Particle Physics and Cosmology

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Edward W Kolb

have the p otential to explain the present state of the Universe In a very real sense the

job of cosmology is to provide a canvas up on which other elds of science including

particle physics can weave their individual threads into the tap estry of our understand

ing of the Universe Nowhere is the inherent unity of science b etter illustrated than in

the interplay b etween cosmology the study of the largest things in the Universe and

particle physics the study of the smallest things

A quick lo ok at the Universe

Before concentrating on the particle physics asp ects of cosmology I will start with a

lo ok at the most imp ortant observational features of the Universe I will then discuss

the Rob ertsonWalker metric and discuss some particle kinematics in the expanding

Universe Then I will develop the dynamics of the FriedmannRob ertsonWalker frw

cosmology The nal part of the intro ductory section will b e a brief review of the

radiationdominated era and primordial nucleosynthesis More details can b e found in

Kolb and Turner

Expansion of the Universe

It was Hubble who discovered a linear relationship b etween the recessional velo cities

of nebulae and their distances The recessional velo city is determined via the Doppler

eect If the relative velo city b etween a source and observer is v then the measured

wavelength of the light will dier from the wavelength of the emitted light

obs emitted

This dierence is expressed in terms of a redshift z dened as

obs emitted

emitted

If one interprets the observation of a redshift of light from distant galaxies as a Doppler

eect then z v c Of course this is a nonrelativistic expression The sp ecial

relativistic expression relating v and z is v c z z If the

R R

relative distance is increasing then z is p ositive

The linear relationship b etween the distance and the redshift Hubbles law can b e

written in several equivalent forms

cz H d

v H d

R L

d h z Mp c h cz Mp c

where a megaparsec Mp c is cm As to the meaning of these symb ols c is the

sp eed of light unless you are an astronomer the redshift z and recessional velo city v

have already b een dened d is the luminosity distance and H is Hubbles constant

Let us p ostp one for a moment questions ab out what exactly the luminosity distance

is and just think of it as the distance to the ob ject without worrying whether it is the

distance when the light was emitted the distance when the light was detected etc

Particle Physics and Cosmology

Sixtyfour years after the discovery of Hubbles law Hubbles constant H is still

not well known It is traditional to express the uncertainty in Hubbles law in terms of

a dimensionless parameter h

H h h

s Mp c

The Hubble constant is the fundamental parameter in cosmology and it is not known to

b etter than a factor of two This uncertainty traces to the oldest and most fundamental

problem of astronomythe distance scale for a review see RowanRobinson

The uncertainty in H will result in a proliferation of factors of h in many of the

equations in subsequent sections

Figure Hubbles data The solid line is a guide to the eye

It is somewhat amusing to lo ok at the original data up on which Hubble based his

claim shown in Figure Clearly it to ok a leap of imagination intuition and genius

to see a linear relationship in the data After all some of the nearby nebulae are

approaching rather than receding With mo dern hop efully more reliable metho ds

for determining distances astronomers are able to extend Hubbles program to much

greater distances All agree on the linear nature of the relationship but do not agree

on the value of H

To orient the particle physicist I have included a small table of extragalactic dis

tances The distance to nearby ob jects in our lo cal group of galaxies like Andromeda

can b e determined by direct means Hence the distance is indep endent of h For more

distant ob jects such as the Virgo cluster of galaxies we can accurately determine the

red shift or equivalently v but not the distance Using the measured z Hubbles

Dep ending up on your law will give the distance in terms of the annoying factor of h

favourite value of h the distance to Virgo is somewhere b etween and Mp c

Edward W Kolb

object z d v

L R

M Andromeda Mp c km s

Virgo Cluster h Mp c km s

Coma Cluster h Mp c km s

Hydra h Mp c km s

Clearly Hubbles law as given in Equation must break down for z Even if one

adopts the sp ecial relativistic Doppler formula we will see in the section on kinematics

that there are imp ortant corrections for z

The expansion of the Universe and Hubbles law will b e discussed further but for

our rst quick view of the Universe it will suce to note that the Universe is expanding

and furthermore the expansion seems to b e isotropic ab out us

The cosmic background radiation

The cosmic background radiation cbr provides fundamental evidence that the Uni

verse b egan from a hot big bang The surface of last scattering for the cbr was the

Universe at an age of ab out years The rst thing to learn ab out the cbr is its

sp ectrum It is a blackb o dy to a remarkable accuracy The b est measurement of the

sp ectrum of the cbr was made with the Cosmic Background Explorer cobe satellite

Mather et al The measurements are summarized in Figure Note that the

true error bars for the measurements are a factor of times smaller than shown in

the gure Clearly the cbr is a blackb o dy with the present temp erature of the Uni

verse T K and deviations from a blackb o dy shap e over the wavelength

interval cm to cm less than

Once the temp erature of the cbr is known the numb er density and energy den

sity of the background photons are also known For a temp erature of T K

eV the numb er density and energy density of the cbr is given by

n cm T

T g cm

After the sp ectrum the next most imp ortant feature of the cbr is its isotropy

Anisotropy is exp ected due to several eects For instance a dip ole moment of the cbr

is exp ected as a result of the motion of our lo cal reference frame with resp ect to the

cbr rest frame Motion with velo city v c through an isotropic blackb o dy radiation

eld of temp erature T results in a frequencyindep endent formula for the temp erature

j j cos distribution T T

The most accurate measurement of the cbr dip ole anisotropy was by cobe an

amplitude of mK corresp onding to a velo city of km s in the general

direction of Hydra for our lo cal group of galaxies cobe has also determined that the

dip ole anisotropy has a thermal sp ectrum

Additional uctuations in the cbr temp erature are also exp ected due to the presence

of density inhomogeneities presumed to have triggered structure formation The search

Particle Physics and Cosmology

Figure The spectrum of the cosmic background radiation

for anisotropies in the cbr b eyond the dip ole anisotropy has o ccupied physicists since

the discovery of the cbr itself in Finally in the long search was rewarded

when the cobe collab oration announced the discovery of anisotropy on angular scales

from ab out to at a magnitude of ab out part in Smo ot et al There

are several metho ds to analyze the anisotropy The cleanest and most reliable indication

of anisotropy is an rms temp erature variation of K on the sky averaged over a

b eam of fwhm cobe also rep orted a quadrup ole anisotropy of K

Figure Anisotropy multipoles of the cosmic background radiation

If one expands the observed temp erature uctuation as a function of angles and

in spherical harmonics

X X

a Y

l m l m

ml l

then measurement of uctuation can b e expressed in terms of the multip ole ampli

tudes a In this expansion the dip ole moment has b een left out since it arises to our l m

Edward W Kolb

p eculiar velo city with resp ect to the cbr rest frame The inferred multip ole amplitudes

from l to l as measured by cobe are shown in Figure

In this rst discussion of the cbr the most imp ortant feature is that the temp erature

of the Universe is well determined T K A dip ole moment of the cbr is also

well known In addition there is now evidence for higher multip ole moments in the

cbr anisotropy However in the excitement and publicity of the discovery of cbr

anisotropy one should not lo ose sight of the most imp ortant asp ect of the cbr its

remarkable isotropy The cbr is isotropic ab out us to b etter than one part in

Homogeneity and isotropy

We live in a hot expanding Universe We also live in a Universe that on large scales

is homogeneous the same at every p oint and isotropic the same in every direction

There is an ample and growing b o dy of evidence for homogeneity and isotropy The

homogeneity and isotropy of the Universe is the most fundamental principle in mo dern

cosmology In fact it is called the Cosmological Principle

The assumption of the isotropy and homogeneity of the Universe in mo dern cos

mology dates back to the work of Einstein who made the assumption not based up on

observational evidence but to simplify the mathematical analysis To day there is am

ple evidence for the isotropy and homogeneity for the part of the Universe we can

observe our present Hubble volume characterized by a length H h Mp c

h cm

The b est evidence for the isotropy of the observed Universe is the uniformity of the

temp erature of the cbr as discussed ab ove If the expansion of the Universe were not

isotropic the expansion anisotropy would lead to a temp erature anisotropy of similar

magnitude Likewise inhomogeneities in the density of the Universe on the surface of

last scattering would lead to temp erature anisotropies In this regard the cbr is a very

p owerful prob e

Additional evidence for the isotropy of the Universe is the isotropy of the xray

background radiation Some large fraction of the xray background is b elieved to b e

from unresolved sources eg qsos at high redshift Likewise a substantial fraction

of faint radio sources are radio galaxies at high redshift z and their distribution

is also isotropic ab out us

Evidence for homogeneity and isotropy from the distribution of galaxies is somewhat

less certain mostly b ecause we are only now mapping the distribution of galaxies on

scales large enough to see homogeneity One can nd some measure of homogeneity of

the Universe by taking a sphere of radius R which contains on average N galaxies and

placing it down at all p oints in the Universe counting the numb er of galaxies inside

it and computing the ro otmeansquare rms numb er uctuations One nds that the

rms numb er uctuations N N decrease with increasing scale and drop b elow unity

for radius R h Mp c This indicates that on scales less than R the Universe is

lumpy and for scales greater than R the Universe b ecomes smo oth

This is not to say that the Universe is structureless on scales greater than R The

b est known example of structures on larger scales comes from the Center for Astro

physics cfa slices of the universe de Laupparent Geller and Huchra shown

Particle Physics and Cosmology

in Figure recall from Equation that d h cz Mp c Some of the struc

ture is the result of the presentation of the data in redshift space which stretches out

things in the radial direction but clearly there is structure on scales much larger than

h Mp c

Clear evidence from the distribution of galaxies for homogeneity of the Universe must

await surveys on scales much larger than the cfa survey These should b e completed

b efore the end of this century Until that time we can only lo ok at larger regions of

the Universe with sparse samples of the lo cation of galaxies ie only a fraction of

galaxies in the sample volume are mapp ed Such a sparse sample for the Automatic

Plate Measuring apm survey Loveday et al is shown in Figure Note that

the apm survey is nearly three times as deep as the cfa survey It do es not seen to

show structures on the size of the survey as the cfa survey do es Just by comparing

the two gures one concludes that the distribution of matter in the Universe is not a

fractal but rather approaches homogeneity on large scales

Figure One of the cfa slices of the Universe containing galaxies

Figure A sparse sample of the apm survey

In conclusion the Universe is lumpy on small scales containing p eople planets

Edward W Kolb

stars galaxies galaxy clusters sup erclusters etc However on large scales the distribu

tion of matter and radiation in the Universe is smo oth We are only now probing the

transition region from lumpy to smo oth in the distribution of galaxies It is exciting

to b e a cosmologist at the time when the largest structures in the Universe are b eing

discovered

The large degree of spatial symmetry in a spatially homogeneous and isotropic Uni

verse will greatly simplify the dynamics of the expansion of the Universe Through

the action of cosmic ination we will b e able to understand why the Universe is ho

mogeneous and isotropic on observable scales as well as understanding why there is

structure in the Universe

The present Universe

I will conclude this quick lo ok at the Universe to day by presenting the parameters that

describ e the present Universe

Expansion The Universe is expanding at a present rate given by Hubbles con

stant expressed in terms of a dimensionless parameter h H h km s Mp c

with h

Temp erature The present temp erature of the Universe is T K

with a dip ole moment of mK and rms temp erature uctuations on a scale of

of ab out K

Homogeneity and Isotropy The Universe is homogeneous and isotropic on large

scales and clumpy on small scales The transition region is ab out R h Mp c

Mass and Energy Density The mass density of the Universe is p o orly known

It is convenient to express mass densities in terms of a critical density formed by

Hubbles constant and Newtons constant

h g cm

Expressed as a fraction of the critical density the matter M photon and radiation

R energy densities of the Universe are

M M C

R R C

where I have included massless neutrino sp ecies at a temp erature of K in addition

to the photons in determining the radiation energy density Clearly to day we live in a

matterdominated Universe since

M R

The Rob ertsonWalker metric

The metric for a space with homogeneous and isotropic spatial sections is the Rob ertson

Walker rw metric which can b e written in the form

r d r sin d ds dt a t

k r

Particle Physics and Cosmology

where r are spatial co ordinates referred to as comoving co ordinates at is

the cosmic scale factor and with an appropriate rescaling of the co ordinates k can b e

chosen to b e or for spaces of constant p ositive negative or zero spatial

curvature resp ectively The co ordinate r in Equation is dimensionless ie at has

dimensions of length and r ranges from to for k Notice that for k

the circumference of a onesphere of co ordinate radius r in the const plane is just

atr and that the area of a twosphere of co ordinate radius r is just a tr

dr k r and however the physical radius of such one and twospheres is at

not atr

The time co ordinate in Equation is the prop er time measured by an observer at

rest in the comoving frame ie r const Observers at rest in the comoving

frame remain at rest ie r remain unchanged and observers initially moving

with resp ect to this frame will eventually come to rest in it Thus if one intro duces

a homogeneous isotropic uid initially at rest in this frame the t const hyp ersur

faces will always b e orthogonal to the uid ow and will always coincide with the

hyp ersurfaces of b oth spatial homogeneity and constant uid density

The dynamical equations that describ e the evolution of the scale factor at follow

R g GT Before pro ceeding we must from the Einstein eld equations R

sp ecify the stressenergy tensor To b e consistent with the symmetries of the metric the

total stressenergy tensor T must b e diagonal and by isotropy the spatial comp onents

must b e equal The simplest realization of such a stressenergy tensor is that of a

p erfect uid characterized by a timedep endent energy density t and pressure pt

T diag p p p The comp onent of the conservation of stress energy

gives the st law of thermo dynamics da pda For equation T

the simple equation of state p w where w is indep endent of time evolves as

a Examples of this simple equation of state we will employ include

radiation p a

matter p a

vacuum energy p const

The comp onent of the Einstein equation gives the Friedmann equation

k G a

a a

A combination of the ii comp onent with the Friedmann equation gives an equation for

the deceleration of the expansion

a G

The expansion rate of the Universe is determined by the Hubble parameter H aa

The Hubble parameter is not constant and in general varies as t The Hubble time

or Hubble radius H sets the time scale for the expansion a roughly doubles in a

Hubble time The Hubble constant H is the present value of the expansion rate The

Friedmann equation can b e recast as

H a H G

Edward W Kolb

Since H a there is a corresp ondence b etween the sign of k and the sign of

k closed

k flat

k open

Particle kinematics

The rst application of particle kinematics with the Rob ertsonWalker metric is a cal

culation of the prop er distance to the horizon ie for a comoving observer with co or

dinates r for what values of r would a light signal emitted at t

reach the observer at or b efore time t A light signal satises the geo desic equation

ds Because of the homogeneity of space without loss of generality we may cho ose

r A light signal emitted from co ordinate p osition r at time t will

reach r in a time t determined by

Z Z

r t

dr dt

k r

and the prop er distance to the horizon measured at time t d t g dr is

H r r

simply at times the ab ove integral

Z Z

t at

dt dat

d t at at

at a t at

We know the b ehaviour of at from the Friedmann equation For the early Universe

we can ignore the curvature term For a radiationdominated Universe a t and

d t t while for a matterdominated Universe a t and d t t If d t is

H H H

nite then our past light cone is limited by a particle horizon which is the b oundary

b etween the visible Universe and the part of the Universe from which light signals have

not reached us The b ehaviour of at near the singularity will determine whether or

not d is nite

The next application of particle kinematics is the redshift The fourvelo city u of

a particle with resp ect to the comoving frame is referred to as its peculiar velo city The

equation of geo desic motion for u is

dx du

d d

where u dx ds and is some ane parameter which we will cho ose to b e the

prop er length ds The comp onent of the geo desic equation is du ds u u

Using the fact that for the Rob ertsonWalker metric the only nonvanishing comp onent

of is aa h where h is the spatial metric the geo desic equation gives

ij ij

jujjuj aa which implies that juj a In an expanding Universe a freely

falling observer is destined to come to rest in the comoving frame even if he has some

mu we see that initial p eculiar velo city Recalling that the fourmomentum is p

the magnitude of the threemomentum of a freelypropagating particle also redshifts

Particle Physics and Cosmology

as a The wavelength of light is inversely prop ortional to the photon momentum

hp If the momentum changes the wavelength of the light also changes The

wavelength at time t denoted as will dier from that at time t denoted as

by z at at This means that the redshift of the wavelength of a

photon is due to the fact that the Universe was smaller when the photon was emitted

Our nal foray into particle kinematics will b e Hubbles law Supp ose a source

eg a galaxy has an absolute luminosity L Its luminosity distance is dened in terms

of the measured ux F by d L F If a source at comoving co ordinate r r

emits light at time t and a detector at comoving co ordinate r detects the light

at t t conservation of energy implies F L a t r z which implies

d a t r z In order to express d in terms of the redshift z the explicit

dep endence up on r must b e removed Since light travels on geo desics

Z Z

r a

dr dat

k r a t at

By use of the Friedmann equation for zero pressure solution is easily found to b e

z z

H R z

Therefore Hubbles law b ecomes

z z z H d q z

where q is the deceleration parameter q at a t at Clearly for

z departures from the linear relationship are exp ected In principle these depar

tures would lead to a value for But in practice evolutionary eects in the brightness

of galaxies for large z have prevented realization of the promise

The radiationdominated era

In a radiationdominated Universe the energy density and pressure can b e expressed

in terms of the photon temp erature T as

g T p g T

R R R

where g counts the total numb er of eectively massless degrees of freedom those

sp ecies with mass m T given by

X X

T T

i i

g g g

i i

T T

ifermions ib osons

The relative factor of accounts for the dierence in Fermi and Bose statistics

Note that g is a function of T since the sum runs over only those sp ecies with mass

m T For T GeV all the sp ecies in the standard mo del gluons W Z

generations of quarks and leptons and complex Higgs doubletshould have b een

relativistic giving g

Edward W Kolb

During the early radiationdominated ep o ch t sec and further

when g const p ie w and at t From this it follows that

R R

the expansion rate and expansion age is

T m T

P l

H g sec t g

m T Mev

P l

Primordial nucleosynthesis

Primordial nucleosynthesis is a most useful prob e of the early Universe and the consis

tency of the big bang mo del The basic idea is that at very high temp eratures T

MeV there were no nuclei but as the Universe expanded and co oled conditions b ecame

hospitable for the formation of nuclei The outcome of primordial nucleosynthesis the

relative abundances of the various nuclei dep end up on the interplay of the expansion

rate of the Universe and the nuclear reactions This has to o ccur in a setting of enor

mous sp ecic entropy In other words primordial nucleosynthesis o ccurs with the ratio

of photons to nucleons of ab out

The outcome of primordial nucleosynthesis is very sensitive to the baryontophoton

ratio usually denoted by The reason for this is simple In nuclear statistical equi

librium nse the fraction of the total baryon mass contributed by a nucleus with A

protons p and neutrons n and binding energy B is

AZ A A A A A Z

expB T X X g A T m X

A A A N

n p

where as usual

is the present baryontophoton ratio g is the numb er of spin degrees of freedom of the

nucleus and m is the mass of a nucleon The sensitive dep endence up on of primordial

nucleosynthesis up on arises from the factor of in this equation Of course

primordial nucleosynthesis represents a departure from nuclear statistical equilibrium

and the nse abundance is not always the actual abundance but nevertheless the nse

abundance sets the value of what the abundance wants to b e

Recent analysis Walker et al of the outcome of primordial nucleosynthesis

suggest that must b e in the range of to times to agree with the inferred

primordial abundances

The formation of structure

Before discussing the physical pro cesses imp ortant in the theory of structure formation

through gravitational instability I will briey review some preliminaries related to a

Fourier analysis of the density eld of the Universe

It is convenient to discuss the density eld of the Universe in terms of the density

contrast where

x x

Particle Physics and Cosmology

and to express the density contrast x in a Fourier expansion

expik x d k x

Here is the average density of the Universe p erio dic b oundary conditions have b een

imp osed and V is a normalization volume Since x is a scalar quantity one can

use either comoving or physical co ordinates in the Fourier expansion unless otherwise

sp ecied I will use comoving co ordinates

A particular Fourier comp onent is characterized by its amplitude j j and its comov

ing wavenumb er k Since x and k are comoving co ordinate quantities the physical

distance and physical wavenumb er are related to the comoving distance and wavenum

b er by dx atdx k k at The wavelength of a p erturbation is related to

phys phys

the wavenumb er by k at

phys

All statistical quantities for gaussian random uctuations can b e sp ecied in terms

of the power spectrum j j In the absence of a b etter idea it is assumed that j j k

k k

that is a featureless p ower law

The rms density uctuation is dened by

h x x i

where h i indicates the average over all space Some manipulation yields

dk k j j

The contribution to from a given logarithmic interval in k is given by

k j j

k V

The uctuation p ower p er logarithmic interval denoted by k will app ear often

Now consider M M the rms mass uctuation on a given mass scale This is

what most p eople mean when they refer to the density contrast on a given mass scale

Mechanically one would measure M as follows Take a volume V which on

r ms W

average contains mass M place it at all p oints throughout space measure the mass

within it and then compute the rms mass uctuation Although it is simplest to cho ose

a spherical volume V with a sharp surface to avoid surface eects one often wishes

to smear the surface This is done by using a window function W r which smo othly

V where denes a volume V and mass M

W W

W r dr r V

V is given in terms of the The rms mass uctuation on the mass scale M

density contrast and window function by

M dk

k jW k j

M V k W

Edward W Kolb

Notice that the rms mass uctuation is given in terms of an integral over k

Taking j j Ak and W r for r r and zero otherwise one nds

k r

for n M M is given by an integral over all wavelengths longer than ab out

r and is roughly equal to the rms value of k r Using this top hat window

function Davis and Peebles nd for the cfai redshift survey that M M

for a sphere of radius of r h Mp c This value of r separates the linear from the

nonlinear regime

It is commonly assumed that the observed structure is a result of the growth of

small seed inhomogeneities In the next section on ination and in Section on phase

transitions I will discuss p ossible origins for the seed p erturbations But b efore doing

that here I will discuss the theory of gravitational instability in an expanding Universe

and discuss the physical nongravitational pro cesses that might aect the p erturbation

sp ectrum First consider the linear theory of gravitational instability

Gravitational instabilitylinear theory

We will start by considering the simplest p ossible form of gravitational instability the

Jeans instability in a nonexpanding p erfect uid The Eulerian equations of Newtonian

motion describing a p erfect uid are

r v

r p r v r v

r G

Here is the matter density p the matter pressure v the lo cal uid velo city and the

gravitational p otential The simplest solution is the static one where the matter is at

rest v and uniformly distributed in space const p const Throughout

we will denote unp erturb ed quantities with the subscript Now consider p erturbations

ab out this static solution expanded as

p p p v v v

We will consider adiabatic p erturbations that is p erturbations for which there are no

spatial variations in the equation of state The adiabatic sound sp eed v is dened

as v p and by assumption there are no spatial variations in the equation of

state v p To rst order the small p erturbation satises the secondorder

dierential equation

v r G

Solutions to Equation are of the form

h i

r t r t A exp ik r i t

Particle Physics and Cosmology

and and k satisfy the disp ersion relation v k G with k jk j

If is imaginary there will b e exp onentially growing mo des if is real the p er

turbations will simply oscillate as sound waves It is clear that for k less than some

critical value will b e imaginary This critical value is called the Jeans wavenumb er

k and is given by

For k k grows exp onentially on the dynamical timescale G

dyn

It is useful to dene the Jeans mass the total mass contained within a sphere of

radius k

J J

Perturbations of mass less than M are stable against gravitational collapse while those

of mass greater than M are unstable

The classical Jeans analysis is not directly applicable to cosmology b ecause the

expansion of the Universe is not taken into account and b ecause the analysis is New

tonian For mo des of wavelength less than that of the horizon ie H a

phys

Newtonian analysis suces so long as the expansion is taken into account

When the expansion of the Universe is taken into account the wave equation b e

comes

v k a

k k k

a a

Again for k k there are unstable growing mo de solutions In the limit k k for a

J J

spatially at matterdominated frw mo del where aa t and Gt

k k

t t

This equation has two indep endent solutions a growing mo de and a decaying mo de

with time dep endence given by

t t

t t t t

i i

t t

i i

Here we see the key dierence b etween the Jeans instability in the static regime and in

the expanding Universe the expansion of the Universe slows the exp onential growth of

the instability and results in p owerlaw growth for unstable mo des

Consider a twocomp onent mo del with nonrelativistic nr sp ecies i eg baryons

or a wimpweakly interacting massive particle and photons during the radiation

dominated era In this case a a t and Consider the evolution of

TOTAL

p erturbations that are Jeans unstable k k If the photons have no p erturbations

then the solution is

i i

In this case the solution is t t a lntt so only a p erturbation with an

i i i i

initial velo city t can actually grow and only logarithmically at that i

Edward W Kolb

The growth of linear p erturbations in a radiationdominated Universe is inhibited

compared to the static situation The physical reason b ehind this fact is easy to see

The classical Jeans instability with exp onential growth is mo derated by the expansion

of the Universe In a matterdominated ep o ch p erturbations grow as a p ower law In

the radiationdominated ep o ch the expansion rate is faster than what it would b e if

there were only matter present and so the growth of p erturbations is further slowed

Damping pro cesses

The theory of gravitational instability discussed so far assumes that the Universe is lled

with a p erfect uid However there are imp ortant departures from this ideal situation

Collisionless damping o ccurs during the radiationdominated era when linear p er

turbations do not grow If the particle sp ecies is collisionless the p erfectuid approx

imation is clearly invalid In this case collisionless phase mixing or Landau damping

will o ccur Perturbations will b e damp ed on length scales smaller than the distance the

particle will travel while decoupled If the sp ecies b ecomes nonrelativistic at time t

then at time t when the Universe b ecomes matterdominated and p erturbations can

start to grow the sp ecies would have freestreamed a distance

Z Z

t t

NR EQ

v t

at dt at dt

at at

where the integral has b een split into two pieces the relativistic regime with v and

the nonrelativistic regime when v Assuming the Universe is radiation dominated

at t

t z lnt t

FS NR NR EQ NR

For a light neutrino sp ecies

Mp c

Perturbations on scales less than are damp ed by free streaming Note that this

length scales is much larger than the length scale asso ciated with galaxies containing a

mass in neutrinos of m eV M where M g is a solar mass

The p erfectuid approximation also breaks down for baryons Before recombi

nation the photons mean free path is small but as the matter particles in Universe

b ecome electrically neutral during recombination the photon mean free path b ecomes

longer and photons can diuse out of dense regions To the degree that the pho

tons are not completely decoupled from the baryons they can drag the baryons along

also damping p erturbations in the baryons This eect is known as Silk damping

A careful calculation of the damping requires solving the Boltzmann equation but

to a go o d approximation the scale for Silk damping is ab out an order of magnitude

smaller than the horizon scale at decoupling Thus baryon p erturbations should b e

h Mp c corresp onding to a mass of damp ed on scales less than

S B

M h M

S B

Particle Physics and Cosmology

Sup erhorizonsize p erturbations

So far the analysis of the evolution of density p erturbations has b een Newtonian For

mo des that are well within the horizon H the Newtonian analysis is ade

phys

quate To treat the evolution of mo des outside the horizon one needs a full general

relativistic analysis Clearly this is b eyond the scop e of these lectures However I will

illustrate the idea in a simple geometric way Consider p erturbations of a spatially at

k frw mo del The Friedmann equation for the unp erturb ed k mo del is

H G k

Now consider a similar frw mo del one with the same expansion rate H but one that

has higher density and is therefore p ositively curved The expansion rate is

G k

H k

If we compare the mo dels when their expansion rates are equal we have made a choice

of gauge in this case the uniform Hubble gauge We immediately see that the density

contrast b etween the two mo dels is given in terms of the curvature of the closed mo del

k a

The evolution of has b een reduced to that of the curvature k a relative to the

energy density As long as is small equivalently k a G the scale factors

for the two mo dels are essentially equal fractional dierence of order In a matter

dominated Universe a while in a radiationdominated Universe a so

a radiation dominated

a matter dominated

Recall that in a matterdominated Universe a t while in a radiationdominated

Universe a t so

tt radiation dominated

tt matter dominated

This simple mo del illustrates several imp ortant p oints ab out the evolution of sup er

horizonsized p erturbations The geometric character of a density p erturbation

which is why density p erturbations are referred to as curvature uctuations What

is actually relevant is the dierence in the evolution of the p erturb ed mo del as compared

to some unp erturb ed reference mo del The freedom in the choice of the reference

mo del is equivalent to a gauge choice so that in general will b e gauge dep endent

As in particle physics when confronted with gauge ambiguity the correct thing to

do is to ask a physical question one whose answer cannot dep end up on the gauge

Here the p erturb ed and reference mo dels are compared by matching their expansion

rates A very useful quantity is p The evolution of is particularly

simple and independent of the background spacetime For sup erhorizonsized mo des

the evolution is const for H

phys

Edward W Kolb

Summary of the evolution of p erturbations

For subhorizonsized p erturbations one can p erform a Newtonian treatment of the

evolution of p erturbations During the radiationdominated ep o ch p erturbations do

not grow During the matterdominated ep o ch p erturbations on scales larger than

the Jeans length grow as at t Perturbations on scales smaller than the

Jeans length oscillate as acoustic waves Before recombination the baryon Jeans mass

is larger than the horizon mass by a factor of ab out after recombination the baryon

Jeans mass drops to ab out M

Collisionless phase mixing damps p erturbations on scales smaller than the free

streaming scale is a few times t a where the subscript nr denotes the

FS NR NR

value of the quantity at the ep o ch when the sp ecies b ecame non relativistic Taking the

particles mass to b e m and the ratio of its temp erature to the photon temp erature

to b e T T the freestreaming scale is roughly Mp c m keV T T For

X FS X X

cold dark matter cdm mo dels this damping scale is smaller than any cosmologically

interesting scale For hot dark matter hdm mo dels the damping scale is larger than

the galactic scale so hdm must b e augmented with some other seeds to grow galaxies

Due to photon diusion adiabatic uctuations in the baryons are strongly damp ed

on scales smaller than the Silk scale This damping o ccurs primarily just as the photons

and baryons decouple The Silk scale is given by h Mp c

S B

For mo des that are sup erhorizon sized the subtleties of the gauge noninvariance

of are imp ortant and a full general relativistic treatment is required There are two

physical mo des a decaying mo de and a growing mo de as well as pure gauge mo des In

the synchronous gauge the growing mo de evolves as a t radiation dominated

and at t matter dominated Alternatively the evolution of sup erhorizon

sized mo des can b e describ ed by the quantity p which is constant

The primeval sp ectrum is mo died by the ab ove physical pro cesses The pro cess

ing of the initial sp ectrum by the damping pro cesses dep end up on the mix of matter

cold hot and baryons It is useful to quantify the pro cessing of the sp ectrum by

sp ecifying a transfer function Since matter uctuations start to grow when the Uni

verse b ecomes matter dominated it is convenient to sp ecify the sp ectrum at this time

t h sec Again in the absence of a b etter idea it is convenient to

sp ecify the unprocessed p ower sp ectrum as a p ower law j j k

The pro cessed sp ectrum for hot dark matter is given by

n k k n

j j Ak Ak expk k

where the neutrino damping scale is k m eVMp c which is equivalent

to m eV Mp c and A provides the overall normalization of the p ower

sp ectrum For cold dark matter the pro cessed sp ectrum is given by

j j

k k k

Mp c h Mp c h Mp c Of course with h

there is an intermediate case known as warm dark matter

Particle Physics and Cosmology

object mass length scale angular scale

Stars M h Mp c h

Globular Clusters M h Mp c h

Galaxies M h Mp c h

Groups of Galaxies M h Mp c h

NonLinear Scale h M h Mp c

Thickness of lss h M h Mp c

Clusters M h Mp c h

Sup erclusters M h Mp c h

Horizon at lss h M h Mp c

Table Length and angular scales in an Universe The mass is related to scale

by M h M and the angle is related to scale by h

Mp c

lss stands for last scattering surface z

Confronting the sp ectrum

In the next two sections we will discuss the generation of p erturbations in ination

and due to defects pro duced in phase transitions He we discuss how we can prob e the

sp ectrum by presentday observations For more details see Liddle and Lyth or

Cop eland Kolb Liddle and Lidsey b

The range of scales of interest stretches from the present horizon scale h Mp c

down to ab out h Mp c the scale which contains roughly enough matter to form a

typical galaxy On the microwave sky an angle of for small enough samples linear

scales of h Mp c For purp oses of discussion it is convenient to split this

range into three separate regions

A Large scales h Mp c h Mp c

These scales entered the horizon after the decoupling of the microwave back

ground Except in mo dels with p eculiar matter contents p erturbations on these

scales have not b een aected by any physical pro cesses and the sp ectrum retains

its original form At present the p erturbations are still very small growing in

the linear regime without mo de coupling Here we are still seeing the primeval

sp ectrum

B Intermediate scales h Mp c h Mp c

These scales remain in the linear regime and their gravitational growth is easily

calculable However they have b een seriously inuenced by the matter content of

the Universe in a way normally sp ecied by a transfer function which measures

the decrease in the density contrast relative to the value it would have had if

the primeval sp ectrum had b een unaected Even in cdm mo dels where the

only eect is the suppression of growth due to the Universe not b eing completely

matter dominated at the time of horizon entry this eect is at the level at

Edward W Kolb

h Mp c To reconstruct the primeval sp ectrum on these scales it is thus

essential to know the matter content of the Universe including dark matter and

of its inuence on the growth of density p erturbations

C Small scales h Mp c h Mp c

On these scales the density contrast has reached the nonlinear regime coupling

together mo des at dierent wavenumb ers and it is no longer easy to calculate

the evolution of the density contrast This can b e attempted either by an ap

proximation scheme such as the Zeldovich approximation Efstathiou or

more practically via N b o dy simulations for example see Davis et al and

references therein Further hydro dynamic eects asso ciated with the nonlinear

b ehaviour can come into play giving rise to an extremely complex problem with

imp ortant nongravitational eects Again the transfer function plays a crucial

role on these scales In hot dark matter mo dels p erturbations on these scales are

most likely almost completely erased by free streaming and hence no information

can b e exp ected to b e available

Let us now consider each range of scales in turn starting with the largest scales and

working down to the smallest scales

A Large scales h Mp c h Mp c

Without doubt the most imp ortant form of observation on large scales for the near

future is largeangle microwave background anisotropies Scales of a couple of degrees

or more fall into our denition of large scales Such measurements are of the purest

form availableanisotropy exp eriments directly measure the gravitational p otential at

dierent parts of the sky on scales where the sp ectrum retains its primeval form Such

measurements also are of interest in that the tensor mo des may contribute

In addition to p erturbations from the scalar density p erturbations the presence of

gravitational waves will lead to temp erature uctuations One can think of gravita

tional waves as propagating mo des asso ciated with transverse traceless tensor metric

FRW

p erturbations of g g h

Tensor mo des do not participate in structure formation and most measurements we

shall discuss are oblivious to them Further tensor mo des inside the horizon redshift

away relative to matter and so tensor mo des also fail to participate in smallangle

microwave background anisotropies

Nevertheless these largescale measurements still exhibit one crucial and ultimately

uncircumventable problem On the largest scales the numb er of statistically inde

p endent sample measurements that can b e made is small Given that the underlying

inationary uctuations are sto chastic one obtains only a limited set of realizations

from the complete probability distribution function Such a subset may insuciently

sp ecify the underlying distribution This eect which has come to b e known as the cos

mic variance is an imp ortant matter of principle b eing a source of uncertainty which

remains even if p erfectly accurate exp eriments could b e carried out At any stage in

the history of the Universe it is imp ossible to accurately sp ecify the prop erties most

Particle Physics and Cosmology

signicantly the mean which is what the sp ectrum sp ecies assuming gaussian statis

tics of the probability distribution function p ertaining to p erturbations on scales close

to that of the observable Universe

Observations other than microwave background anisotropies app ear conned to the

long term future Even such an ambitious pro ject as the Sloan Digital Sky Survey

sdss Gunn and Knap Kron can only reach out to p erhaps h Mp c

which can only touch the lower end of our sp ecied large scales However in order

to sp ecify the uctuations accurately one needs many statistically indep endent regions

seems an optimistic lower estimate which means that the sdss may not sp ecify

the sp ectrum with sucient accuracy ab ove p erhaps h Mp c

A much more crucial issue is that the sdss will measure the galaxy distribution

p ower sp ectrum not the mass distribution p ower sp ectrum In mo dern work it is taken

almost completely for granted that these are not the same and it seems likely to o that

a bias parameter relating the two by a multiplicative constant which remains scale

indep endent over a wide range of scales may b e hop elessly unrealistic Consequently

converting from the galaxy p ower sp ectrum back to that of the matter may require a

detailed knowledge of the pro cess of galaxy formation and the environmental factors

around distant galaxies Once one attempts to reach yet further galaxies with a long

lo okback time one must also understand something ab out evolutionary eects on

galaxies As we shall discuss in the section on intermediate scales it seems likely that

p eculiar velo city data may b e rather more informative than the statistics of the galaxy

distribution

A more useful to ol for large scales is microwave background anisotropies on large

angular scales Our formalism closely follows that of Scaramella and Vittorio

On large angular scales the most convenient to ol for studying microwave background

anisotropies is the expansion into spherical harmonics

X X

a x Y x

l m

ml l

where and are the usual spherical p olar angles and x is the observer p osition

With spherical harmonics dened as in Press et al the reality condition is

a a In the expansion the unobservable monop ole term has b een re

l m

moved The dip ole term has also b een completely subtracted the intrinsic dip ole on

the sky cannot b e separated from that induced by our p eculiar velo city relative to the

comoving frame

With gaussian statistics for the density p erturbations the co ecients a x are

l m

gaussian distributed sto chastic random variables of p osition with zero mean and rota

tionally invariant variance dep ending only on l ha x i hja x j i

l m l m

It is crucial to note that a single observer sees a single realization from the probability

distribution for the a The observed multip oles as measured from a single p oint are

l m

dened as

Q ja j

l m

Edward W Kolb

and indeed the temp erature auto correlation function can b e written in terms of these

T T

C P cos Q

T T

where the average is over all directions on a single observer sky separated by an angle

and P cos is a Legendre p olynomial The exp ectation for the Q averaged over

all observer p ositions is just hQ i l

l l

A given mo del predicts values for the averaged quantities hQ i On large angular

scales corresp onding to the lowest harmonics only the SachsWolfe eect op erates

One has two terms corresp onding to the scalar and tensor mo deswe denote these

contributions by square brackets The scalar term is given in terms of the amplitude of

the scalar density p erturbation when it crosses the Hubble radius

HOR

by the integral

T k S k aH A j

S l l

m k

where j is a spherical Bessel function and the transfer function T k is normalized to

one on large scales

The amplitude of a given Fourier mo de of the dimensionless strain on scale when

it crosses inside the Hubble radius is given by

HOR

k h A

The expression equivalent to S for the tensor mo des contribution to temp erature

uctuations is a rather complicated multiple integral which usually must b e calculated

numerically Abb ot and Wise Starobinsky Lucchin et al Under

many circumstances Lucchin Matarrese and Mollerach suggest n for p ower

law ination there is a helpful approximation which is that the ratio S T is

l l

indep endent of l and given by

A S

S l

T A

l G

On the sky one do es not observe each contribution to the multip oles separately As

uncorrelated sto chastic variables the exp ectations add in quadrature to give

T S

l l l

There are two obstructions of principle These are

Even if one could measure the exactly the last scattering surface b eing closed

means one obtains only a discrete set of informationa nite numb er of the

covering some eective range of scales There will thus b e an uncountably innite

set of p ossible sp ectra which predict exactly the same set of l

Particle Physics and Cosmology

One cannot measure the exactly What one can measure is a single realization

the Q As a sum of l gaussian random variables Q has a probability

l l

distribution which is a distribution with l degrees of freedom The

variance of this distribution is given by

VarQ i hQ

l l

though one should rememb er that the distribution is not symmetric Each Q

is a single realization from that distribution when we really want to know the

mean From a single observer p oint there is no way of obtaining that mean and

one can only draw statistical conclusions based on what can b e measured Thus

a larger set of sp ectra which give dierent sets of can still give statistically

indistinguishable sets of Q The variance falls with increasing l but is signicant

right across the range of large scales

B Intermediate scales h Mp c h Mp c

It is on intermediate scales that determination of the primeval sp ectrum is most

promising Here a range of promising observations are available particularly towards

the small end of the range of scales In terms of technical diculties in interpreting

measurements a tradeo has b een made compared to large scales on the plus side the

cosmic variance is a much less imp ortant player as far more indep endent samples are

available while on the minus side the sp ectrum has b een severely aected by physical

pro cesses and thus has moved a step away from its primeval form

Intermediatescale microwave background anisotropies

In the absence of reionization the relevant angular scales are from ab out down

to ab out arcminutes Should reionization o ccur a lot of the information on these

scales could b e erased or amended in dicult to calculate ways Several exp eriments

are active in this range including the South Pole and max exp eriments

Unlike the largescale anisotropy one cannot write down a simple expression for the

intermediatescale anisotropies even if it is assumed that one has already incorp orated

the eect of dark matter on the growth of p erturbations via a transfer function The

reason is due to the complexity of physical pro cesses op erating A case in p oint is the

exp ected anisotropy sp ecied by the but now for larger l in cdm mo dels n

as calculated in detail by Bond and Efstathiou Bond and Efstathiou

On large scales l is approximately indep endent of l Once we get into the

intermediate regime l exhibits a much more complicated form which is dominated

by a strong p eak at around l This is induced by Thomson scattering from

moving electrons at the time of recombination Bond and Efstathious calculation gives

a p eak height around times as high as the extrap olated SachsWolfe eect Beyond

the rst p eak is a smaller subsidiary p eak at l

In their calculation Bond and Efstathiou assumed b oth the primeval sp ectrum and

the form of the dark matter Of course given the numb er of active and prop osed

Edward W Kolb

dark matter search exp eriments one should b e optimistic that this information will

b e obtained in the not to o distant future However even with this information the

complexity of the calculation makes it hard to conceive of a way of inverting it should

a go o d exp erimental knowledge of the l b e obtained Once again its

much easier to compare a given theory with observation than to extract a theory from

observation

One of the interesting applications of these results might b e a combination with the

largescale measurements The p eak on intermediate scales is due only to pro cesses

aecting the scalar mo des whereas we have p ointed out that the largescale Sachs

Wolfe eect is a combination of scalar and tensor mo des On large scales one cannot

immediately discover the relative normalizations of the two contributions However if

the dark matter is suciently well understo o d the height of the p eak in the intermediate

regime gives this information Should it prove that the tensors do play a signicant

role then this would b e a very interesting result as it immediately excludes slowroll

p otentials for the regime corresp onding to the largest scales Should the tensors prove

negligible then although the conclusions are less dramatic one has an easier inversion

problem on large scales as one can concentrate solely on scalar mo des

Galaxy clustering in the optical and infrared

A Redshift surveys in the optical

Over the last decade enormous leaps have b een made in our understanding of the

distribution of galaxies in the Universe from various redshift surveys Most prominent

is doubtless the ongoing Center for Astrophysics cfa survey Ramella Geller and

Huchra which aims to form a complete catalogue of galaxy redshifts out to

around h Mp c Other surveys of optical galaxies often trading incompleteness

for greater survey depth are also in progress On the horizon is the Sloan Digital Sky

Survey which aims to nd the redshifts of one million galaxies o ccupying one quarter

of the sky with an overall depth of h Mp c and completeness out to h Mp c

The redshifts of galaxies are relatively easy though time consuming to measure and

interpret and so provide one of the more observationally simple means of determining

the distribution of matter in the Universe The main technical problem is to correct

the distribution for redshift distortions which gives rise to the famous ngers of Go d

eect However the distribution of galaxies sp ecied by the galaxy p ower sp ectrum

or correlation function is two steps away from telling us ab out the primeval mass

sp ectrum

We have already discussed that the primeval sp ectrum on intermediate scales

has b een distorted by a combination of matter dynamics and amendments to the

p erturbation growth rate when the Universe is not completely matter dominated

If we know what the dark matter is then this need not b e a serious problem

Galaxies need not trace mass and in mo dern cosmology it is almost always as

sumed they do not This makes the pro cess of getting from the galaxy p ower

sp ectrum to the mass p ower sp ectrum extremely nontrivial Mo dels such as bi

ased cdm rely on the notion of a scaleindep endent ratio b etween the two but

Particle Physics and Cosmology

this to o can only b e an approximation to reality In recent work authors have em

phasised the p ossible inuence of environmental eects on galaxy formation for

instance a nearby quasar might inhibit galaxy formation and indeed it has b een

demonstrated that only very mo dest eects are required in order to profoundly

aect the shap es of measured quantities such as the galaxy angular correlation

function

Despite this attempts have b een made to reconstruct the p ower sp ectrum from var

ious surveys In particular this has b een done for the cfa survey Vogeley et al

and for the Southern Sky Redshift Survey Park et al These reconstructions

remain very noisy esp ecially at b oth large scales p o or sampling and small scales shot

noise and redshift distortions and at present the b est one could do would b e to try and

t simple functional forms such as p owerlaws or parametrized p ower sp ectra to them

Even then the constraints one would get on the slop e of say a tilted cdm sp ectrum are

very weak indeed However these reconstructions go along with the usual claim that

standard cdm is excluded at high condence due to inadequate largescale clustering

without providing any particular constraints on the choice of metho ds of resolving this

conict

Nevertheless with larger sampling volumes such as those which the sdss will p ossess

one should b e able to get a go o d determination of the galaxy p ower sp ectrum across a

reasonable range of scales p erhaps h to h Mp c

B Redshift surveys in the infrared

A rival to redshifts of optical galaxies is those of infrared galaxies based on galaxy

p ositions catalogued by the InfraRed Astronomical Satellite iras pro ject in the mid

eighties The aim here is to sparsesample these galaxies and redshift the subset This

is b eing done by two groups giving rise to the qdot survey Saunders et al

and the Jansky survey Fisher et al Taking advantage of the preexisting

database of galaxy p ositions has allowed these surveys to achieve great depth with

even sampling and reach some interesting conclusions

The main obstacle to comparison with optical surveys is due to the selection metho d

Infrared galaxies are generally young and app ear to p ossess a distribution notably less

clustered than their optically selected counterparts They are thus usually attributed

their own bias parameter which diers from the optical bias The mechanics of pro

ceeding to the p ower sp ectrum are basically the same as for optical galaxies

The most interesting and relevant results here are obtained in combination with

p eculiar velo city information as discussed b elow

C Projected catalogues

As well as redshift surveys one also has surveys which plot the p ositions of galaxies

on the celestial sphere At present the most dramatic is the apm survey encompassing

several million galaxies The measured quantity is the pro jected counterpart of the

correlation function the angular correlation function usually denoted w where is

the angular separation Though arguments remain as to the presence of systematics

one in principle has accurate determinations of the galaxy angular correlation function

The rst aim is to reconstruct the full three dimensional correlation function from this

Edward W Kolb

pro ceeding thence to the galaxy p ower sp ectrum Unfortunately present metho ds of

carrying out this inversion based on inverting Limb ers equation which gives w from

r have proven to b e very unstable and a satisfactory recovery of the full correlation

function has not b een achieved

In its preliminary galaxy identication stage the sdss will provide a huge pro jected

catalogue on which further work can b e carried out

Peculiar velo city ows

Potentially the most imp ortant measurements in largescale structure are those of

the p eculiar velo city eld Because all matter participates gravitationally p eculiar

velo cities directly sample the mass sp ectrum not the galaxy sp ectrum Were one to

know the p eculiar velo city eld this information is therefore as close to the primeval

sp ectrum as is microwave background information

Perhaps the most exciting recent development in p eculiar velo city observations is the

development of the potent metho d by Bertschinger Dekel and collab orators

Using only the assumption that the velo city can b e written as the divergence of a scalar

in gravitational instability theories in the linear regime this is naturally asso ciated

with the p eculiar gravitational p otential they demonstrate that the radial velo city

towardsaway from our galaxy which is all that can b e measured by the metho ds

available can b e used to reconstruct the scalar which can then b e used to obtain the

full three dimensional velo city eld This has b een shown to work very well in simulated

data sets where one mimics observations and then can compare the reconstruction from

those measurements with the original data set So far the noisiness and sparseness of

available real radial velo city data has meant that attempts to reconstruct the elds in

the neighb ourho o d of our galaxy have not yet met with great success however once

b etter and more extensive observational data are obtained one can exp ect this metho d

to yield excellent results

At present potent app ears at its most p owerful in combination with a substantial

redshift survey such as the irasqdot survey As potent supplies information as to

the density eld and the redshift survey to the galaxy distribution the two in com

bination can b e used in an attempt to measure quantities such as the bias parameter

and the density parameter of the Universe Reconstructions of the p ower sp ectrum

have also b een attempted At present the error bars due to cosmic variance b ecause of

small sampling volume due to the sparseness of the data in some regions of the sky and

due to iterative instabilities are large enough that a broad range of sp ectra including

standard cdm are compatible with the reconstructed presentday sp ectrum

With larger data sets and technical developments in the theoretical analysis to ols

potent and indeed velo city data in general app ears to b e a very p owerful means of

investigating the presentday p ower sp ectrum To that one need only add a knowledge

of the dark matter

C Small scales h Mp c h Mp c

Particle Physics and Cosmology

It is worth saying immediately that this promises to b e the least useful range of

scales For many choices of dark matter including the standard hot dark matter sce

nario p erturbations on these scales are almost completely erased by dark matter free

streaming to leave no information as to the primeval sp ectrum Only if the dark matter

is cold do es it seem likely that any useful information can b e obtained

There are several typ es of measurement which can b e made Quite a bit is known

ab out galaxy clustering on small scales such as the twop oint galaxy correlation func

tion However the strong nonlinearity of the density distribution on these scales erases

information ab out the original linearregime structure and the requirement of N b o dy

simulations to make theoretical predictions makes this an unpromising avenue for re

construction even should nature have chosen to leave signicant sp ectral p ower on

these scales There exist very smallscale arcsecondarcminute microwave background

anisotropy measurements though these are susceptible to a numb er of line of sight ef

fects and further the anisotropies are suppressed exp onentially on short scales b ecause

the nite thickness ab out h Mp c of the last scattering surface comes into play

Up to now the most useful constraints on small scales have come from the pairwise

velo city disp ersion the disp ersion of lineofsight velo cities b etween galaxies These

are sensitive to the normalization of the sp ectrum at small scales though unfortunately

susceptible to p ower feeding down from higher scales as well There are certainly

noteworthy constraintsfor instance it is generally accepted that unbiassed standard

cdm generates excessively large disp ersions However the calculations required involve

N b o dy simulations and b ecause a wide range of wavelengths contribute obtaining

knowledge of any structure in the p ower sp ectrum on these scales is likely to prove

imp ossible even if the amplitude can b e determined to reasonable accuracy

Figure Comparison of the measured power spectrum of density perturbations and

the predictions of several models The power spectrum of density perturbations from

galaxies the points on the righthand side is from Fisher et al and the power

spectrum from the cobe dmr measurements the box on the lefthand side is from

Smoot et al The models shown are cold dark matter hot dark matter tilted cold

dark matter and mixed dark matter

Already we are able to test various mo dels for the p ower sp ectrum For example a

Edward W Kolb

galaxy survey gives the p ower sp ectrum on small scales while cbr anisotropies prob e

the p ower sp ectrum on large scales For preliminary mo dels we can take a primordial

p owerlaw sp ectrum j j k n is the HarrisonZeldovich sp ectrum and some

choice for dark matter hot cold or mixed fraction hot plus a fraction cold Pro cessing

the primordial sp ectrum through the transfer function gives the curves of Figure The

simplest mo del is n cdm Clearly the general shap e is correct but the normalization

for the galaxy p oints do es not match the normalization for the cobe p oints

To b etter t the observed sp ectrum several variations on the theme of cdm has b een

prop osed In the mixed dark matter mdm variant a small amount of hot dark matter

is added and The pinch of hot dark matter

HDM CDM B

eg in the form of neutrinos of mass eV or so leads to the suppression of uctuations

on small scales b ecause of the free streaming of neutrinos Another variant involves a

mo dication of the sp ectrum of p erturbations When the sp ectrum of p erturbations is

normalized to the cobe dmr result which xes the sp ectrum on a verylarge scale a

mo dest amount of tilt say n can reduce uctuations on small scales Another

variant involves intro ducing a cosmological constant a p ossibility to o unpalatable to

consider further

Data on largescale structure is accumulating rapidly In a few years we will have

much b etter information ab out the p ower sp ectrum and we will see if any of the simple

mo dels is correct Let us turn now to p ossibilities for generating the p erturbations

Ination

The basic idea of ination is that there was an ep o ch early in the history of the Universe

when p otential or vacuum energy was the dominant comp onent of the energy density

of the Universe During that ep o ch the scale factor grew exp onentially During this

phase known as the de Sitter phase a small smo oth and causally coherent patch of

size less than the Hubble radius H can grow to such a size that it easily encompasses

the comoving volume that b ecomes the entire observable Universe to day

In the original prop osal ination o ccurred in the pro cess of a strongly rstorder

phase transition This mo del was so on demonstrated to b e fatally awed Subse

quent mo dels for ination involved phase transitions that were secondorder or p erhaps

weakly rstorder some even involved no phase transition at all Recently the p ossibil

ity of ination during a strongly rstorder phase transition has b een revived Before

discussing the latest developments in ination I will briey review some of the history

of ination

The art of ination

Prehistory

The years b efore the birth of the inationary Universe contained a rich prehistory of

work in cosmology investigating the cosmological consequences of a Universe dominated

by vacuum energy Vacuum energy is interesting in cosmology b ecause it acts as a

cosmological constant and will drive the Universe in exp onential expansion Recall

Particle Physics and Cosmology

that the expansion of the Universe is determined by the Friedmann equation

a G k

a a

where at is the cosmic scale factor is the energy density of the Universe and the

constant k is or dep ending up on the spatial curvature If the contribution of the

vacuum energy density dominates then is a constant it do es not decrease with a

and for k the solution to the Friedmann equation is

a G

at a exp H t H const

Such a rapid expansion may solve several cosmological problems including the at

nessage problem the homogeneityisotropy problem the problem of the origin of

density inhomogeneities and the monop ole problem

The p ossibility of a Universe dominated by vacuum energy b ecame much more rel

evant with the realization that the Universe may have undergone a series of phase

transitions asso ciated with sp ontaneous symmetry breaking The work of Kirzhnits

and Linde showed that symmetries that are sp ontaneously broken to day should

have b een restored at temp eratures ab ove the energy scale of sp ontaneous symmetry

breaking and as the Universe co oled b elow some critical temp erature denoted as T

there should have b een a phase transition in which the symmetry was broken Thus

phase transitions asso ciated with sp ontaneous symmetry breaking might oer a mecha

nism whereby the early Universe may b e dominated by vacuum energy for some p erio d

of time Kolb and Wolfram

The Classical Era of Old Ination

Although there was a rich prehistory the classical era of ination crystallized with the

pap er of Guth In this classical picture the Universe underwent a strongly rst

order phase transition asso ciated with sp ontaneous symmetry breaking of some Grand

Unied Theory gut Whether the phase transition is rst order or higher order

dep ends up on the details of the Higgs p otential for the scalar eld whose vacuum

exp ectation value is resp onsible for symmetry breaking This theory is now usually

referred to as old ination

In old ination the crucial feature of the p otential was the barrier separating the

symmetric hightemp erature minimum say lo cated at from the lowtemp erature

true vacuum lo cated at If the transition is strongly rst order the transition

from the hightemp erature to the lowtemp erature minimum o ccurs by the quantum

mechanical pro cess of nucleation of bubbles of true vacuum These bubbles of true

vacuum expand at the velo city of light converting false vacuum to true

The bubble nucleation rate p er volume will always b e understo o d when discussing

the bubble nucleation rate dep ends up on the shap e of the p otential but in general it

is written as A exp B where A is a parameter with mass dimension and B

where is the the b ounce action is dimensionless Let us simply assume that A

mass scale of sp ontaneous symmetry breaking ssb

Edward W Kolb

In the classical picture the energy density of the Universe b ecame dominated by

the falsevacuum energy of the Higgs eld and the Universe expanded exp onentially

Sucient ination was never a real concern the problem with the classical picture is

in the termination of the falsevacuum phase usually referred to as the graceful exit

problem

Inside the true vacuum bubble is just what one exp ectsvacuum For successful

ination it is necessary to convert the vacuum energy to radiation The way this is

accomplished in a rstorder phase transition is through the pro cess of collision of

vacuum bubbles In bubble collisions the energy density tied up in the bubble walls

may b e converted to entropy Thus if a rstorder phase transition is to have a graceful

exit there must b e many bubble collisions The decline of the classical era b egan with

the realization that bubbles of true vacuum do not p ercolate and ll the Universe ie

there is no graceful exit The basic reason is that the exp onential expansion of the

background space overwhelms the bubble growth To see this consider the expression

for the coordinate or comoving radius of the bubble Assume that the bubble is

nucleated at time t with zero radius and expands outward at the sp eed of light At

some time t t after nucleation the comoving bubble radius is

exp H t expH t

dt a t r t t

H a

The physical size of the bubble of course is simply R t t atr t t Notice that as

t the comoving bubble size approaches a nite value

expH t

r t

H a

Bubbles nucleated at larger t reach a smaller comoving size than bubbles nucleated

earlier in the transition If a bubble is nucleated at time t at some later time t the

bubble has comoving volume v t t and physical volume V t t given by

expH t

v t t r t t

H a

expH t t

R t t V t t

where here arrows indicate the asymptotic values as t

The probability that a p oint remains in the old falsevacuum phase at time t is

simply

dt V t t exp pt exp H t

Thus the probability that a p oint remains in the falsevacuum phase decreases exp o

nentially in time just as exp ected

Although pt decreases exp onentially the volume of space in the false vacuum is

increasing exp onentially A measure of whether true vacuum regions will p ercolate the

space is the fraction of physical space in false vacuum

H t expH t f t pta t exp

Particle Physics and Cosmology

Clearly whether this fraction increases or decreases in time dep ends up on the comp e

tition b etween the decreasing probability for a p oint to b e in the false vacuum and the

increasing volume of space in the false vacuum A rough estimate of whether f t will

increase or decrease is the criteria that H is much greater or much less than

unity If is much less than one the transition will never b e completed while if is

much greater than one the transition will b e completed but there wont b e a sucient

p erio d of ination So if is small enough to guarantee sucient ination it will b e

to o small for p ercolation to result

This graceful exit problem led to the decline of the classical era of ination and the

dawn of the inationary dark ages

SlowRollover Renaissance of New Ination

So on after the demise of the original mo del ination was revived by the realization that

it was p ossible to have an inationary scenario without recourse to a strongly rstorder

phase transition Linde and Albrecht and Steinhardt prop osed that the

Universe inates in the pro cess of the classical evolution of the vacuum In the classical

evolution of the eld to its true minimum the eld has kinetic energy and p otential

energy If one has a region of the scalar Higgs p otential that is at then the velo city

of the Higgs eld in the evolution to the ground state will b e slow and the p otential

energy of the Higgs eld might dominate the kinetic energy This can b e made more

quantitative by writing the classical equation of motion for a spatially homogeneous

scalar eld called the inaton in an expanding Universe under the inuence of a

p otential V

a dV

a d

If the p otential is at enough that the term can b e neglected the scalar eld will

undergo a p erio d of slow roll The energy density contributed by the scalar eld is

V and in the slowroll region V so the expansion closely

approximates the exp onential solution This theory is sometimes referred to as new

ination

The original prop osal of slowrollover ination was also based up on an SU gut

phase transition The p otential was attened by assuming that it to ok the Coleman

Weinb erg form However it was so on realized that even this p otential was not at

enough If the scalar p otential is approximated by a simple p otential of the form

V in order for density uctuations pro duced in ination to b e small

enough required O Clearly such small numb ers did not arise naturally in

simple unied mo dels and a successful slowrollover ination mo del must b e somewhat

more complicated Unfortunately it was so on discovered that there is no cosmological

upp er b ound on the complexity of mo dels

It was so on realized that the requirement of a small coupling constant could not

easily b e accommo dated in simple particle physics mo dels Of course the usual temp

tation is to mo dify the Higgs sector by adding more representations than required in

the minimal mo del In fact a successful mo del was constructed along these lines by Pi

and by Sha and Vilenkin

Edward W Kolb

For a while it was thought that sup ersymmetric guts could hold the key but they

were so on abandoned for a variety of reasons After sup ersymmetric mo dels some very

interesting sup ergravity mo dels emerged Although many sup ergravity mo dels raised

new problems of their own some sup ergravity mo dels were quite successful and at

least gave a pro of of existence that the inationary scenario might b e implemented in

particle mo dels

All of these Baro que mo dels suered from a low reheat temp erature as a result

of a weakly coupled inaton This made baryogenesis problematical although not im

p ossible All p ostrenaissance ination mo dels involved secondorder transitions and

b ecause ination o ccurred in a smo oth patch of the Universe that originally contained

a single correlation region the observable Universe should contain less than one top o

logical defect pro duced in the transition This is go o d news for the monop ole problem

but bad news for cosmic strings and texture

Ro co co Ination

The complexity of inationary mo dels was again increased as p eople started mo difying

the gravitational sector of the theory In Ro co co ination the identity of the inaton

is up for grabs There are mo dels where the inaton is asso ciated with the radius of

internal dimensions with the extra degree of freedom in fourthorder gravity with the

scalar eld of induced gravity etc Some of these mo dels can b e made to work it might

b e said that none work naturally

Perhaps somewhere along the line as more and more detail was added to make the

mo dels satisfy all of the constraints the message or at least the spirit of ination was

lost

Impressionism

In resp onse to the excesses of Baro que and Ro co co ination there grew up around

Andrei Linde a Russian scho ol of Impressionist ination In the impressionist style

no serious attempt is made to connect the details of inaton with any sp ecic particle

physics mo dels In this way the true essence and b eauty of the inationary Universe is

realized without any of the cluttering details The b est example of this the the chaotic

ination mo del In this mo del the scalar p otential is assumed to b e simply V

What a p erfect example of impressionism This p otential emb o dies features common

to all scalar p otentials without any of the details Of course it is not realistic in the

sense that no one would accept the existence of a scalar eld whose sole purp ose is to

make ination simple but it can b e taken to represent the impressions of every scalar

eld while at the same time representing no scalar eld

As Linde has rep eatedly emphasized it is not even necessary to connect ination

with a phase transition In the chaotic mo del the eld is exp ected to start away

from its minimum at due to chaotic initial conditions From there ination

can b e analyzed as in slow rollover mo dels

Despite the seductive b eauty of the impressionist approach we must demand more

realism Eventually we want a description of the Universe that has the ne details of

Particle Physics and Cosmology

the Baro que or Ro co co but with the simplicity and spirit of impressionism

The Postmo dern Era

One of the most interesting recent developments is p ostmo dernism The p ostmo dern

movement is characterized by an eclectic mixture of classical tradition with some as

p ect of the recent past With this denition it may b e said that rstorder inationary

cosmologies represent a p ostmo dern trend The classical tradition is a rstorder tran

sition while the asp ect of the recent past will b e emb o died by the slow rolling of a

second scalar eld

The key to rstorder ination is the relaxation of the assumption that H

is constant in time There are two ways one might imagine a time dep endence for

The rst way is for H to change Since H G either the eective gravitational

constant G must change or the vacuum energy must change We will see that in

many cases the two p ossibilities are equivalent representations of the same physics

The second way is for to change Of course in general b oth H and might change

If starts small much less than one then there might b e a sucient amount of

ination If grows and eventually b ecomes much greater than one then the bubbles

of true vacuum will p ercolate and collisions b etween the bubble walls might convert the

falsevacuum energy into entropy This is the hop e of rstorder ination

The b est example of a rstorder ination mo del is extended ination La and

Steinhardt The dierence b etween extended ination and Guths mo del is the

theory of gravity JordanBransDicke jbd in extended ination rather than gr in

Guths mo del

In jbd the gravitational constant is set by the value of a scalar eld During

ination this scalar eld evolves and gravity b ecomes weaker as a result the cosmic

scale factor grows as a large p ower of time rather than exp onentially This means that

in extended ination the physical volume of space remaining in the false vacuum grows

only as p ower of time and not exp onentially and unlike Guths original mo del bubble

nucleation can convert all of space to the true vacuum

Scalar eld dynamics

Regardless of the particular mo del of ination scalar eld dynamics plays an imp ortant

role in the cosmology so let us study the scalar eld dynamics in more detail Consider

a minimally coupled spatially homogeneous scalar eld with Lagrangian density

V V L

With the assumption that is spatially homogeneous the stressenergy tensor takes the

form of a p erfect uid with energy density and pressure given by V and

p V The classical equation of motion for is given in Equation All

minimal slowroll mo dels are examples of subinationary b ehaviour which is dened

by the condition H Sup erination where H cannot o ccur here though it is

p ossible in more complex scenarios This allows us to eliminate the timedep endence

Edward W Kolb

in the Friedmann equation and derive the rstorder nonlinear dierential equations

H H V

where G

We can dene two parameters which we will denote as slowroll parameters by

Slowroll corresp onds to f j jg With these denitions the end of ination is

given exactly by A small value of guarantees H V which is often

called the slowroll equation

Density p erturbations arise as the result of quantummechanical uctuations of elds

in de Sitter space First lets consider scalar density uctuations To a go o d approx

imation we may treat the inaton eld as a massless minimally coupled eld Of

course the inaton do es have a mass but ination op erates when the eld is evolving

through a at region of the p otential Just as uctuations in the density eld may

b e expanded in a Fourier series the uctuations in the inaton eld may b e expanded

in terms of its Fourier co ecients x exp ik xd k During ination

k k

there is an event horizon as in de Sitter space and quantummechanical uctuations in

the Fourier comp onents of the inaton eld are given by

k j j H

where H plays a role similar to the Hawking temp erature of black holes Thus

when a given mo de of the inaton eld leaves the Hubble radius during ination it

has impressed up on it quantum mechanical uctuations In analogy to the discussion

of the density p erturbations of the previous section what is called the uctuations

in the inaton eld on scale k is prop ortional to k j j which by Equation is

prop ortional to H Fluctuations in lead to p erturbations in the energy density

Now considering the uctuations as a particular mo de leaves the Hubble radius

during ination we may construct the gauge invariant quantity using the fact that

during ination p

Now using Equation and Equation the amplitude of the density p erturbation

when it crosses the Hubble radius after ination is

HOR

H V m m

jH j m V

P l

Particle Physics and Cosmology

where H and H are to b e evaluated when the scale crossed the Hubble radius

during ination The constant m equals or if the p erturbation reenters during

the matter or radiation dominated eras resp ectively The for radiation is appropriate

to the uniform Hubble constant gauge One o ccasionally sees a value instead which

is appropriate to the synchronous gauge The matter domination factor is the same

in either case Note also that it is exact for matter domination but for radiation

domination it is only strictly true for mo des much larger than the Hubble radius and

there will b e corrections in the extrap olation down to the size of the Hubble radius

Now we wish to know the dep endence of while the righthand side of the

equation is a function of when crossed the Hubble radius during ination We may

nd the value of the scalar eld when the scale go es outside the Hubble radius in

terms of the numb er of efoldings of growth in the scale factor b etween Hubble radius

crossing and the end of ination

It is quite a simple matter to calculate the numb er of efoldings of growth in the

scale factor that o ccur as the scalar eld rolls from a particular value to the end of

ination

Z Z

N H t dt d

The slowroll conditions guarantee a large numb er of efoldings The total amount

of ination is given by N N where is the initial value of at the start

TOT i i

of ination when a rst b ecomes p ositive In general the numb er of efolds b etween

when a length scale crossed the Hubble radius during ination and the end of ination

is given by

lnM GeV lnT GeV N lnMp c

where M is the mass scale asso ciated with the p otential and T is the reheat tem

p erature Relating N and N from Equation results in an expression b etween

and Hop efully this dry formalism will b ecome clear in the example discussed b elow

In addition to the scalar density p erturbations caused by de Sitter uctuations in

FRW

the inaton eld there are gravitational mo de p erturbations g g h caused

FRW

by de Sitter uctuations in the metric tensor Here g is the FriedmannRob ertson

Walker metric and h are the metric p erturbations That de Sitter space uctuations

should lead to uctuations in the metric tensor is not surprising since after all gravitons

are the propagating mo des asso ciated with transverse traceless metric p erturbations

and they to o b ehave as minimally coupled scalar elds The dimensionless tensor metric

p erturbations can b e expressed in terms of two graviton mo des we will denote as h

Performing a Fourier decomp osition of h hx h exp ik x d k we can use the

formalism for scalar eld p erturbations simply by the identication h

k k

with resulting quantum uctuations cf Equation

k jh j H

While outside the Hubble radius the amplitude of a given mo de remains constant

so the amplitude of the dimensionless strain on scale when it crosses the Hubble

Edward W Kolb

radius after ination is given by

HOR

k h A H

P l

where once again H is to b e evaluated when the scale crossed the Hubble radius

during ination

As usual it is convenient to illustrate the general features of ination in the con

text of the simplest mo del chaotic ination which is to inationary cosmology what

drosophila is to genetics In chaotic ination the inaton p otential is usually taken to

have a simple p olynomial form such as V or V For a concrete ex

ample let us consider the simplest chaotic ination mo del with p otential V

This mo del can b e adequately solved in the slowroll approximation yielding

t t

p p

at a exp t t

i i

p p p

t H

as determined by where was dened in with ination ending at

Equation The numb er of efoldings b etween a scalar eld value and the end of

ination is just

N d

Equating Equation and Equation relates and in this mo del for ination

lnMp c

Using Equation and Equation A and A are found to b e

S G

A lnMp c

lnMp c A

We can note three features that are common to a large numb er of but not all

inationary mo dels First A and A have dierent functional dep endences up on

S G

Second A and A increase with Finally A A for scales of interest although

G S S G

not by an enormous factor

The basic picture of the generation of scalar and tensor p erturbations is illustrated in

Figure The main observational information from the cosmic microwave background

arises through the Cosmic Background Explorer cobe satellite and the Tenerife ten

and South Pole sp collab orations Galaxy surveys apm cfa and iras may provide

useful information up to h Mp c while the Sloan Digital Sky Survey sdss should

extend to the lowest scales measured by cobe Peculiar velo city measurements using

the potent P metho ds are imp ortant on intermediate scales The angle measures

angular scales on the cbr in degrees and length scales are in units of h Mp c

Mp c is the d refers to the horizon size to day and at recombination and d h

H NL

scale of nonlinearity See the text for details

Particle Physics and Cosmology

Figure A schematic gure il lustrating the main concepts behind the generation of

scalar and tensor perturbations in ination

Reconstructing the Inaton Potential

Figure illustrates how a knowledge of the p otential allows a prediction of the scalar and

tensor sp ectra Now lets consider the p ossibility of reverse engineering this approach

and try to reconstruct the inaton p otential from knowledge of the scalar and tensor

sp ectra

Reconstruction of the inaton p otential in this manner was rst considered by

Ho dges and Blumenthal Recently this question has b een studied by Cop eland

Kolb Liddle and Lidsey CKLL a b c and also by Turner

CKLL improved up on the Ho dges and Blumenthal HB results in two imp ortant ways

Firstly they considered b oth scalar and tensor mo des whereas HB restricted their

Edward W Kolb

study to the scalars alone This is a vital improvement b ecause as HB realized

the scalar sp ectrum alone is insucient to uniquely determine the inaton p otential

reconstruction is p ossible only up to an undetermined constant and as the reconstruc

tion equations are nonlinear this leads to functionally dierent p otentials giving rise

to the same scalar sp ectrum The tensors even just the tensor amplitude at a single

scale provide just the extra information needed to lift this degeneracy Secondly the

HB analysis made explicit use of the slowroll approximation It is well known that this

approximation breaks down unless b oth the scalar sp ectrum is nearly at and the ten

sor amplitude is negligible CKLL considered the ination dynamics in full generality

However general expressions for the p erturbation sp ectra were studied in a slowroll

expansion

In CKLL a analytic expressions were derived for functionally reconstructing

the p otential in terms of A and A Although more complete expressions were given

S G

see esp ecially CKLL c here I will simply give the expressions to lowest order

in A A The rst result is a consistency equation relating the slop e of the tensor

G S

sp ectrum to A and A

G S

dA A

A d A

This highlights the asymmetry in the corresp ondence b etween the scalar and tensor

sp ectra If one were given the tensor sp ectrum then a simple dierentiation sup

plies the unique scalar sp ectrum However if a scalar sp ectrum is supplied then this

rstorder dierential equation must b e solved to nd the form of A This leaves

an undetermined constant in the tensor sp ectrum and as the consistency equation is

nonlinear this implies that the scalar sp ectrum alone do es not uniquely sp ecify the

functional form of the tensors However knowledge of the amplitude of the tensor

sp ectrum at one scale is sucient to determine this constant and lift the degeneracy

It is the tensor sp ectrum one requires to pro ceed with reconstruction Once the

form of the tensor sp ectrum has b een obtained either directly from observation or

by integrating the consistency equation the p otential as parametrized by may b e

derived

where again this is true to lowest order in A A The reconstruction equations allow

G S

a functional reconstruction of the inaton p otential For suitably simple sp ectra this

can b e done analytically and in CKLL b we illustrated this for wellknown cases

of scalar sp ectra which are exactly scaleinvariant logarithmically corrected from scale

invariance and exact p owerlaws An alternative approach useful for obtaining mass

scales is to concentrate on data around a given length scale and p erturbatively

derive the p otential around its corresp onding scalar eld value If we know

A and A separately then V follows immediately The derivatives of the

G S

p otential can also b e obtained

Let me illustrate the idea by example Within a few years a combination of mi

crowave background anisotropy measurements should give us some information ab out

the scalar and tensor amplitudes at a particular length scale corresp onding to an

angular scale A hyp othetical but plausible data set that this might provide would

b e A A n This would lead to see CKLL

S G

Particle Physics and Cosmology

a b

V GeV V GeV V GeV

In this way cosmology might b e rst to get a piece of the action of GUTscale physics

Cosmological phase transitions

Perhaps the most imp ortant concept in mo dern particle theory is that of sp ontaneous

symmetry breaking ssb The idea that there are underlying symmetries of nature

that are not manifest in the structure of the vacuum app ears to play a crucial role

in the unication of the forces In all unied gauge theoriesincluding the stan

dard electroweak mo delthe underlying gauge symmetry is larger than the unbroken

SU U Of particular interest for cosmology is the theoretical exp ectation that

C EM

at high temp eratures symmetries that are sp ontaneously broken to day were restored

and that during the evolution of the Universe there were phase transitions asso ciated

with sp ontaneous breakdown of gauge and p erhaps global symmetries For example

we can b e reasonably condent that there was such a phase transition at a temp era

ture of order GeV and a time of order sec asso ciated with the breakdown of

SU U U Moreover the vacuum structure in many sp ontaneously bro

L Y EM

ken gauge theories is very rich top ologically stable congurations of gauge and Higgs

elds exist as domain walls cosmic strings and monop oles In addition classical

congurations that are not top ologically stable socalled nontop ological solitons may

exist and b e stable for dynamical reasons Interesting examples include soliton stars

Qballs nontop ological cosmic strings sphalerons and so on

Before discussing the cosmological implications it is useful to review what is meant

by the nitetemp erature p otential

FiniteTemp erature Potential

Lets start with a simple mo del of a real scalar eld with Lagrangian

V V m L

The Lagrangian is invariant under the discrete symmetry transformation The

minima of the classical p otential of Equation are not at zero but at m

The origin is an unstable extremum of the p otential b ecause V where

prime denotes dd Since the quantum theory must b e constructed ab out a stable

extremum of the classical p otential the ground state of the system is either or

and the reection symmetry present in the Lagrangian is broken by the choice

of a vacuum state as is the only p ossible vacuum invariant under

The p otential of Equation is the classical p otential and it is necessary to consider

the eect of quantum corrections Here I will follow the classic pap er of Coleman and

Weinb erg For a general Lagrangian L in the presence of a cnumb er source

J x the vacuumtovacuum amplitude is

Z Z

h j i Z J D exp i d x L J x J

Edward W Kolb

where of course Z J is the generating functional of the Greens functions It is more

useful to consider the generating functional of the connected Greens functions W J

related to Z J by Z J expiW J W J can b e expanded in terms of p owers of J

with the co ecients b eing the connected Greens functions The classical eld is

dened as W J Finally the eective action is W J d xJ x x

c c c

Now the eective action can b e expanded in terms of the oneparticle irreducible

PI Feynman diagrams with n external lines

x x x x d x d x

c n c c n n

Rather than an expansion in p owers of the classical eld one can expand the eective

action in p owers of derivatives of the classical eld

d x V Z

c c c c

Now the constant term V app earing in this expansion is known as the eective

potential By means of a Fourier transform of Equation it is easy to show that the

eective p otential can also b e expressed in terms of a sum of all PI Feynman graphs

with zero momenta

n n

p V

i c

A simple example will illustrate the ab ove dry formalism In this example we follow

Lee and Sciaccaluga and expand the eective p otential ab out rather

than ab out

n n

x x x x d x d x

c n c c n n

V p

c c i

where the co ecients in the expansion are now the generators of the PI diagrams in

the shifted theory where is replaced by Now dV d j which is

c c

simply the tadp ole diagram in the shifted theory up to a factor of i So evaluating

the tadp ole integrating over then setting gives the eective p otential

The shifted theory of Equation gives a p otential with mass squared of m

and a term with coupling i Therefore the tadp ole diagram is

d k

k m

The total p otential to onelo op is the sum of the classical p otential of Equation and

the onelo op correction

Z Z

d k

ln k m V V d V

c c c

Particle Physics and Cosmology

A generalization to arbitrary p otentials would b e to replace the last two terms in the

logarithm by V where V is the classical p otential

A few comments are in order b efore pro ceeding

The integral in Equation is innite This shouldnt scare us We intro duce

a cuto and if the theory is renormalizable all innities can b e absorb ed via

some renormalization prescription

The physical meaning of the onelo op p otential is clear if we integrate over dk to

d k

V V k V

c c c

V corresp onds to the total energy of a uctuation of momentum Since k

k the onelo op correction is clearly the sum of zerop oint energy uctuations

ab out the p oint

If the mo del is generalized to include couplings of to vectors and fermions then

the onelo op p otential will include additional tadp oles where the particle in the

lo op is the vector or the fermion

Now the integral in Equation can b e done by intro ducing a cuto For the

simple mo del that we are studying the p otential to onelo op is

V a a m lnm

c c c c c

where a are cutodep endent constants that will b e determined by renormalization

of the mass and coupling constants

Of course it is the b ehaviour of the theory at nite temp erature that is of interest

to us A simple heuristic derivation of the eects of the thermal bath is to adopt the

realtime formalism in which the propagator D k includes the p ossibility of emission

and absorption from the thermal bath

k m D k

k m i exp E T

Then this propagator is used in the evaluation of the propagator of the tadp ole diagram

The additional temp eraturedep endent part of the propagator leads to an additional

temp eraturedep endent part of the onelo op p otential

dx x ln exp x V T V

c T c

At high temp erature T jmj it is p ossible to expand the logarithm and p erform

the integration V T V T The rst term is sim

T c c

ply the free energy of a massless spin b oson The second term in the expansion is

dep endent with a positive co ecient For instance in the simple theory we have

m Adding V to the classical p otential gives a b een following V

c T c

total p otential with a co ecient of the term quadratic in of m T Clearly c

Edward W Kolb

ab ove some critical temp erature T m the co ecient of the quadratic term

will b e p ositive and b elow T the co ecient will b e negative This is a signal that for

T T the symmetry will b e restored will b e a stable minimum of the p otential

In the evolution from the hightemp erature phase to the lowtemp erature phase there

is a phase transition

Figure The temperature dependence of V for a rstorder phase transition

T c

Only the dependent terms in V are shown

c T c

If also couples to fermions or gauge b osons there will b e additional terms in the

temp eraturedep endent p otential The theory has a secondorder transition The

additional terms in V from gauge b oson contributions can drive the transition to rst

order In general a symmetrybreaking phase transition can b e rst or second order

The temp erature dep endence of V for a rstorder phase transition is shown in

T c

Figure For T T the p otential is quadratic with only one minimum at

C c

When T T a lo cal mininum develops at For T T the two minima

c C

b ecome degenerate and b elow T the minimum b ecomes the global minimum

C c

If for T T the extremum at remains a lo cal minimum there must b e a barrier

C c

b etween the minima at and Therefore the change in in going from

c c c

one phase to the other must b e discontinuous indicating a rstorder phase transition

Moreover the transition cannot take place classically but must pro ceed either through

quantum or thermal tunnelling Finally when T T the barrier disapp ears and the

transition may pro ceed classically For a secondorder transition there is no barrier at

the critical temp erature and the transition o ccurs smo othly

As a nal illustration lets consider the electroweak phase transition In the minimal

electroweak mo del there is a complex SU doublet with p otential

y y

V m Now the complex doublet eld can b e expressed in

terms of real elds

In the standard convention the vacuum exp ectation value of is chosen to lie in the

direction hi and the real eld has a p otential like Equation At tree level

Particle Physics and Cosmology

the Higgs mass is M The Higgs couples to gauge b osons and lead to a mass

for the W and Z of M g M g g The relationship b etween the

W Z

mass of the W and the Fermi constant gives GeV g and g The

Higgs also couples to fermions with Yukawa coupling h M GeV As the

i i

Yukawa coupling is prop ortional to the fermion mass the dominant eect is from the

top quark the most massive fermion Therefore the imp ortant coupling constants

are g g h M GeV and M GeV

T T H

Once the top quark and Higgs masses are determined all the couplings in the minimal

electroweak mo del will b e known

To the classical p otential must b e added the onelo op corrections from the Higgs

the W the Z and all the fermions of course it is a go o d approximation to assume

the fermion contributions are dominated by the top quark At zero temp erature the

onelo op eective p otential is

ln m m V

c c c

c c

ln ln g g g h

c c T

where is an arbitrary mass scale which can b e related to the renormalized coupling

term is the top ln constants The gaugeb oson contribution to the

c c

quark contribution is M GeV and the Higgs b oson contribution

is M GeV A priori all three contributions could b e comparable

Let us consider the case where the Higgs mass is small M GeV and the Higgs

contribution to the onelo op p otential can b e ignored We can then write the p otential

of Equation as

ln B m V

c c c

ln B B

c c c

Here we have used the fact that V implies that m B and

B M GeV The Higgs mass is M V B

Now consider the p otential at nite temp erature As in the previous example the

nitetemp erature p otential will have a temp eraturedep endent piece in addition to the

zerotemp erature part The temp eraturedep endent part receives a contribution from

all particles that couple to the scalar eld including the scalar eld itself The onelo op

p otential at nite temp erature can b e written as a sum of integrals similar to the one

in Equation of the form

h i

F X dx x ln exp x X T

c c

F applies to b oson lo ops and F to fermion lo ops For the electroweak mo del V

T c

is given by

F g g F g V V

T c c

c c

F M F h

H T c

Edward W Kolb

where as b efore V is the onelo op p otential at zero temp erature and for simplicity

we have included only the dep endent terms For T the terms prop ortional to

T are just given by minus the pressure of a gas of the massless fermions and b osons

that couple to

Anderson and Hall showed that a high temp erature expansion of the one

lo op p otential closely approximates the full onelo op p otential for M GeV

and M GeV It is imp ortant to dierentiate b etween the nite temp erature

Higgs mass M T and the zerotemp erature Higgs mass M They obtained for the

H H

p otential

E T V D T T

where D and E are given by

h i h i

D M M M E M M

W Z T W Z

Here T is given by

T M B D

where the physical Higgs mass is given in terms of the onelo op corrected as

M M B with B M M

T Z W H

We use M GeV M GeV and GeV The temp erature

W Z

corrected Higgs selfcoupling is

X X

M M

F B

g c T c T ln M ln M g

F B F T B

B F

F B

where the sum is p erformed over b osons and fermions in our case only the top quark

with their resp ective degrees of freedom g and ln c and ln c

B F B F

The gauge interactions result in an eective attractive term in the p otential so to

oneloop the theory predicts a rstorder electroweak phase transition However the fact

that the phase transition is so weakly rst order implies that the nitetemp erature lo op

expansion is not reliable and the onelo op results cannot b e trusted for the electroweak

transition There is presently a lot of work in nding an improved p otential to describ e

the electroweak transition

Generation of defects

ssb is an intergal part of mo dern particle physics and provided that temp eratures in the

early Universe exceeded the energy scale of a broken symmetry that symmetry should

have b een restored How can we tell if the Universe underwent a series of ssb phase

transitions One p ossibility is that symmetrybreaking transitions were not p erfect

and that false vacuum remnants were left b ehind frozen in the form of top ological

defects domain walls strings and monop oles

As the rst example of a top ological defect asso ciated with sp ontaneous symmetry

breaking consider the domain wall The simple scalar mo del of the previous section can

Particle Physics and Cosmology

b e used to illustrate domain walls The Z reection symmetry ie invariance under

of the Lagrangian of Equation is sp ontaneously broken when takes on a

nonzero vacuum exp ectation So far we have assumed that all of space is in the same

ground state but this need not b e the case Imagine that space is divided into two

regions In one region of space hi and in the other region of space hi

Since the scalar eld must make the transition from to smo othly there

must b e a region where ie a region of false vacuum This transition region

b etween the two vacua is called a domain wall Domain walls can arise whenever a Z

or any discrete symmetry is broken

The solution to the equation of motion sub ject to the b oundary conditions that

describ e a domain wall is z tanhz where the thickness of the wall

is characterized by It should b e clear that the domain wall is

top ologically stable the kink at z can move around or wiggle but it cannot

disapp ear except by meeting up with an antikink and annihilating The stress tensor

for the domain wall is obtained by substitution of the wall solution into the expression

for the stressenergy tensor for a scalar eld T cosh z diag

Note that the z comp onent of the pressure vanishes and that the x and y comp onents

of the pressure are equal to minus the energy density The surface energy density

asso ciated with the wall given by T dz is identical to the

integrated transverse comp onents of the stress T dz That is the surface tension

in the wall is precisely equal to the surface energy density Because of this fact walls

are inherently relativistic and their gravitational eects are inherently nonNewtonian

and very interesting

The existence of largescale domain walls in the Universe to day are ruled out sim

ply based up on their contribution to the total mass density A domain wall of size

H h cm would have a mass of order M H GeV

wall

grams or ab out a factor of GeV times that of the total mass within

the present Hubble volume Walls would also lead to large uctuations in temp erature

of the cbr unless is very small T T G H GeV Appar

ently domain walls are cosmological bad news unless the energy scale andor coupling

constant asso ciated with them are very small

The existence of domain wall solutions for this simple mo del traces to the existence

of the disconnected vacuum states hi The general mathematical criterion for

the existence of top ologically stable domain walls for the symmetrybreaking pattern

G H is that M I where M is the manifold of equivalent vacuum states

M G H and is the homotopy group that counts disconnected comp onents In

the ab ove example G Z H I M Z and M Z I

The next example of a top ological defect is the cosmic string a onedimensional

structure As we shall see cosmic strings are much more palatable to a cosmologist

than domain walls A simple mo del that illustrates the cosmic string is the Ab elian

Higgs mo del a sp ontaneously broken U gauge theory The Lagrangian of the mo del

contains a U gauge eld A and a complex Higgs eld which carries U

charge e

y y

F F L D D

where F A A and D ieA We immediately recognize that the

Edward W Kolb

theory is sp ontaneously broken as V is minimized for hjji The physical

states after ssb are a scalar b oson of mass M and a massive vector b oson of

mass M e

The complex eld can b e written in terms of two real elds i

If the vacuum exp ectation value is chosen to lie in the real direction then the p otential

b ecomes

V where hjji hi However energetics do not determine

the phase of hi since the vacuum energy dep ends only up on jj this fact follows

from the U gauge symmetry Dening the phase of the vacuum exp ectation value

exp i we see that x can b e p osition dep endent However by hi

must b e single valued ie the total change in around any closed path must b e

an integer multiple of Imagine a closed path with As the path is shrunk

to a p oint assuming no singularity is encountered cannot change continuously

from to There must therefore b e one p oint contained within the

path where the phase is undened ie hi The region of false vacuum within

the path is part of a tub e of false vacuum Such tub es of false vacuum must either b e

closed or innite in length otherwise it would b e p ossible to deform the path around

the tub e and contract it to a p oint without encountering the tub e of false vacuum In

most instances these tub es of false vacuum have a characteristic transverse dimension

far smaller than their length so they can b e treated as onedimensional ob jects and are

called strings

The string solution to the equations of motion was rst found by Nielsen and Olesen

At large distances from an innite string in the z direction their solution is

h i

p p

exp iN A ie ln

where is the p olar angle in the xy plane and N is the winding numb er of the

string The stressenergy tensor asso ciated with a long thin straight string is given by

x y diag where is the mass p er unit length of the string which T

dep ends up on the ratio e but generally is of order Note that the pressure is

negativeie it is a string tensionand equal to Like domain walls strings are

intrinsically relativistic

Far from a circular string lo op of radius R the gravitational eld is that of a p oint

particle of mass M R For a lo op of size ab out that of the present horizon

string

M GeV grams As with domain walls there are nonNewtonian grav

string

itational eects asso ciated with strings Recall that for a stress tensor of the form

T diag p p p the Newtonian limit of Poissons equation is r

G p p p For an innite string in the z direction p and p p

and Poissons equation b ecomes r which suggests that space is at out

side of an innite straight string Indeed this is so Vilenkin has solved

Einsteins equations for the metric outside an innite straight cosmic string in the

limit that G In terms of the cylindrical co ordinates r z the metric is

ds dt dz dr G r d By a transformation of the p olar angle

G the metric b ecomes the atspace Minkowski metric as exp ected

spacetime around a cosmic string is that of empty space However the range of the

atspace p olar angle is only G rather than This is

referred to as a conical singularity

Particle Physics and Cosmology

The conical nature of space around a string leads to several striking eects double

images of ob jects lo cated b ehind the string uctuations in the microwave background

and the formation of wakes To understand the formation of double images consider the

simplied situation of an innite string normal to the plane containing the source and

the observer The conical space is at space with a wedge of angular size removed

and p oints along the cuts identied Due to this the observer will see two images of

the source with the angular separation b etween the two images determined by

l l l

sin sin G

d l d l d l

Here l and d are the distances from the string to the source and observer resp ectively

and the second equation is a smallangle approximation The conical metric also leads

to discontinuities in the temp erature of the microwave background Imagine as the

source a p oint on the last scattering surface for the microwave background radiation

An observer at rest with resp ect to the string will see two images of the same p oint

on the last scattering surface separated by an angle for d l Now if the

string and observer are not at rest with resp ect to each other but instead have a relative

velo city v which is p erp endicular to the line of sight the momentum vector of one image

will have a small comp onent order parallel to the direction of v and the other

a small comp onent antiparallel to the direction of v The net eect is a small Doppler

shift of the radiation temp erature T T Gv across the string Based up on this

eect and the observed isotropy of the cbr we can conclude any strings that exist

at present must b e characterized by G A third interesting eect of cosmic

strings are string wakes Consider a long straight string moving through the Universe

with velo city v As the string moves past particles in the Universe the particles will b e

deected and will acquire a wake velo city v Gv transverse to the direction of

motion of the string If the particles have a very small internal velo city disp ersion eg

colddark matter particles or baryons after decoupling then matter on b oth sides of

the passing string will move toward the plane dened by the motion of the string In a

Hubble time a wedgeshap ed sheet of matter with overdensity of order unity op ening

angle G and width v H will form in the wake of the string The mass of the

material within the wakepro duced sheet can b e considerable ab out Gv of that

in the horizon likewise the scale of the thin thicknesswidth G sheets that

are formed is comparable to that of the horizon scale It has b een suggested that the

sheets that form in the wakes of long straight cosmic strings play an imp ortant role in

structure formation

A nal imprint of primordial cosmic strings is gravitational radiation from shrinking

string lo ops While an innite straight string is stable string lo ops are not A curved

string will move so as to minimize its length The motion of a small closed lo op is

particularly simple a lo op of radius R oscillates relativistically with a p erio d R

As it oscillates it will radiate gravitational waves due to its timevarying quadrup ole

moment dimensionally Q R The p ower radiated in gravitational waves is given

by P GQ G where is a numerical constant of order In a

GW GW GW

characteristic time the lo op will radiate away its massenergy shrink to a p oint

and vanish We exp ect R P G R That is a lo op will undergo

GW GW GW

ab out G oscillations b efore it disapp ears

As cosmic strings stand they are cosmologically safe and have several p otentially

Edward W Kolb

interesting consequences they leave b ehind a background of relic gravitational

waves relic string present to day can lead to temp erature uctuations in the cbr

relic string present to day can act as gravitational lenses and string lo ops or

attened structures formed in the wakes of strings can p ossibly serve as seeds to initiate

structure formation in the Universe

In our discussion of cosmic strings we have used the simplest example of a sp on

taneously broken gauge theory for which string solutions exist In general there will

b e string solutions asso ciated with the symmetry breaking G H if the manifold of

degenerate vacuum states M G H contains unshrinkable lo ops ie if the map

ping of M onto the circle is nontrivial This is formally expressed by the statement

that top ologically stable string solutions exist if M I In the ab ove example

G U H I and M U The group U can b e represented by the p oints on

a circle and so U is the mapping of the circle onto itself Such a mapping is

characterized by the winding numb er of the mapping ie N N so

that M Z the set of integers

Domain walls are twodimensional top ological defects strings are onedimensional

defects Pointlike defects also arise in some theories which undergo ssb and remark

ably they app ear as magnetic monop oles A simple mo del that illustrates the magnetic

monop ole solution is an SO gauge theory in which SO is sp ontaneously broken to

U by a Higgs triplet where a is the group space index The Lagrangian density

for this theory is

a a a a a a

L F D D F

c b a a a

A e A A A F

abc

c a a b

D e A

abc

Once again we encounter a theory that undergo es ssb In this mo del two of the three

gauge b osons in the theory acquire a mass through the Higgs mechanism There is also

a physical Higgs particle The masses of the vector and Higgs b osons are M e

The magnitude of h i is xed by the minimization of the p otential jj

However the direction of h i in group space is not This is just a manifestation

of the SO gauge symmetry It should b e clear that the lowest energy solution is

the one where x const x spatial co ordinate since this also minimizes the

kinetic energy spatial gradient term Even if x const the spatial dep endence

a a

of the direction of can often b e gauged away ie D made equal to zero by an

appropriate gauge conguration A x with nite energy However there are Higgs

eld congurations that cannot b e deformed into a conguration of constant by a

niteenergy gauge transformation

An example of a conguration that cannot b e gauged away is the hedgehog cong

b b

uration in which the direction of in group space is prop ortional to r where r is the

unit vector in ordinary space This solution is spherically symmetric and as r tends to

a a

b b

r er Like the domain wall and innity we nd that r t r and A r t

b ab

the cosmic string solutions continuity requires that the Higgs eld vanish as r

The vanishing of the Higgs eld at the origin accounts for the top ological stability of

Particle Physics and Cosmology

the hedgehog There is no way to smo othly deform the hedgehog into a conguration

where hj ji everywhere The size of the monop ole ie the region over which

hj ji is of order The energy of the hedgehog conguration receives contri

butions from b oth the vacuum energy asso ciated with hj ji and spatial gradient

energy asso ciated with the variation of h i

Gauge and Higgs eld congurations corresp onding to a magnetic monop ole exist

if the vacuum manifold M G H asso ciated with the symmetrybreaking pattern

G H contains nonshrinkable surfaces ie if the mapping of M onto the twosphere is

nontrivial Mathematically this is expressed by the statement that monop oles solutions

arise in the theory if M I If G is simply connected then G H H

If G is not simply connected then the generalization of the ab ove expression is

G H H G In the example ab ove G SO H U SO is not

simply connectedit is equivalent to the threesphere with antip o dal p oints identied

and SOU U SO Z Z the integers mo d

We have discussed the three kinds of top ological defects asso ciated with sp onta

neously broken symmetries the monop ole the string and the domain wall The

existence and stability of these ob jects is dictated by top ological considerations

Many sp ontaneously broken gauge theories predict the existence of one or more of the

ab ove top ological defects These ob jects are inherently nonp erturbative and probably

cannot b e pro duced in high energy collisions at terrestrial accelerators It is very likely

that the only place they can b e pro duced is in phase transitions in the early Universe

Although monop oles strings and domain walls are top ologically stable they are not

the minimum energy congurations However their pro duction in cosmological phase

transitions seems unavoidable The unavoidable cosmological pro duction mechanism

is known as the Kibble mechanism

The Kibble mechanism hinges up on the fact that during a cosmological phase tran

sition any correlation length is always limited by the particle horizon The particle

horizon is the maximum distance over which a massless particle could have propagated

since the time of the bang It was given in Equation The correlation length asso ci

ated with the phase transition sets the maximum distance over which the Higgs eld can

b e correlated The correlation length dep ends up on the details of the phase transition

and is temp eraturedep endent It is related to the temp eraturedep endent Higgs mass

T T In any case the fact that the horizon distance is nite in the M

standard cosmology implies that at the time of the phase transition t t T T

C C

the Higgs eld must b e uncorrelated on scales greater than d and thus the horizon

distance sets an absolute maximum for the correlation length

During a ssb phase transition some Higgs eld acquires a vacuum exp ectation

value Because of the existence of the particle horizon in the standard cosmology when

this o ccurs hi cannot b e correlated on scales larger than d H m T There

H P l

fore it should b e clear that the nontrivial vacuum congurations must necessarily b e

pro duced with an abundance of order one p er horizon volume While these top ological

creatures are not the minimumenergy congurations of the Higgs eld they arise as

top ological defects b ecause of the nite particle horizon Since they are stable they

are frozen in as p ermanent defects when they form

So far we have concentrated on the analysis of top ological defects that can arise in

Edward W Kolb

gauge theories However defects can also arise in the sp ontaneous breaking of global

symmetries The analogies of the defects discussed ab ove are global strings and global

monop oles The global eld congurations lo ok like their lo cal counterparts for the

scalar eld but of course there is no vector eld This means that formally the string

and monop ole solutions have innite energy recall for the lo cal defects the energy in

the gauge elds cancels the energy in the Higgs eld far from the defect This is really

not a problem b ecause there the divergence in the energy is only logarithmic and there

are many physical eects to cut it o such as the interdefect separation There are

just two main dierences in the b ehaviour of gauge and global defects the energy

of the global defects are slightly more spread out the global strings can radiate

energy by the emission of NambuGoldstone b osons

However there are new typ es of defects in global symmetry breaking that do not

app ear in the breaking of gauge symmetries For example in the sp ontaneous breaking

of a global ON mo del to ON for N walls app ear for N global strings

result for N global monop oles are pro duced These all have counterparts in lo cal

theories However for N global defects also exist for N the defect is called

global texture and for N they are called Kibble gradients Texture corresp onds

to knots in the Higgs eld that arise when the eld winds around the three sphere

These knots are generally formed by misalignment of the eld on scales larger than the

horizon at the symmetrybreaking phase transition b ecause of the Kibble mechanism

As the knots enter the horizon they collapse at roughly the sp eed of light giving rise to

nearly spherical energy density p erturbations New knots are constantly coming into the

horizon and collapsing leading to a scale invariant sp ectrum of density p erturbations

The magnitude of the p erturbations is set by the scale of the symmetry breaking and

for scenarios of structure formation involving texture the scale of symmetry breaking

must b e ab out GeV

A theory of texture or Kibble gradients b eing resp onsible for the seeds of large

scale structure has b een formulated by Turok Sp ergel and collab orators Pen et al

Texture would provide a very promising alternative to conventional ination

scenarios for generating the primordial density uctuations if indeed they are ubiquitous

in particle physics mo dels In fact texture arises in a variety of theories with non

Ab elian global symmetries that are sp ontaneously broken However even an extremely

small amount of explicit symmetry breaking will sp oil the texture scenario I would

like to close these lectures by discussing how sensitive this theory is to Planckscale

eects This idea was recently discussed by Holman et al and Kamionkowski

et al

To illustrate these p ossibilities consider a theory with a global ON symmetry

sp ontaneously broken to ON by an N vector The theory is describ ed by the

a a

scalar p otential V As mentioned ab ove texture arises for N

There are many arguments suggesting that al l global symmetries are violated at some

level by gravity For example b oth wormholes and black holes can swallow global

charge Virtual black holes or wormholes which should in principle arise in a theory

of quantum gravity will lead to higher dimension op erators which violate the global

symmetry There are two p ossible assumptions one might make ab out the fate of

global symmetries in a Universe that includes gravity The strong assumption is that

despite all indications from lowenergy semiclassical gravitational physics black holes

Particle Physics and Cosmology

wormholes etc it is p ossible to have exact global symmetries in the presence of

gravity This is the assumption made in the standard texture scenario The weak

assumption is that the global symmetry is not a feature of the full theory There

are two p ossible realizations of the weak assumption Either the global symmetry

is approximate in which case one must include the eects of higherdimensional non

renormalizable symmetrybreaking op erators or consistent with indications from semi

classical quantum gravity the global symmetry is never even an approximate symmetry

unless protected by gauge symmetries

If one makes the weak assumption then one must include explicit symmetry break

ing terms If one assumes that gravity do es not resp ect global symmetries at all then

a b

renormalizable op erators like m which explicitly break the global symmetry

P l

should b e included These terms are exp ected for instance by the action of worm

holes swallowing global charge If virtual wormholes of size smaller than the Planck

length are included then we exp ect to b e of order unity In this case it is wrong

to consider an eective lowenergy theory with a global symmetry If one makes the

assumption either that wormholes do not dominate the functional integral or that the

global charge is protected by gauge symmetries then it may b e p ossible to suppress

the renormalizable op erators But even in this case higher dimension op erators should

b e included An example would b e a dimension op erator which would add to V

a b c d e

terms like m Such terms explicitly break the global symmetry

abcde P l

and lead to a mass for the pseudoNambuGoldstone mo de of m m Of course

P l

the mass is suppressed by m but we will show b elow that it still has a drastic eect

P l

on the texture scenario

The implications of the strong and weak assumptions for texture are as follows

With the strong assumption the texture scenario is unaected If one allows un

suppressed wormhole contributions global symmetries and hence texture are a non

starter If all eects of gravitational physics in the lowenergy theory are contained in

nonrenormalizable terms a more careful analysis is required This is the p ossibility

we explore now In this approach we are then required to include all higher dimen

sion op erators consistent with the gauge symmetries of the mo del and suppressed by

appropriate p owers of m

P l

We now consider the eects of the higher dimension op erators These terms will

break the symmetry explicitly generating a complicated p otential for the Nambu

Goldstone mo des In general the vacuum manifold will b e reduced to a p oint though

the p otential will likely have many lo cal minima To see how this works consider the

theory discussed ab ove with N Here the vacuum manifold is the two sphere and

the mo del in two spatial dimensions will have texture In three spatial dimensions

the mo del admits b oth global monop oles and texture although the texture in this case

is not spherically symmetric We express the eld as

sin cos sin sin cos

where and are the angular variables on the twosphere which represent the Nambu

Goldstone mo des of the problem

The eect of the dimension op erators is to intro duce terms to the p otential

for the eld which dep end explicitly on and These are nothing more than the

Edward W Kolb

Y Y and Y spherical harmonics Note that in general the mass of the Nambu

m m m

Goldstone b oson in this p otential is roughly m

P l

As long as the mass of the NambuGoldstone mo de is small compared to the Hubble

parameter the eld will evolve essentially as in the original texture scenario However

once the Compton wavelength of the NambuGoldstone mo de enters the horizon the

eld will b egin to oscillate ab out the minimum or rather the closest lo cal minimum of

its p otential The eld will then align itself on scales larger than the horizon and texture

on all scales quickly disapp ear For texture to b e imp ortant for structure formation

they must p ersist at least until matterradiation decoupling when H eV

The contribution of a dimension d op erator to the NambuGoldstone b oson

mass is m m Given that the texture scenario requires GeV the

P l

requirement that m eV implies that d ie we must b e able to suppress

all op erators up to dimension It is rather dicult to see how this might o ccur

even the use of additional gauge quantum numb ers could not prevent the o ccurrence

of dimension op erators which break a nonAb elian symmetry although they could

protect an Ab elian symmetry We note that if we consider dimension op erators

then the mass b ecomes dynamically imp ortant immediately after the phase transition

texture therefore never exists

In conclusion any mo del which dep ends on the dynamics of NambuGoldstone

mo des will b e extremely sensitive to physics at very high energies Texture can by

no means b e considered a robust prediction of unied theories This is most discourag

ing for the texture scenario On the other hand if texture is discovered then this will

have profound implications not only for theories of structure formation but for Planck

scale physics What b etter way to close lectures on the implications of cosmology for

particle physics

Finally there are other creatures that might b e pro duced in cosmological phase

transitions Nontop ological solitons or Qballs Frieman et al and electroweak

strings Achucarro and Vachaspati

The lesson for cosmological phase transitions is that even with unlimited energy

accelerators are the wrong to ol to prob e the nonp erturbative sector of eld theories

EarlyUniverse phase transitions continue to provide the b est arena for the study of

asp ects of particlephysics theories related to coherent solitonlike ob jects The only

plausible site for the pro duction of ob jects such as monop oles strings walls sphalerons

and the like is an earlyUniverse phase transition All of these can have very signicant

implications for the evolution of the Universe Sphalerons as well as other solitons

pro duced in the electroweak transition have some promise of a cosmological payo

Of course there is an enormous dierence b etween nding a solitonlike solution to the

eld equations and nding solitons in the Universe However even if they are not are

found the techniques develop ed for their study will b e useful additions to the theorists

to olb ox

Particle Physics and Cosmology

Acknowledgements

I am very grateful to several collab orators who greatly inuenced the work recounted

here Ed Cop eland Marcelo Gleiser Rich Holman Andrew Liddle Jim Lidsey Mike

Turner Sharon Vadas Yun Wang and Erick Weinb erg

My work is supp orted in part by the Department of Energy and the National Aer

ounatics and Space Administration Grant nagw at at Fermilab

In times of uncertainty ab out the future of our eld of physics b ecause of p olitical

and nancial trouble in times where the advances of our understanding of the Universe

seemingly meet with hostility from an increasingly large segment of the public in times

when the prosp ects for young scientists seem grim in times when the future of our eld

seems b eyond our p owers to inuence p erhaps it is worthwhile to recall the words of

Johannas Kepler in the last years of his life when his scientic career was caught in

p ersonal and p olitical turmoil caused by the Thirty Years War

When the storms rage around us and the state is threatened by shipwreck

we can do nothing more noble than to lower the anchor of our p eaceful

studies into the ground of eternity

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