the critical region

a purely nonlinear eect is exp ected to b e much weaker

O x where the correlators start to decay In prac

in the largeN limit This is actually the main dierence

tice we cho ose as the endp oint x of the range sampled

max

b etween the two mo dels as can b e seen in Fig

by the simulation the value at which the correlator decays

by ab out two orders of magnitude typically x

max

The eigenvectors that are fed into the Boltzmann co de

IV RESULTS AND COMPARISON WITH DATA

are then interp olated using cubic splines with the condi

tion v k t for k t x

n max

A CMB anisotropies

We use several metho ds to test the accuracy of the

simulation energy momentum conservation of the de

The C s for the standard global texture mo del are

fects co de is found to b e b etter than on all scales

shown in Fig b ottom panel

larger than ab out grid units as is seen in Fig A

comparison with the exact spherically symmetric solution

in nonexpanding space shows very go o d agreement

h

The resulting CMB sp ectrum on Sachs Wolfe scales is

consistent with the line of sight integration of Ref

Furthermore the overall shap e and amplitude of the un

equal time correlators are quite similar to those found in

the analytic largeN approximation see Figs

to The main dierence of the largeN approxima

TABLE I The value of the normalization constant and

tion is that there the eld evolution Eq is ap

the uctuation amplitude are given for the dierent mo d

8

proximated by a linear equation The nonlinearities in

els considered The error in comes from a b est t normal

the largeN seeds which are due solely to the energy mo

ization to the full CMB data set Cosmological parameters

which are not indicated are identical in all mo dels or given

by We consider only spatially at

0 

cdm b

mo dels with and a helium fraction of The

b

3

parameter choice indicated in the top line is referred to as

Julian Borrill suggested to intro duce spherical derivatives

standard texture mo del in the text

that take into account the fact that the vacuum manifold is a

Nsphere and therefore curved and that this curvature should

Vector and tensor mo des are found to b e of the same

b e imp ortant at least in the initial stages of the simulation

order as the scalar comp onent at COBEscales For the

and for unwinding events So far we havent investigated

S V T

this idea suciently to include it into our pro duction co de

standard texture mo del we obtain C C C

in go o d agreement with the predictions of

Refs and Due to tensor and vector con

tributions even assuming p erfect coherence see Fig

panel the total p ower sp ectrum do es not increase fπ=0 top 4

1.5x10 Textures

from large to small scales Decoherence leads to smo oth

]

ing of oscillations in the p ower sp ectrum at small scales 2 4

K)

the nal p ower sp ectrum has a smo oth shap e with 1.0x10 and

µ

a broad low iso curvature hump at and a small [(

l

of the rst acoustic p eak at There is no 3 residual

+1)C 5.0x10

l

structure of p eaks at small scales The p ower sp ectrum (

l

is well tted by the following fourthorder p olynomial in

0.0

x log

10 100 1000

C

l

x x x x

C

4 Large N : 1.5x10 Scalar ] 2 Tensor

K)

The eect of decoherence is less imp ortant for the

µ Vector

4

largeN mo del where oscillations and p eaks are still vis [( 1.0x10

l Textures

ible see Fig b ottom panel This is due to the +1)C

l

fact that the nonlinearity of the largeN limit is only

( 3

l 5.0x10

in the quadratic energy momentum tensor The scalar

eld evolution is linear in this limit in contrast to

0.0

the N texture mo del Since decoherence is inher

10 100 1000

ently due to nonlinearities we exp ect it to b e stronger

l

for lower values of N COBE normalization leads to

FIG Top panel the f mo del Bottom panel The

for the largeN limit

C p ower sp ectrum is shown for the largeN limit b old line

In Fig we plot the global texture C p ower sp ectrum

and for the texture mo del The main dierence is clearly that

for dierent choices of cosmological parameters The

the largeN curve shows some acoustic oscillations which are

variation of parameters leads to similar eects like in the

nearly entirely washed out in the texture case

inationary case but with smaller amplitude At small

scales the C s tend to decrease with increas

ing H and they increase when a cosmological constant

is intro duced Nonetheless the amplitude

m

of the anisotropy p ower sp ectrum at high s remains in

all cases on the same level like the one at low s without

showing the substantial p eak found in inationary mo d

The absence of acoustic p eaks is a stable prediction 13000 h = 0.5 els h = 0.8

11000

of global O N mo dels The mo dels are normalized to 2 h = 1.0 9000

µΚ]

the full CMB data set which leads to slightly larger val [

l 7000

of the normalization parameter G than pure 5000 ues

l(l+1)C

normalization In Table I we give the cosmologi

3000 COBE

parameters and the value of for the mo dels shown

1000 cal

Fig Ω = 0.0 in 13000 Λ

ΩΛ = 0.4

order to compare our results with current exp er 11000 In 2 ΩΛ = 0.8 9000

µΚ]

imental data we have selected a set of dierent [

l 7000

anisotropy detections obtained by dierent exp eriments 5000

l(l+1)C

by the same exp eriment with dierent window func

3000 or

andor at dierent frequencies Theoretical predic 1000 tions

10 100 1000

tions and data of CMB anisotropies are usually compared

l

by plotting the theoretical C curve along with the CMB

FIG The C p ower sp ectrum is shown for dierent val

measurements converted to band p ower estimates We do

ues of cosmological parameters In the top panel we cho ose

this in the top panel of Fig The data p oints show an

and vary h In the b ot

 CDM

b

increase in the anisotropies from large to smaller scales

tom panel we x h and vary We only



b

in contrast to the theoretical predictions of the mo del

consider spatially at universes

0

This fashion of presenting the data is surely correct but

lacks informations ab out the uncertainties in the theoret

ical mo del Therefore we also compare the detected mean

Gaussian which has b een found by simulations on large

scales the C s which are the squares of Gaus

Observed data 4

6.0x10 Textures

sian variables are nonGaussian This eect is however

]

only relevant for relatively low s Keeping this caveat 2 K µ

[

in mind and missing a more precise alternative we nev l 4

+1)C 3.0x10 ertheless indicate the minimal Gaussian error calculated l (

l

according to We add a error from the CMB

normalization The numerical seeds are assumed to b e

accurate

0.0 ab out

Table I I the detected mean square anisotropy

10l 100 1000 In

E xp

with the exp erimental error are listed for

each exp eriment of our data set The corresp onding sky 2 COBE data (Tegmark & Hamilton)

] 4.0x10

2

coverage is also indicated In Fig we plot these data

k Textures µ [

2

p oints together with the theoretical predictions for a tex T

∆ 2.0x102

ture mo del with h and

We nd that apart from the COBE quadrup ole only

0.0

Saskato on exp eriment disagrees signicantly more 1 2 3 4 5 6 7 8 the

Cobe data point (see Table)

than with our mo del But also this disagreement is

b elow and thus not sucient to rule out the mo del

1.6x104

the last column of Table II we indicate Observed data In ] 2 Textures K

µ 3

E xp T h E xp T h

[ 8.0x10

2

T

j

j j j j ∆

0.0

for the j th exp eriment where the theoretical mo del

the standard texture mo del with and h

1 3 5 7 9 11 13 15 17 19 21 23 is

Experiment (see Table)

The ma jor discrepancy b etween data and the

ory comes from the COBE quadrup ole Leaving away

FIG The C sp ectrum obtained in the standard texture

the quadrup ole which can b e contaminated and leads

mo del is compared with data In the top panel exp erimen

to a similar also for inationary mo dels the data

tal results and the theoretical curve are shown as functions

agrees quite well with the mo del with the exception

of In the two lower panels we indicate the value of each

of three Saskato on data p oints Making a rough chi

of the exp erimental data p oints with error bars and

square analysis we obtain excluding the quadrup ole a

the corresp onding theoretical value with its uncertainty The

P

for a total of data p oints value

exp eriments corresp onding to a given numb er are given in

j

j

and one constraint An absolutely reasonable value but

Table I I In the middle panel the COBE data p oints are

one should take into account that the exp erimental data

shown In the b ottom panel other exp eriments are presented

p oints which we are considering are not fully indep en

dent The regions of sky sampled by the Saskato on

E xp

square anisotropy and the exp erimental er

and MSAM or COBE and Tenerife for instance over

E xp

ror directly with the corresp onding theoretical

lap Nonetheless even reducing the degrees of freedom

mean square anisotropy given by

of our analysis to N our is still in the range

p

N N and hence still compatible

X

T h

with the data

C W

This shows that even assuming Gaussian statistics the

mo dels are not convincingly ruled out from present CMB

where the window function W contains all exp erimental

data There is however one caveat in this analysis A

details chop mo dulation b eam etc of the exp eriment

chisquare test is not sensitive to the sign of the discrep

The theoretical error in principle dep ends on the statis

ancy b etween theory and exp eriment For our mo dels

tics of the p erturbations If the distribution is Gaussian

the theoretical curve is systematically lower than the ex

one can asso ciate a samplecosmic variance

p eriments For example whenever the discrepancy b e

tween theory and data is larger than which hap

X

T h

p ens with nearly half of the data p oints in all cases

C W

f

except for the COBE quadrup ole the theoretical value is

smaller than the data If smaller and larger are equally

where f represents the fraction of the sky sampled by a

likely the probability to have or more equal signs is

given exp eriment

This indicates that either

Deviation from Gaussianity leads to an enhancement

the mo del is to o low or that the data p oints are system

of this variance which can b e as large as a factor of

atically to o high The numb er can however not

see Even if the p erturbations are close to

b for IRAS galaxies range b etween b e taken seriously b ecause we can easily change it by

I

and Biasing of IRAS galaxies increasing our normalization on a mo derate cost of

I

is also suggested by measurements of bias in the optical

band For example Ref nds in

o

B Matter distribution

marginal agreement with which obtains

o

A bias for IRAS galaxies is not only p ossible but

even preferred in at global texture mo dels

In Table I we show the exp ected variance of the total

But also with bias our mo dels are in signicant con

mass uctuation in a ball of radius R h Mp c

R

tradiction with the shap e of the p ower sp ectrum at large

for dierent choices of cosmological parameters We

scales As the values of in Table III and Fig clearly

nd h the error coming from the

indicate the mo dels are inconsistent with the shap e of

CMB normalization for a at mo del without cosmolog

the IRAS p ower sp ectrum and they can b e rejected with

ical constant in agreement with the results of Ref

a high condence level The APM data which has the

From the observed cluster abundance one infers

smallest error bars is the most stringent evidence against

and These

texture mo dels Nonetheless these data p oints are not

results which are obtained with the PressSchechter for

measured in redshift space but they come from a de

mula assume Gaussian statistics We thus have to take

pro jection of a D catalog into D space This

them with a grain of salt since we do not know how non

might intro duce systematic errors and thus the errors of

Gaussian uctuations on cluster scales are in the texture

APM may b e underestimated

mo del According to Ref the Hubble constant lies

Mo dels with a cosmological constant agree much b etter

in the interval h Hence in a at

with the shap e of the observed p ower sp ectra the value

CDM cosmology taking into account the uncertainty of

of b eing low for all except the APM data But the

the Hubble constant the texture scenario predicts a rea

values of the bias factors are extremely high for these

sonably consistent value of

mo dels For example IRAS galaxies should have a bias

As already noticed in Refs and unbiased global

b resulting in and in a

I

texture mo dels are unable to repro duce the p ower of

which is to o small even allowing for big variances due to

galaxy clustering at very large scales h Mp c In

nonGaussian statistics

order to quantify this discrepancy we compare our pre

The p ower sp ectra for the largeN limit and for the co

diction of the linear matter p ower sp ectrum with the re

herent approximation are typically a factor to higher

sults from a numb er of infrared and optically

see Fig and the biasing problem is alleviated for

selected galaxy redshift surveys and with the

these cases For we nd h for the large

realspace p ower sp ectrum inferred from the APM pho

N limit and h for the coherent approximation

tometric sample see Fig Here cosmological

This is no surprise since only one source function

s

parameters have imp ortant eects on the shap e and am

the analog of the Newtonian p otential seeds dark matter

plitude of the matter p ower sp ectrum Increasing the

uctuations and thus the coherence always enhances the

Hubble constant shifts the p eak of the p ower sp ectrum

unequal time correlator The second inequality in

to smaller scales in units of hMp c while the inclusion

applies The dark matter Greens function is not oscil

of a cosmological constant enhances large scale p ower

lating so this enhancement translates directly into the

We consider a set of mo dels in h space with linear

p ower sp ectrum

bias as additional parameter In Table III we rep ort

Mo dels which are anticoherent in the sense dened in

for each survey and for each mo del the b est value of the

Section I ID reduce p ower on SachsWolfe scales and en

bias parameter obtained by minimization We also

hance the p ower in the dark matter Anticoherent scal

indicate the value of not divided by the numb er of

ing seeds are thus the most promising candidates which

data p oints The data p oints and the theoretical predic

may cure some of the problems of global O N mo dels

tions are plotted in Fig Our bias parameter strongly

The simple analysis carried out here do es not take into

dep ends on the data considered This is not surprising

account the eects of nonlinearities and redshift distor

since also the catalogs are biased relative to each other

tions Redshift distortions in the texture case should

Mo dels without cosmological constant and with h

b e less imp ortant than in the inationary case since the

only require a relatively mo dest bias b

p eculiar velo cities are rather low see next paragraph

But for these mo dels the shap e of the p ower sp ectrum

Nonlinearities typically set in at k hMp c and

is wrong as can b e seen from the value of which is

should not have a big eect on our main conclusions

much to o large The bias factor is in agreement with our

which come from much larger scales Inclusion of these

prediction for For example our b est t for the IRAS

corrections will result in more smallscale p ower and in

I RAS

data for h is b With

a broadening of the sp ectra which even enhances the

this gives compatible with the direct

conict b etween mo dels and data Furthermore varia

computation

tions of other cosmological parameters like the addition

Whether IRAS galaxies are biased is still under debate

of massive neutrinos hot dark matter which is not con

Published values for the parameter dened as 10000 Textures Textures − coherent approximation LargeN

1000 P(k)

100

10 0.01 0.10 1.00 0.01 0.1 0.01 0.1 1 k [h/Mpc] 106

105 CfA+SSRS2 CfA+SSRS2 105

FIG The dark matter p ower sp ectrum for the texture ] 3

104 104

mo del solid line is compared with the coherent approxima

3 3

short dashed and the largeN limit long dashed The

tion 10 10

are COBE normalized and the cosmological parame sp ectra 2 2

P(k) P(k) [(Mpc/h) 10 10

ters are h  101 101

5 IRAS 5 10 LCRS 10 ]

3

sidered here will result in a change of the sp ectrum on

104 104

small scales but will not resolve the discrepancy at large

103 103 scales

P(k) P(k) [(Mpc/h) 2 2

scale dep endent biasing may exist and

Nonetheless 10 10

to a nontrivial relation b etween the calculated dark

lead 101 101

matter p ower sp ectrum and the observed galaxy p ower 5 IRAS 1.2 Jy APM 105 ] 10

3

sp ectrum We are thus very reluctant to rule out the

4 4

by comparing two in principle dierent things the mo del 10 10

3 3

of which is far from understo o d Therefore we relation 10 10

P(k) P(k) [(Mpc/h)

ould prefer to reject the mo dels on the basis of p ecu w 2

102 10

liar velo city data which is more dicult to measure but 1

101 10

certainly not biased most 0.1 1

5 -1 ] QDOT 3 10 k [(Mpc/h) ]

104 DATA

Bulk velo cities C Ω h=0.5 Λ=0.0 103 Ω h=0.8 Λ=0.0 P(k) P(k) [(Mpc/h) h=1.0 Ω =0.0

102 Λ

o get a b etter handle on the missing p ower on to T Ω

h=0.5 Λ=0.4 Ω

h=0.5 Λ=0.8

h Mp c we investigate the velo city p ower sp ectrum

0.01 0.1 1

h is not plagued by biasing problems The assump

whic k [(Mpc/h)-1]

tion that galaxies are fair tracers of the velo city eld

seems to us much b etter justied than to assume that

FIG Matter Power sp ectrum comparison b etween

they are fair tracers of the mass density We therefore

data and theory References are in the text Data set courtesy

test our mo dels against p eculiar velo city data We use

of M S Vogeley

the data by Ref which gives the bulk ow

Z

H

m

R P k W k R dk

v

in spheres of radii R to h Mp c These data

are derived after reconstructing the dimensional ve

lo city eld with the POTENT metho d see and ref

erences therein

As we can see from Table IV the COBE normalized

texture mo del predicts to o low velo cities on large scales

when compared with POTENT results Recent measure

ments of the bulk ow lead to somewhat lower estimates

just say that nonGaussianity can only weaken the ab ove like R at R h Mp c but

v

constraints still a discrepancy of ab out a factor of in the b est case

Our results naturally lead to the question whether remains

all scaling seed mo dels are ruled out by present data Including a cosmological constant helps at large scales

The main problem of the O N mo del is the missing but decreases the velo cities on small scales

p ower at intermediate scales or R If the observational bulk velo city data is indeed reliable

h Mp c We have briey investigated whether there are some doubts ab out this all global O N

this problem can b e mitigated in a scaling seed mo del mo dels are ruled out

without vector and tensor p erturbations In this case

also scalar anisotropic stresses are reduced by causality

V CONCLUSIONS

requirements see Ref and the comp ensation mech

anism mentioned in Section II is eective For simplicity

we analyze a mo del with purely scalar p erturbations and

We have develop ed a self contained formalism to de

no anisotropic stresses at all f The seed func

termine CMB anisotropies and other p ower sp ectra for

tion is taken from the texture mo del numerical sim

s

mo dels with causal scaling seeds We have applied it to

ulations and we set The resulting CMB

s s

global O N mo dels which contain global monop oles and

anisotropy sp ectrum is shown in Fig top panel A

texture Our main results can b e summarized as follows

smeared out acoustic p eak with an amplitude of ab out

Global O N mo dels predict a at sp ectrum

do es indeed app ear in this mo del This is mainly due

HarrisonZeldovich of CMB anisotropies on large

to the fact that uctuations on large scales are smaller

scales which is in go o d agreement with the COBE

in this mo del as is also evident from the higher value of

results Mo dels with vanishing cosmological con

But also here the dark matter

stant and a large value of the Hubble parameter

density uctuations and bulk velo cities are substantially

give to which is reasonable

lower than observed galaxy density uctuations or the

POTENT bulk ows

Indep endent of cosmological parameters these

Clearly this simple example is not sucient and a

mo dels do not exhibit pronounced acoustic p eaks

more thorough analysis of generic scaling seed mo dels

in the CMB p ower sp ectrum

is presently under investigation So far it is just clear

that contributions from vector and tensor p erturbations

The dark matter p ower sp ectrum from global O N

are severely restricted

mo dels with has reasonable amplitude

but do es not agree in its shap e with the galaxy

p ower sp ectrum esp ecially on very large scales

Acknowledgment

h Mp c

It is a pleasure to thank Andrea Bernasconi Paolo de

Bernardis Roman Juszkiewicz Mairi Sakellariadou Paul

Mo dels with considerable cosmological constant

Shellard Andy Yates and Marc Davis for stimulating dis

agree relatively well with the shap e of the galaxy

cussions Our Boltzmann co de is a mo dication of a co de

p ower sp ectrum but need very high bias b

worked out by the group headed by Nicola Vittorio We

even with resp ect to IRAS galaxies

also thank Michael Vogeley who provided us with the

galaxy p ower sp ectra shown in our gures The numer

The large scale bulk velo cities are by a factor of

ical simulations have b een p erformed at the Swiss sup er

ab out to smaller than the value inferred from

computing center CSCS This work is partially supp orted

by the Swiss National Science Foundation

In view of the still considerable errors in the CMB data

see Fig and the biasing problem for the dark mat

ter p ower sp ectrum we consider the last argument as

the most convincing one to rule out global O N mo d

els Even if velo city data is still quite uncertain ob

servations generally agree that bulk velo cities on the

scale of h Mp c are substantially larger than the

kms obtained in texture mo dels

However all our constraints have b een obtained as

suming Gaussian statistics We know that global defect

mo dels are nonGaussian but we have not investigated

how severely this inuences the ab ove conclusions Such

a study which we plan for the future requires detailed

maps of uctuations the resolution of which is always

limited by computational resources Generically we can

2 2 2 2 2

Exp eriment Data p oint T K K K Sky Coverage Reference

j

COBE

COBE

COBE

COBE

COBE

COBE

COBE

COBE

ARGO Hercules

MSAM

MSAM

MSAM

MSAM

MSAM

MAX HR

MAX PH

MAX GUM

MAX ID

MAX SH

Tenerife

South Pole Q

South Pole K

Python

ARGO Aries

Saskato on

Saskato on

Saskato on

Saskato on

Saskato on

CAT

CAT

TABLE I I The CMB anisotropy detections used in our analysis The and column denote the value of the anisotropy

and the upp er and lower errors resp ectively The references are Tegmark and Hamilton de Bernardis et al

Cheng et al Cheng et al Cheng et al Tanaka et al Gutierrez et al

Gundersen et al Dragovan et al Masi et al Nettereld et al Scott et al

Catalog h Best t bias b Data p oints

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

CfASSRS Mp c

LCRS

LCRS

LCRS

LCRS

LCRS

IRAS

IRAS

IRAS

IRAS

IRAS

IRAS Jy

IRAS Jy

IRAS Jy

IRAS Jy

IRAS Jy

APM

APM

APM

APM

APM

QDOT

QDOT

QDOT

QDOT

QDOT

TABLE I I I Analysis of the matter p ower sp ectrum In the rst column the catalog is indicated Cols and sp ecify the

2 2

mo del parameters In cols and we give the bias parameter inferred by minimization as well as the value of Col

shows the numb er of indep endent data p oints assumed in the analysis

R R h h

v v

TABLE IV Bulk velo cities Observational data from and theoretical predictions estimates the observational

v

uncertainty The uncertainties on the theoretical predictions are around The mo dels with h and h



as well as h are investigated



APPENDIX A COMPLETE DEFINITIONS OF u is xed by the normalization condition u a

GAUGEINVARIANT PERTURBATION

A In the space orthogonal to u we dene the stress

VARIABLES

tensor by

P P T A

In this App endix we give precise denitions of all the

gaugeinvariant p erturbation variables used in this pa

where P u u g is the pro jection onto the subspace

p er These denitions their geometrical interpretation

of T M normal to u It is

and a short derivation of the p erturbation equations can

i

b e found in We restrict the analysis to the spa

i

tially at case K We dene the p erturb ed metric

The p erturbations of pressure and anisotropic stresses

by

can b e parameterized by

g g a h A

j j j

i

p with A

L

i

i i i

where g denotes the standard Friedmann background a

For scalar p erturbations we set

is the scale factor and h denotes the metric p erturbation

j S j

k u

i v u A

u k

Scalar p erturbations

and

S i

i i

Scalar p erturbations of the metric are of the form

k k k

j

j

j

S j i j

Studying the b ehavior of these variables under gauge

h Adt iB k dtdx H H dx dx

j L T ij

transformations one nds that the anisotropic stress p o

i j

k H k k dx dx A

T i j

tential is gauge invariant A gauge invariant velo city

variable is the shear of the velo city eld

Computing the p erturbation of the Ricci curvature scalar

and the shear of the equal time slices we obtain

S m

k k k a V with V v k H

i j ij T

ij

H A R a k R with R H

A

T L

There are several dierent useful choices of gauge in

j

variant density p erturbation variables

k k

i

j

aniso i

K a dx A

j

i

k

D w a ak A

s

with

D w R D w A

g s

k H B A

T

D D w a ak V A

s

The Bardeen p otentials are the combinations

In this work we mainly use D Here w p denotes

g

the enthalpy Clearly these matter variables can b e de

R aa k A

ned for each matter comp onent separately For ideal

A k aa A

uids like CDM or the baryon photon uid long b efore de

coupling anisotropic stresses vanish and c w

L

s

They are invariant under innitesimal co ordinate trans

where c is the adiabatic sound sp eed

s

formations gauge transformations

S

Also scalar p erturbations of the photon brightness

To dene p erturbations of the most general energy mo

are not gauge invariant It has b een show that the

mentum tensor we intro duce the energy density and

combination

the energy ux u as the timelike eigenvalue and normal

ized eigenvector of T

S S j

M R ik n k A

j

u u u T

is gauge invariant This is the variable which we use

here In other work the gauge invariant variable

We then dene the p erturbations in the energy density

M has b een used Since is indep endent of the

and energy velo city eld by

photon direction n this dierence in the denition shows

up only in the monop ole C But clearly as can b e seen

A

from Eq A also the dip ole C is gauge dep endent

The brightness p erturbation of the neutrinos is dened

i

the same way and will not b e rep eated here u u u A

t i

Vector p erturbations where cos n n To nd Eq we use the Fourier

transform normalization

Z

Vector p erturbations of the metric are of the form

f x expik xd x B f k

V j l j

V

h B dx dt ik k H k H dx dx A

j l j j l

with some normalization volume V Assuming that en

where B and H are transverse vector elds The simplest

semble average can b e replaced by volume average then

gauge invariant variable describing the two vectorial de

implies

grees of freedom of metric p erturbations is

Z

T T T T

H B A k

j j j

n n x n x n d x

T T V T T

Z

Vectorial anisotropic stresses are gauge invariant They

T T

k n k n B d k

are of the form

T T

V

ik k k A

T

j l l j

lj

M and using the Inserting our ansatz for

T

addition theorem for spherical harmonics we have

The vector degrees of freedom of the velo city eld are

cast in the vorticity

T T

n n

T T

u u iak k A

lj j l j l l j

X

0



with v B A

j j j

Y n Y n

0 0

m

m

0 0

mm

Vector p erturbations of the photon brightness are gauge

Z

 

invariant To maintain a consistent notation we denote

0 0

k dk d Y k Y k h ik

0

m

m

k

V

them by M

Z

X



P n n k dk h ik B

Tensor p erturbations

from which we can read of Eq

We dene tensor p erturbations of the metric by

For vector and tensor uctuations the ansatz and

must b e taken into account With the same ma

T i j

h H dx dx A

ij

nipulations as ab ove the correlation function of CMB

anisotropies induced by vector mo des reads

where H is a traceless transverse tensor eld

ij

The only tensor p erturbations of the energy momen

T T

n n

tum tensor are anisotropic stresses

T T

Z

X

T

0 0 0 0

A

lj

d k k n n P P B

lj

Tensor p erturbations of the photon brightness are de

T

noted M

where the primes indicate that the quantity is considered

Clearly all tensor p erturbations are gaugeinvariant

in the reference system where k is parallel to the z axis

there are no tensor typ e gauge transformations

and

i

q

APPENDIX B THE CMB ANISOTROPY

0 0

POWER SPECTRUM

V V 

0 0

cos h i cos

Here we derive in some detail Eqs and

V V 

0 0

CMB anisotropies are conveniently expanded in spher

h i sin sin B

P

l

a Y n The co e ical harmonics T nT

lm

m

lm

Assuming statistical isotropy which implies

cients a are random variables with zero mean and ro

lm

tationally invariant variances C hj a j i The mean

lm

V V V V

hj j i hj j i and h i

over the ensemble correlation function of the anisotropy

pattern has the standard expression

we obtain

X

T T

n n C P cos B

T T

T n T n

with

T T

T T T

X

T

B

i B

Z

V V 

The formulas BB and B are used to determine

0 0 0 0

h iP P d k B n n

the CMB anisotropy sp ectrum

where

0 0

n n B

Using the recursion formula for Legendre p olynomials

and the addition theorem for spherical harmonics we

See the web sites

nd after some manipulations

httpastroestecesanlS Agen eral

ProjectsPlanck

Z

V V

hj t k t k j i

and httpmapgsfcnasagov

V

k dk C

W Hu N Sugiyama and J Silk Nature

B

R Durrer M Kunz C Lineweaver and M Sakellari

adou Phys Rev Lett

For the correlation function of the CMB anisotropies

from tensor mo des our ansatz gives

R Durrer and M Kunz Phys Rev D R

T T

U Pen U Seljak and N Turok Phys Rev Lett

n n

T T

Z

X

0 0 0 0

k n n P P d k B

PP Avelino EPS Shellard JHP Wu and B

Allen Phys Rev Lett in print archived under

astroph

with

R Durrer and Z Zhou Phys Rev D

0

i

R Durrer Phys Rev D

T T 

0 0 0

h i cos cos

 

R Durrer Fund of Cosmic Physics

T T 

0 0

h i sin sin B

JM Stewart and M Walker Pro c R So c London

A

Here statistical isotropy leads to

J Bardeen Phys Rev D

T n T n

T T

H Ko dama and M Sasaki Prog Theor Phys Suppl

X

i

R Durrer and M Sakellariadou Phys Rev D

Z

T T 

0 0 0 0

n n h iP P d k B

C Cheung and J Magueijo Phys Rev D

where

0 0 0 0

JP Uzan N Deruelle and N Turok Phys Rev D

n n

B

A Melchiorri and N Vittorio Polarization of the mi

crowave background Theoretical framework in Pro ceed

With straightforward but somewhat cumb ersome manip

ings of the NATO Advanced Study Institute on the

ulations applying the recursion formula for Legendre

Cosmic Background Radiation Strasb ourg archived

p olynomials and the addition theorem for spherical har

under astroph

monics we then obtain

Z

A Albrecht R Battye and J Robinson Phys Rev

T

1

j k j

T

Lett

k dk B C

JO Gundersen et al Astrophys J L A Albrecht D Coulson PG Ferreira and J Magueijo

Phys Rev Lett

M Dragovan et al Astrophys J L

B Allen et al Phys Rev Lett

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