11institutetext:  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow, Russia

# Cosmology and Dark Matter

I. Tkachev

###### Abstract

This lecture course covers cosmology from the particle physicist perspective.
Therefore, the emphasis will be on the evidence for the new physics in
cosmological and astrophysical data together with minimal theoretical
frameworks needed to understand and appreciate the evidence. I review the case
for non-baryonic dark matter and describe popular models which incorporate it.
In parallel, the story of dark energy will be developed, which includes
accelerated expansion of the Universe today, the Universe origin in the Big
Bang, and support for the Inflationary theory in CMBR data.

###### keywords:

Cosmology; dark matter; lectures; dark energy; inflation.

##  1 Introduction

The deeper we dig into microphysics, the deeper connections between smallest
and largest scales are reviled. The world which surrounds us on Earth in
everyday life can be understood in frameworks of electrodynamics with atoms
and molecules in hands. But we need nuclear physics to describe the Sun and
other stars. Moving to even bigger scales, explaining galaxies and the
Universe as a whole, we need good understanding of particle physics.

This relation works the other way around as well. In particular, cosmology
tells us that the Standard Model of particle physics is incomplete. Namely,
that the galaxies and galaxy clusters are made mainly of the Dark Matter,
which overweights usual baryonic matter, and for which there is no room in the
Standard Model of particle physics. It tells also that at the scale of the
Universe the Dark Energy overrules, which can be simple or complicated
substance, but which is not matter. We conclude that there ought to be a new
physics and new particles outside of the Standard Model frameworks. At present
we learn their properties from cosmology only.

The plan of this lectures is as follows. In Section 2, I review the basics of
cosmology: Friedman equations, Hubble expansion and cosmography. Evidence for
the dark energy is also presented in this section. In Section 3 the Hot Big
Bang theory is outlined: relevant thermodynamal facts and relations are
presented and Cosmic Microwave Background Radiation (CMBR) is introduced. ln
Section 4 the study of CMBR anisotropies is described as a tool of precision
cosmology. In Section 5 basics of inflationary theory are given, while some
important technical details of this theory are moved into Appendices. In
Section 6 the evidence for dark matter is described together with dark matter
models and dark matter searches results.

##  2 Basics of Cosmology

There are many excellent recent books and reviews on cosmology in the market,
which readers may consult for missing details. I would especially recommend
the balanced, contemporary and comprehensive book by Gorbunov and Rubakov[1].

###  2.1 Expansion of the Universe

Cosmological dynamics is provided by General Relativity - the Einstein field
equations

$R_{\mu\nu}-{1\over 2}g_{\mu\nu}R=8\pi G\;T_{\mu\nu}\;,$ (1)

where $T_{\mu\nu}$ is a stress energy tensor describing the distribution of
mass in space, $G$ is Newton’s gravitational constant, and the curvature
$R_{\mu\nu}$ is certain function of the metric $g_{\mu\nu}$ and its first and
second derivatives. Immediate consequence of Einstein equations is energy
momentum conservation

$T^{~{}\nu}_{\mu\,\,\,\,;\nu}=0\;.$ (2)

These equations take simple form in important physical situations with special
symmetries. At large scales the Universe as a whole is homogeneous and
isotropic and these symmetries form basis for the construction of cosmological
models. The most general space-time metric describing such universe is the
Robertson-Walker metric

$ds^{2}=dt^{2}-a^{2}(t)\;{\bf dl}^{2}\;,$ (3)

where $a(t)$ is the dimensionless scale factor by which all distances vary as
a function of cosmic time $t$. The scale factor contains all the dynamics of
the Universe as a whole, while the vector product ${\bf dl}^{2}$ describes the
geometry of the space,

${\bf
dl}^{2}=\frac{dr^{2}}{1-k\,r^{2}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\;,$

which can be either Euclidian, or positively or negatively curved. For the
spatial 3-dimensional curvature we find, explicitly

${}^{(3)}\\!R={6k\over a^{2}(t)}\;\;\;\left\\{\begin{array}[]{ll}k=-1&\hskip
14.22636pt{\rm Open}\\\ k=0&\hskip 14.22636pt{\rm Flat}\\\ k=+1&\hskip
14.22636pt{\rm Closed}\end{array}\right.$ (4)

E.g., the space with $k=+1$ can be thought of as a 3-dimensional sphere with a
curvature being inversely proportional to the square of its radius. In this
Section we will model the matter content of the Universe as a perfect fluid
with energy density $\rho$ and pressure $p$, for which the stress-energy
tensor in the rest frame of the fluid is

$\displaystyle T_{\mu}^{~{}\nu}=\left(\begin{array}[]{cccc}{\rho}&0&0&0\\\
0&{-p}&0&0\\\ 0&0&{-p}&0\\\ 0&0&0&{-p}\end{array}\right)$ (9)

With these assumptions the Einstein equations simplify to the Friedmann
equations, which form the dynamical basis of cosmology

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}\,\rho~{}-~{}\frac{k}{a^{2}}\;,$ (10)
$\displaystyle\frac{\ddot{a}}{a}~{}$ $\displaystyle=$
$\displaystyle-~{}\frac{4\pi G}{3}\;(\rho+3\,p)\;.$ (11)

As Alexander Friedmann have shown in 1922, a universe described by such
equations cannot be static, it inevitably expands or collapses. Solution of
Eqs. (10) and (11) can be found in the following way, which also highlights
the physics behind these equations. Differentiating the first Friedmann
equation and combining result with the second one we obtain

$\frac{d{\rho}}{dt}~{}+~{}3\,\frac{\dot{a}}{a}\,({\rho}+{p})=0,$ (12)

which is nothing but the energy-momentum conseravtion
$T^{\mu\nu}_{\,\,\,\,\,\,\,\,\,;\nu}=0$ written for the homogeneous isotropic
medium. On the other hand the result Eq. (12) also corresponds to the First
Law of thermodynamics

$dE~{}+~{}p\,dV~{}=~{}T\,dS,$ (13)

with $dS=0$, where $S$ is entropy, $E=\rho V$, and $V\propto a^{3}$. This is
expected since in derivation of Friedmann equations the energy-momentum tensor
of an ideal fluid was assumed. It turns out that this is valid approximation
most of the time111It is violated though during special moments, in particular
at the initial matter creation after inflation, see below, or at strongly
first order phase transitions, if those existed in the Universe past. and
therefore the expansion of the Universe is adiabatic, $S=\rm const$.

Let $s$ be entropy density, $S=sV$. We know from thermodynamic (and I’ll
derive this in Section 3.2) that in thermal equilibrium the entropy density is
given by

$s={2\pi^{2}\over 45}\,g_{*}\;T^{3},$ (14)

where $g_{*}$ is the number of effective relativistic degrees of freedom

$\footnotesize g_{*}={\footnotesize\sum_{i={\rm bosons}}g_{i}+{7\over
8}\sum_{j={\rm fermions}}g_{j}},$ (15)

and $g_{i}$ is the number of spin states of a given particle, e.g. for
photons, electrons and positrons $g_{\gamma}=g_{e^{-}}=g_{e^{+}}=2$. In this
expression only particles with $m\ll T$ are counted, i.e. $g_{*}$ is a
function of temperature. It is displayed in Fig. 2 by the solid line for the
Standard Model and by the dashed line for the minimal supersymmetric extension
of the SM. Therefore, $S=\rm const$ is equivalent to

$g_{*}^{1/3}\,T\,a={\rm const}.$ (16)

This is very useful relation. In particular, it gives $T\propto a^{-1}$
(neglecting the change in the number of degrees of freedom), and we arrive to
the concept of the Hot Universe right away. Indeed, currently the Universe is
expanding, in the past it was smaller and therefore hotter. In the era of
precision cosmology the change in $g_{*}(T)$ should be counted too, if we go
beyond simple estimates. We do not know all particle content of complete
theory describing Nature, but LHC will fix the actual shape of $g_{*}(T)$ in
the region of highest $T$ of Fig. 2 and, in fact, a sharp MSSM like rise in
the number of degrees of freedom is ruled out already.

Only relativistic particles contribute to entropy, but everything existing
contributes to the energy density, $\rho$. Even vacuum. The equation of state,
$w$, of a substance contributing to $\rho$ is defined as $w\equiv{p}/{\rho}$.
If $w=\rm const$, the energy-momentum conservation, Eq. (12), gives
$\rho=a^{-3(1+w)}\;\rho_{0}$. Using this result, we find from the first
Friedmann equation, Eq. (10), the scale factor as a function of time,
$a=\left({t}/{t_{0}}\right)^{{2}/{3(1+w)}}$.

During the first half of of the last century cosmologists were assuming that
the Universe is filled with a “dust”, $p=0$, while the “dust” particles were
represented by galaxies made of usual matter. Nowadays we know that the
Universe is multicomponent. Its energy density was dominated in turn by
radiation ($p=\rho/3$), by dark matter ($p=0$, as for the ”dust” of galaxies)
and finally by dark energy (equation of state consistent with $p=-\rho$). In
the Table 1 we list: substances known to contribute into energy balance in the
Universe, their defining equations of state, the corresponding scaling of
energy density with expansion, and corresponding solution for $a(t)$ if $k=0$.

Table 1: Substances contributing into the energy balance in the Universe Substance | Equation of state | $\rho(a)$ | $a(t)$
---|---|---|---
Radiation | $w={1}/{3}$ | $\rho=a^{-4}\;\rho_{0}$ | $a=(t/t_{0})^{1/2}$
Matter | $w=0$ | $\rho=a^{-3}\;\rho_{0}$ | $a=(t/t_{0})^{2/3}$
Vacuum | $w=-1$ |  $\rho=\rm const~{}~{}~{}~{}~{}$ |  $a=\exp{(H_{0}t)}$

To parameterize the Fiedmann equations and their solution $a(t)$, cosmologists
introduce cosmological parameters. One of such parameters we have already
encountered, this is $k$ entering Frideman equations explicitly. Despite
paramtrizing spatial geometry of the Universe it was used to predict the
Universe future. Namely, it immediately follows from Eq. (10) that for $k=-1$
or $k=0$ the Universe will expand forever. For $k=+1$ the Universe should
recollapce at the point when r.h.s. of Eq. (10) turns to zero. This should
happen in radiation or matter dominated Universe, but never happens in the
universe dominated by the dark energy with $\rho_{0}>0$. Since nowadays we
know that our Universe is dominated by such dark energy, we already know that
it will expand forever.

Inherently related to the parameter $k$ is critical density. This is the
density at which the Universe is spatially flat, $k=0$

$\rho_{c}\equiv\frac{3}{8\pi G}\;\left(\frac{\dot{a}}{a}\right)^{2}.$ (17)

Critical density can be expressed using a second, directly observable and very
important cosmological parameter, the Hubble constant, $H\equiv{\dot{a}}/{a}$.
To quantify relative contribution, $\rho_{i}$, of each of the components in
the total energy budget of the Universe, $\rho$, the following notations are
introduced, $\Omega_{i}\equiv\rho_{i}/\rho_{c}$ and $\sum_{i}\rho_{i}=\rho$.
The current knowledge of the numerical values of some of these parameters at
$t=t_{0}$ is summarized in the Table 2 below.

Table 2: Cosmological parameters Symbol & Definition | Description | Present value, from Ref. [2]
---|---|---
$t$ | Age of the Universe |  $t_{0}=13.81\pm 0.03$ Gyr
$H=\dot{a}/a$ | Hubble parameter | $H_{0}=67.27\pm 0.66\rm{~{}km~{}s^{-1}~{}Mpc^{-1}}$
$\Omega=\rho/\rho_{c}$ | Spatial curvature | $1-\Omega=0.000\pm 0.005$
$\Omega_{\gamma}=\rho_{\gamma}/\rho_{c}$ | Fraction of photons | $\Omega_{\gamma}=2.48\cdot 10^{-5}\;h^{-2}$
$\Omega_{b}=\rho_{b}/\rho_{c}$ | Baryonic fraction | $\Omega_{b}\,h^{2}=0.02225\pm 0.00016$
$\Omega_{\rm m}=\rho_{\rm m}/\rho_{c}$ | Matter fraction | $\Omega_{\rm m}=0.316\pm 0.009$
$\Omega_{\Lambda}=\rho_{\Lambda}/\rho_{c}$ | Dark Energy fraction | $\Omega_{\Lambda}=0.684\pm 0.009$

Accuracy of numerical values presented in this table should not be over-
appreciated since those were derived with some model assumptions, e.g. that
the dark matter is cold and the equation of state for dark energy is $w=-1$.
Relaxing such assumptions changes presented values somewhat.

###  2.2 The Hubble law

The velocity with which distance $r=a(t)r_{0}$ between two arbitrary galaxies
increases in expanding Universe follows trivially from the definition of the
Hubble parameter, $H\equiv\dot{a}/a$, namely $~{}v=\dot{r}=H\,r$. This
relation, known as the Hubble law, makes the basis for direct observational
determination of the Hubble parameter and was discoverd by Edwin Hubble in
1929. It is also used to set units for measuring $H$. For convenience, in many
cosmological relations, dimensionless “small $h$” is introduced as
$H=100\,h\,\rm~{}km~{}s^{-1}Mpc^{-1}$, then $h\sim O(1)$. Latest value of the
Hubble constant obtained from direct mapping of recession velocities versus
distance corresponds to $H_{0}=(73.00\pm
1.75){\rm~{}km~{}s}^{-1}{\rm~{}Mpc}^{-1}$ [3]. Note that this value is
$3.4\sigma$ higher than the value presented in Table 2 which was derived
indirectly from other cosmological data. This may suggest some unaccounted
systematic uncertainties or may indicate new physics [4].

Looking at cosmologically distant objects we see just the light they emit. One
can wander then, how observables are derived from this limited information?
E.g. how distance and velocity can be measured separately to determine $H$?
The unswear is simple. Velocity is measured by the frequency shift of known
signal, similarly to what police is doing when checking for speeding cars
using Doppler radars. Distance can be derived measuring dimming of objects
with calibrated intrinsic luminosity: objects which are further away are less
bright. Now we shall explain this in more details.

####  2.2.1 Redshift

Photon motion in any metric is described by basic equation, $ds^{2}=0$. In the
Robertson-Walker metric this becomes $ds^{2}=dt^{2}-a^{2}d\chi^{2}=0$, where
$\chi$ is comoving distance along particle trajectory. Define conformal time
$\eta$ as

$d\eta=dt/a.$ (18)

Then $ds^{2}=a^{2}(d\eta^{2}-d\chi^{2})=0$. Remarkably, solution for a photon
world line in conformal coordinates is the same as in Minkowsikan space-time
$\chi=\pm\eta\;+\;{\rm const}$. Therefore, the conformal time lapse between
two events of light emission at one point will be the same as for their
detection at another point, regardless of distance traveled, $d\eta|_{\rm
emission}=d\eta|_{\rm detection}$. Therefore, the proper time lapses for
emissions and detections are related as

$\large\frac{dt}{a}|_{\rm detection}~{}=~{}\frac{dt}{a}|_{\rm emission}.$

Let $dt$ corresponds to the period of some monochromatic signal, then for its
frequency we will have $\omega\,a_{d}~{}=~{}\omega_{0}\,a_{e}$. As a result,
we can say that the wavelength of a signal stretches together with the
expansion of the Universe. An this interpretation is often used. However, it
is incorrect. For example, the space does not stretches inside galaxies, but
$\omega\,a_{d}~{}=~{}\omega_{0}\,a_{e}$ will be always true regardless of how
many times the light signal passed through galaxies between detection and
emission. The signal frequency changes between the point of emission,
$\omega_{0}$, and in the point of detection, $\omega$, because clocks run
differently in those points. In a similar way the gravitational redshift or
blueshift can be also derived and interpreted. At largest cosmological scales
we always have redshift since Universe is expanding.

In measurements, the redshift is quantified as
$z~{}\equiv~{}{(\omega_{0}-\omega)}/{\omega}$ and measured as a frequency
shift of emission or absorption lines of various chemical elements. With this
definition we get

$1+z=\frac{a_{d}}{a_{e}}.$ (19)

In other words, redshift can be used also to label cosmological epoch. For
convenience, we can always normalise the scale factor today as $a_{d}=1$.
Recall relation $a\propto T^{-1}$ (or, more precisely, Eq. 16). This gives
temperature of the Universe as a function of redshift, $T(z)=T_{0}(1+z)$.
Also, using our previous result $\rho=a^{-3(1+w)}\;\rho_{0}$ and Friedmann
equation (10) we can rewrite Hubble parameter as a function of $z$ and
cosmological parameters $H_{0}$ and $\Omega_{i}$ in the following often used
form

$H^{2}(z)=H_{0}^{2}\,\sum_{i}\Omega_{i}\,(1+z)^{3(1+w_{i})}.$ (20)

Differentiating Eq. (19) we find $dz=-Hd\chi$ which is a local form of the
Hubble law, partially expressed through observables already. Instead of $v$ we
have directly measurable redshift z. This is understandable since in the non-
relativistic limit cosmological redshift reduces to the Doppler effect - at
the end, galaxies are receding from us with the Universe expansion. Finally,
integral of the local form of the Hubble law, $dz=-Hd\chi$, gives

$\chi(z)=\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}\,.$ (21)

Now we have to find the way to measure distances to remote objects to complete
construction of the Hubble law generalisation, which would be expressed
through observables only and which would be valid to any redshift.

####  2.2.2 Luminosity distance

Consider two objects with identical luminosities (Standard Candles) placed at
different distances from us. The radiation flux, $F$, scales with distance as
$F^{-1/2}$. Therefore, measuring fluxes from a standard candles, we can
determine the ratio of distances to them. Moreover, the distance to an object
with known luminosity can be defined as Luminosity Distance, $D_{L}$, and
measured via measuring flux from it

$D_{L}^{2}~{}\equiv~{}\frac{L}{4\pi F}.$

Consider now this idea in cosmology. Again, let us write metric in conformal
time, but now we should keep track of changing area with distance at fixed
solid angle, $ds^{2}=a^{2}\;(d\eta^{2}-d\chi^{2}-\chi^{2}\;d\Omega)$. Surface
area at the point of detection is $4\pi\,\chi^{2}$ (we can always normalise
scale factor today as $a=1$). Further, energy and arrival rates of registered
photons are redshifted. This reduces the flux by $(1+z)^{2}$, where $z$ is
redshift of emitter. We get for observed bolometric flux

$F~{}=~{}\frac{L}{4\pi\,\chi^{2}(1+z)^{2}},$

and luminosity distance is

$D_{L}~{}\equiv~{}\sqrt{\frac{L}{4\pi F}}~{}=~{}(1+z)\;\chi.$

Therefore, measuring flux, we can determine comoving distance to a standard
candle. We may not know the value of intrinsic luminosity $L$, but this is not
necessary. It is important only that $L$ should not vary from an object to
object. Then we can compare ratio of fluxes at different redshifts and from
there derive cosmology. For historical reasons astronomers measure flux in
magnitudes, which are defined as $\mu\propto 5\log_{10}F$. Ratio of fluxes
will be difference in magnitudes. Now, we want to see how different are
magnitudes of a standard candles at a given redshift in different cosmologies,
say in cosmology which predicts $\chi(z)$ and in a “base” cosmology which
predicts $\chi(z)_{\rm base}$. We find

$\Delta\mu=\mu-\mu_{\rm base}=5\log_{10}\left[\frac{\chi(z)}{\chi(z)_{\rm
base}}\right]$ (22)

Here $\chi(z)$ and $\chi(z)_{\rm base}$ are given by Eq. (21) with its own
sets of cosmological parameters each.

####  2.2.3 Dark Energy

Figure 1:  Evidence for the dark energy from Supernova data, left panel, and
from the Universe age, right panel.

Supernova of type Ia have been shown to be good standard candles. In
measurements of luminosity distance to them the dark energy has been
discovered. Below I illustrate this result in Fig. 1, left panel, using modern
data and relation Eq. (22). Blue curve corresponds to the Universe without
dark energy, $\Omega_{\Lambda}=0$, while violet curve corresponds to the best
fit over dark energy which gives $\Omega_{\Lambda}=0.7$. As a base cosmology
which was subtracted I used the Universe with $\ddot{a}=0$, but the
subtraction is not important here and is needed for the visualisation purposes
only, to enhance separation of curves on a graph.

In fact, prior to the dark energy discovery in Supernova data, scientists
already suspected for a while that it exists. Several hints existed, I
illustrate the one derived from attempts to determine the Universe age. During
mater dominated expansion $a\propto t^{2/3}$. Therefore, for a matter
dominated Universe without dark energy we would have $H_{0}\,t_{0}=2/3$.
However direct measurements of the Hubble constant gave at the time
$H_{0}=70\pm 7\;{\rm km\;sec^{-1}\;Mpc^{-1}}$, while the age was estimated
(using ages of the oldest stars) as $t_{0}=13\pm 1.5\;{\rm Gyr}$. Therefore,
measurements producing $H_{0}\,t_{0}=0.93\pm 0.15$ were in contradiction with
prediction for the matter dominated Universe.

Let us see what happens if we add dark energy to a matter. The age of the
multicomponent universe can be found integrating $dt=ad\chi$ and using
$dz=-Hd\chi$, which for the universe age at redshift $z$ gives

$t(z)=\int_{z}^{\infty}\frac{dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}\,.$

In particular, if equation of state of dark energy corresponds to a vacuum,
$w=-1$, and universe is spatially flat, $\Omega_{\Lambda}+\Omega_{M}=1$, this
gives

$H_{0}t_{0}=\frac{2}{3\sqrt{\Omega}_{\Lambda}}\ln\left(\frac{1+\sqrt{\Omega}_{\Lambda}}{\sqrt{\Omega}_{M}}\right)\,.$

Such a universe with $\Omega_{\Lambda}\simeq 0.7$ is a good fit to
observations as opposed to a matter dominated universe, see Fig. 1, right
panel and compare to a modern data in Table 2 which give $H_{0}t_{0}\simeq
0.95$.

##  3 Hot Big Bang

So far we have considered cosmography of the late Universe and found that the
Universe should be filled with matter and dark energy. However, the Universe
should contain radiation also. Today its contribution is negligible, but in
the early Universe it was dominant fraction. Indeed, energy density of
radiation is fastest growing fraction when we look back in time, $a\rightarrow
0$, see Table 1. And the Universe was hotter as well in this limit, see Eq. (
16). To reach such conclusions we have to assume also that the Universe was in
thermal equilibrium in the past. But this is inevitable too since in a denser
medium relaxation time is shorter. Universe was indeed in thermal equilibrium
in the past, as we will shortly see. The concept of the Hot Universe is so
natural and so inevitable, that it is hard to imagine nowadays that is was not
widely accepted until relict radiation has beed observed.

###  3.1 Cosmic Microwave Background Radiation

The Universe is filled with radiation which is left-over from the Big Bang.
The name for this first light is Cosmic Microwave Background Radiation (CMBR).
Measurements of tiny fluctuations (anisotropy) in CMBR temperature give a
wealth of cosmological information and became a most powerful probe of
cosmology.

This radiation was predicted by Georgi Gamov in 1946, who estimated its
temperature to be $\sim 5\;K^{\circ}$. Gamov was trying to understand the
origin of chemical elements and their abundances. Most abundant, after
hydrogen, is helium, with its shear being $\sim 25\%$. One possibility which
Gamov considered was nucleo-synthesis of He out of H in stars. Dividing the
total integrated luminosity of the stars by the energy released in one
reaction, he estimated the number of produced He nuclei. This number was too
small in comparison with observations. Gamov assumed then that the oven where
the light elements were cooked-up was the hot early Universe itself. He
calculated abundances of elements successfully and found that the redshifted
relic of thermal radiation left over from this hot early epoch should
correspond to $\sim 5\;K^{\circ}$ at present. In one stroke G. Gamov founded
two pillars (out of four) on which modern cosmology rests: CMBR and Big Bang
Nucleosynthesis (BBN). Hot Big Bang theory was born.

Cosmic microwave background has been accidentally discovered by Penzias and
Wilson [5] at Bell Labs in 1965 as the excess antenna temperature which,
within the limits of their observations, was isotropic, unpolarized, and free
from seasonal variations. A possible explanation for the observed excess noise
temperature was immediately given by Dicke, Peebles, Roll, and Wilkinson and
was published in a companion letter in the same issue [6]. Actually, they were
preparing dedicated CMBR search experiment, but were one month late. Penzias
and Wilson measured the excess temperature as $\sim 3.5\pm 1\;K^{\circ}$. It
is interesting to note that the first (unrecognized) direct measurements of
the CMB radiation was done by T. Shmaonov at Pulkovo in 1955, also as an
excess noise while calibrating the RATAN antenna [7]. He published the
temperature as $(3.7\pm 3.7)\;K^{\circ}$. And even prior to this, in 1940,
Andrew McKellar [8] had observed the population of excited rotational states
of CN molecules in interstellar absorption lines, concluding that it was
consistent with being in thermal equilibrium with a temperature of $\approx
2.7\;K^{\circ}$. Its significance was unappreciated and the result essentially
forgotten. Finally, before the discovery, in 1964 Doroshkevich and Novikov in
an unnoticed paper emphasized [9] the detectability of a microwave blackbody
as a basic test of Gamov’s Hot Big Bang model.

The spectrum of CMBR is a perfect blackbody, with a temperature [10]

$T_{0}=2.7255\pm 0.0006\;K^{\circ},$ (23)

as measured by modern instruments. This corresponds to 410 photons per cubic
centimeter or to the flux of 10 trillion photons per second per squared
centimeter.

###  3.2 Thermodynamics of the Universe

There is no explanation to CMBR but the hot Big Bang. And since CMB is the
radiation with black body spectrum, we know that the Universe was once in the
thermal equilibrium. It immediately follows from Eq. (16) that in the past the
Universe was hotter since it was smaller. We can and should use thermodynamics
describing the early Universe.

For particles in thermal equilibrium the phase-space distribution functions
are:

$f_{i}(k)=\frac{g_{i}}{(2\pi)^{3}}\,\frac{1}{\mathop{\rm
e}\nolimits_{~{}}^{(k_{0}-\mu_{i})/T}\pm 1},$ (24)

where $k_{0}$ is particle energy, $k_{0}=\sqrt{\vec{k}^{2}+m_{i}^{2}}$, $\mu$
is chemical potential and the plus (minus) sign corresponds to fermions
(bosons). Index $i$ refers to different particles species and $g_{i}$ is the
number of their spin states, e.g. for photons, electrons and positrons
$g_{\gamma}=g_{e^{-}}=g_{e^{+}}=2$ correspondingly, for neutrino and
antineutrino $g_{\nu}=g_{\bar{\nu}}=1$. All thermodynamical relations which we
will need are derived using this function. In particular, number density of
i-th particle species and their contribution into energy-momentum tensor are,
correspondingly

$\displaystyle n_{i}$ $\displaystyle=$ $\displaystyle\int d^{3}kf_{i}(k),$
(25) $\displaystyle T_{\mu\nu}(i)$ $\displaystyle=$ $\displaystyle\int
d^{3}k\frac{k_{\mu}k_{\nu}}{k_{0}}f_{i}(k).$ (26)

Equation (26) gives energy density as $\rho=T_{00}$, while pressure is
expressed through the trace over spatial part of the energy-momentum tensor,
$p=-T_{j}^{j}/3$. To find overall energy density and pressure entering
Friedmann equations one has to sum over all particle species, i. Entropy
density is calculated as $s={(\rho+p-\mu n)}/{T}$. Let us consider now
important limits of these expressions.

1\. Relativistic particles. First of all, for relativistic particles,
regardless of particular form of $f(k)$, we have $p=\rho/3$. In other words,
this relation is valid even out of thermal equilibrium and simply follows from
definitions since $k_{0}=|\vec{k}|$ for $m=0$. Further, for relativistic
plasma without chemical potentials, integrals in Eqs. (25) and (26) can be
calculated analytically and are slightly different for bosons and fermions.
Summing out over all particles we find

$\displaystyle n$ $\displaystyle=$ $\displaystyle
g_{*}^{\prime}\,\frac{\zeta(3)}{\pi^{2}}\,T^{3}$ (27) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle g_{*}\,\frac{\pi^{2}}{30}\;T^{4}$ (28)
$\displaystyle s$ $\displaystyle=$ $\displaystyle
g_{*}\,\frac{2\pi^{2}}{45}\,T^{3}$ (29)

where $\zeta(3)\approx 1.2$ is Rieman zeta function and

$\displaystyle g_{*}^{\prime}$ $\displaystyle=$
$\displaystyle{\footnotesize\sum_{{\rm bosons}}g_{i}+{3\over 4}\sum_{{\rm
fermions}}g_{i}}\,.$ $\displaystyle g_{*}$ $\displaystyle=$
$\displaystyle{\footnotesize\sum_{{\rm bosons}}g_{i}+{7\over 8}\sum_{{\rm
fermions}}g_{i}}\,.$

In these expressions particles with $m\ll T$ should be counted only, i. e.
$g_{*}$ and $g_{*}^{\prime}$ are functions of the temperature. Temperature
dependence of $g_{*}$ is shown in Fig. 2. Why it splits on $g_{s}$ and
$g_{\rho}$ at $T\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}\rm MeV$ will be explained later
on.

Figure 2: Left panel. Number of relativistic degrees of freedom, $g_{*}$, as a
function of temperature. Solid line - the Standard Model case, dashed line -
hypothetical behaviour in the Minimal Supersymmetric Standard Model (MSSM).
Important events in the Universe evolutions are also indicated. Right panel.
Increase of photon temperature over neutrino temperature during $e^{+}e^{-}$
annihilation.

2\. Non-relativistic particles. For non-relativistic particles all densities
are exponentially suppressed in thermal equilibrium and, again in the case
without chemical potentials, we find

$\displaystyle n_{i}$ $\displaystyle=$ $\displaystyle
g_{i}\,\left(\frac{m_{i}T}{2\pi}\right)^{3/2}\,\mathop{\rm
e}\nolimits^{-m_{i}/T}$ (30) $\displaystyle\rho_{i}$ $\displaystyle=$
$\displaystyle m_{i}\,n_{i}+\frac{3}{2}T\,n_{i}$ $\displaystyle p_{i}$
$\displaystyle=$ $\displaystyle T\,n_{i}$

Here expression (30) for $n_{i}$ is most important. In particular, it makes
the basis for Saha equation, which will be used repeatedly throughout the
lectures. This equation gives surviving amount of particles when they go out
of equilibrium and will be used to discuss Big Bang nucleosynthesis, hydrogen
recombination and abundance of thermally produced dark matter.

####  3.2.1 Cosmological density of neutrino

In the expanding Universe particle concentrations, $n$, are in equilibrium as
long as reaction rate is sufficiently high $\sigma nv>H$, where $\sigma$ is
corresponding crossection. After that distributions do not change in a
comoving volume, i.e. ”freeze-out”. Weak interaction rate for neutrino
$\sigma^{~{}}_{W}n\sim G_{F}^{2}T^{2}\cdot T^{3}$ matches expansion rate,
$H\sim T^{2}/M_{\rm Pl}$, when $G_{F}^{2}M^{~{}}_{\rm Pl}T^{3}\approx 1$. We
conclude that neutrino are in thermal equilibrium at temperatures $T\gg 1$ MeV
and decouple from the rest of plasma at lower temperatures. Therefore,
Standard Model neutrinos, which have small masses, decouple when they are
still relativistic. The number density of neutrino at this time is given by
Eq. (27) with $g_{*}^{\prime}=2$. Below this temperature, neutrinos are no
longer in thermal equilibrium with the rest of the plasma, and their
temperature simply decreases as $T\propto 1/a$. However, the cosmological
background of photons is heated up by the $e^{+}e^{-}$ annihilations shortly
after neutrino ”freeze-out”. Let us find a relation between $T_{\nu}$ and
$T_{\gamma}$, which will also give the relation between $n_{\nu}$ and
$n_{\gamma}$.

Recall that entropy in the comoving volume conserves, $g_{*}\;T^{3}={\rm
const}$. Before annihilation
$g_{*}=g_{\gamma}+g_{e}\cdot(7/8)=2+4\cdot(7/8)=11/2$. After annihilation
$g_{*}=g_{\gamma}=2$. (Neutrinos are decoupled already and do not participate
in these relations.) Since before annihilation $T_{\nu}=T_{\gamma}$ the
condition $g_{*}\;T^{3}={\rm const}$ gives for the neutrino temperature after
positron annihilation

$T_{\nu}=\left({4}/{11}\right)^{1/3}T_{\gamma}.$ (31)

This can be compared to the result of numerical integration of corresponding
Boltzmann equations which is shown in Fig. 2, right panel. Present day photon
temperature $T_{\gamma}=2.7255~{}K$, therefore, present day neutrino
temperature is $T_{\nu}=1.9454~{}K$. For the number density of one flavour of
left-handed neutrino and antineutrino we find
$n_{\nu}={3}n_{\gamma}/11=115\;{\rm cm}^{-3}$. Here we have used Eq. (27) and
$g_{\gamma}=g_{\nu}=2$. Right-handed neutrino, even if exist and light, are
not in thermal equilibrium and are excited in small amounts, see Ref. [11] and
Section 6.3.4 about ”sterile” neutrino as dark matter.

Assume that by now neutrino became non-relativistic, i.e. their masses are
larger than the present day temperature. In this case, neutrino energy density
is given by $\rho_{\nu}=\sum_{i}m_{\nu i}\,n_{\nu i}$. Since it has to be
smaller than $\Omega_{m}\,\rho_{c}$, we already have the constraint
$\sum_{i}m_{\nu i}<93\;\Omega_{m}h^{2}\;{\rm eV}\approx 10\;{\rm eV}$. Modern
cosmological constraints on neutrino masses are almost two orders of magnitude
stronger actually (see later in the lectures).

Now, using Eqs. (28) and (31) we find that after ”freeze-out” and $e^{+}e^{-}$
annihilation (but at $T>m_{\nu}$) the cosmological radiation background is
parametrized as

$\rho_{r}=\rho_{\gamma}+\rho_{\nu}=\frac{\pi^{2}}{15}\;T_{\gamma}^{4}\,\left[1+\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{eff}\right].$
(32)

At face value $N_{eff}$ stands here for the number of active neutrino flavours
and should be equal to three. But actually according to conventions used by
cosmologists, $N_{eff}\neq 3$, neither it is integer. The reasons are as
follows:

  * •

When $e^{+}e^{-}$ annihilate, neutrino are not decoupled completely yet since
neutrino ”freeze-out” temperature $\approx 1$ MeV is too close to $e^{+}e^{-}$
annihilation temperature. This leads to slight neutrino heating with distorted
distribution (24) and to somewhat larger neutrino energy density $\rho_{\nu}$
[12], which in applications is parametrised simply as larger $N$, to account
only for the increase of $\rho_{\nu}$,

$N_{eff}=3.046.$

  * •

There can be other contributions into radiation, e.g. light sterile neutrinos,
Goldstones, or some other very light particles. These contributions are called
”dark radiation”. They are also included into $N_{eff}$ and ”dark radiation”
is searched for in modern data as a signal that $N_{eff}>3.046$.

Therefore, $N_{eff}$ is another important cosmological parameter, potentially
signalling new physics.

Now we can also understand why $g_{*}$ splits on $g_{s}$ and $g_{\rho}$ at
$T\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}\rm MeV$. At these temperatures,
radiation consists of two fractions with different temperatures each, gas of
photons and gas of neutrino. Therefore, when writing Eqs. (28) and (29) we
have two options. We could have two terms in each of these equations, one for
photons, another for neutrino, each term would have different temperatures. Or
we can do the same way as in Eq. (32), including ratio of temperatures into
$g_{*}$ instead. And this latter approach has been decided to be more
convenient by cosmologists. Since temperature enters in different powers to
energy and entropy densities, we have splitting of $g_{*}$ on $g_{s}$ and
$g_{\rho}$. Asymptotic values of these functions, which can be used at $T<100$
keV, are shown below, assuming there is no dark radiation

$g_{\rho}(0)=2+3.046\;\left(\frac{7}{4}\right)\,\left(\frac{4}{11}\right)^{{4}/{3}}=3.38,$
(33)
$g_{s}(0)=2+3.046\,\left(\frac{7}{4}\right)\,\left(\frac{4}{11}\right)^{{~{}}~{}{~{}}}=3.94.$
(34)

##### Matter-radiation equality

Radiation energy density scales with expansion as $\rho\propto a^{-4}$, while
matter energy density scales as $\rho\propto a^{-3}$. It follows that the
Universe was radiation dominated at the early stages of the evolution. Let us
find now at which cosmological redshift and temperature the very important
event happens: namely, when the energy density of radiation becomes equal to
the energy density of matter. Using Eq. (32) with $N_{eff}=3.046$ and present
day photon temperature, Eq. (23), we obtain $\rho_{r}=4.41\times
10^{-10}\rm\;{GeV}\;{cm^{-3}}$ for the current radiation energy density.
Recall now the value of critical density, $\rho_{c}=1.05\times
10^{-5}\;h^{2}\;\rm{GeV}\;{cm^{-3}}$, Eq. [17), to get $\Omega_{r}=4.2\times
10^{-5}\,h^{-2}$. Since radiation scales as
$\rho_{r}=\rho_{c}\Omega_{r}(1+z)^{4}$ while matter as
$\rho_{m}=\rho_{c}\Omega_{m}(1+z)^{3}$, we find for the redshift of matter-
radiation equality

$1+z_{\rm eq}=\frac{\Omega_{m}}{\Omega_{r}}=3400,$

and for the corresponding temperature $T_{\rm eq}=0.8$ eV. Deriving this I
used values for $\Omega_{m}$ and $h$ from Table 2. Keeping those as free
parameters we have $T_{eq}=5.6\,\Omega_{m}\,h^{2}\;{\rm eV}$.

At higher temperatures the Universe was radiation dominated and its expansion
was governed by the following Hubble parameter

$H=\sqrt{\frac{8\pi
G\rho_{r}}{3}}=\sqrt{\frac{8\pi^{3}g_{*}T^{4}}{90\,M^{2}_{\rm Pl}}}\simeq
1.66\,\sqrt{g_{*}}\,\frac{T^{2}}{M_{\rm Pl}}.$ (35)

Since during radiation dominated stage $H=1/2t$, we obtain the Universe age
(in seconds) as a function of temperature

$t({\rm s})=\frac{2.42}{\sqrt{g_{*}}}\left(\frac{\rm MeV}{T}\right)^{2}.$

Stretching this time-temperature relation to equality temperature, and using
expression (33) for $g_{*}$, we find that at equality the Universe was 65
thousand years old.

###  3.3 Last scattering of light

Baryonic matter is ionized at temperatures higher than the hydrogen ionization
energy $E_{\rm ion}=13.6$ eV and photons are in thermal equilibrium with
primordial plasma. They cannot propagate large distances and the plasma is not
transparent. With expansion the Universe cools down. At some point protons and
electrons of primordial plasma recombined into neutral hydrogen and the
Universe became transparent for radiation. This happens when the mean free
path of photons becomes comparable to the size of the Universe at that time.
Corresponding temperature is called ”last scattering”. After that photons are
travelling without being affected by scattering. We see this light as Cosmic
Microwave Background Radiation (CMBR). More precisely, the CMBR comes from the
surface of the last scattering. We cannot see past this surface. Let us
determine here when the last scattering had occurred in the early Universe.

Fraction of ionized hydrogen as function of temperature can be described by
the Saha equation. It is derived by simply making ratios of number densities,
Eq. (30), of different fractions in question in thermal equilibrium. For the
case of hydrogen recombination

$\frac{n_{e}\,n_{p}}{n^{~{}}_{H}}=\left(\frac{m_{e}T}{2\pi}\right)^{3/2}\mathop{\rm
e}\nolimits^{-E_{\rm ion}/T}.$ (36)

Here $n_{e},~{}n_{p}$ and $n^{~{}}_{H}$ are the number densities of electrons,
protons, and neutral hydrogen respectively. Plasma is electrically neutral,
i.e. $n_{e}=n_{p}$. To find closed relation for the fraction of ionized atoms,
$X\equiv n_{p}/(n_{p}+n^{~{}}_{H})=n_{p}/n^{~{}}_{B}$, we need the relation
between the baryon number density, $n^{~{}}_{B}$, and temperature. This
relation can be parameterized with the help of an important cosmological
parameter called baryon asymmetry

$\eta=\frac{n^{~{}}_{B}}{n_{\gamma}}=\frac{n_{p}+n^{~{}}_{H}}{n_{\gamma}}=(6.1\pm
0.05)\times 10^{-10}\;,$ (37)

where ${n_{\gamma}}$ is the number density of photons, Eq. (27). Baryon
asymmetry can be estimated by an order of magnitude by simply counting the
number of baryons, or comparing element abundances predicted by the theory of
Big Bang Nucleosynthesis to observations. Those are not most precise methods,
though; the value presented in Eq. (37) was obtained from fitting the spectrum
of CMBR fluctuations, see below. Nowadays, this is the most precise
baryometer.

Defining recombination as the temperature when $X=0.1$, we find $T_{\rm
rec}\approx 0.3~{}{\rm eV}$. The Universe became transparent for radiation
when the mean free path of photons became comparable to the size of the
Universe at that time. Photons scatter mainly on electrons and we find that
the Universe became transparent when

$(\sigma_{\gamma e}\,n_{e})^{-1}~{}\sim~{}t\;.$ (38)

Here, $\sigma_{\gamma e}=8\pi\alpha^{2}/3m_{e}^{2}$ is the Compton cross-
section. For the temperature of last scattering we find $T_{\rm ls}\approx
0.26\;{\rm eV}$. Taking the ratio to the current CMBR temeperature we find
$z_{\rm ls}\approx 1000$.

CMBR is the oldest light in the Universe. When registering it, we are looking
directly at the deepest past we can, using photons. These photons had traveled
the longest distances without being affected by scattering, and geometrically
came out almost from the universe Horizon. Therefore the CMBR gives us a
snapshot of the baby Universe at the time of last scattering.

##  4  CMB power spectrum: tool of Precision Cosmology

Figure 3: Sky map of primordial temperature fluctuations in Galactic
coordinates (left panel) and temperature power spectrum (right panel) as
measured by Planck space observatory [2].

The temperature of CMBR is slightly different in different patches of the sky
- to 1 part in 100,000. These temperature deviations are shown in the sky map
Fig. 3, left panel. Measurements of these tiny fluctuations (anisotropy) in
CMBR temperature give us a wealth of cosmological information at an
unprecedented level of precision and became a most powerful probe of
cosmology. The functional form of the CMBR power spectrum is very sensitive to
both the various cosmological parameters and to the shape, strength and nature
of primordial fluctuations. This spectrum is shown in Fig. 3, right panel. In
fact, the values of cosmological parameters listed in Table I largely came out
from fitting model predictions to data as in this figure.

The temperature anisotropy, $T({\bf n})$, as a function of viewing direction
vector ${\bf n}$, as shown in Fig. 3, left panel, is naturally expanded in a
basis of spherical harmonic, $Y_{lm}$

$T({\bf n})=\sum\limits_{l,m}a_{lm}Y_{lm}({\bf n})\;.$ (39)

Coefficients $a_{lm}$ in this decomposition define the angular power spectrum,
$C_{l}$

$C_{l}={\frac{1}{2l+1}}\sum\limits_{m}|a_{lm}|^{2}\;.$ (40)

Assuming random phases, the r.m.s. temperature fluctuation assosiated with the
angular scale $l$ can be found as

$\Delta T_{l}=\sqrt{C_{l}\;l(l+1)/2\pi}\equiv\sqrt{D_{l}}.$ (41)

Spectrum, ${D_{l}}$, as measured by Planck collaboration, is shown in Fig. 3,
right panel. In fact, it was realized already right after the discovery of
CMBR, that fluctuations in its temperature should have fundamental
significance as a reflection of the seed perturbations which grew into
galaxies and clusters. In a pure baryonic Universe it was expected that the
level of fluctuations should be of the order $\delta T/T\sim 10^{-2}-10^{-3}$.
Mesurements of the CMBR anisotropy with ever-increasing accuracy have begun.
Once the temperature fluctuations were shown to be less than one part in a
thousand, it became clear that baryonic density fluctuations did not have time
to evolve into the nonlinear structures visible today. A gravitationally
dominant dark matter component was invoked. For explanations why it is
necessary, see Section 6. Eventually, fluctuations were detected [13] at the
level of $\delta T/T~{}\sim~{}10^{-5}$, consistent with the structure
formation in Cold Dark Matter models with the Harrison-Zel’dovich spectrum of
primordial perturbations motivated by cosmological Inflation, see Section 5
and Appendices.

The foundations of the theory of CMBR anisotropy were set out by Sachs & Wolfe
[14], Silk [15], Peebles & Yu [16], Syunyaev & Zel’Dovich [17]. The measured
spectrum of CMBR power has a characteristic shape of multiple peaks. Positions
of these peaks and their relative amplitudes are sensitive to many
cosmological parameters in a non-trivial way. Fitting the data to model
predictions gives very accurate values for many of these parameters (though
there are some degeneracies between deferent sets). Numerical calculations for
different models were done already in Ref. [18], and power spectra exhibiting
acoustic peaks (similar to those in Fig. 3, right panel) were presented. It
was realized, in particular, that positions of the peaks are shifted with
respect to each other for adiabatic and isentropic primordial fluctuations.

Figure 4: Left panel: Combination of SN Ia (blue contours) and Planck data
(green contours) tell us that the equation of state of the dark energy is
consistent with that of a vacuum, $w=-1$, from Ref [19]. Rigt panel:
Constraints on neutrino mass from combined Planck and BAO data, from Ref [20].

To improve significantly constraints on some cosmological parameters a
combination of CMBR with other data is needed. For example, combining Planck
data with Supernova data we find that the Dark energy equation of state is
close to a vacuum, $w=-1.02\pm 0.06$ [19], while each of these sets alone
would give weak constraints, see Fig. 4, left panel. Combination of Planck
data with data on correlation properties of galaxy clustering, namely data on
Baryon Acoustic Oscillations (BAO) tells us that the Universe is spatially
flat, $1-\sum\Omega_{i}=0.000\pm 0.005$, see Ref. [2]. That same data set
improves many other constraints. An example of constraints on the sum of
neutrino masses in this joint data set is shown in Fig. 4, right panel [20].
Solid blue line corresponds to the case of $\Lambda$CDM model, which means
zero spatial curvature and $w=-1$. Other curves on this figure correspond to a
Dark entry models with equation of state different from that of a vacuum. For
the $\Lambda$CDM the constraint on the neutrino masses is $\sum
m_{\nu}\rm<0.22~{}\rm eV$ with positive $2\sigma$ detection if Dark energy is
more complicated substance than vacuum. However, $\Lambda$CDM is a good model
and is consistent with all cosmological data to date.

Therefore $\Lambda$CDM can be safely assumed, and then other cosmological
parameters can be determined quite well from the CMBR data alone. For example,
parameters presented in Table 2 (except spatial curvature) were obtained from
the CMBR data alone assuming $\Lambda$CDM model. Note that the constraint on
$\Omega_{\Lambda}$ from the supernova luminosity distance relations, Section
2.2.3, I also obtained assuming the $\Lambda$CDM model. Otherwise from the SN
data alone we would only know for sure that the Universe expansion is
accelerating, but the fraction of dark energy, $\Omega_{DE}$, would depend
upon assumed equation of state $w$, as it is shown by blue contours in Fig. 4,
left panel.

###  4.1 Acoustic oscillations

As we could see already, large amount of cosmological information is encoded
in the functional form of the CMBR power spectrum. To get feeling of physics
which is behind, let us give a qualitative picture of why the CMBR power
spectrum has a specific shape of a sequence of peaks, and explain how it
depends on the values of particular cosmological parameters. Insight,
sufficient for the purposes of these lectures, can be gained with the
idealization of a perfect radiation fluid. In complete treatment, one has to
follow the evolution of coupled radiation and metric fluctuations, i.e. to
solve the linearized Einstein equations. However, essential physics of
radiation (or matter) fluctuations can be extracted without going into the
tedious algebra of General Relativity. It is sufficient to consider the
energy-momentum conservation, Eq. (2). To solve for metric perturbations, full
treatment based on Einstein equations, Eq. (1), is needed of course. We will
not do that here, but simply quote results for the evolution of the
gravitational potentials (coincident in some important cases with the
solutions for the Newtonian potentials).

Perturbations of the ideal radiation fluid, $p=\rho/3$, can be separated into
perturbations of its temperature, velocity and gravitational potential. In the
general-relativistic treatment gravitational potential appears as a fractional
perturbation of the scale factor in the perturbed metric

$ds^{2}=a^{2}(\eta)\,[(1+2\Psi)d\eta^{2}-(1-2\Phi)dx^{i}dx_{j}]\;.$ (42)

Two equations contained in the energy-momentum conservation,
$T^{\mu\nu}_{\,\,\,\,\,\,\,\,\,;\nu}=0$ (i.e. temporal $\mu=0$ and spatial
$\mu=i$ parts of this equation), written in metric (42), can be combined to
exclude the velocity perturbations. The resulting expression is simple

$\ddot{\theta}_{k}+\frac{k^{2}}{3}\theta_{k}=-\frac{k^{2}}{3}\Phi_{k}+\ddot{\Phi}_{k}\;.$
(43)

Note that this equation is the exact result for a pure radiation fluid. Here,
$\theta_{k}$ are Fourier amplitudes of $\delta T/T$ with wavenumber $k$, and
$\Phi_{k}$ is a Fourier transform of gravitational potential. Analysis of
solutions of the Einstein equations for $\Phi$ shows that $\Phi_{k}$ do not
depend upon time in two important cases:

  1. 1.

For superhorizon scales, which are defined as $k\eta\ll 1$.

  2. 2.

For all scales in the case of matter dominated expansion, $p=0$.

In these situations the last term in the r.h.s. of Eq. (43), namely,
$\ddot{\Phi}_{k}$, can be neglected. The Einstein equations also restrict the
initial conditions for fluctuations. For the adiabatic mode in the limit
$k\eta\ll 1$ one finds

$\delta_{0k}=-2\Phi_{0k}\;,$ (44)

where $\delta\equiv\delta\rho/\rho$, and subscript $0$ refers to the initial
values. The adiabatic mode is defined as a perturbation in the total energy
density. For the one component fluid, which we consider here, only the
adiabatic mode can exist. Note that fractional perturbation of the scale
factor in metric (42), $a(\eta,{\bf x})=a(\eta)+\delta a(\eta,{\bf x})\equiv
a(\eta)(1-\Phi)$, can be expressed as perturbation of spatial curvature, see
Eq. (4). Therefore, adiabatic perturbations are also called curvature
perturbations. Let us re-write Eq. (44) for temperature perturbations:

  * •

Radiation domination, $\delta=4\,\delta T/T$, and we find

$\theta_{0k}=-\frac{\Phi_{0k}}{2}\;.$ (45)

  * •

Matter domination, $\delta=3\,\delta T/T$, and we find

$\theta_{0k}=-\frac{2\Phi_{0k}}{3}\;.$ (46)

Recall now that in the limit $k\eta\ll 1$ the gravitational potentail is time-
independent, $\Phi={\rm const}$. Therefore, Eq. (43) has to be supplemented by
the following initial conditions in the case of the adiabatic mode:

$\theta_{0k}\neq 0,\hskip 28.45274pt\dot{\theta}_{0k}=0\;.$ (47)

##### Temperature fluctuations on largest scales.

Let us consider the modes which had entered the horizon after matter-radiation
equality, $k\eta_{\rm eq}<1$. For those modes, $\dot{\Phi}=0$ all the way from
initial moments till present, and the solutions of Eq. (43) with adiabatic
inital conditions is

$\theta_{k}+\Phi_{k}=(\theta_{0k}+\Phi_{0k})\,\cos\left(\frac{k\eta}{\sqrt{3}}\right)\;.$
(48)

As gravity tries to compress the fluid, the radiation pressure resists. As in
everyday physics, this leads to acoustic oscillations. But here, it is
important that oscillations are synchronized. All modes have the same phase
regardless of $k$. This is a consequence of $\dot{\theta}_{0k}=0$, which is
valid for all $k$. At the last scattering, the universe becomes transparent
for the radiation and we see a snapshot of these oscillations at
$\eta=\eta_{\rm ls}$.

To get its way to the observer, the radiation has to climb out of the
gravitational wells, $\Phi$, which are formed at the last scattering surface.
Therefore the observed temperature fluctuations are $\theta_{\rm
obs}~{}=\theta(\eta_{\rm ls})+\Phi$, or

$\theta_{k,\rm obs}=\frac{1}{3}\Phi_{0k}\,\cos\left(\frac{k\eta_{\rm
ls}}{\sqrt{3}}\right)\;,$ (49)

where we have used Eq. (46), which relates initial values of $\theta$ and
$\Phi$. Note that overdense regions correspond to cold spots in the
temperature map on the sky, since the gravitational potential is negative.
This is famous Sachs-Wolfe effect [14].

##### Acoustic peaks in CMBR.

Modes caught in the extrema of their oscillation, $k_{n}\eta_{\rm
ls}/\sqrt{3}=n\pi$, will have enhanced fluctuations, yielding a fundamental
scale, or frequency, related to the universe sound horizon,
$s_{*}\equiv\eta_{\rm ls}/\sqrt{3}$. By using a simple geometrical projection,
this becomes an angular scale on the observed sky. In a spatially flat
Universe, the position of the first peak corresponds to $l_{1}\approx 200$,
see below. Both minima and maxima of the cosine in Eq. (49) give peaks in the
CMBR power spectrum, which follow a harmonic relationship,
$k_{n}={n\pi}/s_{*}$, see Fig. 3, right panel.

The amplitudes of the acoustic peaks are recovered correctly after the
following effects are taken into account:

  1. 1.

Baryon loading. The effect of added baryons is exactly the same for the
oscillator equation Eq. (48), as if we had increased the mass of a load
connected to a spring, which oscillates in a constant gravitational field and
with the starting point on the top of an uncompressed coil at rest. The
addition of baryons makes a deeper compressional phase, and therefore
increases every other peak in the CMBR power spectrum. (First, third, fifth,
$\dots$) The CMBR power spectrum is a precise baryometer.

  2. 2.

Time-dependence of $\Phi$ after horizon crossing in radiation dominated
universe. Gravitational potentials are not constant, but decay inside the
horizon during radiation domination. This decay drives the oscillations: it is
timed in such a way that compressed fluid has no gravitational force to fight
with, when the fluid turns around. Therefore, the amplitudes of the acoustic
peaks increase as the cold dark matter fraction decreases, which allows to
measure $\Omega_{m}$.

  3. 3.

Dissipation. This leads to a dumping of higher order peaks in the CMBR power
spectrum.

##### Position of the first peak.

Position of the first peak is determined by the angular size of the sound
horizon at last scattering. Let us calculate here a similar quantity: the
causal horizon (which is larger by a factor of $\sqrt{3}$ in comparison with
the sound horizon). The comoving distance traveled by light, $ds^{2}=0$, from
the “Big Bang” to redshift z is determined by a relation similar to Eq. (21),
but with different integration limits

$\chi(z)=\int_{z}^{\infty}\frac{dz^{\prime}}{H(z^{\prime})}\;,$ (50)

where $H(z)$ is given by Eq. (20). One has to integrate this relation with a
complete set of $\Omega_{i}$. However, from the last scattering to $z\sim 1$,
the Universe was matter dominated. Therefore, the causal horizon in a matter
dominated Universe $\chi(z)={2}/{H_{0}\sqrt{1+z}}$ should give a reasonable
first approximation to the true value of integral in Eq (50). Consider now two
light rays registered at $z=0$ which were separated by a comoving distance
$\chi(z_{\rm ls}^{~{}})$ at the moment of emission. Since both propagate in
the metric $ds^{2}=a^{2}(d\eta^{2}-d\chi^{2}-\chi^{2}d\theta^{2})=0$, we find
for the angular size of horizon at last scattering

$\theta_{h}=\frac{\chi(z_{\rm ls})}{\chi(0)-\chi(z_{\rm
ls})}\approx\frac{1}{\sqrt{1+z_{\rm ls}}}=\sqrt{\frac{T_{0}}{T_{\rm
ls}}}\approx 1.7^{\circ}\;.$ (51)

Note that this is an approximate relation since we had neglected the dark
energy contribution into expansion of the Universe at late stages. To get
sound horizon we have to divide Eq. (51) by $\sqrt{3}$. Observationally, the
sound horizon angular scale is tightly constrained by Planck from the position
of the first peak: $\theta_{*}=0.59648^{\circ}\pm 0.00018^{\circ}$ [2]. This
is important direct observable, which can be used to set constraints on
cosmological parameters entering Eq. (51).

##### Horizon problem.

Relation (51) tells us that regions separated by more than $\;>\;2^{\circ}$ on
the map of microwave sky, Fig. 3, have not been in the causal contact prior to
the last scattering in the standard Friedmann cosmology. We should see
$10^{4}$ causally disconnected regions at the surface of last scattering.
Temperature could vary wildly from point to point which are further away from
each other than $2^{\circ}$. Yet, CMBR temperature is the same to better than
$10^{-4}$ accuracy all over the sky. Observations tell us that all sky regions
were somehow synchronized according to the adiabatic initial conditions, Eq.
(47), with only small initial perturbations present, $\Phi_{i}\ll 1$. This
constitutes the so-called “Horizon problem“ of standard cosmology. In Section
5 we will see how this problem is solved in frameworks of inflationary
cosmology.

##  5 Inflationary Cosmology

In frameworks of “classical” cosmology and assuming no fine-tuning, one
concludes that a typical universe should have had Plankian size, live Plankian
time and contain just a few particles. This conclusion is based on the
observation that Fridmann equations contain single dimension-full parameter
$M_{\rm Pl}\sim 10^{19}$ GeV, while dimensionless parameters naturally are
expected to be of order unity. Yet, the observable Universe contains $10^{90}$
particles in it and had survived $10^{65}$ Plankian times. Where does it all
came from? In other words, why is the Universe so big, flat
($\Omega_{0}\approx 1$) and old ($t>10^{10}$ years), homogeneous and isotropic
($\delta T/T\sim 10^{-5}$), why does it contain so much entropy ($S>10^{90}$)
and does not contain unwanted relics like magnetic monopoles? These puzzles of
classical cosmology were solved with the invention of Inflation [21, 22, 23,
24, 25, 26]. All these questions are related to the initial conditions and one
can simply postulate them. The beauty of Inflation is that it generates these
unnatural initial conditions of Big Bang, while the pre-existing state (which
can be arbitrary to a large extent) is forgotten. Moreover, with development,
Inflationary theory delivered unplanned bonuses. Not only does the Universe
become clean and homogeneous during inflation, but also the tiny perturbations
necessary for the genesis of galaxies are created with the correct magnitude
and spectrum. Below we consider the basics of Inflationary cosmology.

###  5.1 Big Bang puzzles and Inflationary solutions

By definition, Inflation is a period in the Universe evolution when
$\ddot{a}>0$ . Using the second Friedmann equation, Eq. (11), we find that the
inflationary stage is realized when $p<-\rho/3$. In particular, if $p=-\rho$
the energy density remains constant during expansion in accord with the first
law of thermodynamics, Eq. (12), and the physical volume expands exponentially
fast, $a(t)=\mathop{\rm e}\nolimits^{Ht}$, see Eq. (10). Let us see now how
the condition $\ddot{a}>0$ during some early stage solves problems of
classical cosmology.

##### Horizon problem

In Section 4.1 we have found that the angular size of horizon at the moment of
last scattering is $\approx 2^{\circ}$, see Eq. (51), which tells us that we
observe $10^{4}$ causally disconnected regions at the surface of last
scattering. The question then arises, why is the Universe so homogeneous at
large scales?

This problem can be solved if during some period of time the Universe
expansion was sufficiently fast. To find quantitative requirement, let us
consider a power low for the Universe expansion, $a(t)\propto t^{\gamma}$. The
physical size of a given patch grows in proportion to the scale factor,
$R_{\rm P}\propto\;a(t)$. On the other hand, Eq. (50) tells us that the
physical size of a causally connected region (horizon) grows in proportion
with time, $R_{\rm H}=a\chi=t/(1-\gamma)$.

The exponent $\gamma$ depends upon the equation of state, $\gamma=1/2$ for
radiation and $\gamma=2/3$ for the matter dominated expansion. In any case,
for the “classical” Friedmann Universe $\gamma<1$ and the horizon expands
faster than volume. Take the largest visible patch today. It follows that in
the past its physical size should have been larger than the horizon size
(since they are equal today) and therefore this patch should have contained
many casually disconnected regions. On the other hand, if during some period
of evolution $\gamma>1$, the whole visible Universe could have been inflated
from one (“small”) causally connected region. In such cosmology, any given
patch in the Universe passes the boundary of causally connected region twice.
First when it is inflated and becomes bigger than horizon, and second, when
the inflationary stage changes to ”Big Bang” and casually connected region at
some future point in time becomes larger than this given patch. Note that
$\gamma>1$ means $\ddot{a}>0$.

##### Curvature problem

The first Friedmann equation (10) can be re-written as

$k=a^{2}\left(\frac{8\pi
G}{3}\,\rho-H^{2}\right)=a^{2}H^{2}\,\left(\Omega-1\right)=\dot{a}^{2}\,\left(\Omega-1\right).$
(52)

Since $k$ is a constant, we immediately see the problem: during matter or
radiation dominated stages $\dot{a}^{2}$ decreases (this happens for any
expansion stage with $\ddot{a}<0$), and $\Omega$ is driven away from unity.
However, at present we observe $\Omega~{}\approx~{}1$. Therefore, initially
the Universe has to be extremely fine-tuned, say at the epoch of
nucleosynthesis, when temperature was $T\sim 1$ MeV, one should have
$|\Omega(t_{\rm NS})-1|<10^{-15}$, and even stronger tuning is required at
earlier epochs. A possible solution is obvious: accelerated expansion
$\ddot{a}>0$ increases $\dot{a}$ and therefore drives $\Omega(t)$ to unity
prior to radiation dominated stage. A robust prediction of inflationary
cosmology is a flat Universe, $\Omega=1$.

##### The problem of Entropy

As we know already, the energy of a vacuum stays constant despite the
expansion. In this way, room for matter full of energy could have been
created. If there is mechanism to convert vacuum energy into particles and
radiation at some later stage, then the observed huge entropy will be created
and the problem of entropy will be solved. Potentially, this mechanism works
for any inflationary scenario, since the product $\rho a^{3}$ is guaranteed to
grow whenever $\ddot{a}>0$. However, the important question is whether a
graceful exit out of the inflationary stage and successful reheating is indeed
possible. In practice, this issue has killed a number of inflationary models.
Remarkably, the original model by A. Guth [23] had being ruled out precisely
on these grounds [27].

##### For how long the inflationary stage should last?

Inflation has to continue for a sufficiently long time for the problems of
horizon, curvature and entropy to be solved. All these requirements give
roughly the same condition on the number of “e-foldings” of inflation [23] and
we present here a (simplified) derivation based on entropy. Multiplying the
current temperature in the universe by its visible size we find
$T_{0}a_{0}\chi_{0}\sim 10^{30}$, where $\chi_{0}$ is the comoving size of the
present horizon. We also want the whole visible universe to be inflated out of
a single causally connected patch. A given wave mode is in vacuum state when
its wavelength is smaller than the size of Hubble parameter during inflation
and becomes frozen as a classical fluctuation when it becomes larger. This is
just a consequence of the quantum field theory in a universe expanding with
acceleration, $\ddot{a}>0$, see Appendices. Therefore, important inflationary
period, which can be linked to observations, is from the moment when the patch
corresponding to the whole visible Universe goes out of the Hubble scale and
to the moment when inflation ends, i.e. from the moment $a_{i}\chi_{0}=H^{-1}$
until $a_{r}=\mathop{\rm e}\nolimits^{N_{*}}a_{i}$, where the number of
e-foldings, $N_{*}\equiv H\Delta t$, parametrises duration of this
inflationary period. At the end of inflation the vacuum energy goes to
radiation with temperature $T_{r}$ which is related to the present day
temperature as $T_{r}a_{r}=T_{0}a_{0}$, see Eq. (16), and we neglect here the
change in the number of relativistic degrees of freedom from $T_{r}$ to
$T_{0}$. This gives

$\frac{T_{r}}{H}\;\mathop{\rm e}\nolimits^{N_{*}}\approx 10^{30}\;.$ (53)

Number of e-foldings $N_{*}$ depends upon reheating temperature. In popular
models of Inflation the ratio ${T_{r}}/{H_{i}}$ is within a couple orders of
magnitude from unity, and we find $50\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}N_{*}\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}60$. I stress again that $N_{*}$
is not the duration of inflation. The latter cannot be smaller than $N_{*}$,
but inflation can last longer of course, and then our Universe is homogeneous
to scales much much larger than its visible part today.

###  5.2 Models of Inflation

Consider energy-momentum tensor $T_{\mu\nu}$ for a scalar field $\varphi$

$\displaystyle T_{\mu\nu}=\partial_{\mu}\varphi\,\partial_{\nu}\varphi-
g_{\mu\nu}\,{\cal L},$ (54)

with the Lagrangian

$\displaystyle{\cal
L}=\partial_{\mu}\varphi\,\partial^{\mu}\varphi-V(\varphi)\;.$ (55)

In a state when all derivatives of $\varphi$ are zero, the stress-energy
tensor of a scalar field simplifies to
$T_{\mu}^{\nu}=V(\varphi)\,\delta_{\mu}^{\nu}$. This corresponds to a vacuum
state. Indeed, comparing with Eq. (9), we find $V=\rho=-p$. A large number of
inflationary models exists where $\varphi\approx$ const during some period of
evolution and vacuum-like state is imitated. Such field is called inflaton.

1\. False vacuum inflation. Conceptually simple and easily understandable
scenario was suggested by A. Guth [23]. Consider potential $V(\phi)$ which has
a local minimum with a non-zero energy density separated from the true ground
state by a potential barrier. A universe which happened to be trapped in the
meta-stable minimum will stay there for a while (since such a state can decay
only via subbarrier tunneling) and expansion of the universe will diminish all
field gradients. Then the Universe enters a vacuum state and Inflation starts.
Subsequent phase transition into the true minimum ends inflationionary stage
and creates the radiation phase. Today the model of Guth and its variants
based on potential barriers is good for illustration purposes only. It did not
stand up to observations since inhomogeneities which are created during the
phase transition into the radiation phase are too large [27]. But the model
gives easily understandable answer to the frequently asked question: how can
it be that the energy density stays constant despite the expansion?

2\. Chaotic inflation. Andrei Linde was first to realize that things work in
the simplest possible setup [26]. Consider potential

$V(\phi)=\frac{1}{2}m_{\phi}^{2}\phi^{2}.$ (56)

Field equation in an expanding Universe and for the homogeneous mode is
$\ddot{\phi}+3H\dot{\phi}+m_{\phi}^{2}\phi=0.$ If $H\gg m_{\phi}$, the
“friction” $3H\dot{\phi}$ dominates and the field does not move (almost).
Therefore, time derivatives in $\;T_{\mu\nu}$ can be neglected, and inflation
starts (in a sufficiently homogeneous patch of the Universe). A Hubble
parameter in this case is determined by the potential energy, $H\approx
m_{\phi}{\phi}/{M_{\rm Pl}}$, and we see that inflation starts if the initial
field value happened to satisfy ${\phi>M_{\rm Pl}}$. During inflationary stage
the field slowly rolls down the potential hill. This motion is very important
in the theory of structure creation, see Appendices. Inflation ends when
${\phi\sim M_{\rm Pl}}$. At this time, field oscillations start around the
potential minimum and later decay into radiation. In this way all matter
content has been likely created in our Universe. In general, this model
generalises to arbitrary monomial $V(\phi)\propto\phi^{\alpha}$ as field
potential at large $\phi$.

3\. $R^{2}$ \- inflation. Historically, this is the first model of inflation.
It was invented by A. Starobinsky [21, 22]. Einstein-Hilbert action, leading
to Einstein equations (1) should be modified inevitably in quantum field
theory on a curved space-time. In particular, counter-terms proportional to
the squares of different curvature tensors should be added to cancel
divergences. Starobinsky considered the simplest form of extended
gravitational Lagrangian

${\it L}=\frac{M_{\rm Pl}^{2}}{2}R+\beta R^{2},$ (57)

where $R$ \- scalar curvature and $\beta$ \- some dimensionless constant.
Universe inflates in this model. It can be understood as follows. After
conformal rotation this model is equivalent to the usual Einstein gravity plus
a scalar field with potential

$V(\phi)=\Lambda^{4}\left(1-\mathop{\rm e}\nolimits^{-\sqrt{2/3}\,\phi/M_{\rm
Pl}}\right)^{2}.$

This potential has very flat plateau at $\phi>M_{\rm Pl}$, and with such
initial value of $\phi$ the Universe will inflate. The Universe will be heated
up in the same way is in chaotic inflation after $\phi$ will slowly reach
$M_{\rm Pl}$.

###  5.3 Unified theory of Creation

During Inflation the Universe was in a vacuum-like state. We have to figure
out how this “vacuum” had been turned into the matter we observe around us,
and how primordial fluctuations which gave rise to galaxies were created.
Solution to all these problems can be understood in a single unified approach.
Basically, everything reduces to a problem of particle creation in a time-
dependent classical background. On top of every “vacuum” there are
fluctuations of all quantum fields which are present in a given model. This
bath of virtual quanta is indestructible, and even Inflation cannot get rid of
it. Being small, fluctuations obey an oscillator equation

$\ddot{u}_{k}~{}+~{}[k^{2}+m^{2}_{\rm eff}(\eta)]\;u_{k}=0\,,$ (58)

here $u_{k}$ are amplitudes of fluctuating fields in Fourier space. Effective
mass becomes time dependent through the coupling to time-dependent background.
Because $m_{\rm eff}$ is time dependent, it is not possible to keep
fluctuations in a vacuum. If oscillators with momentum $k$ happened to be in
the vacuum at one time, they will not be in the vacuum at a latter time
because positive and negative frequency solutions mix, see Appendices. Several
remarks are in order.

  * •

Eq. (58) is valid for all particle species.

  * •

The equation looks that simple in a conformal reference frame
$ds^{2}=a(\eta)^{2}\;(d\eta^{2}-dx^{2})$. (And a “dot” means derivative with
respect to $\eta$.)

  * •

Of particular interest are ripples of space-time itself: curvature
fluctuations (scalar fluctuations of the metric) and gravity waves (tensor
fluctuations of the metric).

  * •

Effective mass $m_{\rm eff}$ can be non-zero even for massless fields.
Gravitational waves give the simplest example [28], with $m^{2}_{\rm
eff}=-{\ddot{a}}/{a}$. The effective mass for curvature fluctuations has a
similar structure $m^{2}_{\rm eff}=-{\ddot{z}}/{z}$, but with $a$ being
replaced by $z\equiv a\dot{\phi}/H$, see Refs. [29, 30, 31, 32].

  * •

For a scalar field which does not couple to the inflation, the effective mass
is given by Eq. (71). For conformally coupled, but massive scalar it reduces
to $m_{\rm eff}=m_{0}\,a(\eta)$.

Note that creation in Inflationary theory is possible because nature is not
conformally-invariant. Otherwise, $m_{\rm eff}$ would be time-independent and
vacuum would remain vacuum forever. There are two important instances of time
varying classical background in cosmology: expansion of space-time, $a(\eta)$,
and motion of the inflaton field, $\phi(\eta)$. Both can be operational
separately or together at any epoch of creation:

  * •

During inflation. This is when superhorizon size perturbations of metric are
created, which give seeds for the formation of galaxies and Large Scale
Structure in general.

  * •

After inflation while the inflaton oscillates. This is when matter itself is
created out of energy generated from the vacuum.

There are several primary observables which can be calculated out of $u_{k}$
and further used for calculation of quantities of interest. Most useful are:

  * •

The particle occupation numbers, $n_{k}$. Integration over $d^{3}k$ gives the
particle number density.

  * •

The power spectrum of field fluctuations, $P(k)\equiv u_{k}^{*}u_{k}$.
Integration over $d^{3}k$ gives the field variance.

Depending on physical situation, only one or the other may have sense. The
particle number in a comoving volume is useful because it is adiabatic
invariant on sub-horizon scales (or when $m>H$) and allows to calculate the
amount of created matter and abundances of various relics, e.g. dark matter.
But it has no meaning at super-horizon scales when $m<H$. Then the power
spectrum of field fluctuations is used which allows to calculate density
perturbations and gravitational waves generated during inflation. Necessary
details of such calculation are given in Appendices.

####  5.3.1 Testing Inflationionary predictions

Typically, the spectrum of curvature perturbations generated during inflations
has a form $P_{\zeta}(k)=A_{s}k^{n_{s}}$, where $A_{s}$ and ${n_{s}}$ are
constants (i.e. weakly depend upon scale $k$). Similarly, for gravitational
waves $P_{T}(k)=A_{T}k^{n_{T}}$. To the first approximation, the Hubble
parameter $H$ during inflation is constant. Then, power spectra do not depend
on $k$ and $n_{S}=1$, $n_{T}=0$. This case is called the Harrison-Zel’dovich
spectrum [35, 36] of primordial perturbations which has been suggested on
general grounds before inflationary theory was invented. However, in reality,
$H$ is changing and these constants take different, model dependent values.
Nevertheless, there is model independent relation between the slope of tensor
perturbations and the ratio of power in tensor to curvature modes

$r\equiv\frac{P_{T}(k)}{P_{\zeta}(k)}=-8n_{T}\;.$ (59)

This is called the consistency relation to which (simple) inflationary models
should obey. It will be robust and ultimate test of inflationary theory when
imprint of gravitational waves in a CMBR will be discovered.

Figure 5: Constraints on inflationary models in the $(n_{s},r)$ plane, from
Ref. [34]. Coloured line segments with circles at ends correspond to
predictions of different inflationary models with different inflation
potentials. Within each segment $N_{*}$ varies in the interval $50\leq
N_{*}\leq 60$, see Eq. (53).

However, both $A_{s}$ and $n_{s}$ are measured, extracted from CMBR
observations and can be compared to model predictions. The most recent
constraints in the $(n_{s},r)$ plane, obtained by Planck collaboration [34]
are presented in Fig. 5. We see that in chaotic inflationary model, Eq. (56),
the gravitational waves would have been already discovered by Planck, and this
model is ruled out nowdays. Best runner is $R^{2}$ model of A. Starobinsky,
Eq. (57), which is a perfect fit. However, observation of the imprint of
gravity waves in this model will be very difficult task, if possible at all.

To summarise, all predictions of Inflationary cosmology, which could have
being tested so far, have being confirmed. In particular, the Universe is
spatially flat (within experimental errors), see Table I. The primordial
perturbations are of superhorizon size and adiabatic. The spectral index is
close to the Harrison-Zeldovich case, see Fig. 5. Crucial test of inflationary
paradigm would be detection of gravity waves and verification of the
consistency relation.

##  6 Dark Matter

We have seen already in Section 4 that CMBR observations accurately measure
the nature and spectrum of the primordial fluctuations, the geometry of the
Universe, its present expansion rate and its composition, see also Table 2,
which is based on these observations. In particular, these measurements tell
us that in addition to baryonic matter there should be also dark matter which
so far had been seen only through its gravitational influence. This
”sterility” leaves open the possibility that in fact we should look for
modification of gravity, not for dark matter, in order to explain the missing
mass problem. While both possibilities are exciting and beyond contemporary
physics, a successful modified gravity theory was not constructed yet.
Therefore, I will not discuss numerous attempts and various models of modified
gravity here, instead I’ll just give two original references, the early one
[37], and the most recent one [38]. It is difficult to construct such a theory
for several reasons. In particular, the evidence for missing mass exists at
various scales and epochs while modification should explain everything.
Contrary to that, e.g. simple variants of MOND [37] do explain the ”missing
mass” on galactic scales without invoking dark matter, but fail to explain
other evidence. Moreover, MOND is phenomenological, non-relativistic
prescription, not a theory. Therefore, other cosmological tests, beyond CMBR,
are also important. Below we consider cosmological observations that are
independent of the CMB but also point to the existence of non-baryonic dark
matter. At the end of the section I briefly discuss some popular models of
dark matter and present status of dark matter searches in corresponding
models.

###  6.1 Dark Matter: the evidence

Missing mass is seen on all cosmological scales. In particular, it reveals
itself as

  * •

Flat rotational curves in galaxies;

  * •

Gravitational potential which confines galaxies and hot gas in clusters;

  * •

Gravitational lenses in clusters;

  * •

Gravitational potential which allows structure formation from tiny primeval
perturbations;

  * •

Gravitational potential which creates CMBR anisotropies.

In this subsection I shortly review this overwhelming evidence for the unseen,
but gravitating mass.

####  6.1.1 Dark Matter in Galaxies

Figure 6: Left panel: Rotational curve of the galaxy NGC6503. I superimposed
the optical image of corresponding galaxy with the rotational curve,
approximately to scale in radius. Right panel: Scaled cluster density profiles
extracted from X-ray observations of different clusters, from Ref. [45].

Consider a test particle which is orbiting a body of mass $M$ at a distance
$r$. Within the frameworks of Newtonian dynamics the velocity of a particle is
given by

$v_{\rm rot}=\sqrt{G\,M(r)\over r}\;.$ (60)

Outside of the body, the mass does not depend on distance, and the rotational
velocity should obey the Kepler law, $v_{\rm rot}\propto r^{-1/2}$. Planets of
the Solar system obey this law. However, this is not the case for stars or gas
which are orbiting galaxies. Far away from the visible part of a galaxy,
rotational curves are still rising or remain flat. An example is shown in Fig.
6, left panel. An optical image of the NGC6503 galaxy is superimposed with its
rotational curve, approximately to correct scale. The contribution of visible
baryons in the form of stars and hot gas can be accounted for, and the
expected rotational curve can be constructed. The corresponding contributions
are shown in Fig. 6, left panel. One can see that the data-points are far
above the contribution of visible matter. The contribution of missing dark
mass, which should be added to cope with data, is also shown and is indicated
as Dark Halo. For the rotational velocity to remain flat, the mass in the halo
should grow with the radius as $M(r)\propto r$, i.e., the density of dark
matter in the halo should decrease as $\rho(r)\propto r^{-2}$.

####  6.1.2 Dark matter density profiles.

To interpret what is seen in the data, in particular, to interpret the results
of direct and indirect dark matter searches, and to plan for further strategy,
it is important to know the expected phase-space structure of the dark halo
and corresponding dark matter density profiles. For interacting particles a
thermal distribution over energies is eventually established. However, in
conventional cold dark matter models, particles are non-interacting, except
gravitationally. Binary gravitational interactions are negligible for
elementary particles, and resulting phase-space distributions are not unique,
even for a stationary equilibrium states, and even if constraint to the flat
rotational curves is enforced. Below I highlight several such distributions,
which are often discussed in the literature and are used in applications.

1\. The simplest self-gravitating stationary solution which gives flat
rotational curves corresponds to an “isothermal sphere” with Maxwellian
distribution of particles over velocities:

$n(\vec{r},\vec{v})=n(r)\;e^{-v^{2}/v^{2}_{0}}\;.$ (61)

Solution of the equation of hydrostatic equilibrium can be approximated by the
density profile

$\rho(r)=\frac{\rho_{0}}{(1+x^{2})},~{}~{}~{}{\rm where}~{}~{}x\equiv
r/r_{c}\;.$ (62)

It should be stressed that the distribution Eq. (61), in contrast to a
distribution in real thermal equilibrium, depends on particle velocities, not
on their energies. Such distributions may arise in time-dependent
gravitational potential as a result of collisionless relaxation.

2\. There exist several density profiles which are empirical fits to numerical
simulations, most often used is Navarro, Frenk & White (NFW) profile [39].

$\rho(r)=\frac{\rho_{0}}{x\;(1+x)^{2}}.$ (63)

3\. In the CDM model, the distribution of dark matter particles in the phase
space during initial linear stage prior to structure formation corresponds to
thin hypersurface, ${\bf v}=H{\bf r}$ (this is just Hubble law). Since during
collisionless evolution the phase-space density conserves as a consequence of
the Liouville theorem, then even at the non-linear stage the distribution will
still be a thin hypersurface. It can be deformed in a complicated way and
wrapped around, but it cannot tear apart, intersect itself, puff up or
dissolve. The resulting idealised phase-space distribution describing isolated
dark halo has been studied in Ref. [40] for the case of spherical symmetry.
Initial thin hypersurface gets wrapped around indeed, forming large and ever
increasing number of folds in the phase space in the inner galaxy. Existence
of such a folded structure is a topological statement. Corresponding model is
also called ”infall model”. It reproduces flat rotational curves, but there
are several interesting differences to other distributions, though. Rotational
curves of the infall model have several small ripples which appear near
caustics in the dark matter distribution. (Those are places in the phase-space
where particles turn-around and have zero velocity). It is important to note
that caustics may be observable and their discovery in real data will prove
that the missing mass is dark matter indeed, not a modification of gravity,
even if dark matter particles will not be directly identified. The energy
spectrum of dark matter particles at a fixed position also deviates from other
distributions. This may be important for the direct dark matter searches.
Also, the infall model gives the insight [40] on why the empirical fit, the
NFW profile, has this particular form, Eq (63). Observationally, signature of
the infall is seen e.g. in our local group of galaxies [41], but at largest
distances, outside of caustics. Caustics which are furthest away from the
galaxy centre were resolved recently in the high-resolution N-body modelling
of galaxy formation [42]. To understand how far the folded structure of the
infall model continues into the inner halo in a galaxy like our own will
require even larger simulations with better resolution.

####  6.1.3 Dark Matter in Clusters of Galaxies

Already in 1933, F. Zwicky [43] deduced the existence of dark matter in the
Coma cluster of galaxies. Nowadays, there are several ways to estimate masses
of clusters: based on the kinetic motion of member galaxies, on X-ray data,
and on gravitational lensing. These methods are different and independent. In
the dynamical method, it is assumed that clusters are in virial equilibrium,
and the virialized mass is easily computed from the velocity dispersion. In
X-ray imaging of hot intracluster gas, the mass estimates are obtained
assuming hydrostatic equilibrium. Mass estimates based on lensing are free of
any such assumptions. All methods give results which are consistent with each
other, and tell that the mass of the luminous matter in clusters is much
smaller than the total mass. Recent review on basic properties of clusters and
their role in modern astrophysics and cosmology can be found e.g. in [44].

Kinetic mass estimates.  Those are based on the virial theorem, $\langle
E_{\rm pot}\rangle+2\langle E_{\rm kin}\rangle=0$. Here $\langle E_{\rm
kin}\rangle$ is averaged kinetic energy of a constituents in the
gravitationally bound object (e.g. cluster of galaxies) and $\langle E_{\rm
pot}\rangle$ is their averaged potential energy. Measuring the velocity
dispersion of galaxies in the clusters and its geometrical size gives an
estimate of the total mass,

$M\sim\frac{2\langle r\rangle\langle v^{2}\rangle}{G}\;.$ (64)

The result can be expressed as mass-to-light ratio, $M/L$, using the Solar
value of this parameter. For the Coma cluster, which consists of about 1000
galaxies, Zwicky [43] has found

${M}/{L}\sim 300\,h\,\left({M_{\odot}}/{L_{\odot}}\;\right).$ (65)

Modern techniques end up with very much the same answer.

Mass estimates based on X-rays.  Mass is also traced in clusters of galaxies
by the hot gas which is visible in X-rays. Assume hot gas is in thermal
equilibrium in a gravitational well created by a cluster. Then, cluster mass
profiles can be derived from the gas density and temperature as functions of
the distance to a cluster centre. This independent method has its own
advantages and disadvantages. With respect to galaxy dynamics (see above) or
lensing mass estimates (see below), this method has the advantage of being
less sensitive to projection effects along the line of sight through the
cluster. However, validity of the assumptions of ICM hydrostatic equilibrium
and spherical symmetry of the cluster gravitational potential wells may depend
on the evolutionary state of the cluster.

As an example, the radial density profiles derived in Ref. [45] from the
Chandra X-ray satellite data are shown in Fig. 6, right panel. We see that
dark matter density exceeds gas density by an order of magnitude at any value
of the radius. Dark matter density as a function of radius is well fitted by
NFW profile, Eq. (63), which is shown by thick yellow line. Total gas mass
fractions varies between 5 and 15 percent from cluster to cluster and
systematically depends upon cluster mass. These values are somewhat lower than
the Universal baryon fraction suggested by the CMB observations, but
approaches it for the heaviest clusters.

Figure 7: Left panel: Image of the cluster Abell 2218 taken with the Hubble
space telescope (see Ref. [46]). Spectacular arcs resulting from strong
lensing of background galaxies by dark matter in the cluster are clearly seen.
Right panel: Deep Chandra image of the Bullet cluster. The X-ray brightness of
the gas component is coded in yellow, red and blue colours. Distribution of
the gravitating mass, obtained from weak lensing reconstruction, is shown by
green contours, from Ref. [50].

Gravitational Lensing.  As photons travel from a background galaxy to the
observer, their trajectories are bent by mass distributions. This effect of
gravitational lensing allows direct mass measurement without any assumptions
about the dynamical state of the cluster. The method relies on the measurement
of the distortions that lensing induces on the images of the background
galaxies, an example of such distortions is shown in Fig. 7, left panel. A
reconstruction of lens geometry provides a map of the mass distribution in the
deflector. For a review of the method see e.g. Ref [47]. The images of
extended sources are deformed by the gravitational field. In some cases, the
distortion is strong enough to be recognized as arcs produced by a galaxy
cluster serving as a lens, see Fig. 7, left panel. For the cluster A 2218,
shown in this figure, Squires et al. [48] compared the mass profiles derived
from weak lensing data and from the X-ray emission. The reconstructed mass map
qualitatively agrees with the optical and X-ray light distributions. A mass-
to-light ratio of $M/L=(440\pm 80)h$ in solar units has been derived. The gas
to total mass ratio was found to be $M_{\rm gas}/M_{\rm tot}=(0.04\pm
0.02)\,h^{-3/2}$. The radial mass profile agrees with the mass distribution
obtained from the X-ray analysis. For a recent study of mass density profiles
of galaxy clusters derived from the gravitational lensing see e.g. Ref. [49].
A sample of 50 galaxy clusters at 0.15 ¡ z ¡ 0.3 has been studied. Again, dark
matter density as a function of radius is perfectly fitted by the NFW profile,
Eq. (63), but ”isothermal” profile is a bad fit.

Dark matter or modification of gravity?  In principle, the excess
gravitational force, undoubtedly observed in galaxies and clusters of
galaxies, could be not a manifestation of the Dark Matter, but may have origin
in some modification of Einstein gravity. Gravitational lensing studies of the
Bullet Cluster 1E 0657-56 are claimed [50] to provide the best evidence to
date for the existence of dark matter, as opposed do modifications of gravity.
The Bullet Cluster consists of two colliding clusters of galaxies.
Reconstructed distribution of the gas, stars and gravitating matters shown in
Fig. 7, right panel. The X-ray brightness of the hot gas is coded in yellow,
red and blue colours. Distribution of the gravitating mass is shown by green
contours and was obtained from weak lensing reconstruction. It coincides with
distribution of stars, but counts of stars gives small contribution to the
overall mass balance. The hot gas of the two colliding components, seen in
X-rays, represents most of the baryonic, i.e. ordinary, matter in the cluster
pair. The hot gas in this collision was slowed down by a drag force. In
contrast, the dark matter or stars were not slowed by the impact, because they
do not interact strongly with itself or the gas except through gravity. This
produced the separation of gravitating matter and gas seen in the data. If hot
gas was the most massive component in the clusters, and dark matter would be
absent, as proposed by alternative gravity theories, such a separation would
not have been seen. Therefore, dark matter is required to explain what is seen
here.

####  6.1.4 Structure formation and DM

By present time the structures in the Universe (i.e. galaxies and clusters)
are formed already, in other words perturbations in matter have entered non-
linear regime, $\delta\rho/\rho\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$>$}}1$. However, the initial
perturbations were small $\delta\rho/\rho\sim 10^{-5}$, as we know from
measurements of temperature fluctuations in CMBR, see Section 4. Perturbations
do not grow significantly in the radiation dominated epoch, they can start
growing only during matter domination and are growing then in proportion to
the scale factor, $\delta\rho/\rho\sim a=1/z$. Moreover, baryonic plasma is
tightly coupled to radiation, therefore perturbations in baryonic matter start
to grow only after recombination. For the same reason, initial perturbations
in baryons at the time of recombination are equal to fluctuations in CMBR. If
baryons were to constitute the only matter content, then perturbations in
matter at present time would be equal to

$\frac{\delta\rho}{\rho}|_{\rm today}=z_{\rm
rec}\;\frac{\delta\rho}{\rho}|_{\rm rec}\sim 10^{-2}\;,$ (66)

where $z_{\rm rec}\approx 1100$ is the redshift of recombination. This
apparent contradiction is resolved by the dark matter. In our Universe
structure has had time to develop only because perturbations in non-baryonic
dark matter have started their growth prior to recombination. Baryonic matter
then “catch up” simply by falling into already existing gravitational wells.
This is one of the strongest and simplest arguments in favour of non-baryonic
dark matter.

###  6.2 Dark Matter: particle candidates

Cosmology tells us that the Standard Model of particle physics is incomplete.
The model which will extend it should contain particles which would constitute
non-baryonic dark matter. And there should exist some mechanism to produce it
with correct abundance, $\Omega_{\rm DM}\approx 0.27$, see Table 2. Also,
trusted and popular DM candidates appear naturally in the models whose origin
is unrelated to the dark matter problem. There is no shortage of particle
physics models which obey those requirements, with the huge range of DM
paricle masses and very different production mechanisms. Some dark matter
particle candidates are listed in Table 3.

Table 3: Dark Matter particle candidates candidate | mass | some refs
---|---|---
Graviton |  $10^{-21}$ eV | [51]
Axion |  $10^{-5}$ eV | [52]
Sterile neutrino |  $10$ keV | [53]
Mirror matter |  $1$ GeV | [54]
WIMP |  $100$ GeV | [55]
WIMPZILLA |  $10^{13}$ GeV | [56]

Given concrete model of particle physics, a theorist should first calculate
the cosmological abundance of DM produced in the model in hands. Below, in the
subsection 6.3, I give some examples of such calculations to highlight various
mechanisms of DM production. Then, in the subsection 6.4, I briefly describe
vast topic of direct and indirect searches for most popular DM candidates,
with corresponding derived constraints.

###  6.3 Production mechanisms

Depending upon production mechanism, the resulting dark matter can appear as
’cold’, ’warm’ or ’hot’. Loosely speaking, velocities of cold dark matter are
so small that they are not influencing the large scale structure formation at
all. Velocities of hot dark matter particles are too big. Their kinetic energy
does not allow particles to clump galaxy halos and may smear out even clusters
of galaxies. Such DM is ruled out. Warm dark matter is the intermediate case.
It may wash out structure at smallest observable scales of dwarf galaxies but
does not influence formation of big haloes like our Milky Way. Cold dark
matter models have some problems explaining observations at small scales, the
warm dark matter models have some advantages here, see below.

Further, dark matter particle candidates can be divided into several classes
according to a mechanism of their production in the early universe. We start
with popular class of DM candidates referred to as ”thermal relics”.

####  6.3.1 Cosmological abundance of thermal relics

By definition, a thermal relic is assumed to be in thermodynamic equilibrium
at early times. At some point in the evolution particles go out of equilibrium
and after that their number in a comoving volume remains constant. The process
is called ”freeze-out”. For thermal relics it is just the value of particle
mass which determines if it will be hot, warm or cold. To see this, let us
define the free streaming length for a given DM particle species with mass
$M_{X}$ as a horizon size at a temperature when particles are still
relativistic, i.e. at $T\sim M_{X}$. Clearly, structure will be washed out at
all scales smaller than this. Later on, particles are non-relativistic and
cannot move much farther away. Structure is preserved at larger scales.
Horizon size at $T\sim M_{X}$ expanded to present epoch is given by

$L_{fs}\sim\frac{M_{\rm Pl}}{T_{0}M_{X}}.$

For $M_{X}\sim 1~{}{\rm eV}$ this gives $L_{fs}\sim 100~{}{\rm Mpc}$. Clearly,
models with such a big free streaming length are ruled out. On the other hand,
for $M_{X}\sim 1~{}{\rm keV}$ we find $L_{fs}\sim 0.1~{}{\rm Mpc}$. This
corresponds to the size of a dwarf galaxy. Therefore, this gives the lower
bound for the warm DM particle mass:

$M_{X}>1~{}{\rm keV}.$

For thermal relics the resulting dark matter will be definitely cold if
freeze-out occurs when particles are non-relativistic, i.e. at temperatures
smaller than particle mass, $T<M_{X}$. WIMP, and in particular neutralino,
appearing in supersymmetric models, belong to this class of dark matter.

A given particle species will track the equilibrium abundance as long as
reactions which keep them in chemical equilibrium can proceed rapidly enough.
Here, ”rapidly enough” means that the mean free time between interactions is
smaller than the age of the universe, $\tau<t_{u}$. This condition can also be
written as $n\sigma v>H$. In thermal equilibrium, the number density of non-
relativistic particles is given by Eq. (30). In this regime the number density
decreases exponentially fast with decreasing temperature $T$. When the rate of
reactions becomes lower than the expansion rate, the particles can no longer
track the equilibrium value and thereafter particle concentrations in a
comoving volume remain constant. Clearly, the more strongly interacting
particles are, the longer they stay in equilibrium, and the smaller their
freeze-out abundance will be, see Fig. 8. Here we defined particle abundance
as the ratio of particle number to the entropy density, $Y\equiv n/s$.

Figure 8: A schematic view of comoving number density of a stable species as
they evolve through the process of thermal freeze-out.

Freeze-out concentration $n$ is determined by the condition $n_{f}\sigma
v\approx H$, or (neglecting numerical factors)

$n_{f}\approx\frac{H}{\langle\sigma
v\rangle}\approx\frac{T_{f}^{2}}{M_{Pl}\langle\sigma v\rangle}.$

After freeze-out the ratio of $n$ and entropy density $s$ remains constant. In
particular, present density is given by $n_{0}=n_{f}s_{0}/s_{f}$. Therefore

$\Omega_{DM}^{~{}}\equiv\frac{mn_{0}}{\rho_{c}}=\frac{mn_{f}}{s_{f}}\frac{s_{0}}{\rho_{c}}\sim\frac{m}{T_{f}}\frac{1}{\langle\sigma
v\rangle}\frac{T_{0}^{3}}{M_{Pl}\rho_{c}}.$

Freeze-out temperature $T_{f}$ cannot go much below particle mass m, see Fig.
8. One gets $x_{f}\equiv m/T_{f}=20-30$ for all practically interesting values
of annihilation cross-section. Restoring now all numerical factors in the
above estimate we obtain

$\Omega_{DM}^{~{}}=\frac{16\pi^{2}}{3}\,\sqrt{\frac{\pi}{45}}\,\frac{x_{f}^{~{}}g_{0}^{~{}}}{\sqrt{g_{*}(T_{f})}}\,\frac{T_{0}^{3}}{M_{Pl}^{3}H_{0}^{2}}\;\frac{1}{\langle\sigma
v\rangle}.$ (67)

For the $s$ \- wave annihilation $\sigma=\sigma_{0}^{~{}}/v$ and we have
numerically

$\Omega_{DM}^{~{}}\approx 0.2\,\frac{\rm pb}{\sigma_{0}^{~{}}}.$

Note that a picobarn crossections are in the ballpark of the electroweak
scale, ${\rm pb}\approx\alpha^{2}/({100\rm\;GeV})^{2}$. That is why the weakly
interacting massive particles, appearing e.g. in supersymmetric extensions of
the Standard Model are considered to be natural candidates for the dark
matter. Another useful parametrisation of this result is given by

$\Omega_{DM}^{~{}}\approx 0.2\;\,\frac{3\cdot 10^{-26}\;{\rm cm}^{3}\;{\rm
s}^{-1}}{\langle\sigma v\rangle}.$ (68)

This expression is used for the discussion of dark matter direct and indirect
search results and strategies.

####  6.3.2 Cosmological abundance of ultra-light bosons

Dark matter particles can be very light and still very cold if they did not
originated from the thermal bath. Of course this holds for bosons only, since
the phase-space restrictions will not allow light fermions to saturate
required energy density in galaxy halos. Corresponding constraint on fermions
is called Tremaine-Gunn limit [58] and reads
$M_{F}\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$>$}}$ 1 keV.

To illustrate the general idea, let us consider a scalar field with potential
$V(\phi)=m^{2}\phi^{2}/2$. The field equations for the Fourier modes with a
momentum $k$ in an expanding Universe are

$\ddot{\phi}_{k}+3H\dot{\phi}_{k}+(k^{2}+m^{2})\phi_{k}=0\;.$ (69)

Since the term $\propto H$ can be understood as a friction, amplitude of modes
with $9H^{2}\gg(k^{2}+m^{2})$ (almost) does not change with time. Then, the
oscillations of modes with a given $k$ commence when $H$ becomes sufficiently
small, $9H^{2}\ll(k^{2}+m^{2})$. Oscillating modes behave like particles, and
their amplitude decreases with expansion. Since modes with the largest $k$
start oscillations first, they will have the smallest amplitude and the field
becomes homogeneous on a current horizon scale. This holds while mass term is
unimportant, i.e. till $3H>m$. Modes with all $k<3H$ will start oscillations
simultaneously when $3H\approx m$, and will behave like cold dark matter since
then.

Resulting abundance of dark matter will depend upon initial amplitude of modes
with $k<3H$. Why the initial amplitude of such modes is non-zero in the first
place? Such fields are generated during inflation if $m$ is smaller than the
value of the Hubble constant during inflation, see Eq. (79). In this way e.g.
massive gravitons are created as a dark matter, see Ref. [51].

Situation in the case of axions is even easier to understand. Potential for
the axion field $a$ has the following form

$V(a)=m_{a}^{2}f_{a}^{2}\,\left(1-\cos(a/f_{a})\right).$

Axion mass is temperature dependent, $m_{a}=m_{a}(T)$ and at $T\gg 1$ GeV it
is zero. Therefore, at this temperatures $V(a)=0$ and the axion field takes
arbitrary values in the range $0<a/f_{a}<2\pi$. Field oscillations start with
amplitude $a\sim f_{a}$ at $T\approx 1$ GeV when $3H(T)=m_{a}(T)$. Correct
axion abundance is obtained for $10^{-5}\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}m_{a}\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$<$}}10^{-3}$. Note that the field
will be homogeneous on the horizon scale at $T>1$ GeV, but may be
inhomogeneous on larger scales. This may lead to formation of dense clumps,
“axion miniclusters” of the mass $M\sim 10^{-12}\,M_{\odot}$ [59].

####  6.3.3 Cosmological abundance of superheavy particles

Superheavy particles can be created purely gravitationally. As we have seen in
Section 5.3, generically, a quantum field cannot be kept in a vacuum in the
expanding universe. This can be understood on the example of a scalar field,
Eq. (69). In conformal time $\eta$, Eq. (18), and for rescaled field,
$u_{k}\equiv\phi_{k}\,a$, the mode equations take form of an oscillator
equation

$\ddot{u}_{k}~{}+~{}[k^{2}+m^{2}_{\rm eff}(\eta)]\;u_{k}=0\,,$ (70)

with time-dependent mass

$m^{2}_{\rm eff}(\eta)=a^{2}m^{2}-\frac{\ddot{a}}{a}(1-6\xi).$ (71)

This is one particular case of the general situation described by Eq. (58).
The constant $\xi$ describes the coupling to the scalar curvature, the
corresponding term in the Lagrangian is $\xi R\phi$. The value $\xi=0$
corresponds to minimal coupling (Eq. (69) was written for this case), while
$\xi=1/6$ is the case of conformal coupling. Equations for massless,
conformally coupled quanta are reduced to the equation of motion in Minkowski
space-time. Particle creation does not occur in this case. For massive
particles, conformal invariance is broken and particles are created regardless
of the value of $\xi$. Let us consider the case of $\xi=1/6$. It is the
particle mass which couples the system to the background expansion and serves
as the source of particle creation in this case. Therefore, we expect that the
number of created particles in comoving volume is $\propto m^{3}$ and the
effect is strongest for the heaviest particles. In inflationary model (56) the
abundance of created particles, $\Omega_{\rm SH}$, will match observations if
$m\sim 10^{13}$ GeV [56], precise value of required superheavy particle mass
depends upon reheating temperature and the value of $\xi$. Therefore, a dark
matter can be created in the early Universe even if it has no couplings at
all, the only condition reads: be superheavy.

####  6.3.4 Cosmological abundance of sterile neutrino

Active neutrino are massive, this fact signifies a new physics beyond the
Standard Model. Other fermions have masses because they exist as left handed
and right handed states with coupling to the Higgs field H. However, active
neutrinos are left-handed. Therefore, a natural way to generate masses for the
neutrino would be to consider them at the same footing as other fermions and
to add right handed neutrinos, ${N_{j}}$, to the Standard Model Lagrangian,

${\cal L}={\cal
L}_{\rm{SM}}+i\bar{N_{j}}\partial_{\mu}\gamma^{\mu}N_{j}-\left[\lambda_{ji}(\bar{L_{i}}H)N_{j}+\frac{M_{j}}{2}\;\bar{N}_{j}^{c}N_{j}+\rm{h.c.}\right].$
(72)

Flavour indexes $j$ may run from one to three, but not necessarily. In what
follows I omit explicit writing of indexes. In the first term in square
brackets $L_{i}$ stands for a doublet of left-handed leptons. This term
generates Dirac masses for the neutrino, $m_{D}=\lambda\langle H\rangle$. In
general, right handed neutrino may have Majorana masses, $M$, as well. Such
term is forbidden for other fermions since their right-handed components have
charges, but right handed neutrino are neutral.

Right-handed components are also called sterile neutrino since they do not
interact directly with particles of the Standard Model. However, they are not
really sterile since interact with other particles via mixing. Indeed, to get
neutrino mass eigenstates we have to diagonalise mass matrix in square
brackets of Lagrangian (72). This gives mixing of active and sterile neutrino

$\theta=\frac{m_{D}}{M}.$ (73)

Therefore, sterile neutrino interaction matrix elements are the same as for
the active neutrino except they are multiplied by $\theta$. If ${m_{D}}\ll{M}$
the masses of heavy states nearly coincide with $M$ and the lightest among
sterile neutrinos is a good candidate for dark matter if its mass
$M\mathrel{\raisebox{-2.13124pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.54996pt}{$>$}}1$ keV. (But not heavier than 50
keV, otherwise its decays to $\gamma$ will contradict observed X-ray
astrophysical backgrounds, see Section 6.4.)

Sterile neutrino can be produced in the early Universe directly in the
inflation decays [60], or via mixing, Eq. (73), with active neutrino
[Dodelson:1993je]. Production rate of sterile neutrinos in the latter case can
be obtained multiplying production rate for the active neutrinos in primordial
plasma by mixing angle squared

$\Gamma\approx\theta^{2}\sigma^{~{}}_{W}n\sim\theta^{2}G_{F}^{2}T^{2}\cdot
T^{3}.$

Multiplying this rate by time, $t\sim H^{-1}\sim{M^{~{}}_{\rm Pl}}/{T^{2}}$ we
obtain number density of sterile neutrinos produced

$\frac{n_{s}^{~{}}}{n_{\gamma}}\sim\theta^{2}\,G_{F}^{2}T^{3}M^{~{}}_{\rm
Pl}.$

To close this estimate we note that active-sterile neutrino mixing is
temperature dependent [61]

$\theta\rightarrow\theta_{M}=\frac{\theta}{1+2.4(T/200~{}{\rm MeV})^{6}({\rm
keV}/M_{1})^{2}},$

which gives for the production temperature of sterile neutrino

$T\sim 130\left(\frac{M}{1~{}{\rm keV}}\right)^{1/3}~{}{\rm MeV},$

and resulting abundance [Dodelson:1993je]

$\Omega_{s}\sim\Omega_{m}\frac{\sin^{2}(2\theta)}{10^{-7}}\left(\frac{M}{1~{}\rm
keV}\right)^{2},$ (74)

where $\Omega_{m}$ is observed dark matter abundance. Proper calculation
involves solution of Boltzmann equations. Details and the list of references
can be found in the recent review [62]. Quoted result, Eq. (74), corresponds
to zero lepton asymmetry. With maximum asymmetry the required $\theta$ can be
two orders of magnitude smaller [63] at the same mass of sterile neutrino, see
Fig. 10, right panel.

###  6.4 Dark matter searches and constraints

Dark matter particles, in majority of suggested models, can be discovered in
direct dedicated searches in laboratories. Dark matter can also leave trace
and be identified in indirect searches, e.g. analysing data on cosmic ray,
X-ray, gamma-ray and neutrino telescopes.

Figure 9: Left panel: Constraints on neutralino from different underground
experiments are shown by correspondingly marked colored lines. Similarly,
claimed hints of detection are represented by shaded areas. Remaining
expectation for supersymmetric models are shown by shaded area in the lower
right corner, marked as MSSM. Right panel: Constraints on neutralino self-
annihilation cross-section from CMBR. The blue area shows the parameter space
excluded by the Planck data. The yellow line indicates the constraint using
WMAP9 data. The dark grey circles and light grey stars correspond to various
claims of indirect detection as cosmic or $\gamma$-ray excesses. The
horizontal red band corresponds to correct neutrino abundance, Eq. (68). From
Ref. [2].

1\. Neutralino.  WIMP particles have tiny but phenomenologically important
elastic cross-section with usual baryonic matter. For WIMPs heavier than
nuclei, $m_{X}\gg m_{N}$ and a typical velocities in the Galaxy halo, $v\sim
300$ km/s, typical recoil energy is $E_{R}\sim m_{N}v^{2}\sim 1\;-\;100$ keV.
The recoil can be measured studying ionization, scintillation, heat or sound
waves it creates in a detector. Different experiments use different
techniques, or their combinations. But, of course, it should be done deep
underground, in low background laboratories. Current observational bounds on
the scattering cross-section exclude a lot of the WIMP parameter space of MSSM
but do not test the bulk of it, see Fig. 9, left panel. Intriguingly, crystal-
based experiments CDMS Si, CoGeNT ROI, CRESST II and DAMA/LIBRA claim some
hints of a positive dark matter signal. These claims are mutually exclusive,
and cluster in the mass region of tens of GeV and at cross sections between
$10^{−42}$ and $10^{−39}$ cm2, see Fig. 9, left panel. However, the noble-gas
experiments ZEPLIN, XENON and most recent LUX, exclude this parameter region.
Remaining expectation for supersymmetric models, after all constraint are
taken into account, including LHC results, are shown by shaded area in the
lower right corner, marked as MSSM, in the same figure. The uncertainty for
the expected signal arises because the scattering cross-section is not
directly related to the annihilation cross-section.

However, that same self-annihilation that plays a central role in the freeze-
out, see Section 6.3.1, leads also to the dark matter annihilation in the
Galaxy halo. It can give rise to a significant flux of $\gamma$-rays,
neutrinos, and even antimatter such as antiprotons and positrons, especially
from regions with large dark-matter density . This creates prospective signal
for the indirect WIMP detection. It is searched for, as an excess over
conventional astrophysicsl backgrounds, by the orbital cosmic ray
observatories, ground based atmospheric Cherenkov and neutrino telescopes.
Though annihilation cross-section for indirect searches is fixed, some
uncertainty arises here because of certain uncertainty in dark matter density
profiles.

Dark-matter annihilation with a non-vanishing branching ratio into the
electromagnetic channel leads also to distortions of the CMB which has been
probed with WMAP and Planck data [2]. WIMPs lighter than 10 GeV originating in
thermal freeze-out scenario are excluded by these observations and the
advantage of CMB-based limits lies in the absence of astrophysical
uncertainties, see Fig. 9, right panel. Dark matter annihilation
interpretation of the cosmic ray excess detected by AMS, Fermi and Pamela
satellites (shown by dark grey circles) are also excluded now by Planck.
However, the interpretation of the $\gamma$-ray excess from the Galactic
centre measured by Fermi (corresponding parameter regions are indicated by
light grey stars) is still viable. Intriguingly, it intersects with the
horizontal red band which corresponds to the correct neutralino abundance as
thermal relic.

Dark matter particles escape direct detection at colliders such as the Large
Hadron Collider (LHC) at CERN, however, they would produce a characteristic
signal of missing energy. Arising constraints on WIMP-nucleon cross-section
are model dependent, but are powerful and competitive with direct searches in
underground labs, especially in the region of low masses. Recent detailed
review on direct, indirect and collider WIMP searches can be found e.g. in
Ref. [65].

2\. Axions.  Axion interactions with photons and fermions can be parametrised
as

$L_{\rm int}=-\frac{1}{4}g_{a\gamma}\,aF_{\mu\nu}\tilde{F}^{\mu\nu}-\sum_{\rm
fermions}g_{ai}\,a\overline{\psi}_{i}\gamma_{5}\psi_{i}\,,$ (75)

where

$g_{a\gamma}\equiv\frac{\alpha}{2\pi}\frac{C_{a\gamma}}{f_{a}},~{}~{}~{}~{}~{}~{}g_{ai}\equiv\frac{m_{i}}{f_{a}}C_{ai},$

and $C_{a\gamma}$, $C_{ai}$ are model dependent parameters (in simple models
of order unity). Direct axion searches in the laboratory are based on
interactions with $\gamma$. Namely, axions constituting the Milky Way dark
matter halo would resonantly convert into a monochromatic microwave signal in
a high-Q microwave cavity permeated by a strong magnetic field [66]. Such
axion search experiments (ADMX is the most recent one) are called haloscopes.
Similarly, axions or axion-like particles, emitted by the Sun will be
converted in the strong magnetic field into X-ray photons. Axion experiments
which search for this signal (CAST is the most recent one) are called
helioscopes. Constraints obtained by ADMX and CAST experiments are shown in
Fig. 10, left panel.

Figure 10: Left panel: Parameter space for the axion and axion-like dark
matter models. Yellow band corresponds to the correct cosmological abundance
of QCD axions if they make all of dark matter. Regions excluded by ADMX and
CAST are shown by green and blue shaded areas. Dashed lines show prospective
limits of planned experiments. From Ref. [68]. Right panel: Unshaded white
region represents allowed parameter space for DM sterile neutrinos. The upper
and lower thick black lines correspond to correct abundances for zero and
maximal lepton asymmetry. Red region in the upper right corner is forbidden by
the X-ray constraints. The region below 1 keV is ruled out by the phase-space
density arguments. Adapted from Ref. [64].

Interaction Lagrangian Eq. (75) leads also to a new observable astrophysical
phenomena, which may lead to indirect axon detection and give constraints on
axion parameters. Extra energy losses by starts is one of those effects.
Corresponding constraints practically coincide with the bound obtained by
CAST. It is shown by the dotted line marked by the label ”Massive Stars” in
the Fig. 10, left panel. No accident, along the same line we can find models
capable to explain several claims hinting for the axion effects in the
astrophysical data, for a review see e.g. Ref. [67]. Recent detailed review on
direct and indirect axion searches can be found also in Refs. [65, 68].

3\. Sterile neutrino.  Recent detailed review on direct and indirect sterile
neutrino searches can be found in Ref. [62]. In every process where active
neutrino appears, sterile neutrino can appear as well, again via mixing, Eq.
(73). This opens the way for a laboratory sterile neutrino searches. For
example, in the keV mass range, appearance of sterile neutrinos changes
kinematics and the spectrum of nuclear decays. Most recent searches of sterile
neutrinos in tritium $\beta$-decay has started in Troitsk [69], and will be
continued at KATRIN experiment [62].

Also, at one loop level sterile neutrino are decaying into active neutrino and
photon. Loop diagrams for this process are the same as for the electromagnetic
form factors of a massive neutrino in the Standard Model with one external
neutrino leg being connected to sterile neutrino via mixing (73). The decay
width can be easily obtained using e.g. results of Ref. [70] and is given by

$\Gamma_{N\rightarrow\gamma\nu_{a}}=\frac{9\,\alpha\,G_{F}^{2}}{256\cdot
4\pi^{4}}\sin^{2}2\theta\,m_{s}^{5}=5.5\times
10^{-22}\,\theta^{2}\left[\frac{m_{s}}{1~{}\rm keV}\right]^{5}~{}{\rm
s}^{-1}.$ (76)

Because of that, sterile neutrino dark matter is not completely dark. It can
be detected by searching for an unidentified X-ray line, which would appear at
a frequency $\omega=m_{s}/2$. Intensity of this line should follow dark matter
density profiles. Dwarf satellite galaxies are a good places to search for
such a signal because they are dark matter dominated and usual astrophysical
X-ray background is small there [71].

To conclude this section. A large number of various clams exists in the
literature with a hints of indirect dark matter signal for all of the
candidates described above: nutralino, axion-like particles and sterile
neutrino. Do describe these hints in detail would require separate volume,
interested reader can consult recent reviews [65, 67, 62]. As usual, hints
appear at a boundary of allowed parameter space where observational
capabilities are stretched. Moreover, indirect dark matter signal can be
confused with conventional astrophysical backgrounds or effects. Clearly,
these claims are mutually exclusive and it is not possible for all of them to
be precursors of the true signal, since dark matter is either neutrino, or
axions, or sterile neutrino, or something else. On the other hand, one of
those may turn out to be true and it is not excluded that we see already the
tip of a real iceberg.

##  7 Conclusions

As we have seen, cosmology and astrophysics gave us solid evidence that the
Standard Model of particle physics is incomplete. We have to extend it to
explain neutrino masses, baryogenesis, and dark matter. Dark energy can be
explained by the Einstein’s $\Lambda$-term, but we do not know why it exists,
and there seems to be too many coincidences between numerical values of
cosmological parameters. On the other hand, a form of dark energy explains the
Universe origin within inflationary paradigm, which increasingly finds support
in cosmological data.

Cosmology just recently became a precision science but is full of surprises
already, helping to build true model of microphysics. It is up to high energy
physicists to find out what this new physics is. With advances of this program
we, in turn, will have better understanding of the Universe origin, of its
evolution, of its current state, and of its future fate.

## Acknowledgements

I would like to use the opportunity to thank CERN and Dubna for the
organization and support of this annual summer school series for the young
high-energy physicists.

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##  Appendix A Gravitational creation of metric perturbations

As an important and simple example, let us consider quantum fluctuations of a
real scalar field, which we denote as $\varphi$. It is appropriate to rescale
the field values by the scale factor, $\varphi\equiv\phi/a(\eta)$. This brings
the equations of motion for the field $\phi$ into a simple form of Eq. (58).
As usual, we decompose $\phi$ over creation and annihilation operators
$b_{{\bf k}}$ and $b_{{\bf k}}^{\dagger}$

$\phi({\bf
x},\eta)=\int\frac{d^{3}k}{(2\pi)^{3/2}}\;\left[u_{k}(\eta)\,b_{{\bf
k}}\,\mathop{\rm e}\nolimits^{i{\bf k}{\bf x}}+u^{*}_{k}(\eta)\,b_{{\bf
k}}^{\dagger}\,\mathop{\rm e}\nolimits^{-i{\bf k}{\bf x}}\right]\;.$ (77)

Mode functions $u_{k}$ satisfy Eq. (58). In what follows we will assume that
$\varphi$ is the inflaton field of the “chaotic” inflationary model, Eq. (56).
During inflation $H\gg m$ and $H\approx{\rm const}$. So, to start with, we can
assume that $\varphi$ is a massless field on the constant deSitter background.
(The massive case can be treated similarly, but analytical expressions are
somewhat more complicated and do not change the result in a significant way.
Corrections due to change of $H$ can also be taken into account, and we do
that later for the purpose of comparison with observations.) With a constant
Hubble parameter during inflation the solution of Friedmann equations in
conformal time is

$a(\eta)=-\frac{1}{H\eta}$ (78)

and the equation for mode functions of a massless, conformally coupled to
gravity $(\xi=0)$, scalar field takes the form

$\ddot{u}_{k}+k^{2}{u}_{k}-\frac{2}{\eta^{2}}\;{u}_{k}=0\;.$ (79)

Solutions which start as vacuum fluctuations in the past
($\eta\rightarrow-\infty$) are given by

${u}_{k}=\frac{\mathop{\rm e}\nolimits^{\pm
ik\eta}}{\sqrt{2k}}\left(1\pm\frac{i}{k\eta}\right)\;.$ (80)

Indeed, at $\eta\rightarrow-\infty$ the second term in the parentheses can be
neglected and we have the familiar mode functions of the Minkowski space time.
The wavelength of a given mode becomes equal to the horizon size (or “crosses”
the horizon) when $k\eta=1$. Inflation proceeds with $\eta\rightarrow 0$, so
the modes with progressively larger $k$ cross the horizon. After horizon
crossing, when $k\eta\ll 1$, the asymptotics of mode functions are

$u_{k}=\pm\frac{i}{\sqrt{2}k^{3/2}\eta},{\rm~{}~{}~{}~{}or~{}~{}~{}}\varphi_{k}=\frac{u_{k}}{a(\eta)}=\mp\frac{iH}{\sqrt{2}k^{3/2}}\;.$
(81)

The field variance is given by

$\langle
0|\phi^{2}(x)|0\rangle=\int\frac{d^{3}k}{(2\pi)^{3}}\;|\varphi_{k}|^{2}\;.$
(82)

and we find in the asymptotic (the careful reader will recognize that this is
already regularized expression with zero-point fluctuations being subtracted)

$\langle\varphi^{2}\rangle=\frac{H^{2}}{(2\pi)^{2}}\int\frac{dk}{k}\;.$ (83)

Defining the power spectrum of the field fluctuations as a power per decade,
$\langle\varphi^{2}\rangle\equiv\int P_{\varphi}(k)\;{d\ln k}$, we find

$P_{\varphi}(k)=\frac{H^{2}}{(2\pi)^{2}}\;.$ (84)

###  A.1 Curvature perturbations

According to Eq. (4), the three-dimensional curvature of space sections of
constant time is inversely proportional to the scale factor squared,
${}^{(3)}R\propto{a^{-2}}$. Therefore, the perturbation of spatial curvature
is proportional to ${\delta a}/{a}$, and this ratio can be evaluated as

$\zeta\equiv\frac{\delta a}{a}=H\delta
t=H\,\frac{\delta\varphi}{\dot{\varphi}}\;.$ (85)

This allows to relate the power spectrum of curvature perturbations to the
power spectrum of field fluctuations

$P_{\zeta}(k)=\frac{H^{2}}{\dot{\varphi}^{2}}\;\,P_{\varphi}(k)\;,$ (86)

and we find for the power spectrum of curvature perturbations

$P_{\zeta}(k)=\frac{1}{4\pi^{2}}\;\frac{H^{4}}{\dot{\varphi}^{2}}\;.$ (87)