Overview of Cosmology- High Level but mostly accesible

Overview of Cosmology- High Level but mostly accesible
Cosmology
arXiv:1504.03587v1 [astro-ph.CO] 14 Apr 2015
V.A. Rubakov
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia
Abstract
In these lectures we first concentrate on the cosmological problems which,
hopefully, have to do with the new physics to be probed at the LHC: the nature
and origin of dark matter and generation of matter-antimatter asymmetry. We
give several examples showing the LHC cosmological potential. These are
WIMPs as cold dark matter, gravitinos as warm dark matter, and electroweak
baryogenesis as a mechanism for generating matter-antimatter asymmetry. In
the remaining part of the lectures we discuss the cosmological perturbations
as a tool for studying the epoch preceeding the conventional hot stage of the
cosmological evolution.
1
Introduction
The more we learn about our Universe, the better we understand that it is full of mysteries. These fall
into three broad classes. One major mystery is dark energy, which deserves a separate class. We briefly
discuss dark energy in Section 5, although, honestly speaking, we do not have much to say about it. The
second class most likely has to do with the early hot epoch of the cosmological evolution, and the third
one with an even earlier stage which preceeded the hot epoch. Along with dark energy, we encounter
mysteries of the second class when studying the present composition of the Universe. It hosts matter
but not antimatter, and after 40 years after it was understood that this is a problem, we do not have an
established theory explaining this asymmetry. The Universe hosts dark matter, and we do not know what
it is made of. In this context, one of the key players is the Large Hadron Collider. Optimistically, the LHC
experiments may discover dark matter particles and their companions, and establish physics behind the
matter-antimatter asymmetry. Otherwise they will rule out some very plausible scenarios; this will also
have profound impact on our understanding of the early Universe. Let us mention also exotic hypotheses
on physics beyond the Standard Model, like TeV scale gravity; their support by the LHC will have an
effect on the early cosmology, which is hard to overestimate.
In the first part of these lectures we concentrate on a few examples showing the LHC cosmological
potential. Before coming to that, we briefly introduce the basic notions of cosmology that are useful for
our main discussion. We then turn to dark matter, and present the WIMP scenario for cold dark matter,
which is currently the most popular one — for good reason. We also consider light gravitino scenario
for warm dark matter. Both are probed by the LHC, as they require rather particular new physics in
the LHC energy range. We then discuss electroweak baryogenesis — a mechanism for the generation
of matter-antimatter asymmetry that may have operated at temperature of order 100 GeV in the early
Universe. This mechanism also needs new physics at energies 100 − 300 GeV, so it will be confirmed or
ruled out by the LHC.
The third class of mysteries is related to cosmological perturbations, i.e., inhomogeneities in energy density and associated gravitational potentials and, possibly, relic gravity waves. As we explain in
the second part of these lectures, the observed properties of density perturbations show that they were
generated at some epoch that preceeded the hot stage of the cosmological evolution. Obviously, the very
fact that we are confident about the existence of such an epoch is a fundamental result of theoretical and
observational cosmology. The most plausible hypothesis on that epoch is cosmological inflation, though
the observational support of this hypothesis is presently not particularly strong, and alternative scenarios
have not been ruled out. We will briefly discuss the potential of future cosmological observations in
discriminating between different options.
These lectures are meant to be self-contained, but we necessarily omit numerous details, while
trying to make clear the basic ideas and results. More complete accounts of particle physics aspects of
cosmology may be found in reviews [1]. Dark matter, including various hypotheses about its particles,
is reviewed in Ref. [2]. Electroweak baryogenesis is discussed in detail in reviews [3]. For reviews on
dark energy, see, e.g., Ref. [4]. The theory and observations of cosmological perturbations are presented
in Ref. [5].
Homogeneous isotropic Universe
2
2.1
Friedmann–Lemaître–Robertson–Walker metric
Two basic facts about our visible Universe are that it is homogeneous and isotropic at large spatial scales,
and that it expands.
There are three types of homogeneous and isotropic three-dimensional spaces. These are1 threesphere, flat (Euclidean) space and three-hyperboloid. Accordingly, one speaks about closed, flat and
open Universe; in the latter two cases the spatial size of the Universe is infinite, whereas in the former
the Universe is compact.
The homogeneity and isotropy of the Universe mean that its hypersurfaces of constant time are
either three-spheres or Euclidean spaces or three-hyperboloids. The distances between points may (and,
indeed, do) depend on time, i.e., the interval has the form
ds2 = dt2 − a2 (t)dx2 ,
(1)
where dx2 is the distance on unit three-sphere/Euclidean space/hyperboloid. The metric (1) is usually
called Friedmann–Lemaître–Robertson–Walker (FLRW) metric, and a(t) is called the scale factor. In
our Universe ȧ ≡ da
dt > 0, which means that the distance between points of fixed spatial coordinates x
grows, dl2 = a2 (t)dx2 . The space stretches out; the Universe expands.
The coordinates x are often called comoving coordinates. It is straightforward to check that x =
const is a time-like geodesic, so a galaxy put at a certain x at zero velocity will stay at the same x.
Furthermore, as the Universe expands, non-relativistic objects loose their velocities ẋ, i.e., they get
frozen in the comoving coordinate frame.
Observational data set strong constraints on the spatial curvature of the Universe. They tell that to
a very good approximation our Universe is spatially flat, i.e., our 3-dimensional space is Euclidean. In
what follows dx2 is simply the line interval in the Euclidean 3-dimensional space.
2.2
Redshift
Like the distances between free particles in the expanding Universe, the photon wavelength increases
too. We will always label the present values of time-dependent quantities by subscript 0: the present
wavelength of a photon is thus denoted by λ0 , the present time is t0 , the present value of the scale factor
is a0 ≡ a(t0 ), etc. If a photon was emitted at some moment of time t in the past, and its wavelength at
the moment of emission was λe , then we receive today a photon whose physical wavelength is longer,
λ0
a0
=
≡1+z .
λe
a(t)
Here we introduced the redshift z. The redshift of an object is directly measurable. λe is fixed by
physics of the source, say, it is the wavelength of a photon emitted by an excited hydrogen atom. So,
1
Strictly speaking, this statement is valid only locally: in principle, homogeneous and isotropic Universe may have complex
global properties. As an example, spatially flat Universe may have topology of three-torus. There is some discussion of
such a possibility in literature, and fairly strong limits have been obtained by the analyses of cosmic microwave background
radiation [6].
2
one identifies a series of emission or absorption lines, thus determining λe , and measures their actual
wavelengths λ0 . These spectroscopic measurements give accurate values of z even for distant sources.
On the other hand, the redshift is related to the time of emission, and hence to the distance to the source.
Let us consider a “nearby” source, for which z 1. This corresponds to relatively small (t0 − t).
Expanding a(t), one writes
a(t) = a0 − ȧ(t0 )(t0 − t) .
(2)
To the leading order in z, the difference between the present time and the emission time is equal to the
distance to the source r (the speed of light is set equal to 1). Let us define the Hubble parameter
H(t) =
ȧ(t)
a(t)
and denote its present value by H0 . Then Eq. (2) takes the form a(t) = a0 (1 − H0 r), and we get for the
redshift, again to the leading non-trivial order in z,
1+z =
1
= 1 + H0 r .
1 − H0 r
In this way we obtain the Hubble law,
z = H0 r ,
z1.
(3)
Traditionally, one tends to interpret the expansion of the Universe as runaway of galaxies from each
other, and redshift as the Doppler effect. Then at small z one writes z = v, where v is the radial velocity
of the source with respect to the Earth, so H0 is traditionally measured in units “velocity per distance”.
Observational data give [8]
H0 = [71.0 ± 2.5]
km/s
≈ (14 · 109 yrs)−1 ,
Mpc
(4)
where 1 Mpc = 3 · 106 light yrs = 3 · 1024 cm is the distance measure often used in cosmology.
Traditionally, the present value of the Hubble parameter is written as
H0 = h · 100
km
.
s · Mpc
(5)
Thus h ≈ 0.71. We will use this value in further estimates.
Let us point out that the interpretation of redshift in terms of the Doppler shift is actually not
adequate, especially for large enough z. In fact, there is no need in this interpretation at all: the “radial velocity” enters neither theory nor observations, so this notion may be safely dropped. Physically
meaningful quantity is redshift z itself.
A final comment is that H0−1 has dimension of time, or length, as indicated in Eq. (4). Clearly,
this quantity sets the cosmological scales of time and distance at the present epoch.
2.3
Hot Universe
Our Universe is filled with cosmic microwave background (CMB). Cosmic microwave background as
observed today consists of photons with excellent black-body spectrum of temperature
T0 = 2.726 ± 0.001 K .
(6)
The spectrum has been precisely measured by various instruments and does not show any deviation from
the Planck spectrum [7].
3
Thus, the present Universe is “warm”. Earlier Universe was warmer; it cooled down because of
the expansion. While the CMB photons freely propagate today, it was not so at early stage. When the
Universe was hot, the usual matter (electrons and protons with rather small admixture of light nuclei)
was in the plasma phase. At that time photons strongly interacted with electrons due to the Thomson
scattering and protons interacted with electrons via Coulomb force, so all these particles were in thermal
equilibrium. As the Universe cooled down, electrons “recombined” with protons into neutral hydrogen
atoms, and the Universe became transparent to photons. The temperature scale of recombination is, very
crudely speaking, determined by the ionisation energy of hydrogen, which is of order 10 eV. In fact,
recombination occured at lower temperature2 , Trec ≈ 3000 K. An important point is that the duration of
the period of recombination was considerably shorter than the Hubble time at that epoch; to a reasonable
approximation, recombination occured instantaneously.
The importance of the recombination epoch (more precisely, the epoch of photon last scattering;
we will use the term recombination for brevity) is that the CMB photons travel freely after it: the density
of hydrogen atoms was so small (about 250 cm−3 right after recombination) that the gas was transparent
to photons. So, CMB photons give the photographic picture of the Universe at recombination, i.e., at
redshift and age
zrec = 1090 ,
trec = 370 000 years .
(7)
It is worth noting that even though after recombination photons no longer were in thermal equilibrium with anything, the shape of the photon distribution function has not changed, except for overall
redshift. Indeed, the thermal distribution function for ultra-relativistic particles, the Planck distribution,
depends only on the ratio of frequency to temperature, fP lanck (p, T ) = f (ωp /T ), ωp = |p|. As the Unirec )
verse expands, the photon momentum gets redshifted, p(t) = p(trec ) · a(ta(t)
, the frequency is redshifted
in the same way, but the shape of the spectrum remains Planckian, with redshifted temperature. Hence,
the Planckian form of the observed spectrum is no surprise. Generaly speaking, this property does not
hold for massive particles.
At even earlier times, the temperature of the Universe was even higher. The earliest time at the hot
stage which has been observationally probed so far is the Big Bang Nucleosynthesis epoch; that epoch
began at temperature of order 1 MeV, when the lifetime of the Universe was about 1 s. At that time the
weak processes like
e− + p ←→ n + νe
switched off, and the comoving number density of neutrons freezed out. Somewhat later, these neutrons
combined with protons into light elements in thermonuclear reactions
2
3
p+n →
H +p →
2
He + H →
2
H +γ ,
3
He + γ ,
4
He + p ,
(8)
etc., up to 7 Li. Comparison of the Big Bang Nucleosynthesis theory with the observational determination
of the composition of cosmic medium gives us confidence that we understand the Universe at that epoch.
Notably, we are convinced that the cosmological expansion was governed by General Relativity.
2.4
Properties of components of cosmic medium
Let us come back to photons. Their effective temperature after recombination scales as
T (t) ∝ a−1 (t) .
2
(9)
The reason is that the number density of electrons and protons is small compared to the number density of photons. At
temperature above 3000 K, a hydrogen atom formed in an electron-proton encounter is quickly destroyed by absorbing a
photon from the high energy tail of the Planck distribution, and after that the electron/proton lives long time before it meets
proton/electron and forms a hydrogen atom again. In thermodynamical terms, at temperatures above 3000 K there is large
entropy per electron/proton, and recombination is not thermodynamically favourable because of entropy considerations.
4
This behaviour is characteristic of ultra-relativistic free species (at zero chemical potential). The same
formula is valid for ultra-relativistic particles (at zero chemical potential) which are in thermal equilibrium. Thermal equilibrium means adiabatic expansion; during adiabatic expansion, the temperature of
ultra-relativistic gas scales as the inverse size of the system, according to usual thermodynamics. The
energy density of ultra-relativistic gas scales as ρ ∝ T 4 , and pressure is p = ρ/3.
Both for free photons, and for photons in thermal equilibrium, the number density behaves as
follows,
nγ = const · T 3 ∝ a−3 ,
and the energy density is given by the Stefan–Boltzmann law,
ργ =
π2
· 2 · T 4 ∝ a−4 ,
30
(10)
where the factor 2 accounts for two photon polarizations. The present number density of relic photons is
nγ,0 = 410 cm−3 ,
(11)
and their energy density is
ργ,0 = 2.7 · 10−10
GeV
.
cm3
(12)
An important characteristic of the early Universe is the entropy density of cosmic plasma in thermal equilibrium. It is given by
2π 2
s=
g∗ T 3 ,
(13)
45
where g∗ is the number of degrees of freedom with m . T , that is, the degrees of freedom which are
relativistic at temperature T ; each spin state counts as an independent degree of freedom, and fermions
contribute to g∗ with a factor of 7/8. The point is that the entropy density scales exactly as a−3 ,
sa3 = const ,
(14)
while temperature scales approximately as a−1 . The property (14) is nothing but the reflection of the fact
that the Universe expands relatively slowly, and the evolution is adiabatic (barrig fairly exotic scenarios
with strong entropy generation at some early cosmological epoch). The temperature would scale as
a−1 if the number of relativistic degrees of freedom would be independent of time. This is not the
case, however. Indeed, the value of g∗ depends on temperature: at T ∼ 10 MeV relativistic species
are photons, neutrinos, electrons and positrons, while at T ∼ 1 GeV four flavors of quarks, gluons,
muons and τ -leptons are relativistic too. The number of degrees of freedom in the Standard Model at
T & 100 GeV is
g∗ (100 GeV) ≈ 100 .
The present value of the entropy density (taking into account neutrinos as if they were massless) is
s0 ≈ 3000 cm−3 .
(15)
The parameter g∗ determines not only the entropy density but also the energy density of the cosmic
plasma in thermal equilibrium. The Stefan–Boltzmann law gives
ρrad =
π2
g∗ T 4 ,
30
(16)
where subscript rad indicates that we are talking about the relativistic component (radiation in broad
sense).
5
Let us now turn to non-relativistic particles: baryons, massive neutrinos, dark matter particles,
etc. If they are not destroyed during the evolution of the Universe (that is, they are stable and do not
annihilate), their number density merely gets diluted,
n ∝ a−3 .
(17)
This means, in particular, that the baryon-to-photon ratio stays constant in time (we consider for definiteness the late Universe, T . 100 keV),
ηB ≡
nB
= const ≈ 6.1 · 10−10 .
nγ
(18)
The numerical value here is determined by two independent methods: one is Big Bang Nucleosynthesis
theory and measurements of the light element abundances, and another is the measurements of the CMB
temperature anisotropy. It is reassuring that these methods give consistent results (with comparable
precision).
The energy density of non-relativistic particles scales as
ρ(t) = m · n(t) ∝ a−3 (t) ,
(19)
in contrast to more rapid fall off (10) characteristic of relativistic species.
Finally, dark energy density does not decrease in time as fast as in Eqs. (10) or (19). In fact, to a
reasonable approximation dark energy density does not depend on time at all,
ρΛ = const .
(20)
Dark energy with exactly time-independent energy density is the same thing as the cosmological constant,
or Λ-term.
2.5
Composition of the present Universe
The basic equation governing the expansion rate of the Universe is the Friedmann equation, which we
write for the case of spatially flat Universe,
2
ȧ
8π
2
H ≡
=
Gρ ,
(21)
a
3
where dot denotes derivative with respect to time t, ρ is the total energy density in the Universe and G
19
is Newton’s gravity constant; in natural units G = MP−2
l where MP l = 1.2 · 10 GeV is the Planck
mass. The Friedmann equation is nothing but the (00)-component of the Einstein equations of General
Relativity,
1
R00 − g00 R = 8πT00 ,
2
specified to FLRW metric.
Let us introduce the parameter
ρc =
3
GeV
H 2 ≈ 5 · 10−6
.
8πG 0
cm3
(22)
According to Eq. (21), it is equal to the sum of all forms of energy density in the present Universe. As
a side remark, we note that the latter statement would not be true if our Universe were not spatially flat.
However, according to observations, spatial flatness holds to a very good precision, corresponding to less
than 1 per cent deviation of the total energy density from ρc [9].
As we now discuss, the cosmological data correspond to a very weird composition of the Universe.
6
Before proceeding, let us introduce a notion traditional in the analysis of the composition of the
present Universe. For every type of matter i with the present energy density ρi,0 , one defines the parameter
ρi,0
Ωi =
.
ρc
P
Then Eq. (21) tells that i Ωi = 1 where the sum runs over all forms of energy. Let us now discuss
contributions of different species to this sum.
We begin with baryons. The result (18) gives
ρB,0 = mB · nB,0 ≈ 2.4 · 10−7
GeV
.
cm3
(23)
Comparing this result with the value of ρc given in Eq. (22), one finds
ΩB = 0.045 .
Thus, baryons constiute rather small fraction of the present energy density in the Universe.
Photons contribute even smaller fraction, as is clear from Eq. (12), namely Ωγ ≈ 5 · 10−5 . From
electric neutrality, the number density of electrons is about the same as that of baryons, so electrons contribute negligible fraction to the total mass density. The remaining known stable particles are neutrinos.
Their number density is calculable in Hot Big Bang theory and these calculations are nicely confirmed
by Big Bang Nucleosynthesis. The present number density of each type of neutrinos is
nνα ,0 = 110 cm−3 ,
where να = νe , νµ , ντ (more appropriately, να are neutrino mass eigenstates). Direct limit on the mass of
electron neutrino, mνe < 2 eV, together with the observations of neutrino oscillations suggest that every
type of neutrino has mass smaller than 2 eV (neutrinos with masses above 0.05 eV must be degenerate,
according to neutrino oscillation data). The energy density of all types of neutrinos is thus smaller than
ρc :
X
1
GeV
ρν,total =
mνα nνα < 3 · 2 eV · 110
∼ 6 · 10−7
,
3
cm
cm3
α
which means that Ων,total < 0.12. This estimate does not make use of any cosmological data. In fact,
cosmological observations give stronger bound
Ων,total . 0.014 .
(24)
This bound is mostly due to the analysis of the structures at relatively small length scales, and has to
do with streaming of neutrinos from the gravitational potential wells at early times when neutrinos were
moving fast. In terms of the neutrino masses the bound (24) reads [10, 11]
X
mνα < 0.6 eV ,
so every neutrino must be lighter than 0.2 eV. It is worth noting that the atmospheric neutrino data, as
well as K2K, Minos and T2K experiments tell us that the mass of at least one neutrino must be larger than
about 0.05 eV. Comparing these numbers, one sees that it may be feasible to measure neutrino masses by
cosmological observations (!) in the future.
Coming back to our main topic here, we conclude that most of the energy density in the present
Universe is not in the form of known particles; most energy in the present Universe must be in “something
unknown”. Furthermore, this “something unknown” has two components: clustered (dark matter) and
unclustered (dark energy).
7
Clustered dark matter consists presumably of new stable massive particles. These make clumps
of energy (mass) which constitute most of the mass of galaxies and clusters of galaxies. There are
various ways of estimating the contribution of non-baryonic dark matter into the total energy density of
the Universe (see Ref. [2] for details):
– Composition of the Universe affects the angular anisotropy of cosmic microwave background.
Quite accurate measurements of the CMB anisotropy, available today, enable one to estimate the total
mass density of dark matter.
– Composition of the Universe, and especially the density of non-baryonic dark matter, is crucial
for structure formation of the Universe. Comparison of the results of numerical simulations of structure
formation with observational data gives reliable estimate of the mass density of non-baryonic clustered
dark matter.
The bottom line is that the non-relativistic component constitutes about 27 per cent of the total
present energy density, which means that non-baryonic dark matter has
ΩDM ≈ 0.22 ,
(25)
the rest is due to baryons.
There is direct evidence that dark matter exists in the largest gravitationally bound objects – clusters of galaxies. There are various methods to determine the gravitating mass of a cluster, and even mass
distribution in a cluster, which give consistent results. To name a few:
– One measures velocities of galaxies in galactic clusters, and makes use of the gravitational virial
theorem,
1
Kinetic energy of a galaxy = Potential energy .
2
In this way one obtains the gravitational potential, and thus the distribution of the total mass in a cluster.
– Another measurement of masses of clusters makes use of intracluster gas. Its temperature obtained from X-ray measurements is also related to the gravitational potential.
– Fairly accurate reconstruction of mass distributions in clusters is obtained from the observations
of gravitational lensing of background galaxies by clusters.
These methods enable one to measure mass-to-light ratio in clusters of galaxies. Assuming that
this ratio applies to all matter in the Universe3 , one arrives at the estimate for the mass density of clumped
matter in the present Universe. Remarkably, this estimate agrees with Eq. (25).
Finally, dark matter exists also in galaxies. Its distribution is measured by the observations of
rotation velocities of distant stars and gas clouds around a galaxy.
Thus, cosmologists are confident that much of the energy density in our Universe consists of new
stable particles. We will see that there is good chance for the LHC to produce these particles.
Unclustered dark energy. Non-baryonic clustered dark matter is not the whole story. Making
use of the above estimates, one obtains an estimate for the energy density of all particles, Ωγ + ΩB +
Ων,total + ΩDM ≈ 0.27. This implies that 73 per cent of the energy density is unclustered. This
component is called dark energy; it has the properties similar to those of vacuum. We will briefly discuss
dark energy in Section 5.
All this fits nicely all cosmological observations, but does not fit to the Standard Model of particle
physics. It is our hope that the LHC will shed light at least on some of the properties of the Universe.
3
This is a fairly strong assumption, since only about 10 per cent of galaxies are in clusters.
8
2.6
Regimes of cosmological expansion
The cosmological expansion at the present epoch is determined mostly by dark energy, since its contribution to the right hand side of the Friedmann equation (21) is the largest,
ΩΛ = 0.73 .
Non-relativistic matter (dark matter and baryons) is also non-negligible,
ΩM = 0.27 ,
(26)
while the energy density of relativistic matter (photons and neutrinos, if one of the neutrino species is
massless or very light) is negligible today. This was not always the case. Making use of Eq. (10) for
photons and relativistic neutrinos, Eq. (17) for non-relativistic matter, and assuming for definiteness that
dark energy density is constant in time, we can rewrite the Friedmann equation (21) in the following
form
8π
[ρΛ + ρM (t) + ρrad (t)]
3MP2 l
"
3
4 #
a
a
0
0
= H02 ΩΛ + ΩM
+ Ωrad
.
a(t)
a(t)
H 2 (t) =
(27)
It is appropriate for our purposes to treat neutrinos as massless particles; including their contribution to
Ωrad one has
Ωrad = 8.4 · 10−5 .
(28)
Equation (27) tells that at early times, when the scale factor a(t) was small, the expansion was dominated
by relativistic matter (“radiation”), later on there was long period of domination of the non-relativistic
matter, and in future the expansion will be dominated by dark energy,
. . . =⇒ Radiation domination =⇒ Matter domination =⇒ Λ–domination .
Dots here denote some cosmological epoch preceding the hot stage of evolution; as we discuss in Section 6, we are confident that such an epoch existed, but do not quite know what it was. Making use of
(26) and (28), it is straightforward to find the redshift at radiation–matter equality, when the first two
terms in (27) are equal,
a0
ΩM
≈ 3000 ,
1 + zeq =
=
a(teq )
Ωrad
and using the Friedmann equation one finds the age of the Universe at equality
teq ≈ 60 000 years .
Note that recombination occured at matter domination, but rather soon after equality, see (7).
It is useful for what follows to find the evolution of the scale factor at the radiation domination
epoch. At that time the energy density is given by Eq. (16), so that the Friedmann equation can be written
as follows
T2
H= ∗ ,
(29)
MP l
√
where MP∗ l = MP l /(1.66 g∗ ). Now, we neglect for simplicity the dependence of g∗ on temperature,
and hence on time, and recall that in this case the temperature scales as a−1 , see Eq. (14). Hence, we
obtain
ȧ
const
= 2 .
a
a
9
This gives the desired evolution law
a(t) = const ·
√
t.
(30)
The constant here does not have physical significance, as one can rescale the coordinates x at some fixed
moment of time, thus changing the normailzation of a.
There are several points to note regarding the result (30). First, the expansion decelerates:
ä < 0 .
This property holds also for the matter dominated epoch, but, as we see momentarily, it does not hold for
domination of the dark energy.
Second, time t = 0 is the Big Bang singularity (assuming erroneously that the Universe starts
being radiation dominated). The expansion rate
H(t) =
1
2t
diverges as t → 0, and so does the energy density ρ(t) ∝ H 2 (t) and temperature T ∝ ρ1/4 . Of course,
the classical General Relativity and usual notions of statistical mechanics (e.g., temperature itself) are
not applicable very near the singularity, but our result suggests that in the picture we discuss (hot epoch
right after the Big Bang), the Universe starts its classical evolution in a very hot and dense state, and
its expansion rate is very high in the beginning. It is customary to assume for illustrational purposes
that the relevant quantities in the beginning of the classical expansion take the Planck values, ρ ∼ MP4 l ,
H ∼ MP l , etc.
Third, at a given moment of time the size of a causally connected region is finite. Consider signals
emitted right after the Big Bang and travelling with the speed of light. These signals travel along the
light cone with ds = 0, and hence a(t)dx = dt. So, the coordinate distance that a signal travels from the
Big Bang to time t is
Z t
dt
x=
≡η.
(31)
0 a(t)
In the radiation dominated Universe
η = const ·
√
t.
The physical distance from the emission point to the position of the signal is
Z t
dt
lH,t = a(t)x = a(t)
= 2t .
0 a(t)
As expected, this physical distance is finite, and it gives the size of a causally connected region at time
t. It is called the horizon size (more precisely, the size of particle horizon). A related property is that
an observer at time t can see only the part of the Universe whose current physical size is lH,t . Both at
radiation and matter domination one has, modulo numerical constant of order 1,
lH,t ∼ H −1 (t) .
To give an idea of numbers, the horizon size at the present epoch is
lH,t0 ≈ 15 Gpc ' 4.5 · 1028 cm .
One property of the Universe that starts its expansion from rafiation domination is puzzling. Using
Eq. (31) one sees that the size of the observable Universe increases in time. For example, the coordinate
size of the present horizon is about 50 times larger that the coordinate size of the horizon at recombination. Hence, when performing CMB observations we see 502 regions on the sphere of last scattering
10
Fig. 1: Causal structure of space-time in hot Big Bang theory
which were causally disconnected at the recombination epoch, see Fig. 1. Yet they look exactly the same!
Clearly, this is a problem for the hot Big Bang theory, which is called horizon problem. We will see in
Section 6 that this problem has a somewhat different side, which unambiguously shows that the hot Big
Bang theory is not the whole story: the hot epoch was preceeded by some other, very different epoch of
the cosmological evolution.
To end up this section, let us note that the properties of the Universe dominated by dark energy
are quite different. Assuming for definiteness that ρΛ is independent of time, we immediately find the
solution to the Friedmann equation for the Λ-dominated Universe:
a(t) = const · eHΛ t ,
where HΛ =
(32)
q
8πρΛ /3MP2 l . The cosmological expansion accelerates,
ä > 0 .
The dark energy was introduced precisely for explaining the accelerated expansion of the Universe at the
present epoch.
3
Dark matter
Dark matter is characterized by the mass-to-entropy ratio,
ρ
DM
s
0
=
ΩDM ρc
0.22 · 5 · 10−6 GeV · cm−3
≈
= 4 · 10−10 GeV .
s0
3000 cm−3
(33)
This ratio is constant in time since the freeze out of dark matter density: both number density of dark
matter particles nDM (and hence their mass density ρDM = mDM nDM ) and entropy density dilute
exactly as a−3 .
Dark matter is crucial for our existence, for the following reason. Density perturbations in baryonelectron-photon plasma before recombination do not grow because of high pressure, which is mostly
due to photons; instead, perturbations are sound waves propagating in plasma with time-independent
11
amplitudes. Hence, in a Universe without dark matter, density perturbations in baryonic component
would start to grow only after baryons decouple from photons, i.e., after recombination. The mechanism
of the growth is pretty simple: an overdense region gravitationally attracts surrounding matter; this matter
falls into the overdense region, and the density contrast increases. In the expanding matter dominated
Universe this gravitational instability results in the density contrast growing like (δρ/ρ)(t) ∝ a(t).
Hence, in a Universe without dark matter, the growth factor for baryon density perturbations would be at
most4
a(t0 )
Trec
≈ 103 .
(34)
= 1 + zrec =
a(trec )
T0
The initial amplitude of density perturbations is very well known from the CMB anisotropy measurements, (δρ/ρ)i = 5 · 10−5 . Hence, a Universe without dark matter would still be pretty homogeneous:
the density contrast would be in the range of a few per cent. No structure would have been formed, no
galaxies, no life. No structure would be formed in future either, as the accelerated expansion due to dark
energy will soon terminate the growth of perturbations.
Since dark matter particles decoupled from plasma much earlier than baryons, perturbations in
dark matter started to grow much earlier. The corresponding growth factor is larger than Eq. (34), so that
the dark matter density contrast at galactic and sub-galactic scales becomes of order one, perturbations
enter non-linear regime and form dense dark matter clumps at z = 5 − 10. Baryons fall into potential wells formed by dark matter, so dark matter and baryon perturbations develop together soon after
recombination. Galaxies get formed in the regions where dark matter was overdense originally. The
development of perturbations in our Universe is shown in Fig. 2. For this picture to hold, dark matter
particles must be non-relativistic early enough, as relativistic particles fly through gravitational wells
instead of being trapped there. This means, in particular, that neutrinos cannot constitute a considerable
part of dark matter, hence the bound (24).
δDM
δB
δγ
Φ
teq
trec
tΛ
t
Fig. 2: A sketch of the time dependence, in the linear regime, of density contrasts of dark matter, baryons and photons, δDM ≡ δρDM /ρDM , δB and δγ , respectively, as well as the Newtonian potential Φ. teq and tΛ correspond
to the transitions from radiation domination to matter domination, and from decelerated expansion to accelerated
expansion, trec refers to the recombination epoch.
4
Because of the presence of dark energy, the growth factor is even somewhat smaller.
12
Depending on the mass of the dark matter particles and mechanism of their production in the
early Universe, dark matter may be cold (CDM) and warm (WDM). Roughly speaking, CDM consists
of heavy particles, while the masses of WDM particles are smaller,
CDM :
mDM & 100 keV ,
(35a)
WDM :
mDM = 3 − 30 keV .
(35b)
This assumes that the dark matter particles were in thermal (kinteic) equilibrium at some early times, or,
more generally, that their kinetic energy was comparable to temperature. This need not be the case for
very weakly interacting particles; a well known example is axions which are cold dark matter candidates
despite their very small mass. Likewise, very weakly interacting warm dark matter particles may be
much heavier than Eq. (35b) suggests.
We will discuss warm dark matter option later on, and now we move on to CDM.
3.1
WIMPS: best guess for cold dark matter
There is a simple mechanism of the dark matter generation in the early Universe. It applies to cold dark
matter. Because of its simplicity and robustness, it is considered by many as a very likely one, and the
corresponding dark matter candidates — weakly interacting massive particles, WIMPs — as the best
candidates. Let us describe this mechanism in some detail.
Let us assume that there exists a heavy stable neutral particle Y , and that Y -particles can only be
destroyed or created via their pair-annihilation or creation, with annihilation products being the particles
of the Standard Model. We will see that the overall cosmological behaviour of Y -particles is as follows.
At high temperatures, T mY , the Y -particles are in thermal equilibrium with the rest of cosmic
plasma; there are lots of Y -particles in the plasma, which are continuously created and annihilate. As
the temperature drops below mY , the equilibrium number density decreases. At some “freeze-out”
temperature Tf the number density becomes so small, that Y -particles can no longer meet each other
during the Hubble time, and their annihilation terminates. After that the number density of survived Y ’s
decreases like a−3 , and these relic particles contribute to the mass density in the present Universe. Our
purpose is to estimate the range of properties of Y -particles, in which they serve as dark matter.
Assuming thermal equilibrium, elementary considerations of mean free path of a particle in gas
give for the lifetime of a non-relativistic Y -particle in cosmic plasma, τann ,
σann · v · τann · nY ∼ 1 ,
where v is the velocity of Y -particle, σann is the annihilation cross section at velocity v and nY is the
equilibrium number density given by the Boltzmann law at zero chemical potential,
nY = gY ·
mY T
2π
3/2
e−
mY
T
,
where gY is the number of spin states of Y -particle. Note that we consider non-relativistic regime,
mY T . Let us introduce the notation
σann v = σ0
(in fact, the left hand side is to be understood as thermal average). If the annihilation occurs in s-wave,
then σ0 is a constant independent of temperature, for p-wave it is somewhat suppressed at T mY .
−1 with the
One should compare the lifetime with the Hubble time, or annihilation rate Γann ≡ τann
expansion rate H. At T ∼ mY , the equilibrium density is of order nY ∼ T 3 , and Γann H for not
too small σ0 . This means that annihilation (and, by reciprocity, creation) of Y -pairs is indeed rapid, and
Y -particles are indeed in thermal equilibrium with the plasma. At very low temperature, on the other
(eq)
hand, the equilibrium number density nY is exponentially small, and the equilibrium rate is small,
13
(eq)
Γann H. At low temperatures we cannot, of course, make use of the equilibrium formulas: Y particles no longer annihilate (and, by reciprocity, are no longer created), there is no thermal equilibrium
with respect to creation–annihilation processes, and the number density nY gets diluted only because of
the cosmological expansion.
The freeze-out temperature Tf is determined by the relation
−1
τann
≡ Γann ∼ H ,
where we can still use the equilibrium formulas, as Y -particles are in thermal equilibrium (with respect
to annihilation and creation) just before freeze-out. Making use of the relation (29) between the Hubble
parameter and temperature at radiation domination, we obtain
σ0 (Tf ) · nY (Tf ) ∼
or
σ0 (Tf ) · gY ·
mY Tf
2π
3/2
e
Tf2
,
MP∗ l
−
mY
Tf
∼
(36)
Tf2
MP∗ l
.
The latter equation gives the freeze-out temperature, which, up to loglog corrections, is
Tf ≈
mY
ln(MP∗ l mY σ0 )
(37)
(the possible depndence of σ0 on temperature is irrelevant in the right hand side: we are doing the
calculation in the leading-log approximation anyway). Note that this temperature is somewhat lower than
mY , if the relevant microscopic mass scale is much below MP l . This means that Y -particles freeze out
when they are indeed non-relativistic, hence the term “cold dark matter”. The fact that the annihilation
and creation of Y -particles terminate at relatively low temperature has to do with rather slow expansion
of the Universe, which should be compensated for by the smallness of the number density nY .
At the freeze-out temperature, we make use of Eq. (36) and obtain
nY (Tf ) =
Tf2
MP∗ l σ0 (Tf )
.
Note that this density is inversely proportional to the annihilation cross section (modulo logarithm). The
reason is that for higher annihilation cross section, the creation–annihilation processes are longer in
equilibrium, and less Y -particles survive.
Up to a numerical factor of order 1, the number-to-entropy ratio at freeze-out is
nY
1
'
.
∗
s
g∗ (Tf )MP l Tf σ0 (Tf )
(38)
This ratio stays constant until the present time, so the present number density of Y -particles is nY,0 =
s0 · (nY /s)f reeze−out , and the mass-to-entropy ratio is
ln(MP∗ l mY σ0 )
ln(MP∗ l mY σ0 )
ρY,0
mY nY,0
p
=
'
'
,
s0
s0
g∗ (Tf )MP∗ l σ0 (Tf )
g∗ (Tf )MP l σ0 (Tf )
where we made use of Eq. (37). This formula is remarkable. The mass density depends mostly on one
parameter, the annihilation cross section σ0 . The dependence on the mass of Y -particle is through the
logarithm and through g∗ (Tf ); it is very mild. The value of the logarithm here is between 30 and 40,
depending on parameters (this means, in particular, that freeze-out occurs when the temperature drops
30 to 40 times below the mass of Y -particle). Plugging in other numerical values (g∗ (Tf ) ∼ 100,
14
MP∗ l ∼ 1018 GeV), as well as numerical factor omitted in Eq. (38), and comparing with Eq. (33) we
obtain the estimate
σ0 (Tf ) ≡ hσvi(Tf ) = (1 ÷ 2) · 10−36 cm2 .
(39)
This is a weak scale cross section, which tells us that the relevant energy scale is TeV. We note in passing
that the estimate (39) is quite precise and robust.
If the annihilation occurs in s-wave, the annihilation cross section may be parametrized as σ0 =
α2 /M 2 where α is some coupling constant, and M is a mass scale (which may be higher than mY ). This
parametrization is suggested by the picture of Y pair-annihilation via the exchange by another particle
of mass M . With α ∼ 10−2 , the estimate for the mass scale is roughly M ∼ 1 TeV. Thus, with
very mild assumptions, we find that the non-baryonic dark matter may naturally originate from the TeVscale physics. In fact, what we have found can be understood as an approximate equality between the
cosmological parameter, mass-to-entropy ratio of dark matter, and the particle physics parameters,
1
TeV 2
mass-to-entropy '
.
MP l αW
Both are of order 10−10 GeV, and it is very tempting to think that this is not a mere coincidence. If it is
not, the dark matter particle should be found at the LHC.
Of course, the most prominent candidate for WIMP is neutralino of the supersymmetric extensions of the Standard Model. The situation with neutralino is somewhat tense, however. The point is
that the pair-annihilation of neutralinos often occurs in p-wave, rather than s-wave. This gives the suppression factor in σ0 ≡ hσann vi, proportional to v 2 ∼ Tf /mY ∼ 1/30. Hence, neutralinos tend to be
overproduced in most of the parameter space of MSSM and other models. Yet neutralino remains a good
candidate, especially at high tan β.
3.2
Warm dark matter: light gravitinos
The cold dark matter scenario successfully describes the bulk of the cosmological data. Yet, there are
clouds above it. First, according to numerical simulations, CDM scenario tends to overproduce small
objects — dwarf galaxies: it predicts hundreds of satellite dwarf galaxies in the vicinity of a large galaxy
like Milky Way, whereas only dozens of satellites have been observed so far. Second, again according
to simulations, CDM tends to produce too high densities in galactic centers (cusps in density profiles);
this feature is not confirmed by observations either. There is no strong discrepancy yet, but one may be
motivated to analyse a possibility that dark matter is not that cold.
An alternative to CDM is warm dark matter whose particles decouple being relativistic. Let us
assume for definiteness that they are in kinetic equilibrium with cosmic plasma when their number density freezes out (thermal relic). After kinetic equilibrium breaks down, and WDM particles decouple
completely, their spatial momenta decrease as a−1 , i.e., the momenta are of order T all the time after
decoupling. WDM particles become non-relativistic at T ∼ m, where m is their mass. Only after
that the WDM perturbations start to grow5 : as we mentioned above, relativistic particles escape from
gravitational potentials, so the gravitational potentials get smeared out instead of getting deeper. Before
becoming non-relativistic, WDM particles travel the distance of the order of the horizon size; the WDM
perturbations therefore are suppressed at those scales. The horizon size at the time tnr when T ∼ m is
of order
M∗
M∗
l(tnr ) ' H −1 (T ∼ m) = P2 l ∼ P2 l .
T
m
Due to the expansion of the Universe, the corresponding length at present is
l0 = l(tnr )
5
a0
T
MP l
∼ l(tnr )
∼
,
a(tnr )
T0
mT0
The situation in fact is somewhat more complicated, but this simplified picture will be sufficient for our estimates.
15
(40)
where we neglected (rather weak) dependence on g∗ . Hence, in WDM scenario, structures of sizes
smaller than l0 are less abundant as compared to CDM. Let us point out that l0 refers to the size of the
perturbation as if it were in the linear regime; in other words, this is the size of the region from which
matter collapses into a compact object.
The present size of a dwarf galaxy is a few kpc, and the density is about 106 of the average density
in the Universe. Hence, the size l0 for these objects is of order 100 kpc ' 3 · 1023 cm. Requiring that
perturbations of this size, but not much larger, are suppressed, we obtain from Eq. (40) the estimate (35b)
for the mass of WDM particles.
Among WDM candidates, light gravitino is probably the best motivated. The gravitino mass is of
order
m3/2 '
F
,
MP l
√
where F is the supersymmetry breaking scale.
√ Hence, gravitino masses are in the right ballpark for
rather low supersymmetry breaking scales, F ∼ 106 − 107 GeV. This is the case, e.g., in gauge
mediation scenario. With so low mass, gravitino is the lightest supersymmetric particle (LSP), so it is
stable in many supersymmetric extensions of the Standard Model. From this viewpoint gravitinos can
indeed serve as dark matter particles. For what follows, important parameters are the widths of decays
of other superpartners into gravitino and the Standard Model particles. These are of order
ΓS̃ '
MS̃5
F2
'
MS̃5
m23/2 MP2 l
,
(41)
where MS̃ is the mass of the superpartner.
One mechanism of the gravitino production in the early Universe is decays of other superpartners.
Gravitino interacts with everything else so weakly, that once produced, it moves freely, without interacting with cosmic plasma. At production, gravitinos are relativistic, hence they are indeed warm dark
matter candidates. Let us assume that production in decays is the dominant mechanism and consider
under what circumstances the present mass density of gravitinos coincides with that of dark matter.
The rate of gravitino production in decays of superpartners of the type S̃ in the early Universe is
d(n3/2 /s)
n
= S̃ ΓS̃ ,
dt
s
where n3/2 and nS̃ are number densities of gravitinos and superpartners, respectively, and s is the entropy
density. For superpartners in thermal equilibrium, one has nS̃ /s = const ∼ g∗−1 for T & MS̃ , and
nS̃ /s ∝ exp(−MS̃ /T ) at T MS̃ . Hence, the production is most efficient at T ∼ MS̃ , when the
number density of superpartners is still large, while the Universe expands most slowly. The density of
gravitinos produced in decays of S̃’s is thus given by
n3/2
'
s
d(n3/2 /s)
· H −1
dt
T ∼MS̃
'
M5
ΓS̃ −1
M∗
1
H (T ∼ MS̃ ) '
· 2 S̃ 2 · P2l .
g∗
g∗ m3/2 MP l M
S̃
This gives the mass-to-entropy ratio today:
MS̃3
m3/2 n3/2 X
'
,
3/2
s
g∗ MP l m3/2
(42)
S̃
where the sum runs over all superpartner species which have ever been relativistic in thermal equilibrium.
The correct value (33) is obtained for gravitino masses in the range (35b) at
MS̃ = 100 − 300 GeV .
16
(43)
Thus, the scenario with gravitino as warm dark matter particle requires light superpartners, which are to
be discovered at the LHC.
A few comments are in order. First, decays of superpartners is not the only mechanism of gravitino
production: gravitinos may also be produced in scattering of superpartners. To avoid overproduction of
gravitinos in the latter processes, one has to assume that the maximum temperature in the Universe
(reached after post-inflationary reheating stage) is quite low, Tmax ∼ 1 − 10 TeV. This is not a particularly plausible assumption, but it is consistent with everything else in cosmology and can indeed
be realized in some models of inflation. Second, existing constraints on masses of strongly interacting
superpartners (gluinos and squarks) suggest that their masses exceed Eq. (43). Hence, these particles
should not contribute to the sum in Eq. (42), otherwise WDM gravitinos would be overproduced. This
is possible, if masses of squarks and gluinos are larger than Tmax , so that they were never abundant in
the early Universe. Third, gravitino produced in decays of superpartners is not a thermal relic, as it was
never in thermal equilibrium with the rest of cosmic plasma. Nevertheless, since gravitinos are produced
at T ∼ MS̃ and at that time have energy E ∼ MS̃ ∼ T , our estimate (40) does apply. Finally, the
decay into gravitino and the Standard Model particles is the only decay channel for the next-to-lightest
superpartner (NLSP). Hence, the estimate for the total width of NLSP is given by Eq. (41), so that
cτN LSP = a few · mm − a few · 100 m
for m2/3 = 3−30 keV and MN LSP = 100−300 GeV. Thus, NLSP should either be visible in a detector,
or fly it through.
Needless to say, the warm gravitino scenario is a lot more contrived than the WIMP option. It is
reassuring, however, that it can be ruled out or confirmed at the LHC.
3.3
Discussion
If dark matter particles are indeed WIMPs, and the relevant energy scale is of order 1 TeV, then the Hot
Big Bang theory will be probed experimentally up to temperature of (a few) · (10 − 100) GeV and down
to age 10−9 − 10−11 s in relatively near future (compare to 1 MeV and 1 s accessible today through Big
Bang Nucleosynthesis). With microscopic physics to be known from collider experiments, the WIMP
density will be reliably calculated and checked against the data from observational cosmology. Thus,
WIMP scenario offers a window to a very early stage of the evolution of the Universe.
If dark matter particles are gravitinos, the prospect of probing quantitatively so early stage of the
cosmological evolution is not so bright: it would be very hard, if at all possible, to get an experimental
handle on the gravitino mass; furthermore, the present gravitino mass density depends on an unknown
reheat temperature Tmax . On the other hand, if this scenario is realized in Nature, then the whole picture
of the early Universe will be quite different from our best guess on the early cosmology. Indeed, gravitino
scenario requires low reheat temperature, which in turn calls for rather exotic mechanism of inflation.
The mechanisms discussed here are by no means the only ones capable of producing dark matter,
and WIMPs and gravitinos are by no means the only candidates for dark matter particles. Other dark
matter candidates include axions, sterile neutrinos, Q-balls, very heavy relics produced towards the end
of inflation, etc. Hence, even though there are grounds to hope that the dark matter problem will be
solved by the LHC, there is no guarantee at all.
4
Baryon asymmetry of the Universe
In the present Universe, there are baryons and almost no antibaryons. The number density of baryons
today is characterized by the ratio ηB , see Eq. (18). In the early Universe, the appropriate quantity is
∆B =
nB − nB̄
,
s
17
where nB̄ is the number density of antibaryons, and s is the entropy density. If the baryon number
is conserved, and the Universe expands adiabatically, ∆B is constant, and its value, up to a numerical
factor, is equal to η (cf. Eqs. (11) and (15)). More precisely,
∆B ≈ 0.8 · 10−10 .
At early times, at temperatures well above 100 MeV, cosmic plasma contained many quark-antiquark
pairs, whose number density was of the order of the entropy density,
nq + nq̄ ∼ s ,
while the baryon number density was related to densities of quarks and antiquarks as follows (baryon
number of a quark equals 1/3),
1
nB = (nq − nq̄ ) .
3
Hence, in terms of quantities characterizing the very early epoch, the baryon asymmetry may be expressed as
nq − nq̄
.
∆B ∼
nq + nq̄
We see that there was one extra quark per about 10 billion quark-antiquark pairs! It is this tiny excess
that is responsible for the entire baryonic matter in the present Universe: as the Universe expanded and
cooled down, antiquarks annihilated with quarks, and only the excessive quarks remained and formed
baryons.
There is no logical contradiction to suppose that the tiny excess of quarks over antiquarks was built
in as an initial condition. This is not at all satisfactory for a physicist, however. Furthermore, inflationary
scenario does not provide such an initial condition for the hot Big Bang epoch; rather, inflation theory
predicts that the Universe was baryon-symmetric just after inflation. Hence, one would like to explain
the baryon asymmetry dynamically.
The baryon asymmetry may be generated from initially symmetric state only if three necessary
conditions, dubbed Sakharov’s conditions, are satisfied. These are
(i) baryon number non-conservation;
(ii) C- and CP-violation;
(iii) deviation from thermal equilibrium.
All three conditions are easily understood. (i) If baryon number were conserved, and initial net
baryon number in the Universe was zero, the Universe today would still be symmetric. (ii) If C or CP
were conserved, then the rate of reactions with particles would be the same as the rate of reactions with
antiparticles, and no asymmetry would be generated. (iii) Thermal equilibrium means that the system
is stationary (no time dependence at all). Hence, if the initial baryon number is zero, it is zero forever,
unless there are deviations from thermal equilibrium.
There are two well understood mechanisms of baryon number non-conservation. One of them
emerges in Grand Unified Theories and is due to the exchange of super-massive particles. It is similar,
say, to the mechanism of charm non-conservation in weak interactions, which occurs via the exchange of
heavy W -bosons. The scale of these new, baryon number violating interactions is the Grand Unification
scale, presumably of order MGU T ' 1016 GeV. It is rather unlikely that the baryon asymmetry was
generated due to this mechanism: the relevant temperature would be of order MGU T , while so high
reheat temperature after inflation is difficult to obtain.
Another mechanism is non-perturbative [12] and is related to the triangle anomaly in the baryonic
current (a keyword here is “sphaleron” [13, 14]). It exists already in the Standard Model, and, possibly
with slight modifications, operates in all its extensions. The two main features of this mechanism, as
18
applied to the early Universe, is that it is effective over a wide range of temperatures, 100 GeV < T <
1011 GeV, and that it conserves (B − L).
Let us pause here to discuss the physics behind electroweak baryon and lepton number nonconservation in little more detail, though still at a qualitative level. A detailed analysis can be found
in the book [15] and in references therein.
The first object to consider is the baryonic current,
Bµ =
1 X µ
·
q̄i γ qi ,
3
i
where the sum runs over quark flavors. Naively, the baryonic current is conserved, but at the quantum
level its divergence is non-zero, the effect called triangle anomaly (similar effect goes under the name of
axial anomaly in the context of QED and QCD),
∂µ B µ =
1
g 2 µνλρ a a
Fµν Fλρ ,
· 3colors · 3generations ·
3
32π 2
a and g are the field strength of the SU (2)
where Fµν
W gauge field and the SU (2)W gauge coupling,
respectively. Likewise, each leptonic current (n = e, µ, τ ) is anomalous,
∂µ Lµn =
g2
a
a
.
· µνλρ Fµν
Fλρ
32π 2
a (x, t) ∝ g −1 which have
A non-trivial fact is that there exist large field fluctuations, Fµν
Q≡
Z
d3 xdt
g2
a
a
· µνλρ Fµν
Fλρ
6= 0 .
32π 2
Furthermore, for any such fluctuation the value of Q is integer. Suppose now that a fluctuation with
non-vanishing Q has occured. Then the baryon numbers in the end and beginning of the process are
different,
Z
d3 xdt ∂µ B µ = 3Q .
Bf in − Bin =
(44)
Likewise
Ln, f in − Ln, in = Q .
(45)
This explains the selection rule mentioned above: B is violated, (B − L) is not.
At zero temprature, the large field fluctuations that induce baryon and lepton number violation
are vacuum fluctuations, called instantons, which to a certain extent are similar to virtual fields that
emerge and disappear in vacuum of quantum field theory at the perturbative level. The peculiarity is that
instantons are large field fluctuations. The latter property results in the exponential suppression of the
probability of their emergence, and hence the rate of baryon number violating processes, by a factor
e
− 16π2
g
2
∼ 10−165 .
On the other hand, at high temperatures there are large thermal fluctuations (“sphalerons”) whose rate
is not necessarily small. And, indeed, B-violation in the early Universe is rapid as compared to the
cosmological expansion at sufficiently high temperatures, provided that
hφiT < T ,
where hφiT is the Higgs expectation value at temperature T .
19
(46)
One may wonder how baryon number may be not conserved even though there are no baryon
number violating terms in the Lagrangian of the Standard Model. To understand what is going on, let
us consider a massless left handed fermion field in the background of the SU (2) gauge field A(x, t),
which depends on space-time coordinates in a non-trivial way. As a technicality, we set the temporal
component of the gauge field equal to zero, A0 = 0, by the choice of gauge. One way to understand the
behavior of the fermion field in the gauge field background is to study the system of eigenvalues of the
Dirac Hamiltonian {ω(t)}. The Hamiltonian is defined in the standard way
HDirac (t) = iαi (∂i − igAi (x, t))
1 − γ5
,
2
where αi = γ 0 γ i , so that the Dirac equation has the Schrödinger form,
i
∂ψ
= HDirac ψ .
∂t
We are going to discuss the eigenvalues ωn (t) of the operator HDirac (t), treating t as a parameter. These
eigenvalues are found from
HDirac (t)ψn = ωn (t)ψn .
At A = 0 the system of levels is shown schematically in Fig. 3. Importantly, there are both positiveand negative-energy levels. According to Dirac, the lowest energy state (Dirac vacuum) has all negative
energy levels occupied, and all positive energy levels empty. Occupied positive energy levels (three of
them in Fig. 3) correspond to real fermions, while empty negative energy levels describe antifermions
(one in Fig. 3). Fermion-antifermion annihilation in this picture is a jump of a fermion from a positive
energy level to an unoccupied negative energy level.
As a side remark, this original Dirac picture is, in fact, equivalent to the more conventional (by
now) procedure of the quantization of fermion field, which does not make use of the notion of negative
energy levels. The discussion that follows can be translated into the conventional language; however, the
original Dirac picture turnd out to be a lot more transparent in our context. This is a nice example of the
complementarity of various approaches in quantum field theory.
Let us proceed with the discussion of the fermion energy levels in gauge field backgrounds. In
weak background fields, the energy levels depend on time (move), but nothing dramatic happens. For
adiabatically varying background fields, the fermions merely sit on their levels, while fast changing
fields generically give rise to jumps from, say, negative- to positive-energy levels, that is, creation of
fermion-antifermion pairs. Needless to say, fermion number (Nf − Nf¯) is conserved.
The situation is entirely different for the background fields with non-zero Q. The levels of lefthanded fermions move as shown in the left panel of Fig. 4. Some levels necessarily cross zero, and the
net number of levels crossing zero from below equals Q. This means that the number of left-handed
fermions is not conserved: for adiabatically varying gauge field A(x, t) the motion of levels shown in
the left panel of Fig. 4 corresponds to the case in which the initial state of the fermionic system is vacuum
(no fermions at all) whereas the final state contains Q real fermions (two in the particular case shown).
If the evolution of the gauge field is not adiabatic, the result for the fermion number non-conservation is
the same: there may be jumps from negative energy levels to positive energy levels, or vice versa. These
correspond to creation or annihilation of fermion-antifermion pairs, but the net change of the fermion
number (number of fermions minus number of antifermions) remains equal to Q. Importantly, the initial
and final field configurations of the gauge field may be trivial, A = 0 (up to gauge transformation), so
that fermion number non-conservation may occur due to a fluctuation that begins and ends in the gauge
field vacuum6 . This is precisely an instanton-like vacuum fluctuation. At finite temperatures, processes
of this type occur due to thermal fluctuations, sphalerons.
6
A subtlety here is that in four-dimensional gauge theories, this is impossible for Abelian gauge fields, so fermion number
non-conservation is inherent in non-Abelian gauge theories only.
20
ω
Fig. 3: Fermion energy levels at zero background gauge field.
ω
ω
t
t
Fig. 4: Motion of fermion levels in background gauge fields with non-vanishing Q (shown is the case Q = 2).
Left panel: left-handed fermions. Right panel: right-handed fermions.
If the same gauge field interacts also with right-handed fermions, the motion of the levels of the
latter is opposite to that of left-handed fermions. This is shown in the right panel of Fig. 4. The change in
the number of right-handed fermions is equal to (−Q). So, if the gauge interaction is vector-like, the total
fermion number Nlef t + Nright is conserved, while chirality Nlef t − Nright is violated even for massless
fermions. This explains why there is no baryon number violation in QCD. On the other hand, nonperturbative violation of chirality in QCD in the limit of massless quarks has non-trivial consequences,
which are indeed confirmed by phenomenology. In this sense anomalous non-conservation of fermion
quantum numbers is an experimentally established fact.
In electroweak theory, right-handed fermions do not interact with SU (2)W gauge field, while
left-handed fermions do. Therefore, fermion number is not conserved. Since fermions of each SU (2)W doublet interact with the SU (2)W gauge bosons (essentially W and Z) in one and the same way, they
are equally created in a process involving a gauge field fluctuation with non-zero Q. This again leads to
the relations (44) and (45), i.e., to the selection rules ∆B = ∆L, ∆Le = ∆Lµ = ∆Lτ .
It is tempting to use this mechanism of baryon number non-conservation for explaining the baryon
asymmetry of the Universe. There are two problems, however. One is that CP-violation in the Standard
Model is too weak: the CKM mechanism alone is insufficient to generate the realistic value of the baryon
21
asymmetry. Hence, one needs extra sources of CP-violation. Another problem has to do with departure
from thermal equilibrium that is necessary for the generation of the baryon asymmetry. At temperatures
well above 100 GeV electroweak symmetry is restored, the expectation value of the Higgs field φ is
zero7 , the relation (46) is valid, and the baryon number non-conservation is rapid as compared to the
cosmological expansion. At temperatures of order 100 GeV the relation (46) may be violated, but the
Universe expands very slowly: the cosmological time scale at these temperatures is
H −1 =
MP∗ l
∼ 10−10 s ,
T2
(47)
which is very large by the electroweak physics standards. The only way in which strong departure from
thermal equilibrium at these temperatures may occur is through the first order phase transition.
Vef f (φ)
Vef f (φ)
φ
φ
Fig. 5: Effective potential as function of φ at different temperatures. Left: first order phase transition. Right:
second order phase transition. Upper curves correspond to higher temperatures.
The property that at temperatures well above 100 GeV the expectation value of the Higgs field is
zero, while it is non-zero in vacuo, suggests that there may be a phase transition from the phase with
hφi = 0 to the phase with hφi =
6 0. The situation is pretty subtle here, as φ is not gauge invariant, and
hence cannot serve as an order parameter, so the notion of phases with hφi = 0 and hφi =
6 0 is vague.
In fact, neither electroweak theory nor most of its extensions have a gauge-invariant order parameter, so
there is no real distinction between these “phases”. This situation is similar to that in liquid-vapor system,
which does not have an order parameter and may or may not experience vapor-liquid phase transition as
temperature decreases, depending on other parameters characterizing this system, e.g., pressure. In the
Standard Model the role of such a parameter is played by the Higgs self-coupling λ or, in other words,
the Higgs boson mass.
Continuing to use somewhat sloppy terminology, we observe that the interesting case for us is
the first order phase transition. In this case the effective potential (free energy density as function of
7
There are subtleties at this point, see below.
22
φ) behaves as shown in the left panel of Fig. 5. At high temperatures, there exists one minimum of
Vef f at φ = 0, and the expectation value of the Higgs field is zero. As the temperature decreases,
another minimum appears at finite φ, and then becomes lower than the minimum at φ = 0. However,
the probability of the transition from the phase φ = 0 to the phase φ 6= 0 is very small for some time,
so the system gets overcooled. The transition occurs when the temperature becomes sufficiently low, as
shown schematically by an arrow in Fig. 5. This is to be contrasted to the case, e.g., of the second order
phase transition with the behavior of the effective potential shown in the right panel of Fig. 5. In the
latter case, the field slowly evolves, as the temperature decreases, from zero to non-zero vacuum value,
and the system remains very close to the thermal equilibrium at all times.
φ=0
φ 6= 0
φ=0
φ 6= 0
φ 6= 0
φ=0
Fig. 6: First order phase transition: boiling Universe.
The first order phase transition occurs via spontaneous creation of bubbles of the new phase inside
the old phase. These bubbles then grow, their walls eventually collide, and the new phase finally occupies
entire space. The Universe boils, as shown schematically in Fig. 6. In the cosmological context, this
process happens when the bubble nucleation rate per Hubble time per Hubble volume is of order 1,
Γnucl ∼ H −4 . The velocity of the bubble wall in the relativistic cosmic plasma is roughly of the order
of the speed of light (in fact, it is somewhat smaller, from 0.1 c to 0.01 c), simply because there are no
relevant dimensionless parameters characterizing the system. Hence, the bubbles grow large before their
walls collide: their size at collision is roughly of order of the Hubble size. While at nucleation the bubble
is microscopic — its size is dictated by the elecroweak scale and is roughly of order (100 GeV)−1 ∼
10−16 cm — its size at collision of walls is macroscopic, H −1 ∼ a few cm, as follows from Eq. (47).
Clearly, boiling is a highly inequilibrium process, and one may hope that the baryon asymmetry may be
generated at that time. And, indeed, there exist mechanisms of the generation of the baryon asymmetry,
which have to do with interactions of quarks and leptons with moving bubble walls. The value of the
resulting baryon asymmetry may well be of order 10−10 , as required by observations, provided that there
is enough CP-violation in the theory.
A necessary condition for the electroweak generation of the baryon asymmetry is that the inequality (46) must be violated just after the phase transition. Indeed, in the opposite case the electroweak
baryon number violating processes are fast after the transition, and the baryon asymmetry, generated
23
during the transition, is washed out afterwards. Hence, the phase transition must be of strong enough
first order. This is not the case in the Standard Model. To see why this is so, and to get an idea in which
extensions of the Standard Model the phase transition may be of strong enough first order, let us consider
the effective potential in some detail. At zero temperature, the Higgs potential has the standard fom,
V (φ) = −
Here
m2 2 λ 4
|φ| + |φ| .
2
4
1/2
|φ| ≡ φ† φ
(48)
is the length of the Higgs doublet φ, m2 = λv 2 and v = 247 GeV is the Higgs expectation value in
vacuo. The Higgs boson mass is related to the latter as follows,
√
mH = 2λv .
(49)
Now, to the leading order of perturbation theory, the finite temperature effects modify the effective potential into
α
β
λ
Vef f (φ, T ) = |φ|2 − T |φ|3 + |φ|4 ,
(50)
2
3
4
with α(T ) = −m2 + ĝ 2 T 2 , where ĝ 2 is a positive linear combination of squares of coupling constants
of all fields to the Higgs field (in the Standard Model, a linear combination of g 2 , g 0 2 and yi2 , where g
and g 0 are gauge couplings and yi are Yukawa couplings), while β is a positive linear combination of
cubes of coupling constants of all bosonic fields to the Higgs field. In the Standard Model, β is a linear
3 /v 3 and M 3 /v 3 ,
combination of g 3 and g 0 3 , i.e., a linear combination of MW
Z
β=
3 + M3
1 2MW
Z
.
2π
v3
(51)
The cubic term in Eq. (50) is rather peculiar: in view of Eq. (48) it is not analytic in the original Higgs
field φ. Yet this term is crucial for the first order phase transition: for β = 0 the phase transition would
be of the second order. The origin of the non-analytic cubic term can be traced back to the enhancement
of the Bose–Einstein thermal distribution at low momenta, p, m T ,
fBose (p) = √
e
1
p2 +m2
a
T
−1
T
'p
,
p2 + m2a
where ma ' ga |φ| is the mass of the boson a that is generated due to the non-vanishing Higgs field, and
ga is the coupling constant of the field a to the Higgs field. Clearly, at p ga |φ| the distribution function
is non-analytic in φ,
T
fBose (p) '
.
ga |φ|
It is this non-analyticity that gives rise to the non-analytic cubic term in the effective potential. Importantly, the Fermi–Dirac distribution,
fF ermi (p) = √
e
1
p2 +m2
a
T
,
+1
is analytic in m2a , and hence φ† φ, so fermions do not contribute to the cubic term.
With the cubic term in the effective potential, the phase transition is indeed of the first order:
at high temperatures the coefficient α is positive and large, and there is one minimum of the effective
24
potential at φ = 0, while for α small but still positive there are two minima. The phase transition ocurs
at α ≈ 0; at that moment
βT 3 λ 4
Vef f (φ, T ) ≈ −
|φ| + |φ| .
3
4
We find from this expression that immediately after the phase transition the minimum of Vef f is at
φ'
βT
.
λ
Hence, the necessary condition for successfull electroweak baryogenesis, φ > T , translates into
β>λ.
(52)
According to Eq. (49), λ is proportional to m2H , whereas in the Standard Model β is proportional to
3 + M 3 ). Therefore, the relation (52) holds for small Higgs boson masses only; in the Standard
(2MW
Z
Model one makes use of Eqs. (49) and (51) and finds that this happens for mH < 50 GeV, which is ruled
out8 .
This discussion indicates a possible way to make the electroweak phase transition strong. What
one needs is the existence of new bosonic fields that have large enough couplings to the Higgs field(s),
and hence provide large contriutions to β. To have an effect on the dynamics of the transition, the new
bosons must be present in the cosmic plasma at the transition temperature, T ∼ 100 GeV, so their masses
should not be too high, M . 300 GeV. In supersymmetric extensions of the Standard Model, the natural
candidate for long time has been stop (superpartner of top-quark) whose Yukawa coupling to the Higgs
field is the same as that of top, that is, large. The light stop scenario for electroweak baryogenesis would
indeed work, as has been shown by the detailed analysis in Ref. [16].
Yet another issue is CP-violation, which has to be strong enough for successfull electroweak baryogenesis. As the asymmetry is generated in the interactions of quarks and leptons (and their superpartners
in supersymmetric extensions) with the bubble walls, CP-violation must occur at the walls. Recall now
that the walls are made of the Higgs field(s). This points towards the necessity of CP-violation in the
Higgs sector, which may only be the case in a theory with more than one Higgs fields.
To summarize, electroweak baryogenesis requires a considerable extension of the Standard Model,
with masses of new particles in the range 100 − 300 GeV. Hence, this mechanism will definitely be ruled
out or confirmed by the LHC. We stress, however, that electroweak baryogenesis is not the only option at
all: an elegant and well motivated competitor is leptogenesis [17]; several other mechanisms have been
proposed that may be responsible for the baryon asymmetry of the Universe.
5
Dark energy
Dark energy, the famous “substance”, does not clump, unlike dark matter. It gives rise to the accelerated
expansion of the Universe. As we see from Eq. (32), the Universe with constant energy density should
expand exponentially; if the energy density is almost constant, the expansion is almost exponential. Let
us make use of the first law of thermodynamics, which for the adiabatic expansion reads
dE = −pdV ,
and apply it to comoving volume, E = ρV , V = a3 . We obtain for dark energy
dρΛ = −3
da
(ρΛ + pΛ ) ,
a
8
In fact, in the Standard Model with mH > 114 GeV, there is no phase transition at all; the electroweak transition is smooth
crossover instead. The latter fact is not visible from the expression (50), but that expression is the lowest order perturbative
result, while the perturbation theory is not applicable for describing the transition in the Standard Model with large mH .
25
or
da
dρΛ
= −3 (1 + w) ,
ρΛ
a
where we introduced the equation of state parameter w such that
pΛ = wρΛ .
Thus, (almost) time-independent dark energy density corresponds to w ≈ −1, i.e., effective pressure of
dark energy is negative. We emphasize that pressure is by definition a spatial component of the energymomentum tensor, which in the homogeneous and isotropic situation has the general form
Tµν = diag (ρ, p, p, p) .
Dark energy density does not depend on time at all, if pΛ = −ρΛ , i.e.,
Tµν = ρΛ ηµν ,
where ηµν is the Minkowski tensor. This is characteristic of vacuum, whose energy-momentum tensor
must be Lorentz-covariant. Observationally, w is close to −1 to reasonably good precision. The most accurate detrmination, which, however, does not include systematic errors in supernovae data and possible
time-dependence of w, is [9]
w = −0.98 ± 0.05 .
(53)
So, the dark energy density is almost time-independent, indeed.
The problem with dark energy is that its present value is extremely small by particle physics
standards,
ρDE ≈ 4 GeV/m3 = (2 × 10−3 eV)4 .
In fact, there are two hard problems. One is that particle physics scales are much larger than the scale
relevant to the dark energy density, so the dark energy density is zero to an excellent approximation.
Another is that it is non-zero nevertheless, and one has to understand its energy scale. To quantify the
first problem, we recall the known scales of particle physics and gravity,
ΛQCD ∼ 1 GeV ,
MW ∼ 100 GeV ,
Mpl ∼ 1019 GeV .
Strong interactions :
Electroweak :
Gravitational :
In principle, vacuum should contribute to ρΛ , and there is absolutely no reason for vacuum to be as light
as it is. The discrepancy here is huge, as one sees from the above numbers.
To elaborate on this point, let us note that the action of gravity plus, say, the Standard Model has
the general form
Z
√
−g d4 x ,
S = SEH + SSM − ρΛ,0
R
√
where SEH = −(16πGN )−1 R −g d4 x is the Einstein–Hilbert action of General Relativity, SSM
is the action of the Standard Model and ρΛ,0 is the bare cosmological constant. In order that the vacuum
energy density be almost zero, one needs fantastic cancellations between the contributions of the Sandard
Model fields into the vacuum energy density, on the one hand, and ρΛ,0 on the other. For example, we
know that QCD has a complicated vacuum structure, and one would expect that the energy density of
QCD combined with ρΛ,0 should be of order (1 GeV)4 . Nevertheless, it is not, so at least for QCD,
one needs a cancellation on the order of 10−44 . If one goes further and considers other interactions, the
numbers get even worse.
26
What are the hints from this “first” cosmological constant problem? There are several options,
though not many. One is that the Universe could have a very long prehistory. Extremely long. This
option has to do with relaxation mechanisms. Suppose that the original vacuum energy density is indeed
large, say, comparable to the particle physics scales. Then there must be a mechanism which can relax
this value down to an acceptably small number. It is easy to convince oneself that this relaxation could
not happen in the history of the Universe we know of. Instead, the Universe should have a very long
prehistory during which this relaxation process might occur. At that prehistoric time, the vacuum in the
Universe must have been exactly the same as our vacuum, so the Universe in its prehistory must have
been exactly like ours, or almost exactly like ours. Only in that case could a relaxation mechanism work.
There are concrete scenarios of this sort [18]. However, at the moment it seems that these scenarios are
hardly testable, since this is prehistory.
Another possible hint is towards anthropic selection. The argument that goes back to Weinberg and
Linde [19, 20] is that if the cosmological constant were larger, say, by a factor of 100, we simply would
not exist: the stars would not have formed because of the fast expansion of the Universe. So, the vacuum
energy density may be selected anthropically. The picture is that the Universe may be much, much
larger than what we can see, and different large regions of the Universe may have different properties. In
particular, vacuum energy density may be different in different regions. Now, we are somewhere in the
place where one can live. All the rest is empty of human beings, because there the parameters such as
vacum energy density are not suitable for their existence. This is disappointing for a theorist, as this point
of view allows for arbitrary tuning of fundamental parameters. It is hard to disprove this option, on the
other hand. We do exist, and this is an experimental fact. The anthropic viewpoint may, though hopefully
will not, get more support from the LHC, if no or insufficient new physics is found there. Indeed, another
candidate for an environmental quantity is the electroweak scale.
Let us recall in this regard the gauge hierarchy problem: the electroweak scale MW ∼ 100 GeV is
much lower than the natural scale in gravitational physics, the Planck mass, MP l ∼ 1019 GeV. The electroweak scale in the Standard Model is unprotected from large contributions due to high energy physics,
and in this sense it is very similar to the cosmological constant. There are various anthropic arguments
showing that the electroweak scale must be small. A simple example is that if one makes it larger without
touching other parameters, then quarks would be too heavy. Neutron would be the lightest baryon, and
proton would be unstable. There would be no stable hydrogen, and that is presumably inconsistent with
our existence. Hence, one of the “solutions” to the gauge hierarchy problem is anthropic.
An interesting part of the story is that unlike the cosmological constant, there are natural ways
to make the electroweak scale small and render it small in extensions of the Standard Model, like low
energy supersymmetry. All these extensions require new physics at TeV energies. So we are in a situation
where the experiment has to say its word. If it says that none of these extensions is there in Nature, then
we will have to take the anthropic viewpoint much more seriously than before.
Turning to the “second” cosmological constant problem, we note that the scale 10−3 eV may be
associated with some new light field(s), rather than with vacuum. This implies, in general, that ρΛ
depends on time, i.e., w 6= −1 and w may well depend on time itself. “Normal” field (called quitessence
in this context) has w > −1, but there are examples (rather contrived) of fields with w < −1 (called
phantom fields). Current data are compatible with time-independent w equal to −1, but their precision
is not particularly high. We conclude that future cosmological observations may shed new light on the
field content of fundamental theory.
6
Cosmological perturbations and the very early Universe
With Big Bang nucleosynthesis theory and observations, we are confident of the theory of the early
Universe at temperatures up to T ' 1 MeV, that corresponds to age of t ' 1 second. With the LHC, we
hope to be able to go up to temperatures T ∼ 100 GeV and age t ∼ 10−10 second. The question is: are
27
we going to have a handle on even earlier epoch?
The key issue in this regard is cosmological perturbations. These are inhomogeneities in the energy
density and assoiated gravitational potentials, in the first place. This type of inhomogeneities is called
scalar perturbations, as they are described by 3-scalars. There may exist perturbations of another type,
called tensor; these are primordial gravity waves. We will mostly concentrate on scalar perturbations,
since they are observed; tensor perturbations are important too, and we comment on them later on. It
is worth pointing out that perturbations of the present size below ten Megaparsec have large amplitudes
today and are non-linear, but in the past their amplitudes were small, and they can be described within the
linearized theory. Indeed, CMB temperature anisotropy tells us that the perturbations at recombination
epoch were roughly at the level
δρ
δ≡
= 10−4 − 10−5 .
ρ
Thus, the linearized theory works very well before recombination and somewhat later.
Properties of scalar perturbations are mesured in various ways. Perturbations of large spatial scales
leave their imprint in CMB temperature anisotropy and polarization, so we have very detailed knowledge
of them. Shorter wavelength perturbations are studied by analysing distributions of galaxies and quasars
at present and in relatively near past. There are several other methods, some of which can probe even
shorter wavelengths. As we discuss in more detail below, scalar perturbations in the linear regime are
actually Gaussian random field, and the first thing to measure is its power spectrum. Overall, independent
methods give consistent results, see Fig. 7.
Cosmic medium in our Universe has several components that interact only gravitationally: baryons,
photons, neutrinos, dark matter. Hence, there may be and, in fact, there are perturbations in each of these
components. As we pointed out in the beginning of Section 3, electromagnetic interactions between
baryons, electrons and photons were strong before recombination, so these species made single fluid,
and it is appropriate to talk about perturbations in this fluid. After recombination, baryons and photons
evolved independently.
The main point of this part of lectures is that by analysing the density perturbations, we have already learned a number of very important things. To appreciate what they are, it is instructive to consider
first the baryon-electron-photon fluid before recombination. Perturbations in this fluid are nothing but
sound waves; they obey a wave equation. So, let us turn to the wave equation in the expanding Universe.
6.1
Wave equation in expanding Universe. Subhorizon and superhorizon regimes.
The actual system of equations for density perturbations in the baryon-electron-photon fluid and associated gravitational potentials is fairly cumbersome. So, let us simplify things. Instead of writing and then
solving the equations for sound waves, let us consider a toy example, the case of massless scalar field.
The general properties of density perturbations are similar to this case, although there are a few places in
which they differ; we comment on the differences in due course.
The action for the massless scalar field is
Z
Z
1
1 2
1
4 √
µν
3
3
2
S=
d x −g g ∂µ φ∂ν φ =
d xdt a
φ̇ − 2 (∂i φ) ,
2
2
2a
where we specified to FLRW metric in the second expression. The field equation thus reads:
−
d 3
(a φ̇) + a∂i ∂i φ = 0,
dt
i.e.,
φ̈ + 3H φ̇ −
1 2
∂ φ=0,
a2 i
28
(54)
Fig. 7: Power spectrum of density perturbations, measured by various methods and translated to the present epoch
by using the linearized theory [21]. k is the present wavenumber, and h ≈ 0.7 is the dimensionless Hubble
parameter at the present epoch.
where H ≡ ȧ/a is again the Hubble parameter. This equation is linear in φ and homogeneous in space,
so it is natural to represent φ in terms of the Fourier harmonics,
Z
φ(x, t) =
eikx φk (t) d3 k.
Clearly, the value of k for a given Fourier mode is constant in time. However, k is not the physical
wavenumber (physical momentum), since x is not the physical distance. k is called conformal momentum, while physical momentum equals q ≡ 2π/λ = 2π/(a(t)∆x) = k/a(t). ∆x here is timeindependent comoving wavelength of perturbation, and λ is the physical wavelength; the latter grows
due to the expansion of the Universe. Accordingly, as the Universe expands, the physical momentum of
a given mode decreases (gets redshifted), q(t) ∝ a−1 (t). For a mode of given conformal momentum k,
Eq. (54) gives:
k2
φ̈ + 3H φ̇ + 2 φ = 0.
(55)
a
Besides the redshift of momentum, the cosmological expansion has the effect of inducing the second
term, “Hubble friction”.
Equation (55) has two time-dependent parameters of the same dimension: k/a and H. Let us
consider two limiting cases: k/a H and k/a H. In cosmological models with conventional
equation of state of the dominant component (e.g., matter-dominated or radiation-dominated Universe),
29
H −1 is of the order of the size of the cosmological horizon, see Section 2.6. So, the regime k/a H
is the regime in which the physical wavelength λ = 2πa/k is greater than the horizon size (this is
called superhorizon regime), while for k/a H the physical wavelength is smaller than the horizon
size (subhorizon regime). The time when the wavelength of the mode coincides with the horizon size is
called horizon crossing. In what follows we denote this time by the symbol ×. Both at radiation- √
and
matter-dominated epochs, the ratio
k/(aH)
grows.
Indeed,
in
the
radiation-dominated
epoch
a
∝
t,
√
while H ∝ t−1 , so k/(aH) ∝ t. This means that every mode was at some early time superhorizon,
and later on it becomes subhorizon, see Fig. 8. It is straightforward to see that for all cosmlogically
interesting wavelengths, horizon crossing occurs much later than 1 s after the Big Bang, i.e., at the time
we are confident about. So, there is no guesswork at this point.
H(t)
q1 (t) = k1 /a(t)
q2 (t)
t×
t
Fig. 8: Physical momenta (solid lines, k2 < k1 ) and Hubble parameter (dashed line) at radiation- and matterdominated epochs. t× is the horizon entry time.
Now we can address the question of the origin of density perturbations. By causality, any mechanism of their generation that operates at the radiation- and/or matter-dominated epoch, can only work
after the horizon entry time t× . Indeed, no physical process can create a perturbation whose wavelength
exceeds the size of an entire causally connected region. So, in that case the perturbation modes were
never superhorizon. On the other hand, if modes were ever superhorizon, they have to exist already in
the beginning of the hot epoch. Hence, in the latter situation one has to conclude that there existed another
epoch before the hot stage: that was the epoch of the generation of primordial density perturbations.
Observational data, notably (but not only) on CMB temperature anisotropy and polarization, disentangle these two possibilities. They unambiguously show that density perturbations were superhorizon
at radiation and matter domination!
To understand how this comes about, let us see what is special about a perturbation which was
superhorizon at the hot stage. For a superhorizon mode, we can neglect the term φ · k 2 /a2 in Eq. (54).
Then the field equation, e.g., in the radiation–dominated Universe (a ∝ t1/2 , H = 1/2t), becomes
φ̈ +
3
φ̇ = 0 .
2t
30
(56)
The general solution to this equation is
B
φ(t) = A + √ ,
t
(57)
where A and B are constants. This behavior is generic for all cosmological perturbations at the hot
stage: there is a constant
mode (A in our case) and a mode that decays in time. If we extrapolate the
√
decaying mode B/ t back in time, we get very strong (infinite in the limit t → 0) perturbation. For
density perturbations (and also tensor perturbations) this means that this mode corresponds to strongly
inhomogeneous early Universe. Therefore, the consistency of the cosmological model dictates that the
decaying mode has to be absent for actual perturbations. Hence, for given k, the solution is determined
by a single parameter, the initial amplitude A of the mode φk .
After entering the subhorizon regime, the modes oscillate — these are the analogs of conventional
sound waves. In the subhorizon regime one makes use of the WKB aproximation to solve the complete
equation
3
k2
φ̈ + φ + 2 φ = 0 .
(58)
2t
a (t)
The general solution in the WKB approximation reads
Z t
A0
k
0
φ(t) =
cos
dt + ψ0 ,
0
a(t)
0 a(t )
(59)
with the two constants being the amplitude A0 and the phase ψ0 . The amplitude A0 of these oscillations
is determined by the amplitude A of the superhorizon initial perturbation, while the phase ψ0 of these oscillations is uniquely determined by the condition of the absence of the decaying mode, B = 0. Imposing
this condition yields
Z t
k
a×
0
sin
dt ,
(60)
φ(t) = cA
0
a(t)
0 a(t )
where the constant c is of order 1 and can be evaluated by solving the complete equation (58). The
decreasing amplitude of oscillations φ(t) ∝ 1/a(t) and the particular phase ψ0 = −π/2 in Eq. (59) are
peculiar properties of the wave equation (54), as well as the radiation-dominated cosmological expansion.
However, the fact that the phase of oscillations is uniquely determined by the requirement of the absence
of the superhorizon decaying mode is generic.
The perturbations in the baryon–photon medium before recombination — sound waves — behave
in a rather similar way. Their evolution is as follows:

const,
outside horizon,

t
δργ
R
δγ ≡
=
(61)
vs a(tk 0 ) dt0 ,
inside horizon,
 const · cos
ργ
0
p
where vs ≡ dp/dρ is the sound speed. The baryon–photon
medium before recombination is almost
√
relativistic9 , since ρB < ργ . Therefore, vs ≈ 1/ 3. Let us reiterate that the phase of the oscillating
solution in (61) is uniquely defined.
6.2
Oscillations in CMB angular spectrum
CMB gives us the photographic picture of the Universe at recombination (photon last scattering), see
Fig. 9. Waves of different momenta k are at different phases at recombination. At that epoch, oscillations in time in Eq. (61) show up as oscillations in momentum. This in turn gives rise to the observed
oscillations in the CMB angular spectrum.
9
This does not contradict the statement that the Universe is in matter-dominated regime at recombination. The dominant
component at this stage is dark matter.
31
6.2
Oscillations in CMB angular spectrum
CMB gives us the photographic picture of the Universe at recombination (photon last scattering), see Fig. 9. Waves of different momenta k are at different
phases at recombination. At that epoch, oscillations in time in Eq. (61) show
up as oscillations in momentum. This in turn gives rise to the observed oscillations in the CMB angular spectrum.
Fig. 9: CMB sky as seen by WMAP experiment.
Fig. 9: CMB sky as seen by WMAP experiment.
9
This does not contradict the statement that the Universe is in matter-dominated regime at recombination. The dominant
component at this stage is dark matter.
In more detail, at the time of last scattering trec we have
Z trec
δργ
k
0
37
δγ ≡
vs 0 dt = A(k) · cos krs ,
= A(k) · cos
ργ
a(t )
0
(62)
where A(k) is linearly related to the initial amplitude of the superhorizon perturbation and is a nonoscillatory function of k, and
Z trec
dt0
vs 0
rs =
a(t )
0
is the comoving size of the sound horizon at recombination, while its physical size equals a(trec )rs . So,
we see that the density perturbation at recombination indeed oscillates as a function of wavenumber. The
period of this oscillation is determined by rs , which is a straighforwardly calculable quantity.
Omitting details, the fluctuation of the CMB temperature is partially due to the density perturbation
in the baryon-photon medium at recombination. The relevant place is the point where the photons last
scatter before coming to us. This means that the temperature fluctuation of photons coming from the
direction n in the sky is, to a reasonable accuracy,
δT (n) ∝ δγ (xn , ηrec ) + δTsmooth (n) ,
where Tsmooth (n) corresponds to the non-oscillatory part of the CMB angular spectrum, and
xn = −n(η0 − ηrec ) .
Here the variable η is defined in (31), and η0 is its present value, so that (η0 − ηrec ) is the coordinate
distance to the sphere of photon last scattering, and xn is the coordinate of the place where the photons
coming from the direction n scatter last time. Tsmooth (n) originates from the gravitational potential
generated by the dark matter perturbation; dark matter has zero pressure at all times, so there are no sound
waves in this component, and there are no oscillations at recombination as a function of momentum.
One expands the temperature variation on celestial sphere in spherical harmonics:
X
δT (n) =
alm Ylm (θ, φ).
lm
32
The multipole number l characterizes the temperature fluctuations at the angular scale ∆θ = π/l. The
sound waves of momentum k are seen roughly at an angle ∆θ = ∆x/(η0 − ηrec ), where ∆x = π/k is
coordinate half-wavelength. Hence, there is the correspondence
l ←→ k(η0 − ηrec ) .
Oscillations in momenta in (62) thus translate into oscillations in l, and these are indeed observed, see
Fig. 10.
Fig. 10: The angular spectrum of the CMB temperature anisotropy [22]. The quantity in vertical axis is Dl defined
in Eq. (65).
To understand what is shown in Fig. 10, we note that all observations today support the hypothesis
that alm are independent Gaussian random variables. Gaussianity means that
2
P (alm )dalm
a
1
− lm
e 2Cl dalm ,
=√
2πCl
(63)
where P (alm ) is the probability density for the random variable alm . For a hypothetical ensemble of
Universes like ours, the average values of products of the coefficients alm would obey
halm al0 m0 i = Cl δll0 δmm0 .
(64)
This gives the expression for the temperature fluctuation:
2
h[δT (n)] i =
X 2l + 1
l
4π
Cl ≈
Z
dl
Dl ,
l
where
l(l + 1)
Cl .
(65)
2π
It is the latter quantity that is usually shown in plots, in particular, in Fig. 10. Note the unconventional
scale on the horizontal axis, aimed at showing both small l region (large angular scales) and large l
region.
Dl =
The fact that the CMB angular spectrum has oscillatory behavior unambiguously tells us that
density perturbations were indeed superhorizon at hot cosmological stage. If these perturbations were
33
generated by some causal mechanism after horizon entry, there would be no reason for the phase ψ0 in
(59) (better to say, in the analog of (59) for density perturbations) to take a very definite value. Instead,
one would expect that this phase is a random function of k, so there would be no oscillations in l in the
CMB angular spectrum at all. This is indeed the case in concrete causal models aimed at generating
the density perturbations at the hot stage, which make use, e.g., of topological defects (strings, textures,
etc.), see Fig. 11.
6000
2
l(l+1)Cl / 2π [µK ]
5000
4000
WMAP
Abelian Higgs strings
Semilocal strings
Textures
3000
2000
1000
0
1
10
2
10
multipole moment: l
3
10
Fig. 11: The angular spectrum of the CMB temperature anisotropy in causal models that generate the density
perturbations at the hot stage (non-oscillatory lines) versus data (sketched by oscillatory line) [23].
Another point to note is that the CMB measurements show that at recombination, there were
density perturbations which were still superhorizon at that time. These correspond to low multipoles,
l . 50. Perturbations of these wavelengths cannot be produced at the hot stage before recombination,
and, indeed, causal mechanisms produce small power at low multipoles. This is also seen in Fig. 11.
6.3
Baryon acoustic oscillations
Another manifestation of the well defined phase of sound waves in baryon-photon medium before recombination is baryon acoustic oscillations. Right after recombination, baryons decouple from photons,
the sound speed in the baryon component becomes essentially zero, and the spatial distribution of the
baryon density freezes out. Since just before recombination baryons, together with photons, have energy
distribution (61) which is oscillatory function of k, there is oscillatory component in the Fourier spectrum
of the total matter distribution after recombination. This oscillatory component persists until today, and
shows up as oscillations in the matter power spectrum P (k). This is a small effect, since the dominant
component at late times is dark matter,
ρM (k) = ρDM (k) + ρB (k) ,
and only ρB oscillates as function of k
δρB (k) ≈ ρB δγ (k) = ρB · A(k) · cos krs
34
(66)
(as we already noticed, dark matter has zero pressure at all times, so there are no sound waves in this
component). Nevertheless, this effect has been observed in large galaxy surveys, see Fig. 12.
Fig. 12: Baryon acoustic oscillations in matter power spectrum detected in galaxy surveys [24].
There is a simple interpretation of the effect. As we discuss below, the overdensities in the baryonphoton medium and in the dark matter are at the same place before horizon entry (adiabatic mode). But
before recombination the sound speed in baryon-photon plasma is of the order of the speed of light, while
the sound speed in dark matter is basically zero. So, the overdensity in baryons generates an outgoing
density wave after horizon crossing. This wave propagates until recombination, and then freezes out. On
the other hand, the overdensity in the dark matter remains in its original place. The current distance from
the overdensity in dark matter to the front of the baryon density wave equals 150 Mpc. Hence, there is
an enhanced correlation between matter perturbations at this distance scale, which shows up as a feature
in the correlation function10 , see Fig. 13. In the Fourier space, this feature produces oscillations (66).
6.4
“Side” remarks
Before proceeding to further discussion of primordial perturbations, let us make a couple of miscellaneous remarks.
10
Notice that the separation at Fig. 13 is given in h−1 Mpc, where h = H0 /100 km/(s · Mpc) ≈ 0.7. Hence, 100 h−1 Mpc
roughly corresponds to 150 Mpc.
35
Fig. 13: Baryon acoustic oscillations in galaxy correlation function [25].
6.4.1
Cosmic variance.
We can measure only one Universe, and the best one can do is to define the angular spectrum Cl obtained
from the data as
l
X
1
|alm |2 .
Cl =
2l + 1
m=−l
This is not the same thing as Cl defined in (64), as the latter definition involves averaging over an
ensemble of Universes. For given l, there are (2l+1) independent
coefficients alm only, so there exists an
√
irreducible statistical uncertainty of order δCl /Cl ∼ 1/ 2l + 1, called cosmic variance. It is particularly
pronounced at small l and, indeed, it is much larger than the experimental errors in this part of the angular
spectrum (as an example, error bars in the left part of Fig. 10 are precisely due to the cosmic variance).
6.4.2
Measuring the cosmological parameters.
The angular spectrum of CMB temperature anisotropy and polarization, as well as other cosmological
data, encodes information on the cosmological parameters. As an example, the sound horizon at recombination is a good standard ruler back at that epoch. It is seen at an angle that depends on the geometry
of 3-dimensional space (an interval is seen at larger angle on a sphere than on a plane) and on the dark
energy density (since dark energy affects the distance to the sphere of photon last scattering). This is
shown in Fig. 14.
Likewise, the baryon acoustic oscillations provide a standard ruler at relatively late times (low
redshifts z ∼ 0.2 − 0.4). A combination of their measurement with CMB anisotropy give quite precise
determination of both spatial curvature (which is found to be zero within error bars) and dark energy
density. Notably, this determination of ρΛ is in good agreement with various independent data, notably,
with the data on SNe 1a, which were the first unambiguous evidence for dark energy [27, 28].
There are many other ways in which the cosmological parameters, including ΩB and ΩDM , affect
36
Fig. 14: Effect of spatial curvature (left) and dark energy (right) on the CMB temperature angular spectrum [26].
Ωk = ±(RH0 )2 is the relative contribution of spatial curvature to the Friedmann equation, with R being the radius
of spatial curvature. Negative sign corresponds to 3-sphere. As Ωk decreases, the curves in the left plot move left.
Likewise, the curves on the right plot move left as ΩΛ increases.
the CMB anisotropies. In particular, the hights of the acoustic peaks in the CMB temperature angular
spectrum are very sensitive to the baryon-to-photon ratio ηB (and hence to ΩB ), the overall shape of the
curve in Fig. 10 stronfly depends on ΩDM , etc. By fitting the CMB data and combining them with the
results of other cosmological observations, one is able to obtain quite precise knowledge of our Universe.
6.5
Properties of primordial density perturbations — hints about the earliest cosmological epoch
As we emphasized above, the density perturbations were generated at a very early, pre-hot epoch of the
cosmological evolution. Obviously, it is of fundamental importance to figure out what precisely that
epoch was. One of its properties is clear right away: it must be such that the cosmologically relevant
wavelengths, including the wavelengths of the present horizon scale, were subhorizon early at that epoch.
Only in that case the perturbations of these wavelengths could be generated in a causal manner at the
pre-hot epoch. Notice that this is another manifestation of the horizon problem discussed in Section 2.6:
we know from the observational data on density perturbtions that our entire visible Universe was causally
connected by the beginning of the hot stage.
An excellent hypothesis on the pre-hot stage is inflation, the epoch of nearly exponential expansion,
a(t) = e
R
Hdt
H ≈ const .
,
Originally [29], inflation was designed to solve the problems of the hot Big Bang cosmology, such as
the horizon problem, as well as the flatness, entropy and other problems. It does this job very well: the
horizon size at inflation is at least
Z t
dt0
lH (t) = a(t)
= H −1 eH(t−ti ) ,
0)
a(t
ti
where ti is the time inflation begins, and we set H = const for illustrational purposes. This size is huge
for t − ti H −1 , so the entire visible Universe is naturally causally connected.
From the viewpoint of perturbations, the physical momentum q(t) = k/a(t) decreases (gets redshifted) at inflation, while the Hubble parameter stays almost constant. So, every mode is first subhorizon
(q(t) H(t)), and later superhorizon (q(t) H(t)) at inflation. This situation is opposite to what happens at radiation and matter domination, see Fig. 15; this is precisely the pre-requisite for generating
the density perturbations. In fact, inflation does generate primordial density perturbations [30], whose
properties are consistent with everything we know about them.
37
inside
horizon
outside
horizon
H(t)
q(t)=
inside
horizon
k
a(t)
inflation
te
RD, MD epochs
t
Fig. 15: Physical momentum and Hubble parameter at inflation and later. te is the time of the inflation end.
Inflation is not the only hypothesis proposed so far, however. One option is the bouncing Universe
scenario, which assumes that the cosmological evolution begins from contraction, then the contracting
stage terminates at some moment of time (bounce) and is followed by expansion. A version is the
cycling Universe scenario with many cycles of contraction–bounce–expansion. Another scenario is that
the Universe starts out from nearly flat and static state and then speeds up its expansion. Theoretical
realizations of these scenarios are more difficult than inflation, but they are not impossible, as became
clear recently. So, one of the major purposes of cosmology is to choose between various hypotheses on
the basis of observational data. The properties of cosmological perturbations are the key issue in this
regard.
There are several things which we already know about the primordial density perturbations. By
“primordial” we mean the perturbations deep in the superhorizon regime at the radiation-domination
epoch. As we already know, perturbations are time-independent in this regime. They set the initial
conditions for further evolution, and this evolution is well understood, at least in the linear regime. Hence,
using observational data, one is able to measure the properties of primordial perturbations. Of course,
since the properties we know of are established by observations, thery are valid within certain error bars.
Conversely, deviations from the results listed below, if observed, would be extremely interesting.
First, density perturbations are adiabatic. This means that there are perturbations in the energy
density, but not in composition. More precisely, the baryon-to-entropy ratio and dark matter-to-entropy
ratio are constant in space,
n n
B
DM
δ
= const ,
δ
= const .
(67)
s
s
This is consistent with the generation of the baryon asymmetry and dark matter at the hot cosmological
epoch: in that case, all partciles were at thermal equilibrium early at the hot epoch, the temperature
completely characterized the whole cosmic medium at that time, and as long as physics behind the
baryon asymmetry and dark matter generation is the same everywhere in the Universe, the baryon and
dark matter abundance (relative to the entropy density) is necessarily the same everywhere. In principle,
there may exist entropy (or isocurvature) perturbations, such that at the early hot epoch energy density
(dominated by relativistic matter) was homogeneous, while the composition was not. This would give
initial conditions for the evolution of density perturbations, which would be entirely different from those
characteristic of the adiabatic perturbations. As a result, the angular spectrum of the CMB temperature
38
anisotropy would be entirely different, see Fig. 16. No admixture of the entropy perturbations have
been detected so far, but it is worth emphasizing that even small admixture will show that the most
popular mechanisms for generating dark matter and/or baryon asymmetry (including those discussed in
Sections 3 and 4) have nothing to do with reality. One would have to think, instead, that the baryon
asymmetry and/or dark matter were generated before the beginning of the hot stage.
Fig. 16: Angular spectrum of the CMB temperature anisotropy for adiabatic perturbations (left) and entropy perturbations (right) [26].
Second, the primordial density perturbations are Gaussian random field. Gaussianity means that
the three-point (and all odd) correlation function vanishes, while the four-point function and all higher
order even correlation functions are expressed through the two-point function via Wick’s theorem:
hδ(k1 )δ(k2 )δ(k3 )i = 0
hδ(k1 )δ(k2 )δ(k3 )δ(k4 )i = hδ(k1 )δ(k2 )i · hδ(k3 )δ(k4 )i
+ permutations of momenta ,
while all odd correlation functions vanish. A technical remark is in order. As a varibale characterizing
the primordial adiabatic perturbations we use here
δ ≡ δρrad /ρrad = δρ/ρ
deep at the radiation-dominated epoch. This variable is not gauge-invariant, so we implicitly have chosen
the conformal Newtonian gauge. In cosmological literature, other, gauge-invariant quantities are commonly in use, ζ and R. In the conformal Newtonian gauge, there is a simple relationship, valid in the
superhorizon regime at radiation domination:
3
R=ζ= δ.
4
We will continue to use δ as the basic variable.
Coming back to Gaussianity, we note that this property is characteristic of vacuum fluctuations of
non-interacting (linear) quantum fields. Hence, it is quite likely that the density perturbations originate
from the enhanced vacuum fluctuations of non-interacting or weakly interacting quantum field(s). Free
quantum field has the general form
Z
(+)
(−)
φ(x, t) = d3 ke−ikx fk (t)a†k + eikx fk (t)ak ,
39
where a†k and ak are creation and annihilation operators. For the field in Minkowski space-time one has
(±)
fk (t) = e±iωk t , while enhancement, e.g. due to the evolution in time-dependent background, means
(±)
that fk are large. But in any case, Wick’s theorem is valid, provided that the state of the system is
vacuum, ak |0i = 0.
Inflation does the job very well: fluctuations of all light fields get enhanced greatly due to the fast
expansion of the Universe. This is true, in particular, for the field that dominates the energy density at
inflation, called inflaton. Enhanced vacuum fluctuations of the inflaton are nothing but perturbations in
the energy density at inflationary epoch in the simplest inflationary models, which are reprocessed into
perturbations in the hot medium after the end of inflation. The generation of the density perturbations is
less automatic in scenarios alternative to inflation, but there are various examples showing that this is not
a particularly difficult problem.
Non-Gaussianity is an important topic of current research. It would show up as a deviation from
Wick’s theorem. As an example, the three-point function (bispectrum) may be non-vanishing,
hδ(k1 )δ(k2 )δ(k3 )i = δ(k1 + k2 + k3 ) G(ki2 ; k1 k2 ; k1 k3 ) 6= 0 .
The shape of G(ki2 ; k1 k2 ; k1 k3 ) is different in different models, so this shape is a potential discriminator. In some models the bispectrum vanishes, e.g., due to symmetries. In that case the trispectrum
(connected 4-point function) may be measurable instead. Non-Gaussianity is very small in the simplest
inflationary models, but it can be sizeable in more contrived models of inflation and in alternatives to
inflation. It is worth emphasizing that non-Gaussianity has not been detected yet.
Another important property is that the primordial power spectrum of density perturbations is flat
(or almost flat). A convenient definition of the power spectrum for homogeneous and anisotropic Gaussian random field is11
1
hδ(k)δ(k0 )i =
P(k)δ(k + k0 ) .
(68)
4πk 3
The power spectrum P(k) defined in this way determines the fluctuation in a logarithmic interval of
momenta,
Z ∞
dk
2
hδ (x)i =
P(k) .
k
0
By definition, the flat spectrum is such that P is independent of k. It is worth noting that the flat spectrum
was conjectured by E. Harrison [31] and Ya. Zeldovich [32] in the beginning of 1970’s, long before
realistic mechanisms of the generation of density perturbations have been proposed.
In view of the approximate flatness, a natural parametrization is
ns −1
k
P(k) = As
,
k∗
(69)
where As is the amplitude, (ns −1) is the tilt and k∗ is a fiducial momentum, chosen at one’s convenience.
The flat spectrum in this parametrization has ns = 1. Cosmological data favor the value ns ≈ 0.96 (i.e.,
slightly smaller than 1), see below, but it is fair to say that ns = 1 is still consistent with observations.
The flatness of the power spectrum calls for some symmetry behind this property. In inflationary
theory this is the symmetry of the de Sitter space-time, which is the space time of constant Hubble rate,
ds2 = dt2 − e2Ht dx2 ,
H = const .
This metric is invariant under spatial dilatations supplemented by time translations,
x → λx ,
11
t→t−
1
log λ .
2H
Note that the the definition of the power spectrum used in Figs. 7 and 12 is different from (68).
40
At inflation, H is almost constant in time, and the de Sitter symmetry is an approximate symmetry. For
this reason inflation automatically generates nearly flat power spectrum.
The de Sitter symmetry is not the only candidate symmetry behind the flatness of the power spectrum. One possible alternative is conformal symmetry [33, 34]. The point is that the conformal group
includes dilatations, xµ → λxµ . This property indicates that the relevant part of the theory possesses no
scale, and has good chance for producing the flat spectrum. Model-building in this direction has begun
recently [34].
6.6
What’s next?
Thus, only very basic facts about the primordial denisty perturbations are observationally established.
Even though very suggestive, these facts by themselves are not sufficient for unambiguously establishing
the properties of the Universe at the pre-hot epoch of its evolution. In coming years, new properties
of cosmological perturbations will hopefully be discovered, which will shed much more light on this
pre-hot epoch. Let us discuss some of the potential observables.
6.6.1
Tensor perturbations = relic gravity waves
The simplest, and hence most plausible models of inflation predict sizeable tensor perturbations, which
are perturbations of the metric independent of perturbations in the energy density. After entering the
horizon, tensor perturbations are nothing but gravity waves. The reason for their generation at inflation is
that the exponential expansion of the Universe enhances vacuum fluctuations of all fields, including the
gravitational field itself. In inflationary theory, the primordial tensor perturbations are Gaussian random
field with nearly flat power spectrum
nT
k
PT = AT
,
(70)
k∗
where the inflationary prediction is nT ≈ 0 (the reason for different definitions of the tensor spectral
index nT in (70) and scalar spectral index ns in (69) is purely historical).
On the other hand, there seems to be no way of generating nearly flat tensor power spectrum
in alternatives to inflation. In fact, most, if not all, alternative scenarios predict unobservably small
amplitude of tensor perturbations. Thus, the discovery of tensor modes would be the strongest possible
argument in favor of inflation. It is worth noting that non-observation of tensor perturbations would
not rule inflation out: there are numerous models of inflation which predict tensor modes of very small
amplitude.
The tensor power is usually characterized by the tensor-to-scalar ratio
r=
AT
.
As
The simplest inflationary models predict, roughly speaking, r ∼ 0.1 − 0.3. The current situation is
summarized in Fig. 17. Clearly, there is an indication for the negative scalar tilt (ns − 1) or non-zero
tensor amplitude, or both, though it is premature to say that the flat scalar spectrum with no tensor modes
(the Harrison–Zeldovich point) is ruled out.
For the time being, the most sensitive probe of the tensor perturbations is the CMB temperature
anisotropy. However, the most promising tool is the CMB polarization. The point is that a certain class
of polarization patterns (called B-mode) is generated by tensor perturbations, while scalar perturbations
are unable to create it. Hence, the Planck experiment, and especially dedicated experiments aiming at
measuring the CMB polarization may well discover the tensor perturbations, i.e., relic gravity waves.
Needless to say, this would be a profound discovery. To avoid confusion, let us note that the CMB
polarization has been already observed, but it belongs to another class of patterns (so called E-mode) and
is consistent with the existence of the scalar perturbations only.
41
Fig. 17: Allowed regions (at 68% and 95% CL) in the plane (ns , r), where ns is the scalar spectral index and r is
the tensor-to-scalar ratio. Rectangle shows the Harrison–Zeldovich point (flat scalar spectrum, no tensor modes).
Points on the lines show predictions of popular inflationary models.
6.6.2
Scalar tilt.
Inflationary models and their alternatives will be constrained by the precise determination of the scalar
tilt (ns − 1) and its dependence on momentum k. It appears, however, that the information on ns (k)
that will be obtained in reasonably near future will be of limited significance from the viewpoint of
discriminating between different (and even grossly different) scenarios.
6.6.3
Non-Gaussianity.
As we pointed out already, non-Gaussianity of density perturbations is very small in the simplest inflationary models. Hence, its discovery will signalize that either inflation and inflationary generation of
density perturbations occured in a rather complicated way, or an alternative scenario was realized. Once
the non-Gaussianity is discovered, and its shape is revealed even with moderate accuracy, many concrete
models will be ruled out, while at most a few will get strong support.
6.6.4
Statistical anisotropy.
In principle, the power spectrum of density perturbations may depend on the direction of momentum,
e.g.,
ki kj
P(k) = P0 (k) 1 + wij (k) 2 + . . .
k
where wij is a fundamental tensor in our part of the Universe (odd powers of ki would contradict commutativity of the Gaussian random field δ(k), see Eq. (68)). Such a dependence would definitely imply
that the Universe was anisotropic at the pre-hot stage, when the primordial perturbations were generated.
This statistical anisotropy is rather hard to obtain in inflationary models, though it is possible in inflation
with strong vector fields [35]. On the other hand, statistical anisotropy is natural in some other scenarios,
including conformal models [36]
The statistical anisotropy would show up in correlators [37]
halm al0 m0 i
with l0 6= l and/or m0 6= m
At the moment, the situation with observational data is controversial [38], and the new data, notably from
the Planck experiment, will hopefully clear it up.
42
6.6.5
Admixture of entropy perturbations.
As we explained above, even small admixture of entropy perturbations would force us to abandon the
most popular scenarios of the generation of baryon asymmetry and/or dark matter, which assumed that
it happened at the hot epoch. The WIMP dark matter would no longer be well motivated, while other,
very weakly interacting dark matter candidates, like axion or superheavy relic, would be prefered. This
would make the direct searches for dark matter rather problematic.
7
Conclusion
We are at the eve of new era not only in particles physics, but also in cosmology. There is reasonably
well justified expectation that the LHC will shed light on long-standing cosmological problems of the
origin of the baryon asymmetry and nature of dark matter in our Universe. The ideas we discussed in
these lectures in this regard may well be not the right ones: we can only hypothesize on physics beyond
the Standard Model and its role in the early Universe.
In fact, the TeV scale physics may be dramatically different from physics we get used to. As an
example, it is not excluded that TeV is not only electroweak, but also gravitational scale. This is the case
in models with large extra dimensions, in which the Planck scale is related to the fundamental gravity
scale in a way that involves the volume of extra dimensions, and hence the fundamental scale can be
much below MP l (for a review see, e.g., Ref. [39]). If the LHC will find that, indeed, the fundamental
gravity scale is in the TeV range, this would have most profound consequences for both microscopic
physics and cosmology. On the microscopic physics side, this would enable one to study at colliders
quantum gravity and its high-energy extension — possibly string theory, while on the cosmological side,
the entire picture of the early Universe would have to be revised. Inflation, if any, would have to occur
either at low energy density or in the regime of strong quantum gravity effects. The highest temperatures
in the usual expansion history would be at most in the TeV range, so dark matter and baryon asymmetry
would have to be generated either below TeV temperatures or in quantum gravity regime. Even more
intriguing will be the study of quantum gravity cosmological epoch, with hints from colliders gradually
coming. This, probably, is too bright a prospective to be realistic.
It is more likely that the LHC will find something entirely new, something theorists have not
thought about. Or, conversely, find so little that one will have to get serious about anthropic principle.
In any case, the LHC results will definitely change the landscape of fundamental physics, cosmology
included.
The observational data unequivocally tell us that the hot stage of the cosmological evolution was
preceeded by some other epoch, at which the cosmological perturbations were generated. The best
guess for this epoch is inflation, but one should bear in mind that there are alternative possibilities. It
is fascinating that with new observational data, there is good chance to learn what precisely that pre-hot
epoch was. It may very well be that in this way we will be able to probe physics at the energy, distance
and time scales well beyond the reach of the LHC.
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