astro-ph/9404040 19 Apr 94 could ments There marriage phase just ary sance troweak School Based In In Introduction scenario a of meaningfully recent the few up on transitions has extrap olating provide p ossible discuss ing the cosmological of These duction including Particles gauge of the with late b een observational lectures particle THE years lectures standard the the an to role s the theory an and the mo dels overview expanding notion and in ationary given in explosion mo del Department Cosmology sp eculate observational physics in ationary concepts provide seeding STANDARD Fermi and cosmology NASA Fermilab predictions to evidence at for that and of JOSHUA very TASI early Friedmann Robertson Walker and the the the National of an reviewing Baksan Batavia pro duction ab out from top ological University information high of universe which early Chicago generation introductory large scale s cosmology the cosmology Astronomy for Boulder particle ABSTRACT the energies universe Kabardino Bulkaria horizon Accelerator has A non baryonic IL its Astrophysics theoretical of evolution b eginning IL of basic defects led FRIEMAN structure quantum of led physics on CO Chicago review COSMOLOGY has and to the and The a prop erties the to June its USA framework b een atness could baryon a Laboratory with of Astrophysics large scale nal adoption of of eld cosmology number in particle Center the the big particular undergoing the Russia FERMILAB Conf A universe lectures b e uctuations very Universe problems asymmetry Subsequent bang and motivating created as of emerged dark April clustering at early underwent remarkable the give cosmology summarizing Baksan the matter standard its and in and an universe lectures standard April puzzles in the cosmological own intro their end International of which in ation a develop to galaxies I rebirth renais name This elec one
from redshift p eculiar velocity and photometric surveys gathered by ground based
telescop es see the lectures by Kirshner in this volume Studies of rich clusters of
galaxies via their gravitational lensing e ects as well as X ray emission from hot
intracluster gas have started to provide new clues to the distribution of dark matter
In addition the recent detection of large angle anisotropies in the cosmic microwave
background radiation by the COBE satellite provides the rst prob e of structure
on very large scales while a series of anisotropy exp eriments on smaller scales now
have tantalizing results On the scale of the universe itself there has b een steady
progress in attempts to measure the cosmological parameters in particular the age
expansion rate and mean density as well as the light element abundances more
precisely see the lectures by Schramm and Walker
As a consequence of these observational advances cosmology is b ecoming data
driven in an unprecedented way theorists no longer have the luxury of untethered
sp eculation but must now confront their mo dels with an impressive array of ob
servations There is still debate ab out the reliability and interpretation of much
of the data but we are de nitely entering the scienti c age of cosmology those
theories which are su ciently worked out are b ecoming increasingly falsi able and
many will stand or fall in the coming years This is a very healthy development
for the eld It is safe to say that at present the big bang framework for the large
scale evolution of the universe remains healthy but that we still lack a standard
tested mo del for the origin and evolution of structure within this framework see
the lectures by Scherrer on structure formation
In these introductory lectures I cover only a very small p ortion of the many
topics of recent interest in cosmology The rst chapters provide a general overview
of the standard cosmology the hot big bang mo del fo cusing on its kinematics and
dynamics the observational evidence in its favor and the current status of mea
surements of the global cosmological parameters Subsequent chapters review the
early universe exploring asp ects of the cosmic microwave background radiation and
particle relics The nal section presents a brief introduction to the in ationary sce
nario for the very early universe and some of its observational implications The
in ationary scenario has not yet received the empirical backing to justify its inclu
sion in the standard cosmological mo del it is discussed here b ecause it nevertheless
provides a comp elling theoretical framework for the very early universe one which
should b e tested by observations in coming years
Readers wishing to delve more fully into these topics would do well to move
on to two textb o oks which cover them in substantial detail Principles of Physical
Cosmology by P J E Peebles and The Early Universe by E W Kolb and M S
Turner
Before plunging in I note that the app endix contains a brief discursion on
notation and units
The Standard Cosmology
Homogeneity Isotropy and the Cosmological Principle
The standard hot Big Bang mo del based on the homogeneous and isotropic
Friedmann Robertson Walker FRW spacetimes is a remarkably successful op er
ating hypothesis describing the evolution of the Universe on the largest scales It
provides a framework for such observations as the Hubble law of recession of galax
ies interpreted in terms of the expansion of the universe the abundances of the light
elements in excellent agreement with the predictions of primordial nucleosynthesis
and the thermal sp ectrum and angular isotropy of the cosmic microwave background
radiation CMBR as exp ected from a hot dense early phase of expansion
While homogeneity and isotropy are strictly sp eaking assumptions of the mo del
they rest on a strong and growing foundation of observational supp ort The evi
dence for angular isotropy on large scales comes from the smallness of the CMBR
large angle anisotropy detected by COBE and FIRS the quadrup ole anisotropy
detected in the rst year of COBE data is T T from the isotropy
l
of radiation backgrounds at other wavelengths as well as from the isotropy of deep
galaxy and radio source counts Note that these di erent sources carry information
ab out quite disparate scales and types of matter in the Universe For example
the APM and EDSGC surveys measured the angular p ositions of more than one
million galaxies over an area covering ab out of the southern sky out to an
e ective depth of roughly h Mp c The galaxy surveys give us information
ab out the lo cal distribution of luminous matter in the universe On the other hand
through the Sachs Wolfe e ect the large angle CMBR measurements directly prob e
the gravitational p otential over length scales of order h Mp c and are thus
sensitive to the mass distribution itself the distinction is imp ortant to keep in
mind since the evidence for dark matter should lead us to b e wary ab out identify
ing the distribution of light with that of mass We will return to the CMBR b elow
and rst fo cus on the galaxy distribution
In a survey of galaxy angular p ositions to some limiting apparent brightness
the joint probability of nding two galaxies in elements of solid angle d and d
separated by an angle is given by
dP N d d w
g g
where N is the mean surface density of galaxies in the survey and w the
g g
galaxy two point angular correlation function measures the excess probability over
random of nding a galaxy pair with this separation If galaxies are distributed
isotropically on large scales we should nd w at large angles and the
g g
correlation function should scale in a de nite way with survey depth b oth of which
are observed The APM data for w are shown in Fig from The correlation
g g
o
function is a p ower law w for but breaks b elow this p ower law
g g
o o
at for jw j and b ecomes lost in the noise This b ehavior
g g
Figure Galaxy angular correlation function w measured in the APM survey
g g
for galaxies in the apparent magnitude interval b
J
implies that the variance in the number of galaxies in a patch of xed angular size
b ecomes small for large patch size in other words averaged over su cently large
angular scales we see roughly the same number of galaxies brighter than a xed
limit p er steradian in di erent parts of the sky
Evidence for large scale homogeneity comes in part from galaxy redshift surveys
However compared to the angular photometric surveys which currently give two
dimensional information for several million galaxies currently complete redshift
surveys yield three dimensional information for typically several thousand galaxies
in a more lo cal neighborho o d In the redshift surveys one can verify directly that
the rms uctuations in the spatial number density of galaxies b ecome small when
averaged over large enough scales For example in the full sky surveys selected from
infrared sources in the IRAS catalog the Jy survey of Fisher etal and the
in QDOT survey of Efstathiou etal shown in Fig the rms uctuation in the
number of galaxies in cubical volumes of side L h Mp c is of order N N
g al g al
and decreases with increasing cell volume This approach to homogeneity is
roughly consistent with that seen in the much deep er but two dimensional angular
surveys In fact the approach to homogeneity as a function of increasing scale
observed in Fig is somewhat more gradual than was exp ected in the p opular
cold dark matter mo del for galaxy formation this is the famous problem of extra
large scale p ower which we will comment up on later Larger structures such as
sup erclusters great attractors great voids and long laments do exist and have
Figure An equal area pro jection of the IRAS QDOT galaxies op en circles at
distances in the range h Mp c Solid squares are regions not surveyed and
the black band is an excluded region encompassing the Galactic disk from ref
received considerable attention Of particular note in this regard was the discovery
of the Great Wall extending roughly h Mp c by Geller and Huchra
in the Center for Astrophysics CfA survey extension But in a statistical sense
the net uctuations in galaxy number b ecome small on the largest scales where they
have b een reliably counted in large area redshift surveys This is consistent with
the visual impression from Fig
This trend is con rmed by the visual app earance of large scale structure in the
ongoing Las Campanas redshift survey going considerably deep er than the CfA
survey they do not nd evidence for coherent structures larger than the Great
Wall The Las Campanas survey uses a multi ber sp ectrograph to simultaneously
measure many redshifts in the same eld an ingenious technological development
which makes p ossible the extension of complete large area redshift surveys to greater
depths in a nite survey time Advances in multi ber sp ectroscopy will b e further
exploited by the Sloan Digital Sky Survey which aims to measure one million galaxy
redshifts over a contiguous area of sr in the northern sky This survey will use
a b er sp ectrograph to accumulate redshifts at an unprecedented rate A con
current photometric survey will measure angular p ositions for roughly million
galaxies
From the CMBR and galaxy observations discussed ab ove we infer that on
large scales the universe app ears to b e distributed in a statistically homogeneous
and isotropic way around us To step from this evidence to a cosmological mo del
we must add to it a further assumption b ecause we observe the universe from only
a single vantage p oint and have direct information only ab out conditions on our
past light cone This assumption often called the Copernican Principle states that
we do not o ccupy a sp ecial p osition in the Universe the conditions we observe are
typical of those seen by observers on distant galaxies By itself the Cop ernican
Principle would allow a large range of p ermissible cosmological mo dels e g an
inhomogeneous fractal universe with self similar structure on all scales or homo
geneous mo dels with anisotropic expansion in di erent directions However when
combined with the observations of isotropy ab ove the Cop ernican principle b ecomes
quite p owerful for it implies that the Universe should app ear isotropic ab out every
p oint By a straightforward geometric argument isotropy ab out every p oint or
more precisely ab out every lo cal fundamental observer comoving with the cosmic
uid in turn implies that the Universe is spatially homogeneous i e that there
exist spacelike surfaces of uniform energy momentum which evolve according to a
universal time t This more general assumption of global homogeneity and isotropy
is called the Cosmological Principle a symmetry principle that is the foundation
of the standard cosmology
At this p oint one is naturally tempted to ask why the universe should have such
strong symmetries of isotropy and homogeneity The attempt to provide a dynam
ical answer to this question is one of the motivating forces b ehind the in ationary
scenario which we shall discuss later on For now we will adopt the cosmological
principle as a working hypothesis and explore some of its consequences
Hubble s Law and Particle Kinematics
The cosmological principle leads directly to Hubble s law of the expanding uni
verse Consider a triangle formed by three fundamental observers each of whom
de nes the lo cal cosmic rest frame at some initial time If the universe remains
homogeneous and isotropic over time then the triangle at all later times must b e
similar to the original one the length of each side scales up by the same factor a t
Extending this argument to all other fundamental observers we see that a t must
b e a universal scale factor such that the distance t b etween any two fundamental
observers satis es t a t where is the initial separation Then the relative
sp eed of one observer with resp ect to the other is given by
v t d dt a a a t H t t
Note that H is in general time dep endent but for observers su ciently nearby
that the light travel time c is small compared to the time over which H changes
appreciably we can replace H by its present value H t H where subscript
denotes the present ep o ch Relation then repro duces Hubble s observation in
that the recession sp eed of a galaxy is prop ortional to its distance from us
The prop ortionality constant H a a is now known as the Hubble parameter
A recent example of this is shown in Fig taken from ref each circle indi
cates a cluster of galaxies with distance estimated using the Tully Fisher relation
for spiral galaxies and recession velocity inferred from the redshift see b elow the
horizontal error bars indicate the spread in inferred distances for a number
of galaxies p er cluster The slop e of the straight line t to the p oints is H
km sec Mp c other metho ds of estimating H yield values in the range
km sec Mp c and it is conventional to write this as H h km sec Mp c with
h The closed circles around Mp c indicate clusters thought to b e
Figure Hubble s law galaxy recession sp eed is shown against distance inferred
from the Tully Fisher relation
falling into the Great Attractor region an inhomogeneity that lo cally p erturbs the
Hubble ow
The cosmological principle with eqn tells us ab out the kinematics of par
ticle motions in the universe First consider non relativistic particles A massive
free particle passes a fundamental observer FO at time t with relative sp eed v c
p
since the FO is at rest with resp ect to the lo cal uid v is the p eculiar velocity of
p
the particle with resp ect to the lo cal rest frame After a time interval dt the par
ticle has moved a distance d v dt and overtakes a second fundamental observer
p
FO who has a sp eed dv H t d H v dt relative to FO At that time FO
p
measures the particle s p eculiar velocity to b e v t dt v t dv Thus the
p p
p eculiar velocity satis es the equation of motion
a dv dv
p
v H v
p p
dt dt a
The solution is
v
p
a
that is in a homogeneous and isotropic universe p eculiar velocities decrease with
the expansion Now consider a dilute non relativistic gas in thermal equilibrium
By equipartition the mean kinetic energy is mhv i k T where T is
B g g
the gas temp erature by eqn the gas temp erature satis es T v a
g
This applies for example to baryons well after they decouple from the photon
background b efore decoupling the baryon temp erature is tied by scattering to the
photon temp erature which scales di erently see b elow
The same reasoning can b e applied to study the evolution of relativistic particles
Consider a photon with initial frequency t as measured by FO After the interval
dt it has travelled a distance d cdt and passes FO who has a sp eed v H d
H cdt with resp ect to FO The photon frequency t dt observed by FO is given
by the rst order Doppler shift
v
d t dt t H dt
c
The frequency thus satis es
d a
dt a
with solution
t
a t
The photon frequency is redshifted with the expansion and its wavelength is stretched
with the scale factor t a For a photon gas in thermal equilibrium
the temp erature thus scales inversely with the scale factor T hE i a
r r
Generalizing the argument one nds that the De Broglie wavelengths of all free
particles follow the scale factor in this way This leads us to de ne the redshift z
a t
obs
z t
e
a t
em e
is the ratio of the photon wavelength at emission t to its wavelength observed at
e
the present t This expression holds more generally than this derivation suggests
it is valid even when the p erio d is comparable to or longer than t although
its op erational meaning is then less transparent The redshift z thus plays several
roles through the Doppler shift it is a measure of recession velocity v cz through
the Hubble law it is a measure of distance v cz H and through eqn
the redshift z t can b e used to characterize a cosmological ep o ch t The most
distant ob jects directly observed are luminous quasars which have b een seen out to
redshifts approaching z By comparison the photons in the cosmic microwave
background radiation were probably emitted or more precisely last scattered at
a redshift z
The Metric Friedmann Robertson Walker Mo dels
In the context of general relativity the cosmological principle severely restricts
the form of the spatial geometry a uniform stress energy momentum tensor implies
that the constant time surfaces have uniform spatial curvature The metric on
such surfaces has the form
dr
d sin d r ds a t
k r
where a t is the global scale factor which describ es the overall expansion or con
traction r are the xed comoving co ordinates carried by the fundamental
observers and k is the sign of the spatial curvature The case k cor
resp onds to at Euclidean space R written in spherical co ordinates k cor
resp onds to the geometry of the three sphere S and k to the hyperb oloid
H the three dimensional analogue of a hyperb olic saddle Thus mo dels with
k are spatially in nite op en while those with k are spatially nite
closed Note that the apparent singularity at r for k is the usual
co ordinate singularity at the p oles of a sphere By homogeneity and isotropy the
full spacetime metric then takes the Friedmann Robertson Walker FRW form
F RW
dx dx dt ds ds g
where t is prop er time measured on the clo cks carried by the fundamental comoving
observers
We can now b e somewhat more precise ab out the assumption of homogeneity and
isotropy clearly this is meant to b e taken in some average sense over large scales It
is useful to think of this p erturbatively the actual spacetime metric can b e written
F RW
h x t In an appropriate as the FRW metric plus a p erturbation g g
gauge h satis es an equation of motion analogous to the Newtonian p otential
Through the Sachs Wolfe e ect the COBE observations roughly indicate that
F RW
T T h g the departure from the FRW metric on large
scales is very small at least in the chosen gauge This argument extends to smaller
scales as well although galaxies represent highly non linear condensations of the
mass density their asso ciated gravitational p otential is small Thus at
least when averaged over scales larger than individual galaxies the p erturbation to
the FRW spacetime metric asso ciated with inhomogeneities is very small and the
homogeneity assumption is an excellent rst approximation
The FRW mo dels are characterized by the global scale factor a t whose dynam
ics is determined by the matter content of the universe through Einstein s equations
R g R g GT
Treating the matter as a p erfect uid lo cally at rest and using the FRW metric
these b ecome the Friedmann equations
a G k
H
a a
and
G p a
a
Here T is the mean energy density of matter p T is its pressure and is the
ii
cosmological constant i e the e ective contribution to the energy momentum from
the vacuum state Although it follows from the Einstein equations it is also useful
to separately write down the energy momentum conservation equation r T
for the uid in the FRW universe
d
H p
dt
Observations suggest that the uid energy density of the universe is currently
dominated by non relativistic matter m while the early universe was dominated
by ultrarelativistic particles or radiation r Non relativistic matter is also some
times called pressureless dust b ecause its uid pressure is dynamically negligible
p hv i
m m m
In this case from eqn the energy density scales as
a
m
This is consistent with conservation of particle number in a xed comoving volume
n a since the particle energy is dominated by its rest mass mn For
m m m
radiation the equation of state is
p
r r
and eqn then yields
a
r
The faster fall o of with the scale factor is consistent with the Doppler redshift of
r
the frequency of a photon emitted by a receding observer in the expanding universe
eqn For massless particles with average energy hE i this implies hE i a
r r
and thus hE in a
r r r
Two features of the solutions to the Friedmann equation with
are worth noting In this case there is a one to one corresp ondence b etween the
spatial geometry and the fate of the universe op en mo dels k expand forever
while closed mo dels k eventually recollapse b ecause the energy density for
matter and radiation fall o faster than a Conversely in the early universe
a a the matter and radiation terms dominate over the spatial curvature and
the dynamics of the mo del is well approximated by setting k In this limit for
a matter dominated universe eqns and give the solution
a t t MD k
m
Gt
while for radiation domination from and
a t t RD k
r
Gt
This clearly generalizes for a uid with equation of state p w
f f
w w
a a t t
f f
Gt w
In the next few sections we fo cus on the cosmologically recent matter dominated
era and subsequently turn to the radiation dominated early universe
Cosmological Parameters H and t
The principal observable cosmological parameters of the FRW mo dels are the
Hubble parameter H a a the age of the Universe t the present non
relativistic mass density relative to the critical density of the spatially at Einstein
de Sitter k mo del
G H
cr it
the deceleration parameter q aa a which measures the rate at which the
gravitational attraction of the matter is slowing down the expansion and the con
tribution of the cosmological constant to the present expansion rate H
From the Friedmann equations these parameters are related by
k
a H
q
For vanishing cosmological constant determines the sign of the spatial cur
vature for the spatially at mo del k and it is less than one for op en
mo dels
The age of the universe is related to the other parameters through an expression
of the form H t h t yr f where f is a function of order
unity Thus the Hubble time H h yr sets the timescale for the age of
the universe while the Hubble length cH h Mp c sets the lengthscale for
the present observable universe For a matter dominated universe with f
falls monotonically with increasing and two useful limits are f a t
for and f since a t for More generally over the
range k an excellent approximation is
q
sinh
a a
p
H t
a
where
a
Figure Evolution of the cosmic scale factor scaled to its present value
Some examples of the evolution of the scale factor a t are shown in Fig
including three cases with empty at and closed
and one example of a at k mo del dominated by a cosmological
constant This gure displays the decrease in the expansion age
H t from unity as is increased from zero It also demonstrates that a signi cant
cosmological constant term can make H t large The solution for the spatially at
k mo del with non zero has the form
a t H t
sinh
a t
where At late times it approaches the exp onential de Sitter solution
p
t
3
a t e
Much early e ort was sp ent trying to measure or constrain the parameters q
and through the classical cosmological tests such as the Hubble diagram an
gular size as a function of redshift and galaxy counts as a function of redshift and
apparent brightness For example to construct the Hubble diagram one measures
the apparent brightness of a well de ned sample of ob jects say the brightest galax
ies in clusters as a function of the ob ject s redshift for galaxies of xed intrinsic
luminosity the scaling of apparent magnitude with redshift is a function of the
cosmological parameters Unfortunately galaxies at large distances where the dis
tinction b etween mo del parameters b ecomes observable are seen when they were
much younger than their nearby counterparts so a mo del for galaxy luminosity
evolution must b e used to interpret the results Signi cant progress has b een made
in understanding galaxy evolution and there is hop e that the e ects of evolution
and cosmology might b e disentangled in coming years but these tests currently do
not place stringent universally accepted constraints on the cosmological parameters
Recently it has b een p ointed out that the probability that a quasar at a given
redshift is gravitationally lensed by a foreground galaxy is a sensitive test for the
cosmological constant in mo dels with one generally exp ects a higher lens
ing probability Based on surveys for lensed quasars the b ound has b een
inferred in the case of a spatially at k universe For more on the standard
cosmological tests the reader is referred to Peebles
H and the Distance Scale
The Hubble parameter relates the observed recession velocity v or redshift z of
r
a galaxy to its distance d for v c the recession velocity is v cz H d v
r r p
where v is the p eculiar radial velocity of the galaxy with resp ect to the Hubble
p
ow usually assumed to arise from gravitational clustering Galaxy redshifts can b e
measured quite accurately so all the di culty in determining H resides in nding
reliable distance indicators for extragalactic ob jects at distances large enough that
the Hubble term dominates over the p eculiar motion Observed p eculiar velocities
are typically of order km sec so that distance measurements b eyond Mp c or
more recession velocities ab ove km sec are required for reasonable accuracy
A wide variety of techniques has b een used to establish an extragalactic dis
tance scale and this is re ected in the spread of results for H roughly
km sec Mp c Distance estimates made using metho ds such as the Tully Fisher
relation b etween cm rotation sp eed and infrared luminosity for spiral galaxies
calibrated by observations of Cepheid variable stars in several nearby galaxies have
yielded high values for the expansion rate roughly H km sec Mp c Two
newer metho ds planetary nebula luminosity functions and galaxy surface bright
ness uctuations yield values for H in this range as well and are b eing further
developed On the other hand metho ds using Type Ia sup ernovae as standard can
dles have yielded low values H km sec Mp c SNe Ia are thought to b e
the explosions of white dwarfs which accrete matter from binary companions until
they reach the Chandrasekhar mass and there is some evidence that they form a
homogeneous class they also have the advantage that they can b e observed to great
distances In the future Hubble Space Telescope observations of Cepheids in other
nearby galaxies and p erhaps as far as the Virgo cluster which are hosts to SNe Ia
or which can b e used as Tully Fisher calibrators should help improve the situation
The recent discovery of a probable Type Ia sup ernova at z also raises hop es
that a sample of SNe Ia at z could signi cantly constrain q provided the
disp ersion in SNe Ia luminosities is su ciently narrow
There are also a variety of metho ds b eing employed to measure the distances of
extragalactic ob jects directly bypassing the extragalactic distance ladder built up
from Cepheids Using the expanding photosphere metho d Schmidt Kirshner and
Eastman have determined the distances to type I I sup ernovae at large distances
and nd go o d agreement with the Tully Fisher distances for these galaxies Other
direct metho ds which hold future promise include measurement of the Sunyaev
Zel dovich e ect due to the Compton upscattering of CMBR photons by hot gas
in rich clusters and the di erential time delay b etween images in gravitationally
lensed quasars
The Age of the Universe t
Three metho ds have traditionally b een used to infer the age of the Universe
t Nuclear cosmo chronology is based on radioactive dating of r pro cess elements
that is heavy elements formed by rapid neutron capture most probably in sup er
novae The element ratios Re Os and Ur Th generally indicate t Gyr
with a large uncertainty due to the unknown element formation history e g the
star formation rate over time A second metho d involves the co oling of white
dwarfs when low mass stars exhaust their nuclear fuel they b ecome degenerate
white dwarfs gradually co oling and b ecoming fainter The number of white dwarfs
as a function of luminosity drops dramatically for L L suggesting
w d
that there has not b een su cient time for them to fade b elow this value Coupled
with mo dels of white dwarf co oling this implies that the age of the galactic disk is
ab out t Gyr
The most extensively studied technique for constraining t is the determina
tion of the ages of the oldest globular clusters in the galaxy When stars nish
burning hydrogen they turn o the main sequence characteristically reddening
and brightening By observing the color magnitude color vs apparent bright
ness diagram for a cluster one can determine the apparent brightness of stars
in the cluster that are now leaving the main sequence Knowing the distance to
the cluster then gives the absolute luminosity of stars at the turn o On the
other hand stellar evolution theory relates stellar luminosity to the time a star
sp ends on the main sequence crudely a star turns o the main sequence when
a xed fraction f of its initial hydrogen mass M X M has b een con
H H
verted into helium here X is the initial H mass fraction of the star
H
The conversion of H to He releases erg for every gm of H burned
Thus over the course of the main sequence the total energy radiated by a star
is E f X M f X M M erg and its evolutionary lifetime on the main
H H
sequence is E L f X M M L L yr Now stellar mo dels indi
ms H
cate that the relationship b etween mass and main sequence luminosity is roughly
L M yielding a main sequence lifetime of L L years
ms
Thus the measurement of the main sequence turn o luminosity gives an estimate
of the age of the cluster The turn o luminosity in the oldest globular clusters in the
galaxy is slightly b elow that of the sun yielding the age estimate t
g c
Gyr This argument is a gross oversimpli cation of the complex pro cess of stellar
mo delling and cluster iso chrone tting but it gives a heuristic understanding of the
resulting age estimate The largest source of error is apparently the uncertainty
in the distances to the globular clusters It is hop ed that observations with the
corrected Hubble Space Telescope mirror successfully repaired in late could
reduce the uncertainty in t to as little There may also b e residual systematic
g c
mo del uncertainties asso ciated with the initial hydrogen or helium fraction X
H
opacities metallicity etc
We now turn to the nal cosmological parameter of interest the density of the
universe
Cosmological Parameters and Dark Matter
It is convenient to parameterize the mass density of the universe in terms of
the mass to light ratio say in the V band hM Li M L Dividing the
T
present critical density
G h gm cm h M Mp c H
c
by the observed mean luminosity density j hL Mp c the critical
V
T
mass to light ratio for the universe is h and the cosmic density
c
parameter can b e expressed as h The mass to light ratio in
the solar neighborho o d is approximately while the central cores of ellipti
cal galaxies yield h so the density of luminous matter that is of matter
asso ciated with typical stellar p opulations is inferred to b e How
l um
ever it is well known that the luminous parts of galaxies are not the whole story
there is strong evidence from at spiral galaxy rotation curves from a variety of
observations of galaxy clusters and from large scale p eculiar motions that there is
a substantial amount of dark matter in the universe
Dark matter in galaxies
An example of this phenomenon for galaxies is shown in Fig which shows a
mo del for the rotation sp eed of the galaxy as a function of radius The three lower
curves show the contributions from the luminous comp onents two bulge comp o
nents and the disk Clearly they cannot account for the observed rotation sp eed
b eyond a few kp c from the galactic center To match the observations taken from
the compilation in ref one assumes in addition a quasi spherical distribution of
dark matter called the halo with a density pro le of the form
r
h
a r
In Fig the adopted halo core radius is a kp c On large scales the halo mass
scales as M r r yielding a at rotation curve v GM r r km sec
h
c
Figure Rotation curve for our galaxy an average over a number of di erent
observations solid p oints is shown with a mo del that includes luminous and dark
matter
for the mo del shown in Fig Clearly eqn cannot extend to arbitrarily large
radii b ecause the enclosed mass would diverge the halo is presumably truncated by
tidal or other e ects at large r The observation of high prop er motion stars in the
solar neighborho o d presumably b ound to the Galaxy implies that the lo cal value
of the galactic escap e velocity exceeds km sec and this can b e used to
place a lower b ound on the halo truncation radius If the halo is sharply truncated
at r the escap e sp eed v at r r satis es
t e t
ln r r v v
t
c e
Using v km sec at the solar radius r R kp c and v yields
e c
r kp c and a total galaxy mass M M Comparing with the
t g al
luminosity of the disk plus the bulge L L in the V band this
g al
implies that the total mass to light ratio for the Milky Way is at least
MW
This is consistent with the requirement that distant globular clusters and satellite
galaxies are b ound to the Galaxy as well as with mass to light ratios inferred from
more extended at rotation curves in other spiral galaxies If these systems are
typical of the universe we infer h for the matter asso ciated with galaxy
halos
It is interesting to compare these values with the baryon density inferred from
B
primordial nucleosynthesis see the lectures by Schramm and Walker which
has b een restricted to the range h Comparison with
B l um
ab ove suggests that some or most of the baryons are dark or in underluminous
p opulations Furthermore comparison with the escap e sp eed and rotation curve
b ound h shows that baryons could constitute some or all of the dark
matter in galaxy halos dep ending on their extent
One p ossibility for dark baryons in halos would b e degenerate brown dwarfs
substellar M M ob jects which did not reach su ciently high temp era
ture to burn hydrogen Currently several indep endent groups are searching for halo
dwarfs which have b een dubb ed MACHOS for massive compact halo ob jects the
signature is a microlensing event in which a background star say in the LMC or the
bulge of the Milky Way symmetrically brightens and fades as a MACHO passes
near its line of sight Although they are distinguishable from ordinary variable
and are stars by the time symmetry and achromaticity of the light curve such
microlensing events would b e intrinsically rare the probability that a given back
ground star is lensed is v c so a large number of stars must
c
b e accurately monitored The American Australian MACHO pro ject currently ob
tains CCD photometry for several million stars p er night and has discovered many
p erio dic variable stars The French EROS and Polish OGLE groups are also mak
ing progress In September and Octob er all three groups rep orted the rst
evidence for microlensing events by ob jects in the Galaxy The most prob
able masses for the lensing ob jects are roughly in the range M As
data is accumulated over the next few years we will b e able to infer information
ab out the contribution of MACHOs to the Milky Way halo It is also worth noting
that microlensing in external galaxies has b een sought previously by monitoring
the brightness of gravitationally lensed QSO images over time In particular a mi
crolensing event in the lensed quasar has b een established but in this
case it is di cult to establish the MACHO mass since the microlensing probability
is closer to unity
Dark matter in clusters
Moving to larger scales for clusters of galaxies the traditional dynamical metho d
of estimating cluster masses and mass to light ratios rst used by Zwicky who
discovered the missing mass problem in the s has b een to apply the virial
theorem to measured cluster velocity disp ersions M hv i hR i where hv i
tot
ij
is the velocity disp ersion of the galaxies in a cluster and R is the separation
ij
b etween them This metho d assumes that galaxies trace the cluster mass and
that the galaxy velocity distribution is isotropic b oth of which may not b e go o d
approximations Indep endent information on the dark matter distribution in the
inner cores of clusters comes from the giant luminous arcs and arclets high redshift
galaxies gravitationally lensed by foreground clusters These arcs are formed
when a galaxy is nearly imaged into an Einstein ring Measurements of the cluster
and background galaxy redshifts yield an estimate of the cluster mass within the
impact parameter of the lens for most of the cases studied these estimates are in
reasonable agreement with the mass to light ratios inferred from the virial theorem
but it should b e noted that the arc observations prob e only the inner few hundred
kp c of the clusters Work is presently b eing done to extend this idea to map clusters
on larger scales by lo oking for more minute statistical distortions in the alignments
of background galaxies Typical inferred cluster values are h which
would imply Since clusters are rare ob jects o ccupying a very small
fraction of the universe one exp ects this to b e a lower b ound in which case some
form of non baryonic dark matter is probably required given the limits on ab ove
B
X ray observations most recently by the ROSAT satellite have b egun to map
the density and temp erature pro les of the hot gas which p ermeates many clusters
Since the gas is in hydrostatic equilibrium it can b e used to trace the cluster mass
distribution including dark matter directly
k T r dlnT dlnn
M r
tot
G m dlnr dlnr
p
where is the mean molecular weight of the gas and n r and T r are the gas
density and temp erature Cluster masses inferred from X ray observations are gen
erally comparable to but less than virial estimates Consistent with this
the X ray measurements indicate that clusters are surprisingly baryon rich in the
sense that the gas constitutes typically h of the inferred binding mass
within approximately h Mp c of the center of a rich cluster like Coma More
over the dark matter tends to b e more centrally concentrated than the gas out to
this scale suggesting that the gas fraction of the whole cluster is at least as large
as the value ab ove If this ratio is representative of the baryon mass fraction of
the universe then the nucleosynthesis b ound on would imply the upper limit
B
h Including the baryon stellar comp onent in cluster galaxies only
strengthens this limit This has b een taken as evidence against the universe having
closure density and would require advocates of in ation which implies
k to fall back on a cosmological constant or some other smo othly distributed
comp onent The other p ossibility would b e to lo osen the nucleosynthesis b ounds
on through some non standard scenario such as inhomogeneous nucleosynthesis
B
but the upp er b ound on is not raised su ciently in this mo del to get around
B
the argument ab ove One should p erhaps b e cautious ab out inferring the universal
baryon fraction from a rich cluster like Coma In particular clusters with higher
X ray temp eratures and therefore larger total masses may have a larger fraction
of their total mass in gas This trend coupled with the steeply falling distribution
function of cluster temp eratures dn dT T for k T keV suggests that
c
the mean gas and baryon fraction may b e substantially b elow the value for Coma
since the mean is dominated by the more numerous co oler clusters k T
C oma
keV This would raise the derived upp er b ound on closer to unity and suggests
that massive clusters like Coma may not b e representative of the baryon fraction
of the Universe On the other hand N b o dy simulations of cold dark matter with
baryons indicate that the baryon fraction in a Coma size cluster should b e repre
sentative of the mean so the present situation is somewhat confusing
Large scale ows
Moving to still larger scales the deviations from the Hubble ow have b een
used to infer the cosmic density over scales of order h Mp c The basic idea is
to compare samples of the density p erturbation eld and the p eculiar velocity eld
covering the same volume assuming they arise gravitationally the prop ortionality
b etween them dep ends on the rate of growth of the density uctuations which in
turn dep ends on On very large scales as discussed ab ove the rms density
uctuations are small so linear p erturbation theory away from the FRW spacetime
is a reasonable rst approximation see the lectures by Scherrer In this case one
nds the relation
r v H
p
where the density eld x x and the p erturbation growth rate
enters through dln dlna If one expresses distances d in terms of their
equivalent Hubble velocities v H d then H drops out of Eqn so the
uncertainty in the Hubble parameter do es not undermine this metho d A number
of di erent approaches have b een used to extract in this way and they have
all given consistently high answers The top ographic correlation b etween
the observed galaxy eld and the density eld inferred from galaxy p eculiar velocity
surveys suggests that galaxies do broadly trace the mass distribution on large scales
in the sense that more galaxies are found in regions of high mass density that is
high r v but the galaxy distribution may b e biased with resp ect to the mass
p
In the simplest linear bias mo del the smo othed galaxy and mass density elds
are assumed to b e prop ortional x b x where b is the bias factor
g al g al g al
taken to b e constant for a given class of galaxies Since is what is observed
g al
the comparisons based on Eqn actually constrain the combination b
g al
where the bias factor refers e g to galaxies selected from the IRAS catalog Recent
determinations have found b For a bias factor of order unity this is
g al
consistent with which is pleasing to theorists and also buttresses the case
for non baryonic dark matter It is well to keep in mind however that biasing
is presumably a complex pro cess asso ciated with the non linear stages of galaxy
formation so that the prop ortionality of the galaxy and density elds may b e non
linear and or scale dep endent In addition even with substantial smo othing of
the density eld the p erturbation amplitude in many regions is not small and
corrections to the linear theory must b e taken into account
A summary of typical estimates of the density parameter on di erent scales is
given in Table
Non baryonic Dark Matter
If as the dynamical observations on scales larger than clusters sug
B
gest some form of non baryonic matter must b e invoked to make up the balance
Moreover since this matter must b e dark it is natural to consider
l um
weakly interacting particles as candidates It is convenient to distinguish two broad
Table Estimates of the cosmic density parameter
System
Luminous matter in galaxies h
Galaxy halos h
Clusters h
Large scale ows
classes of non baryonic dark matter hot and cold on the basis of their clustering
prop erties The prototypical hot dark matter candidate is a light neutrino with
mass m eV see Sec Since they are relativistic m T until rela
tively recent ep o chs light neutrinos would free stream out of and damp out density
p erturbations up to the scale of galaxy clusters Galaxies would form after clusters
via fragmentation top down Due to phase space constraints light neutrinos
would not cluster signi cantly on the scale of galaxies baryons would constitute
the predominant dark matter in galaxy halos while neutrinos would dominate in
clusters Cold dark matter on the other hand is de ned to have negligible free
streaming length it clusters on all scales In cold dark matter mo dels structure
generally forms hierarchically with smaller clumps merging to form larger ones
b ottom up For this reason many but not all theorists since the early s
have tended to prefer cold over hot dark matter but it is worth noting the recently
surging p opularity of a mix n match scenario a combination of cold and
hot dark matter say with m eV pro duces a favorable sp ectrum of large scale
density p erturbations in the context of in ation with some apparent advantages
over pure cold dark matter
The theoretically favorite candidates for cold dark matter are weakly interacting
massive particles WIMPs with masses generally in the range GeV and the
axion an ultra light pseudoscalar with a mass of order eV The most attractive
WIMP candidate is the neutralino the lightest sup ersymmetric fermionic partner of
the standard mo del b osons its weak annihilation rate in the early universe naturally
leaves it with an abundance comparable to the present critical density The axion
is the pseudo Nambu Goldstone b oson asso ciated with sp ontaneous breakdown of
a global U symmetry the Peccei Quinn symmetry introduced to explain why
the strong interactions conserve CP The global symmetry is sp ontaneously broken
at some large mass scale f through the vacuum exp ectation value of a complex
PQ
p
scalar eld h i f exp ia f At energies b elow the scale f the only
PQ PQ PQ
relevant degree of freedom is the massless axion eld a the angular mo de around
the b ottom of the p otential At a much lower energy scale MeV the
QC D
symmetry is explicitly broken when QCD b ecomes strong and the axion obtains a
p erio dic p otential of height In invisible axion mo dels with Peccei Quinn
QC D
scale f GeV the resulting axion mass is m f eV
PQ a PQ
QC D
and Although light invisible axions interact so weakly with cross section
a
f that they were never in thermal equilibrium they form as a cold Bose
PQ
condensate
Accelerator searches for neutrino mixing and b eta decay exp eriments on neu
trino mass should provide useful constraints on the p ossibility of neutrino dark
matter Active exp erimental e orts are also underway to detect b oth WIMPs and
axions Direct WIMP detection lo oks for the signals pro duced when a halo WIMP
collides with a nucleus in a kg size cryogenic crystal dep ositing of order keV
in ionization and phonons detector schemes based on scintillation and excitations
of sup er uids and sup erconductors are also b eing developed Indirect WIMP de
tectors search for high energy neutrinos pro duced when WIMPs annihilate in the
Sun and the Earth large underground or underwater detectors currently in place
or under development with sensitivity to WIMP annihilations include Kamiokande
MACRO AMANDA and DUMAND Accelerator searches for sup ersymmetry also
will constrain the neutralino parameter space A large scale axion search based at
Livermore exp ected to come on line in will search for resonant conversion of
halo axions to microwave photons in a cavity with a strong magnetic eld A scaled
up version of an idea originally prop osed by Sikivie this detector should approach
the cosmologically interesting region of axion couplings for the rst time
Finally given the paucity of direct evidence for dark matter it is probably
healthy to keep an op en mind to alternatives While it is natural to ascrib e at
galaxy rotation curves and large cluster velocity disp ersions to unseen matter
Milgrom and others have argued that they may instead signal a breakdown of
Newton s law of inertia at very low acceleration The extent to which Milgrom s
mo di ed Newtonian dynamics MOND accounts for all the phenomena normally
imputed to dark matter is controversial and a full theory with which one could
explore cosmology has b een lacking Nevertheless at a minimum it provides a use
ful challenge to the accepted dogma On a relatively more conservative side the
p ossibility that the dark matter interacts by other long range forces in addition to
gravity has recently b een explored Such interactions are signi cantly constrained
by galaxy and cluster observations but could nevertheless have interesting impli
cations for structure formation and biasing
Cosmological Parameters Taking Sto ck
With this brief survey in hand it is useful to pause and place these numbers for
the cosmological parameters in theoretical context If an extended p erio d of in ation
to ok place in the early universe see b elow then the spatial geometry should now b e
observationally indistinguishable from k If the cosmological constant vanishes
from Eqn spatial atness implies with the concomitant requirement
of non baryonic dark matter and thus t H h yr This is
uncomfortably low compared to globular cluster ages unless h t Gyr
and de nitely problematic unless h t Gyr still on the low side of the
H observations However a non vanishing is certainly allowed at some level by
observations and has sp oradically come into vogue most recently to alleviate b oth
this age problem and the large scale p ower problem for in ationary cosmology For
example in a at mo del with Eqn gives t H
h Gyr consistent with the lower b ound t Gyr for the entire observed
g c
range of H and yielding a healthy t Gyr old universe for h This
lower value of is consistent with the dynamical mass estimates from clusters but
somewhat b elow the recent estimates from large scale ows Since there is currently
little theoretical guidance as to why j j is as small as it is and no rm pro of that it
should vanish it is probably b est to keep an op en mind although the fact that we
would b e living just at the ep o ch when is comparable to might seem to b eg for
explanation The third p ossibility is that theoretical prejudice is wrong and that
we live in an op en low density p erhaps purely baryonic universe with negligible
in which case the globular cluster age range is also compatible with somewhat
larger values of H The challenge in this case is to explain the large scale ows
or attribute them to systematic distance errors and to form large scale structure
without violating CMBR anisotropy constraints In this case it also would b e an
o dd coincidence that we live just at the ep o ch when the curvature term in is
b ecoming appreciable compared to the matter term
The Early Universe
We have seen that the FRW mo dels provide a natural framework for under
standing observations of the universe on the largest scales in particular the Hubble
law Aside from the Hubble expansion two other observational pillars supp ort the
temple of the standard cosmology the thermal sp ectrum of the CMBR and the light
element abundances Both direct our attention backward to much earlier ep o chs in
the history of the universe the hot big bang
The Cosmic Microwave Background
The K cosmic microwave background radiation discovered thirty years ago
has since b een shown to b e remarkably isotropic to roughly one part in on angu
lar scales larger than a few arc minutes and to have an exactly thermal blackbo dy
sp ectrum to within the exp erimental errors The sp ectral measurements cover the
range of photon wavelengths from roughly cm while the most impressive
veri cation of the Planck sp ectrum comes from the recent COBE FIRAS observa
tions in the range cm The measured FIRAS ux vs frequency is shown
in Fig it is indistinguishable from a Planck sp ectrum with a temp erature of
T K The error bars on the p oints are to o small to see so they have
b een multiplied by a factor of in the gure
The blackbo dy nature of the CMBR is indicative of a medium that has relaxed
to thermal equilibrium However there are strong grounds for b elieving that the
recent universe is transparent to CMBR photons that is that the mean free path
for microwave photon scattering is much longer than the present Hubble length i
we observe radio galaxies as p oint sources out to redshifts of order z which re
quires microwave transparency out to this scale ii even p ostulating a fully ionized
Figure CMBR brightness vs frequency inverse wavelength measured by COBE
FIRAS
intergalactic medium with closure density the mean free time for Comp
ig m
ton scattering is longer than the Hubble time back to a redshift z Therefore
the CMBR is not now in thermo dynamic equilibrium and could not have b een ther
malized at recent cosmological ep o chs The standard explanation is that the CMBR
is a remnant of a much earlier hot dense phase of the universe when the matter
and radiation were in equilibrium Indeed running the Hubble expansion backward
in time this is just what one would exp ect as the matter is compressed it heats
up b ecoming a dense ionized plasma which e ciently scatters and thermalizes the
radiation on a timescale much shorter than the expansion timescale
This argument suggests that the early universe can b e accurately describ ed as a
dilute adiabatically expanding gas of particles and radiation in lo cal thermal equi
librium uniquely characterized by the instantaneous temp erature T t While this
may re ect conditions over much of the early history of the universe we know it is
not the full story b ecause a gas in thermal equilibrium is featureless and structure
less and the observed universe is not so b oring Although thermal equilibrium is a
go o d starting p oint the interesting ep o chs in cosmic history the ones which leave
p otentially observable signatures or relics such as the light elements the baryon
asymmetry and particle dark matter are those in which a particle sp ecies i go es out
of thermal equilibrium b ecause the rate of interactions keeping it in equilibrium
falls b elow the expansion rate H t This happ ens at a temp erature T de ned
i F
by
H
T
F
T is called the freeze out temp erature at temp eratures T T the particle s
F F
comoving number density is xed n a Thus the study of the early universe
i
requires b oth equilibrium and non equilibrium thermo dynamics the Boltzmann
equation For completeness it is worth noting that there may also b e particle relics
such as magnetic monop oles and axions which were never in thermal equilibrium
It is helpful to have some characteristic numbers for the CMBR For ultra
relativistic particles in kinetic equilibrium at temp erature T the energy density
is
b osons
g T
i i
fermions
and the number density
b osons
g T n
i i
fermions
where g is the number of internal degrees of freedom and the Riemann zeta function
i
For blackbo dy radiation at T K the present CMBR energy
density is gm cm and the photon number density is n
cm For comparison the present baryon density is n m h
B B p B
cm and the baryon to photon ratio is
n
B
h
B
n
In the early universe baryons and photons are kept in thermal contact by Comp
ton scattering e e As the universe expands and co ols eventually it b e
comes energetically favorable for the ionized plasma to recombine to form neutral
hydrogen e p H When this happ ens the density of free electrons drops
precipitously and the Compton scattering rate p er photon n falls b e
e C omp
low the expansion rate H as a result the photons decouple freeze out from the
matter In the absence of subsequent reionization of the matter the photons will
have travelled freely since that ep o ch which is therefore called the surface of last
scattering
It is a useful exercise to estimate when hydrogen recombination and photon
decoupling to ok place Since the recombination rate is initially rapid compared to
the expansion rate the baryons and photons maintain ionization equilibrium The
densities of the non relativistic baryons in thermal equilibrium are
m m T
i i i
exp n g
i i
T
where i e p H and are chemical p otentials De ning the ionization fraction
i
x n n n and using chemical equilibrium to relate the chemical p otentials
p p H
we arrive at the Saha equation for the equilibrium ionization fraction
p e H
p
x B T
eq
p
exp
x m T
e
eq
where B m m m eV is the binding energy of neutral hydrogen
e p H
Using for and taking h from nucleosynthesis one nds that
B
x drops rapidly away from unity that is the hydrogen e ectively recombines
eq
at a temp erature T eV Since the temp erature scales as T z
r ec
K this corresp onds to the ep o ch z This result is just what one exp ects
r ec
recombination takes place when there are to o few photons left in the high energy tail
of the thermal distribution to ionize the hydrogen This happ ens at a temp erature
somewhat b elow the binding energy of hydrogen b ecause the number of photons p er
baryon is so large Once x drops appreciably b elow unity the Compton
eq
rate p er photon x n so on falls b elow the expansion rate and the
C omp
photons decouple at a redshift z From this corresp onds to a
dec
decoupling time t h sec h yr for
dec
If photons decoupled at such an early ep o ch why do es the CMBR still have such
a precisely thermal sp ectrum The simple reason is that the expansion preserves
the blackbo dy thermal sp ectrum even in the absence of thermalizing interactions
At temp eratures T T the photons are in kinetic equilibrium with a Bose
dec
distribution function
f p t t
dec
exp E T
where E jpj After decoupling the photons are free particles their energies just
redshifting with the expansion E t E t a t a t However since they are
dec dec
no longer interacting the o ccupation number must b e conserved for each momen
tum state As a result the distribution function maintains the form with
e ective temp erature given by T t T t a t a t since this holds for each
dec dec
momentum state the sp ectrum remains thermal
In fact the issue of the preservation of the thermal sp ectrum is more subtle than
this argument suggests and the sp ectrum observation by FIRAS a more p owerful
to ol b ecause we have not taken into account the p ossibility that some physical
pro cess could distort the sp ectrum away from the form For example annihi
lating or decaying elementary particles or explo ding black holes could release energy
into the photon baryon plasma that is not completely thermalized by the time of
photon decoupling Alternatively after decoupling an early generation of stars
quasars or active galactic nuclei might release enough energy to heat up and reion
ize the baryons which would then scatter o the CMBR and distort the sp ectrum
The kind of sp ectral distortion pro duced dep ends on the ep o ch when such a pro cess
releases energy into the photon baryon system Energy released into the baryons at
z is completely thermalized by Compton scattering and bremmstrahlung
r el
emission the photons relax to a new equilibrium distribution at a higher temp er
ature leaving no distortion Energy released in the interval z
r el
cannot b e completely thermalized b ecause photon pro duction b ecomes ine cient
in this regime multiple Compton scattering leads to a Bose Einstein photon distri
bution
f p
E
exp
T
where the chemical p otential is related to the energy density injected by
T The FIRAS results shown in Fig can b e translated into the
b ound j T j severely restricting the energy density injected at such
times If energy is released into the electrons later at z z e
dec r el
scattering pro duces a Compton y distortion which can b e expressed as a frequency
dep endent temp erature
T
T e
y for T
y
T
y T for T
T T e
The y parameter is related to the energy release by y Eqn states
that Compton scattering by hot electrons at T T moves photons from low to
e
high frequency depleting the sp ectrum in the low frequency Rayleigh Jeans region
and enhancing it in the Wien part of the sp ectrum Comparison with the data in
Fig yields the observational constraint y These b ounds together
severely constrain any mo del in which energy is released b etween roughly one and
yr after the big bang
The Radiation Era
Summing eqn over all e ectively massless sp ecies in equilibrium we can
de ne the total radiation energy density by
g T
r
where g T counts the number of e ective relativistic Bose B and Fermi F
degrees of freedom which may have di erent e ective temp eratures
X X
T T
i i
g T g g
i i
T T
i B i F
As the universe expands and co ols the function g T decreases whenever T drops
b elow the mass m of a particle sp ecies For the photon and assuming massless
i
neutrinos it is currently g T ab ove the e e mass threshold g
MeV and at temp eratures ab ove the masses of all particles in the
standard electroweak mo del g T GeV
From and assuming there are no exotic heretofore unknown relativistic
particles the present radiation energy density is t gm cm
r
Comparing with the critical density in eqn the radiation makes a negligible
contribution to the present density
h
r
con rming our assumption in Sec that the present universe is matter dominated
On the other hand since a the universe was radiation dominated at
r m
early times From the transition from radiation to matter domination
m
o ccurred when
r
a T
eq
z h
eq
a T
eq
which corresp onds to a photon temp erature T h eV For this
eq
happ ened when the age of the universe was t h sec Note that for
eq
the universe b ecomes matter dominated b efore but not long b efore
B
recombination
During the radiation dominated RD ep o ch t t the spatial curvature term
eq
in the Friedmann equation can b e neglected since it scales with a more slowly than
or and we know that it is at most comparable to the matter density term
m r
to day The RD era solution is therefore describ ed by Combining and
we can relate the temp erature and expansion rate in the RD ep o ch
T
H g
M
P l
Since H t from we can also relate temp erature and age
M T
P l
sec t
MeV
g T g
Table Thermal history of the universe
T GeV t sec Event
Planck era quantum gravity
Grand uni cation in ation top ological defects
Sup ersymmetry technicolor
Electroweak transition SU U U
em
Quark hadron transition chiral symmetry breaking
Neutrino decoupling
e e annihilation
Matter radiation equality
H recombination photon decoupling
Non linear structures form
Chicago Bulls three p eat
note this was given as a prediction at TASI
A timeline for the early universe with some imp ortant ep o chs delineated is shown
in Table The CMBR prob es conditions back to the time of photon decoupling
while the light element abundances prob e conditions at temp eratures comparable
to nuclear binding energies T MeV Going backward in time QCD predicts that
chiral symmetry should b e broken at a temp erature of order MeV At ab out that
time quarks should also b ecome con ned inside hadrons but it is not clear if this is
a smo oth second order transition or a rst order transition involving the release of
latent heat and the nucleation of hadron bubbles If the quark hadron transition is
rst order the resulting inhomogeneous baryon distribution might leave observable
signatures in the light element abundances
Going to earlier times the electroweak symmetry should b e restored at a tem
p erature of order GeV The dynamics of this transition is currently an area of
active investigation if it is a rst order transition it is p ossible that the resulting
non equilibrium conditions were rip e for the generation of the baryon asymmetry
If baryons and anti baryons had b een present in exactly equal numbers in the early
universe their annihilation would eventually have driven the baryon to photon ra
tio down to a value many orders of magnitude smaller than Thus when the
baryons were relativistic there must have b een a small asymmetry b etween the
density of baryons and anti baryons n n n n n As rst p ointed out
B b b b
b
by Sakharov the generation of such a baryon asymmetry requires baryon number
and CP violating interactions as well as a departure from thermal equilibrium since
n n in equilibrium It is currently thought that baryogenesis takes place at the
b
b
GUT or electroweak eras
Going back earlier than sec we must invoke physics b eyond the stan
dard electroweak mo del and the events b ecome increasingly sp eculative Particle
physics mo dels suggest there may b e new physics lurking at the TeV scale p erhaps
sup ersymmetry technicolor or various extensions of the standard mo del In the
simplest grand uni ed theories the strong and electroweak interactions are uni ed
the symmetry b etween them restored at an energy scale of order GeV A cos
mological phase transition at that ep o ch might lead to in ation or to the generation
of top ological defects such as monop oles cosmic strings or textures the resulting
density and gravitational wave p erturbations pro duced could provide the seeds for
large scale structure and leave a signature in the CMBR anisotropy Classical cos
mology runs into a wall at the Planck era t sec at that ep o ch quantum
uctuations in the spacetime metric are exp ected to b e large and a quantum theory
of gravity is required If sup erstrings provide the fundamental description of nature
inherently stringy e ects would b ecome imp ortant around that scale
Relic Neutrinos and Hot Dark Matter
As an interesting example of interactions freezing out and leaving a relic sp ecies
consider light neutrinos The cross section for e e scattering through W
e e
and Z exchange is of order p p M where is the weak ne structure
w
w
constant M M M is the gauge b oson mass and p are the fermion momenta in
Z W
the center of mass First consider the very early universe at temp eratures T M
In this case the momenta p T M resulting in a cross section h v i T
w
Since the particle density n T the interaction rate is T Comparing
w
with the expansion rate H T M we nd that the interaction rate is larger
P l
H at temp eratures T M GeV If we b egin our consideration
w P l
w
of cosmology at the Planck era T GeV this tells us that gauge interactions
are initially out of equilibrium but lo cal thermal equilibrium is a go o d working
assumption b elow ab out GeV As a result inferences ab out physics near the
Planck scale which rely on equilibrium thermo dynamics should b e viewed with
skepticism
The situation changes when the temp erature falls b elow the gauge b oson masses
T M In this regime we have M T p m and the e cross section
e
b ecomes h v i T M G T where G is the Fermi constant G
F F
w F
GeV Comparison with the expansion rate now gives H G M T which
w P l
F
falls b elow unity at the neutrino freeze out temp erature T MeV For light
F
neutrinos we have T m so the neutrinos are ultrarelativistic with a density
F
comparable to photons when they freeze out such particles are called hot relics as
opp osed to cold relics which are non relativistic T m when they freeze out
F
Consider the implications of neutrino decoupling for the present neutrino density
We will make use of the fact that in thermal equilibrium the total entropy in a
comoving volume is conserved by the rst law of thermo dynamics S sa
constant where the entropy density is
p
s
T
So on after the neutrinos decouple the electrons and p ositrons annihilate e e
at T m MeV converting their entropy into photons Since this takes
e
place after freeze out the neutrinos do not partake of the entropy b o ost Before
annihilation the e e entropy is s T and the photon entropy
e
s T After annihilation the photon entropy has b een increased by
the factor
s
s
which means the photons are heated with resp ect to the neutrinos
T s
T s
Using the present CMBR temp erature from COBE this implies a present neutrino
temp erature of T K and a neutrino density for a single neutrino sp ecies
n T T n n cm
We can also use this to calculate the present energy density in radiation assuming
all three neutrino sp ecies are massless we have
g T
which leads directly to
It is p ossible that one or more of the neutrinos is massive the current exp eri
MeV If a keV m eV m mental neutrino mass limits b eing m
e
neutrino is massive and stable or long lived compared to the age of the universe
it could contribute more substantially to the density of the universe From
and one nds
m n m
h eV
h
c
From sec neutrinos could provide the dark matter and close the universe
if h which would require a mass m eV While larger than the mass
e
limit either the muon and tau neutrinos could have masses in this range moreover
even an electron neutrino at the upp er end of the exp erimental mass limit would
contribute substantially to the energy density
If neutrinos are the dark matter an interesting argument due to Tremaine and
Gunn shows that they cannot cluster in galaxies so they would not provide the
dark halos of the kind shown in Fig To see the p oint supp ose neutrinos were
clustered in the halo of a galaxy like the Milky Way Their number density would
R
b e n d pf p f hp i f m v where the halo velocity disp ersion is
h
v km sec By the Pauli exclusion principle the neutrino o ccupation number
h
v must satisfy f p which implies the constraint n m v m
h
h
The lo cal halo density in the mo del of Fig is of order GeV cm
h
eV for this to b e comp osed of neutrinos would require m v
h
h
Extending this argument to dwarf galaxies leads to the tighter constraint m
several hundred eV These numbers are signi cantly higher than the closure
density neutrino mass turning the argument around we conclude that in a neutrino
dominated universe galaxy halos must b e baryonic
Relic WIMPS and Cold Dark Matter
In the discussion ab ove we assumed that the neutrinos are light compared to
their freeze out temp erature m T MeV If neutrinos or some other
F
weakly interacting particle were much heavier the situation changes the particle
b ecomes non relativistic b efore freeze out so its abundance relative to radiation
is depleted by annihilations such particles are collectively known as cold relics or
WIMPs Examples of cold relics are sup ersymmetric neutralinos fermionic partners
of standard mo del b osons with masses in the range m GeV TeV If R
parity is conserved the lightest sup ersymmetric particle LSP is stable and is a
prime candidate for cold dark matter with
LS P
Accurate calculations of relic WIMP abundances generally require numerical
solution of the Boltzmann equation which describ es how the WIMP abundance
relative to its equilibrium value evolves over time in the expanding universe How
ever one can get an order of magnitude estimate from eqn which states that
the relic particle abundance relative to the entropy density is approximately the
equilibrium abundance at freeze out Using the annihilation rate at equilibrium
n h v i with n given by the Boltzmann form here assuming
eq eq
and the expansion rate in the freeze out condition b ecomes dropping factors
of
m T
F
mT e h v iM
F P l
H T
T
F
F
Thus the relic WIMP to entropy ratio is
m T
F
n mT e m
eq F
s T M mh v i T
T
P l F
F
F
For weakly interacting particles with masses in the GeV TeV range the ratio
m T at freeze out For concreteness consider a massive neutrino
F
other WIMPs are qualitatively similar but di er in quantitative detail for non
relativistic neutrinos of mass m the four momentum p m and the annihilation
cross section estimated at the b eginning of the last section b ecomes G m
F
Substitution into gives n s m GeV Using eqns and
this corresp onds to a neutrino mass density of order h m GeV
This tells us that a particle with mass in the GeV range and annihilation cross
section typical of the weak interactions will have a relic density comparable to the
closure density of the universe Note that such cold particles have no trouble tting
into galaxy halos unlike light neutrinos
The In ationary Universe
Motivation the Horizon and Flatness problems
The in ationary scenario originally arose out of the attempt to solve several
puzzles which arise when one extrap olates the standard cosmology back to the very
early universe The most imp ortant of these puzzles are the horizon and atness
problems which we consider in turn
Consider a light signal emitted from the origin at time t from eqn it
e
will reach a co ordinate distance r at a time t given by
h
Z Z
t r
h
dt dr
p
a t
t
k r
e
If the second integral converges as t then there exists a particle horizon two
e
fundamental observers separated by a co ordinate distance larger than r t
h e
will not yet b e in causal contact at time t The prop er distance from the origin to
the co ordinate r t at time t is called the particle horizon radius d
h e h
Z
t
dt
d t a t
h
a t
In the standard cosmology in the early radiation dominated phase a t t and
the particle horizon d ct Note that this is also comparable to the Hubble length
h
H t t More generally d is nite if the scale factor a t grows more slowly
h
than t at early times from the criterion for this is p a reasonable
condition for most uids Now recall that in the absence of early reionization the
CMBR photons last scattered at a time t h sec If photons
dec
emitted from two p oints separated by the particle horizon distance at the time of
last scattering would arrive with an angular separation of order degree at the
observer to day Therefore in the standard cosmology when we lo ok at a map of
the microwave sky we would exp ect it to lo ok very anisotropic on angular scales
larger than a degree b ecause we are seeing photons emitted from regions which were
not yet in causal contact with each other at the time of last scattering Yet the
microwave sky is remarkably isotropic over all scales larger than a few arcminutes
to of order one part in This is the horizon problem the CMBR is isotropic
over scales which were not yet in causal contact when the photons last scattered
This argument might app ear circular for we have done this calculation using the
homogeneous and isotropic FRW mo del so by assumption the CMBR must app ear
isotropic over all scales and one might conclude that there is no problem This
ob jection is spurious we could easily rep eat the derivation of the particle horizon
in a cosmological mo del which is nearly but not exactly homogeneous and isotropic
say with temp erature uctuations which are of order instead of and
end up with essentially the same answer That is we do not need to assume exact
homogeneity and isotropy for there to b e a horizon problem
The second puzzle of the standard cosmology is the atness problem In essence
this is the puzzle of why the spatial curvature term in do es not presently
dominate over the matter density term by a large margin if k and we
can rewrite as
a t H t
j t j
From a H a is a decreasing function of time if w Consequently
in the standard cosmology with non negative uid pressure the observational fact
that is within an order of magnitude of unity means that t must have b een
extremely close to unity at earlier times at the Planck time for example it requires
In the absence of other prejudices one might have guessed that j t j
P l
the universe would emerge from the quantum gravity Planck era with the matter
density and curvature terms having comparable amplitudes but the result ab ove
implies that they must have b een identical at that time to one part in Another
way to state this is if conditions had b een just slightly di erent at the Planck time
the universe would have gone into free expansion a t or have recollapsed long
b efore the present ep o ch In other words the natural timescale for the universe to
sec yet our universe b ecome curvature dominated is the Planck time t
P l
is still not strongly curvature dominated at t sec t
P l
It is imp ortant to emphasize that these two puzzles are not inconsistencies of
the standard cosmological mo del Rather they p oint to features of the observed
universe which the standard mo del do es not explain and which moreover app ear
highly unlikely when the standard cosmology is embedded in a somewhat larger
class of cosmological mo dels That is they are b oth problems of initial conditions
in the phase space of initial conditions the set of initial data that evolve to a
Universe like ours is very small A more physical way to say this is that homogeneity
and spatial atness are unstable prop erties of the mo del so the present nearly
homogeneous and spatially at state of the Universe app ears to b e very sensitive to
the initial conditions In ation was designed to reduce this sensitivity by widening
the class of initial conditions which evolve to a nearly homogeneous and spatially
at state within the observable universe This is the sense in which in ation is said
to solve the horizon and atness problems
In ation necessary ingredients
The discussion of the horizon and atness problems implicitly p ointed to their
solution what is wanted is an early era when the scale factor accelerates a for
this implies that a is increasing with time i e that a grows faster than t Because
of the rapid growth of the scale factor when it is accelerating this was dubb ed
in ation by Guth From acceleration requires that the energy density b e
dominated by some uid with equation of state p How long must such an
in ationary ep o ch last Supp ose the accelerated expansion b egins at an ep o ch t
i
and ends at t Then from we can compare the present and initial curvature
e
terms
a H a H j j
i i e e
N
e
e
j j a H a H
i e e
In the last expression I have assumed for the sake of argument that the universe is
radiation dominated for t t that the in ation ep o ch ends around the GUT scale
e
T GeV and that H H Here N is the number of e folds of growth of
e e i e
the scale factor during in ation
Z
t
e
N
e
a t a t exp H t dt e
e i
t
i
We would like the ratio on the left hand side of to b e of order unity so that
i
do es not have to b e extremely close to one The right hand side of then implies
that the scale factor must grow by at least N e folds during the in ationary
e
ep o ch If this is identical to the condition that our observable universe have
i
emerged from a region that was causally connected at the onset of in ation thus
solving the horizon problem as well
The pleasing feature of in ation is that it drives the spatial curvature to zero
via rapid expansion of the spatial hypersurfaces After in ation the curvature term
again grows in imp ortance but it has b een reset by in ation to such a tiny value
that it is still catching up to the matter term to day In fact if the universe enters an
extended in ationary p erio d it is quite plausible that it lasts considerably longer
than the required e folds of expansion in which case the spatial curvature term
would still b e exp onentially small to day In other words unless N is almost exactly
e
a su ciently long p erio d of in ation implies that the present universe should b e
observationally indistinguishable from b eing spatially at i e
There is a second necessary ingredient for in ation to b e a viable scenario for
the very early universe The accelerated expansion not only drives the spatial
curvature to zero but it also drives the density of matter and radiation to zero
At the end of N e folds of expansion the radiation temp erature would b e at
e
most T T e which is less than eV if T M This would invalidate the
e i i P l
argument in eqn where we assumed T M and among other things
e GU T
make it imp ossible to account for the light element abundances through primordial
nucleosynthesis Thus a p erio d of accelerated expansion by itself is not enough to
solve the horizon and atness problems at the end of in ation the energy density in
the uid that drives the in ationary expansion must b e converted into radiation to
rep opulate and reheat the universe A minimal condition for successful in ation is
that the reheat temp erature after in ation b e high enough for the baryon asymmetry
to b e generated if baryogenesis takes place at the electroweak scale this implies
T GeV
e
In ation Scalar Field Dynamics
In Guth prop osed that a scalar eld trapp ed in a false vacuum state
with non zero p otential energy could act as the driving force of in ation and thereby
solve the horizon and atness problems It is easy to see that this ts the bill for
accelerated expansion the energy momentum comp onents for a classical scalar eld
x t with p otential V are
r
p V
and
r
V
If the eld is trapp ed in a lo cal minimum of the p otential it will relax to a static
c
con guration with p V constant which satis es the criterion
c
for accelerated expansion In this case eqn b ecomes H GV
c
constant and the solution is the exp onential de Sitter expansion of In
other words a eld dominated by constant p otential energy acts as an e ective
cosmological constant
While a promising idea false vacuum in ation did not satisfactorily incorp orate
the second necessary ingredient of reheating once the eld is trapp ed for su ciently
long in the false vacuum state it must tunnel through the p otential barrier which
held it there in order to reach the true vacuum where V In this case
t
all the p otential energy of the false vacuum ends up in the walls of the nucleated
bubbles of true vacuum and is not converted into radiation by bubble collisions
e ciently enough The remedy was so on provided by Linde and by Albrecht
and Steinhardt instead of the eld b eing trapp ed in a metastable minimum of
the p otential it can b e classically evolving on a very at p otential if it is initially
displaced from the p otential minimum If the p otential is su ciently at the kinetic
and p otential energy terms in eqns will redshift away with the expansion
leaving the p otential term to dominate as b efore In ation ends when the eld
reaches a steep er part of the p otential the eld sp eeds up and eventually oscillates
ab out its p otential minimum If is coupled to lighter particles these coherent eld
oscillations lead to particle creation and thereby reheat the universe the scalar eld
energy is converted to radiation This new in ation mo del thus solves the graceful
exit problem of the original false vacuum scenario
In these early in ation mo dels it was assumed that the scalar eld driving
in ation the in aton was a Higgs eld asso ciated with the sp ontaneous breakdown
of the grand uni ed gauge symmetry The eld would b e initially displaced from
its global minimum by nite temp erature e ects It was so on realized however
that the in aton must b e extremely weakly self coupled in order for its quantum
uctuations to generate an acceptable amplitude of density p erturbations Since
Higgs elds are coupled to gauge elds however radiative corrections typically
generate large self couplings As a result the concept of in ation was divorced
from Higgs elds and gauge symmetry breaking and a pandora s b ox of particle
physics mo dels was op ened up Nevertheless the di erent mo dels of in ation share
many common dynamical features and it is those which I will fo cus on
To see how in ation works in a little more detail consider the dynamics of the
evolving scalar eld Within each Hubble volume i e ignoring spatial gradients
the evolution of the eld is describ ed by the classical equation of motion for a
homogeneous eld t
H V
where is the decay width of the in aton eld a phenomenological term which has
b een introduced to roughly mo del the back reaction e ects of particle creation and
reheating on the scalar eld The expansion rate H a a is determined by the
Friedmann equation
H V
M
P l
and it is also useful to have the second order Friedmann equation
h i
a
V
a M
P l
From the condition that the universe b e in ating a requires that the
kinetic energy by sub dominant V A su cient although not necessary
condition for this is that the eld b e slow ly rolling SR in its p otential In this
context slowly rolling is a technical term the eld is said to b e slowly rolling
when its motion is overdamped i e
H
so that the term can b e dropp ed in the equation of motion we assume
H during this phase From the de ning SR condition implies that
V thus if the SR condition is well satis ed the universe is in ating
Slow rollover is analytically convenient for it implies two consistency conditions
on the slop e and curvature of the p otential
p
V M
P l
j V j H and
V
where the slow roll equation of motion has b een used in the last equality eqn
without the second derivativer term Supp ose in ation b egins when the eld value
is t and the SR ep o ch ends when reaches a value at which one of
i i e
the inequalities is violated To solve the cosmological puzzles we demand
that the scale factor of the universe in ates by at least e foldings during the SR
regime
Z Z
t
e e
V
d N ln a a H dt
i e e i
V
M t
i i
P l
Once the p otential is chosen determines in terms of the parameters in V
e
The condition then constrains the initial value of the eld
i
Once grows b eyond the SR condition breaks down and the eld evolution is
e
more appropriately describ ed in terms of oscillations ab out the p otential minimum
These coherent oscillations excite the creation of particles to which is coupled
which in turn damps the oscillations and reheats the universe once the created
particles thermalize by scattering By energy conservation the phenomenological
damping term in the scalar equation of motion the scalar decay rate acts as a
source for radiation
H
r r
We can identify two qualitative regimes for reheating i if H reheating is
very e cient nearly all the scalar p otential energy is converted to particles leading
to a reheat temp erature given by T V ii on the other hand if H
e
RH
reheating is ine cient and the scalar eld oscillations partially redshift away b efore
converting to particles In this case the reheating temp erature is lower and is given
by
q
M g T
P l RH
where g is the number of relativistic degrees of freedom Clearly the reheating
e ciency is determined by the couplings of the in aton to other elds E cient
reheating say with a reheat temp erature high enough to allow baryogenesis at
the GUT scale T GeV requires relatively strong couplings In many
RH
mo dels however such couplings induce lo op corrections to the scalar self interaction
which would upset the requisite atness of its p otential As a result a high reheat
temp erature is not always p ossible
Quantum Fluctuations and Density Perturbations
So on after the arrival of false vacuum in ation it was appreciated that in ation
could in principle provide one of the holy grails of cosmology a causal mecha
nism for the origin of density uctuations that later grow to form large scale struc
ture The problem can b e seen by rst considering the standard cosmology without
in ation in this case the physical wavelength asso ciated with a density p erturba
tion mo de grows with the scale factor a t while the Hubble length scales
phy s
up more rapidly H t recall the current Hubble distance is d H h
h
Mp c Consequently at a redshift larger than z where z H p er
turbations with present scale were larger than the instantaneous Hubble radius
which sets the lengthscale for causal pro cesses For p erturbations on cosmologi
cally interesting scales to day say h Mp c this implies z
or T eV here for the sake of simplicity I have assumed matter dominated
expansion throughout the reader can insert the appropriate radiation dominated
corrections to make the numbers more rigorous In the standard cosmology how
ever curvature p erturbations cannot b e causally generated on scales larger than the
instantaneous Hubble radius b ecause of lo cal energy conservation Thus p ertur
bations on scales of galaxies and larger must have b een generated by some physical
eV after these scales crossed inside the Hubble ra pro cess acting at T
dius While some work has b een done on a late time phase transition at such a
scale such mo dels do require new physics at surprisingly low energy scales and
have not b een met with overwhelming enthusiasm Moreover the resulting CMBR
anisotropies in such scenarios may b e intolerably large If one do es not p ostulate a
late time origin for large scale p erturbations then they must simply b e p ostulated
as initial conditions b ecause they could not have had a causal origin at ep o chs
z z Note that this argument applies to curvature p erturbations an imp ortant
lo ophole is the early generation of large scale iso curvature p erturbations
An early ep o ch of in ation alters this situation during accelerated expansion
the Hubble radius H grows more slow ly than the scale factor in a de Sitter ep o ch
H is constant Thus a p erturbation of comoving wavelength could b e created
causally on a physical scale less than H during in ation expand outside the
Hubble radius at some time t during in ation and then eventually re enter
A
the Hubble radius in the recent radiation or matter dominated era at a time t
B
This hypothetical p ossibility was turned into a physical plausibility when it was
realized that quantum uctuations of the slowly rolling eld in new in ation could
provide a mechanism for generating density p erturbations In a nutshell one
treats the spatial uctuations of the eld ab out its homogeneous classical exp ec
tation value as an approximately massless quantum eld in de Sitter space in its
vacuum state such a eld has a sp ectrum of zero p oint uctuations with an rms
amplitude H These spatial variations in corresp ond to uctuations in
the in aton energy density which are converted into adiabatic density uctuations
in all sp ecies during reheating The resulting density p erturbation amplitude when
a comoving wavelength re enters the Hubble radius is given approximately by
H
t t
A B
It is straightforward to show that scales corresp onding to the range from the galaxy
scale to the present Hubble radius h Mp c crossed outside the Hubble
radius in the interval e folds b efore the end of in ation During slow rollover
the in aton is e ectively at its terminal velocity which changes slowly over time
since the p erio d of ten e folds of the scale factor is brief the right hand side of
changes very little over this interval As a result the density p erturbation
amplitude at Hubble radius crossing t is nearly constant
B
Now the density p erturbation amplitude at Hubble radius crossing is essentially
the gravitational p otential p erturbation on that scale G M G
H where we have used the Friedmann equation to relate G
phy s
H Thus in ation predicts that the gravitational p otential uctuations are nearly
indep endent of scale constant Such a scale invariant sp ectrum of p otential
uctuations was rst prop osed on di erent grounds by Harrison Zel dovich and
Peebles and Yu ten years b efore in ation Note that the gravitational p otential
also sets the scale for the large angle CMBR anisotropy through the Sachs Wolfe
e ect photons climbing out of deep er p otential wells at the time of last scattering
su er a larger gravitational redshift resulting in the temp erature shift T T
1
this result holds for the Einstein de Sitter mo del
H
Considerable excitement was generated when analysis of maps of the CMBR sky
from the rst year of COBE DMR data yielded a large angle anisotropy consistent
with the scale invariant sp ectrum
To get a feel for the result I sketch out the steps of the calculation here
for a particularly complete discussion see The idea is to treat small uctuations
semi classically by expanding the scalar eld and metric p erturbatively around their
homogeneous solutions e g
x t t x t
o
where t is the classical homogeneous solution to the scalar equation of motion in
o
the FRW background eqn and is a quantum eld op erator with equal time
commutation relation
x t x t t i x x a t
A similar expansion is p erformed for the metric It is convenient to consider the
Fourier transform
Z
h i
d k
ik x
a t e h c x t
k
k
where a is the annihilation op erator for mo de k t is the classical mo de function
k
k
and k ak is the xed comoving wavenumber of the mo de We will consider
phy s
the p ower sp ectrum P k of the eld the contribution p er logarithmic wavenumber
interval to the variance of the uctuation
Z
dk
h i P k
k
The p ower sp ectrum is prop ortional to the Fourier transform of the two point cor
relation function of the eld in the vacuum
Z
k k
ik x
d xe h j x j i j j P k
k
In general the mo de equation for can b e complicated since it involves coupled
k
scalar and gravitational degrees of freedom However at least in the synchronous
gauge g the term involving metric p erturbations can b e neglected yielding
k
H V
k k k o k
a
The slow rollover conditions imply that the p otential term in is also negligible
reducing the equation of motion to that of a free massless eld in de Sitter space
The normalized solution is
R
exp ik dt a
p
for k aH
k a
t
k
constant for k aH
The mo de crosses outside the Hubble radius during in ation when H
phy s
i e when k aH The solution states that the mo de oscillates inside the
Hubble radius and then freezes out when it crosses outside This is the mathematical
statement of the physical intuition that microphysics only op erates coherently on
scales less than Hubble radius For mo des inside the Hubble radius from
and the p ower sp ectrum can b e written P k H k H a When a
mo de crosses outside the Hubble radius the rms uctuation on that scale is given
by
H
P k aH
k aH
To nd the resulting density p erturbation amplitude one must solve the p er
turb ed Einstein equations due to the gauge freedom in the choice of time co ordi
nate that is in the way spacetime can b e sliced up into spacelike hypersurfaces one
chooses a gauge to simplify the problem at hand It is particularly convenient to
make use of a gauge invariant measure of the p erturbation amplitude for adiabatic
curvature p erturbations
p
o o
where is an analog of the Newtonian p otential that completely sp eci es the in
trinsic Ricci curvature of the p erturb ed surfaces The variable is useful b ecause
it generally satis es constant for p erturbations far outside the Hubble radius
k aH b oth during and after in ation Moreover the p otential is negligible
at Hubble radius crossing so de ning the p ower sp ectrum of by analogy with
we have
H V
k aH o
P k
t t
B A
where we have used and the slow roll equation of motion H V
This completes the sketch of the derivation of
Using the slow roll equation of motion the p erturbation amplitude can b e ex
pressed in terms of the height and slop e of the p otential
p
V
P
V M
P l
A useful way to characterize the scale dep endence of the p erturbations is to consider
the density p erturbation p ower sp ectrum P k j j where is the Fourier
k k
transform of x and we relate di erent conventions for what is called the
p ower sp ectrum by using the notation P k P The density sp ectrum can b e
n
s
written as a p ower law in wavenumber P k k where n corresp onds
s
to the scale invariant sp ectrum i e to P constant To see this recall that
H and that the contribution p er logarithmic wavenumber
interval to the rms uctuation on scale is k j j Small deviations
k
from scale invariance can b e expressed in terms of the index n
s
One can carry through an argument similar to that ab ove for gravitational wave
p erturbations in de Sitter space Expanding the metric ab out the de Sitter FRW
solution the tensor mo des h satisfy the massless scalar eld equation
with no p otential term De ning a canonically normalized scalar eld for each
p
p olarization state h G the tensor amplitude at Hubble crossing is
H V
p
h
k H a
M M
P l
P l
The induced Sachs Wolfe anisotropy on large scales is T T h
k H a
Chaotic In ation a worked example
The result for the p erturbation amplitude provides a severe constraint on
the form of the in aton p otential To get a feeling for the numbers consider one
of the simplest forms for the p otential a pure quartic V an example
of Linde s chaotic in ation In this case the eld rolls down to the origin from its
p
initial value the slow roll conditions are satis ed for M
e P l
and the number of in ation e folds is given by
i
N
e i e
e i
M M
P l P l
Thus the condition of su cient in ation N requires M and the
e i P l
density p erturbation amplitude on the scale of the present Hubble radius is
i
P
M
P l
The p erturbation sp ectrum is almost exactly scale invariant in this mo del n
s
b ecause changes very little in the interval b etween and e folds b efore the
end of in ation The COBE observation of CMBR anisotropy indicates P
which implies the in aton self coupling must satisfy
Naturalness and the Small Coupling Problem
The preceding example illustrates a general conclusion that applies to all in
ation mo dels the in aton eld must b e very weakly self coupled In di erent
mo dels this constraint app ears in di erent guises but it is alway present in one
form or another the p otential must contain a very small dimensionless number of
order
Attitudes concerning this problem vary widely among in ation theorists it is
often said that constitutes unacceptable ne tuning but that is a
misapplication of the ne tuning concept in this context To others it is not an issue
of great concern b ecause we know there exist other small numbers in physics such
as lepton and quark Yukawa couplings g and the ratio M M
Y w eak P l
Partly as a consequence of the latter view in recent years it has b ecome customary
to decouple the in aton completely from particle physics mo dels to sp ecify an
in aton sector with the requisite prop erties with little regard for its physical
origin
Nevertheless it is meaningful and imp ortant to ask whether such a small value
for is in principle unnatural Clearly the answer dep ends on the particle physics
mo del within which is embedded and on one s interpretation of naturalness A
small parameter is said to b e technically natural if it is protected against large
radiative corrections by a symmetry i e if setting increases the symmetry
of the system For example in this way low energy sup ersymmetry might protect
the small ratio M M by cancelling b oson and fermion lo ops However in
w eak P l
technically natural mo dels the small coupling while stable against radiative
corrections is itself unexplained and is generally p ostulated i e put in by hand
solely in order to generate successful in ation Technical naturalness is a useful
concept for low energy e ective Lagrangians like the electroweak theory and its
sup ersymmetric extensions but it p oints to a more fundamental level of theory for
its origin For example the underlying origin of the mass hierarchy M M in
w eak P l
sup ersymmetric theories is thought to b e asso ciated with hidden sector physics in
sup ergravity theories which start out with no small dimensionless couplings at the
Planck scale Since in ation takes place relatively close to the Planck scale it would
b e preferable to nd the in aton in such particle physics mo dels which are strongly
natural that is which have no small numbers in the fundamental Lagrangian
In a strongly natural gauge theory all small dimensionless parameters ulti
mately arise dynamically e g from renormalization group or instanton factors
like exp where is a gauge coupling In particular in an asymptotically
free theory the scale M at which a logarithmically running coupling constant
b ecomes unity is M M e where M is the fundamental mass scale in the
theory In some mo dels the in aton coupling arises from a ratio of mass scales
q
M M As a result in such mo dels is naturally exp onentially sup
q
pressed e
One example of a class of such mo dels is natural in ation in which the role
of the in aton is played by a pseudo Nambu Goldstone boson in particle physics
mo dels an example of this is the axion In its simplest incarnation consider a
global U symmetry sp ontaneously broken at a high energy scale f by a non
zero exp ectation value for a complex scalar eld the p otential for which takes the
familiar Mexican hat or wine b ottle form At scales much b elow f the only
relevant degree of freedom is the massless angular variable around the b ottom of
the p otential At an energy scale f the symmetry is explicitly broken the
Mexican hat is tilted generating a p otential for which is generally of the form
V cos N f
In axion mo dels characterizes the scale at which a running gauge coupling
q
b ecomes strong and is related to the fundamental scale f by f e For
f M GeV and M GeV this eld can drive in ation in
P l GU T
this case the e ective quartic coupling is f as required In the
standard mo del and in typical grand uni ed theories the only strong gauge coupling
is that of QCD with MeV However in sup ergravity and sup erstring
QC D
mo dels there is a hidden gauge sector which interacts only gravitationally with
ordinary quarks and leptons It has b een suggested that the hidden sector gauge
interactions b ecome strong at a scale comparable to that ab ove GeV in
order to break sup ersymmetry at the TeV scale in the observable sector
An interesting p ossibility in this case is that the p erturbation sp ectrum can
deviate signi cantly from scale invariance the sp ectral index is given by
M
P l
n
s
f
The constraint that the universe reheat after in ation to a temp erature T
RH
GeV leads to the constraint f M which implies n allowing a small
P l s
but signi cant break from scale invariance n Contrary to some erroneous
s
statements in the literature such a breaking from scale invariance do es not require
any ne tuning of the parameters in this mo del Arguments from the formation
of structure combined with the COBE observations at any rate indicate that the
sp ectral index n
s
A Brief Lo ok at Structure Formation
Let us nish with a cursory lo ok at how primordial p erturbations e g from
in ation end up lo oking to day As discussed ab ove the primordial sp ectrum from
n
s
in ation is generally a p ower law in wavenumber P k j t j Ak and
k i
the present sp ectrum is related to the primordial one through a transfer function
j t j T k j t j The transfer function T k enco des the scale dep endence
k k i
of the linear gravitational evolution of the p erturbation mo des it dep ends
on the nature of the dark matter hot or cold its density as well as and
DM B
H On scales which enter the Hubble radius after the universe b ecomes matter
dominated k k h Mp c all p erturbations undergo the same growth
eq
rate so the transfer function T k on smaller scales k k it b ends over
eq
to T k k as k re ecting the suppressed growth of uctuations which
cross inside the Hubble radius while the universe is still radiation dominated For
standard cold dark matter CDM we have h and assume negligible baryon
density leading to the characteristic scale k Mp c A
B col d eq
reasonable analytic t to the linear transfer function for CDM mo dels is given by
k k k
T k
h h h
in the approximation For hot dark matter on the other hand
B col d
in the absence of seed p erturbations such as cosmic strings the transfer function is
also exp onentially damp ed by relativistic neutrino free streaming for wavenumbers
larger than k m eV Mp c Finally the present density sp ectrum is
related to the galaxy p ower sp ectrum by a bias prescription for the simplest linear
bias mo del
k j t j T P k b P k b
k i g al
g al g al
As mentioned in Sec galaxy catalogs such as the CfA the IRAS Jy and
QDOT and the angular APM and EDSGC catalogs suggest that the sp ectrum
Figure The APM galaxy angular correlation function of Fig is shown with two
mo del predictions standard h CDM and a b est t low density CDM mo del
h from ref The upp er and lower bracketing curves for each mo del
indicate the exp ected spread due to cosmic variance
predicted in and with standard CDM h linear bias and n
s
may not have exactly the right shap e to match the observed galaxy sp ectrum hav
ing relatively to o little to o much p ower on large small scales Solutions to this
extra p ower problem can b e classi ed according to which element on the right hand
side of Eqn one tinkers with In all cases the aim is to atten the shap e of the
sp ectrum at intermediate wavenumbers k h Mp c by increasing the relative
p ower on these scales compared to smaller wavelengths For example as indi
cates one can abandon standard CDM and increase the characteristic wavelength
where the transfer function T k b ends down providing more relative large scale
p ower by reducing from unity in the context of in ation this requires either
a sp ecial choice of the number of e folds N or the introduction of a cosmological
e
constant to ensure An example of this is shown in Fig which again
shows the APM galaxy angular correlation function plotted against the prediction
for w b see b elow in linear theory for standard and low density CDM
g al
b oth with n Recall that lowering also increases the expansion age H t
s
Another recently p opular alternative is to employ a mixture of hot and cold
dark matter with from eqn this can b e achieved
hot col d
with a light neutrino with mass m eV Due to free streaming the neutrino
admixture partially suppresses p ower on small scales which is what is wanted A
third p ossibility is to retain standard CDM but consider primordial p erturbation
sp ectra with n from in ation but it app ears that such tilting of the sp ectrum
s
when normalized to COBE do es not by itself solve the p ower problem without
violating other constraints Alternatively one can admit a more complex scheme for
biasing in which the bias factor in is scale dep endent b b k and increases
g g
at large scales Unlike the other xes in this case the extra p ower on large scales
is an optical illusion a prop erty of the galaxy eld but not the underlying density
eld This di erence leads to qualitatively di erent b ehavior of the higher order
e g three p oint galaxy correlation functions an e ect which should b e testable
against observations
Since all in ation mo dels predict a deviation from scale invariance at some level
it is interesting to see what limits on the sp ectral index n follow from COBE in the
s
context of standard CDM For these mo dels the density p ower sp ectra form a two
parameter family characterized by the sp ectral index n and the normalization A
s
Instead of A it is common to normalize sp ectra by the rms linear mass uctuation
where in spheres of radius h Mp c h M M i
1
R h Mp c
Z
dk k P k W k R
R
and the window function
sin k R k R cos k R W k R
k R
lters out the contribution from small scales Redshift surveys of optically selected
galaxies in particular the CfA and APM surveys indicate that the variance in
galaxy counts on this scale is of order unity Thus in a linear bias mo del and
ignoring non linear gravitational e ects the bias factor for these galaxies would b e
b for other galaxy p opulations b may di er from unity
opt g al
From the Sachs Wolfe e ect for a given normalization and sp ectral index we
can calculate the exp ected large angle CMBR anisotropy The rms temp erature
o o
uctuation on the scale of observed by COBE
T
n
s
then yields a relation b etween them e Thus mo dels which
obtain more relative large scale p ower by tilting the sp ectrum to n reduce
s
the p ower on galaxy scales If the p ower on small scales is reduced to
the p oint that galaxy formation would o ccur to o recently in the context of these
mo dels this leads to the constraint n This b ound assumes that
s
gravitational waves make a negligible contribution to the CMBR signal While
this is a go o d approximation for mo dels such as natural in ation there are other
in ation p otentials for which the gravity wave contribution to the large angle CMBR
anisotropy b ecomes appreciable when n An example is p ower law in ation
s
pro duced by an exp onential p otential for in this case the relative contribution
of the tensor mo des compared to the scalar density p erturbation mo des to the
large angle anisotropy is R n However contrary to some statements in
s
the literature this relation is not generic to in ation but only holds approximately
for a restricted class of mo dels Clearly for mo dels in which it do es apply the e ect
is to lower the COBE constraint on the p erturbation amplitude for xed n For
s
mo dels with n this makes the ep o ch of structure formation more recent and
s
the corresp onding lower b ound on n even tighter As this sub ject has b ecome a
s
b o oming theoretical industry in the last year I refer the reader to recent issues of
Physical Review D for more details
Acknowledgements
I would like to thank the TASI organizers T Walker S Raby and K T
Mahanthappa for putting together an enjoyable school the students for asking go o d
questions without throwing anything and the Chicago Bulls for exp erimentally
verifying my prediction This work was supp orted in part by the DOE and by
NASA grant NAGW at Fermilab
App endix A Notation and Units
It is useful to b e able to translate back and forth b etween astronomical units
and high energy physics units The fundamental unit of distance in extragalactic
astronomy is the parsec p c p c light years cm The typical
galactic scale is measured in kiloparsec kp c for example the solar system is
approximately R kp c from the center of the Milky Way The nearest large
galaxy to our own is Andromeda at a distance of roughly kp c and the typical
distance b etween neighboring large galaxies outside clusters is a few Megaparsec
Mp c Absolute extragalactic distances are uncertain by a factor of order and
are usually given in terms of h Mp c where the Hubble parameter H h
km sec Mp c and observations indicate h see b elow Cosmological
distances are often estimated using redshifts via the Hubble law v cz H d and
so are sometimes expressed in terms of recession velocity km sec h Mp c
Astronomical masses and luminosities are measured in solar units M
gm L erg sec Large galaxies commonly have a luminosity of order
L and mass of order M In particle physics it is often
convenient to calculate in natural units in which h c k In this case
B
all dimensionful quantities can b e expressed in units of energy say in GeV for
example Length Energy At the end of a calculation to convert an expression
to physical units insert appropriate p owers of hc GeV cm and convert
lengths to times by dividing by c cm sec When gravity is involved use
the Planck mass energy M G GeV Finally when converting
P l
energy to temp erature E k T and vice versa use eV K To
B
go from particle physics to astronomical units it is sometimes handy to remember
that the sun contains ab out protons i e GeV M
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