11institutetext: ICREA and ICC, Barcelona, Spain

# A Taste of Cosmology

L. Verde<EMAIL_ADDRESS>
###### Abstract

This is the summary of two lectures that aim to give an overview of cosmology.
I will not try to be too rigorous in derivations, nor to give a full
historical overview. The idea is to provide a “taste” of cosmology and some of
the interesting topics it covers. The standard cosmological model is presented
and I highlight the successes of cosmology over the past decade or so. Keys to
the development of the standard cosmological model are observations of the
cosmic microwave background and of large-scale structure, which are
introduced. Inflation and dark energy and the outlook for the future are also
discussed. Slides from the lectures are available from the school website:
physicschool.web.cern.ch/PhysicSchool/CLASHEP/CLASHEP2011/.

##  0.1 Cosmology: Why should you care?

Cosmology aims at the study of the Universe as a whole, but, in particular, at
getting an understanding of the Universe’s origin, structure, composition,
evolution and fate. In general, and for many applications, for cosmologists
galaxies can be treated as “points”. This gives an idea of the scales
involved. Suitable units to measure scales must be introduced; cosmological
distances are measured in megaparsecs, 1 Mpc = $3.86\times 10^{22}$ m; masses
are often expressed in solar masses, $1\leavevmode\nobreak\
\mathrm{M}_{\odot}=1.99\times 10^{30}$ kg.

Cosmology in the past 15 years has made the transition to precision science
and has moved from a data-starved science to a data-driven science. This
development has been driven by new technology, which has made it possible to
survey the Universe farther than before. As a result of the “avalanche of
data” of the past 20 years or so, cosmology now has a standard model. The
“standard cosmological model” or the “standard model of cosmology” only needs
a few parameters to describe the origin, composition and evolution of the
Universe on large scales. However, there is a great difference between
modelling and understanding. As we will see, the standard cosmological model
leaves many deep open questions, for example: What is dark matter? Do we
understand gravity on very large scales? Is there energy associated with the
vacuum? These questions show that there are deep connections between cosmology
and fundamental physics, and that cosmological observations can be crucial to
understanding open questions in physics.

Testing fundamental physics by looking up at the sky is not new. The interplay
between astrophysics and fundamental physics has already produced spectacular
findings (as an example, think about the solar neutrino problem). Cosmological
observations are key to testing fundamental physics in four areas, in
particular: dark matter, neutrinos, inflation and dark energy.

###  0.1.1 Assumptions

There are some fundamental assumptions at the basis of cosmology.

  1. 1.

Physics as we know it can describe the Universe as a whole, i.e. it is the
same in all corners of the Universe.

  2. 2.

On the largest scales – those mostly of interest for cosmology – the driving
force is gravity, and we can neglect complicated effects of baryonic physics.

  3. 3.

Gravity is described by general relativity (GR) – although for many
applications the Newtonian limit works fine.

  4. 4.

The Universe on large scales is homogeneous and isotropic. This is called the
cosmological principle. We know that on small scales there are inhomogeneities
and anisotropies – think about galaxies, stars, the Solar System, us …. But on
very large scales, one assumes that these inhomogeneities get smoothed,
averaged out.

  5. 5.

Therefore, the metric is Friedmann–Lemaître–Robinson–Walker (FLRW). Locally,
it cannot be, of course! Think about the metric next to a black hole, for
example. This statement says that on very large scales the global metric is
FLRW.

Note that the puzzling aspects and the open questions inherent to the standard
cosmological model are leading many researchers to challenge these basic
assumptions. While most people would feel uncomfortable touching the first two
assumptions, an active area of research is to explore the observational
consequences of dropping the other assumptions.

###  0.1.2 Parameters of the model

The parameters of the standard model for cosmology can be divided into three
types: (i) parameters describing the smooth Universe, i.e. parameters that
govern the global geometry of space-time and parameters that govern the
expansion rate; (ii) parameters that describe the inhomogeneous Universe, i.e.
that characterize the inhomogeneities; and (iii) parameters that parametrize
our ignorance – for complicated, nonlinear physical processes that we cannot
properly describe and model, one resorts to hiding all those and describing
the effects with extra parameters.

Of course, it is important to consider models more complex than the standard
cosmological model by introducing “extra parameters”. Finding no evidence for
these extra parameters would further support the standard model.

If the metric is FLRW, then there are only three possible (global) geometries:
closed, flat or open. The parameter that gives the global geometry is
$\Omega=\rho/\rho_{\rm crit}$, where $\rho$ denotes the density. The critical
density ($\rho_{\rm crit}$) is such that the geometry is spatial flat.
$\Omega>1$ indicates a Universe with positive curvature and $\Omega<1$ one
with negative curvature. $\Omega$ can be decomposed into several components,
given by the composition of the Universe. For example, everything that shines
(e.g. stars) is composed by baryonic matter. There is also, of course, dark
baryonic matter such as gas, dust, etc.

However, there is more than meets the eye. The mass of galaxies, for example,
can be computed from galaxy rotation curves (angular velocity plotted against
distance from the centre). The rotation curve of a spiral galaxy can be
measured by tracing the gas through specific atomic or molecular transitions,
usually in the radio part of the electromagnetic spectrum. The rotation curves
of galaxies remain flat well beyond the extent of the optical light, as first
shown by Vera Rubin in the 1970s. In the Solar System, where most of the mass
is concentrated in the Sun, the inner planets have higher circular velocity
than the outer planets. A flat rotation curve indicates the existence of an
unseen dark component called a halo that extends well beyond the luminous part
of a galaxy. It is useful to introduce a concept called “mass-to-light ratio”
to quantify the unseen (dark) matter component. Our Sun has by definition a
mass-to-light ratio of unity. Galaxies have a mass-to-light ratio of order 10.

Galaxies tend not to be isolated, but to cluster in galaxy clusters
(gravitationally bound groups of hundreds of galaxies). In clusters of
galaxies, the depth of the potential well (and thus the total mass) can be
estimated from the velocity dispersion of the galaxies (as Zwicky did in the
1930s) or from the temperature of the intra-cluster gas, which is so hot that
it emits X-rays. This gives a mass-to-light ratio for clusters of order 100,
indicating even more dark matter. More spectacularly, and on even larger
scales, the dark matter can be seen through its gravitational lensing: the
deep potential well of the dark matter in clusters of galaxies bends the light
path from background sources, distorting them into in arcs and making multiple
images. From gravitational lensing observations, the density profile of the
dark matter in clusters (and in galaxies) has been mapped out. The dark matter
“halo” extends well beyond the stars/galaxy component and has a steep density
profile towards the centre.

More recently, new direct evidence has emerged on dark matter and its nature
[1]. The so-called “bullet cluster” shows the aftermath of an impact where a
small cluster (the bullet) hit and passed through a larger cluster (the
target). Combined observations of the optical (stars), X-rays (gas) and
gravitational lensing (all, but mostly dark matter) show that the gas
(baryonic matter, which is collisional), feeling drag due to electromagnetic
interactions, lags behind the collisionless component (dark matter and
stars/galaxies – which on these large scales for all purposes act as a
collisionless component).

Dark matter should be not only collisionless but also cold, i.e. have small
velocity. Dark matter must be cold to give steep halo density profiles (see
e.g. [2]) and to describe the clustering properties of large-scale structure
(see section 0.3).

Let us move on to consider the expansion of the Universe. Hubble in the 1920s
realized that there was a linear relation between the distance of galaxies and
the wavelength shift of their spectra (mostly to longer wavelengths, thus
called redshift). By interpreting the galaxy redshift as recession velocity,
we obtain the Hubble law $cz=v=H_{0}d$, where $d$ denotes the distance, $z$
the redshift and $H_{0}$ the Hubble constant. Owing to the finite speed of
light, redshift is therefore an indicator not only of distance but also of
time: light from an object at “high” $z$ comes not only from far away but also
from the past.

Hubble showed that the Universe is not static but expanding (something that
today we take for granted). As a consequence, the Universe must have had a
beginning. There are more consequences of the expansion. If we run the “movie”
of the expansion backwards, we obtain that, far enough back in the past, all
the content of the Universe was concentrated in a singularity (the Big Bang).
Also, during expansion, the Universe cooled, so the early Universe was hot.
The hot Big Bang scenario is extremely successful in explaining some key
observations: the abundance of light elements and the cosmic microwave
background (CMB). The CMB is discussed in section 0.2. The abundance of light
elements is explained by the so-called Big Bang nucleosynthesis. I will not
discuss nucleosynthesis here, although it is a fascinating subject. I will
just mention that primordial nucleosynthesis took place minutes after the Big
Bang and that, through the hot starting temperature and the gradual cooling of
the Universe, the nuclei of elements from deuterium, helium isotopes, lithium
and beryllium were made. Elements heavier than that were subsequently made by
stars.

Note that, in the expansion of the Universe, what is expanding really is
space-time. It is not that galaxies are running away from one another through
a static space-time. At first approximation, galaxies are fixed in space and
moving with it. Galaxies in reality do move through space-time, as we will see
later, but this motion is small compared to the Hubble expansion and is
sourced by small inhomogeneities. As the Hubble expansion is a stretching of
space-time, there is no contradiction to find that at high redshift things
seems to move at speeds comparable to the speed of light: no information is
really moving through space-time at that speed (or faster).

If we consider the expansion of the Universe and the homogeneity and isotropy
properties, then it is useful to define a scale factor to describe the
expansion. The distance $r$ between any two points as a function of time $t$
can be factorized as $r(t)=r(t_{0})a(t)$, where $a$ denotes the scale factor
and $r(t_{0})$ is the comoving coordinate. It is easy to see that the Hubble
parameter is given by $\dot{a}/a$, where the dot denotes the time derivative.
Also, it is easy to see that $a=1/(1+z)$.

It is easy to conclude that, if the Universe is expanding and its content is
matter (dark or not) – and possibly curvature – gravity should slow down the
expansion over time. Not only that, but by measuring the deceleration, one
should be able to “weigh” the Universe! So it was a big surprise when, in
1998, two teams [3, 4] analysing their data came to the conclusion that the
expansion was actually accelerating. Their measurements relied on surveying a
particular type of exploding star and measuring their redshifts. These
exploding stars are “standard candles”, i.e. they all have the same intrinsic
luminosity.111To be completely precise, these are “standardizable” candles;
they all have approximately the same intrinsic brightness, but any variation
correlates with other observables and therefore can be effectively corrected
for. The redshifts give the recession velocity, and the observed brightness of
the standard candle (the intrinsic luminosity of which is known, and the same
for all standard candles) gives an estimate of the distance. Thus, one can map
the recession velocity as a function of distance, and trace the expansion
history of the Universe.

The principal investigators of the two teams received the Nobel Prize in
Physics in 2011. The discovery of the accelerated expansion has changed the
course of cosmology, with most major observational efforts today being devoted
to understanding the acceleration.

As an interesting historical note, recall that Einstein added a cosmological
constant term ($\Lambda$) to his equations motivated by the need to make the
Universe static. When Hubble discovered the expansion of the Universe,
Einstein deemed $\Lambda$ the biggest blunder of his life. However, the
discovery of the accelerated expansion brought $\Lambda$ back. Matter (which
is pressureless if it is dark matter or has a positive pressure) makes the
expansion slow down; a component with negative pressure (such as a
cosmological constant) is needed to make the expansion accelerate. So, in the
standard cosmological model, there is a component to $\Omega$ made by baryonic
matter ($\Omega_{\rm b}$), a component made by dark matter ($\Omega_{\rm
dm}$), a component made by radiation, which is unimportant in the energy
budget today, and a component given by $\Lambda$ ($\Omega_{\Lambda}$).
However, other “stuff” than a simple cosmological constant (fluids with
negative pressure, slowly rolling scalar fields, more exotic things such as
deviations from general relativity) could fit the bill. All this goes
generically under the name of “dark energy”. The Nobel Prize presentation
speech says: “The discovery of accelerating expansion through studies of
distant supernovae has thus changed our image of the Universe in an
unexpected, dramatic way. We have realised that we live in a Universe which
largely consists of components that are unknown to us. Understanding dark
energy is a challenge for scientists all over the world.”

The evolution of the Universe is driven by these parameters ($\Omega_{\rm m},\
\Omega_{\Lambda},\ H_{0}$) through the Friedmann equations222See class slides.
and the properties of the various components. Note that the different
components of the Universe scale differently with the expansion, i.e. dilute
differently as the Universe expands. For example, the cosmological constant
does not dilute, whereas matter dilutes as $(1+z)^{3}$ .

So, let us move on to introduce the parameters describing clustering. We
concentrate on clustering on very large scales, say from a few Mpc and above,
called large-scale structure. In the standard cosmological model, all
structures in the Universe arise from small primordial perturbations (see
section 0.5) that grew under gravity. No theory in cosmology will be able to
predict that a particular galaxy will form in a specific place; a cosmological
model will be able to predict the statistical properties of the fluctuations.
The statistical tool most used is the power spectrum, which is the Fourier
counterpart of the correlation function. The way to characterize the
primordial power spectrum is as a power law with an amplitude (usually called
$A$)333The numerical value of the amplitude is dependent on the convention
used, i.e. the choice of the pivot point, which is the $k$ value where the
amplitude is set. and a slope ($n_{\rm s}$). Here the subscript “s” stands for
scalar perturbations. In the standard cosmological model, scalar perturbations
are all there effectively is, but beyond the (minimal) standard model there
may be other type of perturbations (e.g. isocurvature, tensors, vorticity,
etc., depending on how imaginative or non-standard one wants to be).

Finally, there is one more parameter in the model, which is one of those
characterizing our ignorance: this is the “optical depth” parameter. It
encloses information about the epoch of formation of the first stars and how
powerful they were at emitting radiation that ionizes the gas in the Universe.

So, there are six parameters in the standard cosmological model, also called
$\Lambda$CDM or LCDM (CDM standing for “cold dark matter”). These six
parameters fit observations of the Universe from $z=1100$ (corresponding to
380 000 years after the Big Bang or, in the LCDM model, 13.7 billion years
ago) to today. The Universe is spatially flat, made of baryonic matter, cold
dark matter, cosmological constant and radiation, which is unimportant today
(but much more important in the past). One more parameter describes the
present-day expansion rate (the Hubble constant). In this model, inflation
(see section 0.5) generated the primordial perturbations that grew under
gravity. Two parameters characterize the primordial perturbations: their power
spectrum power-law amplitude and slope. This is an extremely successful model
and the values of the six basic parameters are known with per cent accuracy.

It is interesting to note that the model was developed when the datasets
available were much smaller and of much lower quality than those available
today. But the model has survived the scrutiny of time and the stringent test
imposed by comparison to the new data. This does not mean that there is no
space for small deviations from this standard cosmological model, and in fact
much of the research in cosmology these days is focused on constraining
deviations from this model. This model leaves many open questions, such as the
following: What is $\Lambda$? Is it really a constant? What is dark matter?
Why are $\Omega_{m}$ and $\Omega_{\Lambda}$ comparable today? What is the
physics behind inflation? We will revisit these issues later on.

##  0.2 The cosmic microwave background

The Universe cools as it expands – the redshift results in a reduction of
temperature (think of Wien’s law). If the Universe is cool, large and
expanding today, it must have been smaller and warmer in the past. This leads
to the cosmic microwave background (CMB), the relict radiation from the Big
Bang that pervades the whole Universe. Also, while looking far away
corresponds to looking back in time, the CMB also gives a “surface” behind
which we cannot see directly.

Regular hydrogen gas lets light pass through more or less unimpeded. This is
the case today, where the hydrogen gas is either cold and atomic, or very
thin, hot and ionized. But in the early Universe, when it was much warmer, the
gas would have been ionized, and the Universe opaque to light. As the Universe
cooled, the electrons and protons “recombined” into normal hydrogen, and the
Universe suddenly became transparent. This gives the last scattering surface:
a snapshot of the early Universe.

As a historical note, the CMB was discovered “accidentally” in 1965 by two
engineers (Penzias and Wilson) at Bell Labs in New Jersey, USA, as a uniform
glow across the sky in the radio. At the same time a few miles away at
Princeton University, researchers (Dicke, Peebles, Roll and Wilkinson) were
looking for the “primordial fireball”, which would be an unavoidable
consequence of the hot Big Bang model. It should be the blackbody emission of
hot, dense gas (temperature $T\sim 3000$ K, peak wavelength
$\lambda_{\max}\sim 1000$ nm) redshifted by a factor of 1000 to a peak
wavelength of 1 mm and $T\sim 3$ K radiation. When they heard of the discovery
at Bell Labs, Dicke was reported to say “Well boys, we have been scooped!”
Penzias and Wilson received the Nobel Prize in 1978. Two papers (the discovery
one and the theoretical explanation) by the two groups were published back to
back in the Astrophysical Journal in 1965 [5, 6].

We had to wait until 1992 for the results of the COBE satellite to confirm
that the radiation was truly a blackbody and that small anisotropies (about
one part in $10^{5}$) were present. The small anisotropies correspond to the
primordial fluctuations that grew under gravitational instability to form all
the structures we see today. The COBE team was awarded the Nobel Prize in
Physics in 2006. The prize dedication mentions “These measurements … marked
the inception of cosmology as a precise science.” In fact, accurate
measurements of the CMB fluctuations were crucial in establishing the standard
cosmological models and constraining its parameters at the per cent level.

We see density perturbations as temperature fluctuations: photons need to
climb out of potential wells before they can travel to us. Thus, from
potential hills, the radiation is blueshifted and they look hotter; from
potential wells, the radiation is redshifted and they look colder. See Fig. 1
for a map of these temperature fluctuations as seen by the WMAP satellite. It
is from the statistical properties of these fluctuations that it was possible
to constrain cosmological parameters so well.

Recall that the last scattering surface (which we see as the CMB) is a
snapshot of the early Universe. As the inverse was hot and photons were
coupled to baryons, it gives us a snapshot of the photon–baryon fluid. On
large scales, we see primordial ripples. On smaller scales, there are two
competing effects: the radiation pressure from photons, and the compression
from gravity. Thus sound waves are generated, which stop oscillating at
recombination, and we see in the CMB a snapshot of when the oscillations
stopped. There is another important ingredient, which is the horizon size at
the last scattering surface. The Universe back then was so dense that the
speed of sound was very close to the speed of light. Pressure can
counterbalance perturbations only on scales smaller than how far the sound
could have travelled from the Big Bang to the epoch of the CMB (sound
horizon). The horizon corresponds to a fundamental mode and there should be an
imprint in the CMB sound waves of the sound horizon as a fundamental model and
its overtones. Another interesting fact in the history of cosmology
(intertwined with political history) is the prediction of the properties of
the CMB perturbations: on both sides of the Iron Curtain (Sunyaev and
Zeldovich in Russia, and Peebles and Yu in the USA), the same prediction was
being worked out, but the West was not aware of the developments published in
Russian until they were translated (see [7, 8]).

Figure 1: Cosmic microwave background temperature anisotropies as seen by the
WMAP satellite. Figure courtesy of the WMAP science team.

A CMB map such as the one shown in Fig. 1 must be compressed to study
cosmology: the temperature fluctuation is expanded in spherical harmonics and
the temperature angular power spectrum is computed. If the anisotropy is a
Gaussian random field (as predicted by inflation, to a good approximation),
then all the statistical information of the map is enclosed in the angular
power spectrum $C_{\ell}$ (Fig. 2).

Figure 2: Angular power spectrum of the cosmic microwave background
temperature anisotropies as seen by the WMAP satellite (black) and the South
Pole telescope (blue points).

Few features are evident from the $C_{\ell}$ plot. On large scales, larger
than the sound horizon, we see the primordial ripples. On smaller scales,
acoustic peaks are seen, which corresponds to extrema of the photon–baryon
fluid oscillations. A peak is associated with the main compression mode or the
fundamental tone corresponding to the sound horizon size. Smaller peaks are
associated with the maximum rarefactions and compressions of the overtones.
The fundamental mode acts as a standard ruler, as the sound horizon size is
well determined from first principles and depends very little on cosmology. By
observing such a standard ruler, we are constructing the largest possible
triangle which can then be used to determine the global geometry of space-time
between us and the last scattering surface. Also, the heights of the peaks
corresponding to the compression give information about the depth of the
potential wells and thus the total amount of matter. The relative heights of
the peaks corresponding to compression (odd peaks) and rarefaction (even
peaks) yield information about the amount of baryonic matter. Finally, all
these processes originated from a primordial spectrum of fluctuations and the
$C_{\ell}$ also carries information on that. (For a more rigorous derivation
of all this, see e.g. [9, 10].)

A key role in establishing the current standard model of cosmology was played
by the WMAP satellite. WMAP produced the highest-resolution full-sky map of
the CMB anisotropies to date. A detailed analysis of the shape of the angular
temperature power spectrum from the WMAP data constrains the parameters of the
LCDM model at the per cent level (see [11, 12, 13] and references therein).
For example, the composition of the Universe is summarized in Fig. 3.

Figure 3: The Universe composition in the standard model for cosmology. The
CMB data from WMAP were crucial to establish this model.

Before concluding this section, we should mention that the Planck satellite
has been measuring the CMB sky with even better precision than WMAP and should
release its results in a year or so. There is more information in the CMB
radiation than the temperature fluctuations: the CMB light is also polarized.
This will be discussed in section 0.5.1.

##  0.3 Large-scale structure

Large-scale structure denotes the large-scale distribution of matter that is
traced by galaxies (and gas). When we see the primordial fluctuations in the
CMB (albeit with the processing due to the fact that we see the radiation,
i.e. processed through the radiation transfer function), we see the initial
conditions. When we observe the large-scale structure, we see the result of
the action of gravity. Of course, if we trace large-scale structure using the
galaxy distribution (or the intergalactic hydrogen gas via observations of the
so-called Lyman-alpha forest), we must bear in mind that galaxies may not be
perfectly faithful tracers of the dark matter distribution, i.e. they could be
biased tracers. On the other hand, it has recently become possible to trace
the dark matter distribution directly via gravitational lensing. At present,
gravitational lensing measurements on large-scale structure scales still have
large associated error bars, but in the near future specifically designed
surveys will make gravitational lensing an invaluable tool to map the large-
scale structures in an unbiased way.

Again, cosmology deals with statistical properties of the large-scale
structure, and the most widely used tools are the power spectrum and the
corresponding two-point correlation function. Note that, even if the initial
conditions were perfectly Gaussian, nonlinear gravitational evolution would
make the distribution non-Gaussian, 444In fact, if $\delta=\delta\rho/\rho$,
where $\rho$ stands for the density, in a Gaussian distribution the average
value of $\delta$ is zero and the distribution is symmetric around zero. But
as perturbations grow under gravity, overdense regions can grow almost without
limit (e.g. you, as a human being with an average density close to the one of
water, do correspond to $\delta\sim 2\times 10^{30}$), but underdense regions
can never go below $\delta=-1$; therefore, a skewness develops. and thus the
power spectrum of large-scale structure does not include all the statistical
properties of the distribution. Nevertheless, the power spectrum of large-
scale structure is a powerful tool to constrain cosmology. Similarly to the
CMB transfer function, the primordial spectrum also gets processed through the
so-called matter transfer function. I will not go into further details here,
but I will just say that, for a given model, the matter transfer function can
be computed very accurately and thus there is (within a model) a one-to-one
correspondence between the primordial spectrum and the matter power spectrum.
To date, the power spectrum has been probed from scales of
$k=0.001\leavevmode\nobreak\ h$/Mpc to highly nonlinear scales of $\sim
5\leavevmode\nobreak\ h$/Mpc using different tracers and different measurement
approaches. The standard cosmological model as constrained by, for example,
CMB-only observations fits extremely well the large-scale structure
observations. The combination of the two datasets can be used to break some
parameter degeneracies but, more importantly, to test for deviations from the
standard model.

The CMB data alone in particular are not too stringent in measuring the value
of $\Omega_{\Lambda}$ once the spatial flatness assumption is dropped or even
imposing flat geometry if constraining the properties of dark energy (whether
it is truly a cosmological constant or some dynamical quantity). The
limitation of the CMB in constraining dark energy is that the CMB is situated
at a given redshift ($z=1090$ in the standard cosmological model to be
precise). As discussed in section 0.6, dark energy is seen through the
acceleration of the expansion rate, so one really needs to be able to look at
more than a single snapshot of the Universe. The large-scale structure allows
one to do that. In fact, the fundamental mode that we see in the CMB, given by
the sound horizon at last scattering, is also imprinted in the large-scale
structure clustering. In fact, imagine a perturbation with dark matter,
baryonic matter and photons starting all in the same phase. Photons freely
stream out of the dark matter potential well carrying the baryons with them.
At decoupling, baryons find themselves offset with respect to the dark matter
and fall into the dark matter potential wells. But, as baryons are one-seventh
of the dark matter, the dark matter “feels” the baryons also through gravity.
Thus the same acoustic oscillations present in the CMB radiation should be
present in the dark matter distribution, albeit with a much smaller amplitude.
If, then, galaxies form in the dark matter potential wells, the galaxy
distribution should also carry the same imprint. This signature goes under the
name baryon acoustic oscillations (BAO) [14] and was first detected in galaxy
surveys in 2005 [15, 16]. Of course, if the BAO signature can be detected in a
large-scale structure survey at different redshifts, one would have a standard
ruler with which to trace the expansion history of the Universe. More
precisely, a measurement of the radial BAO signal would give a combination of
the standard ruler and the Hubble parameter as a function of redshift, and the
angular BAO signal would yield a combination of the standard ruler and the
angular diameter distance. This is what current, forthcoming and planned
surveys are aiming to measure. In addition, the shape of the large-scale
structure power spectrum encloses information not only about the primordial
fluctuations but also about cosmological parameters. Of course, there are many
challenges in this measurement beside the technical difficulty of surveying
the position of galaxies over a sizeable part of the observable Universe;
among them, modelling the effects of nonlinear evolution, and the fact that
galaxies may not be faithful tracers of the underlying dark matter
distribution. To date, the most recent constraints on cosmology from such a
signal are presented in Refs. [17, 18, 19].

Figure 4: (left) The large-scale structure power spectrum as measured from the
Digital Sky Survey data release 7. (right) The same power spectrum divided by
a smooth one to highlight the BAO feature. The solid line is the theory
prediction for a standard LCDM model. Figure reproduced from Ref. [17].

One aspect that the combination of CMB and large-scale structure can measure
well and for which future data promise to be particularly constraining and of
interest to this audience is neutrino properties.

##  0.4 Neutrino properties and cosmology

Neutrino oscillations indicate that neutrinos have mass, although neutrino
mass eigenstates are not the same as flavour eigenstates. The standard model
has three neutrinos and, combined with oscillation constraints (one mass
splitting much larger than the other one), one concludes that three neutrino
mass hierarchies are possible: normal (one heavy neutrino and two light ones –
NI), inverted (two heavy and one light – IH) and degenerate (where the mass
splitting is small compared to the overall neutrino mass scale). While
particle physics experiments are extremely sensitive to flavour but much less
sensitive to the absolute mass, cosmology is insensitive to flavour but
sensitive to the absolute mass scale. Cosmological constraints on neutrino
properties are highly complementary to particle physics experiments for
several reasons, as we explain below.

  1. 1.

Relict neutrinos.
Relict neutrinos produced in the early Universe are hardly detectable by weak
interactions, making it impossible with foreseeable technology to detect them
directly. But cosmological probes offer the opportunity to detect (albeit
indirectly) relict neutrinos. The hot Big Bang model predicts a background of
relict neutrinos in the Universe with an average number density of $\sim
100N_{\nu}$ cm-3. These neutrinos decouple from the CMB at redshift $z\sim
10^{10}$ when the temperature was about a few MeV, but remain relativistic
down to much lower redshifts depending on their mass.

Massive neutrinos affect cosmological observations in different ways. Primary
CMB data alone can constrain the total neutrino mass, $\Sigma$, if it is above
$\sim 1$ eV ([13] find $\Sigma<1.3$ eV at 95% confidence) because these
neutrinos become non-relativistic before recombination, leaving an imprint in
the CMB. Neutrinos with masses $\Sigma<1$ eV become non-relativistic after
recombination, altering matter–radiation equality for fixed $\Omega_{m}h^{2}$;
this effect is degenerate with other cosmological parameters from primary CMB
data alone. After neutrinos become non-relativistic, their free streaming
damps the small-scale power and modifies the shape of the matter power
spectrum below the free-streaming length. The free-streaming length of each
neutrino family depends on its mass.

Current cosmological observations do not detect any small-scale power
suppression and break many of the degeneracies of the primary CMB, yielding
constraints of $\Sigma<0.3$ eV [17]. A detection of such an effect, however,
would provide a detection, although indirect, of the cosmic neutrino
background. The fact that oscillations predict a minimum total mass
$\Sigma\sim 0.54$ eV implies that forthcoming surveys have the statistical
power to detect the cosmic neutrino background.

  2. 2.

Cosmology remains a key avenue to determine the absolute neutrino mass scale.
Particle physics experiments will be able to place lower limits on the
effective neutrino mass which depend on the hierarchy, with no rigorous limit
achievable in the case of normal hierarchy. Neutrinos free streaming suppress
the small-scale clustering of large-scale cosmological structures by an amount
that depends on neutrino mass. Planned surveys should yield an error on
$\Sigma$ of 0.04 (for details see Ref. [20]) and therefore will not only
detect the effect of massive neutrinos on clustering but also determine the
absolute neutrino mass scale.

  3. 3.

“What is the hierarchy (normal, inverted or degenerate)?”
Neutrino oscillation data are unable to resolve whether the mass spectrum
consists of two light states with mass $m$ and a heavy one with mass $M$ (NH)
or two heavy states with mass $M$ and a light one with mass $m$ (IH) in a
model-independent way. Cosmological observations can help to determine the
hierarchy. Since cosmology is insensitive to flavour, one might expect that
cosmology may not help in determining the neutrino mass hierarchy. However,
for $\Sigma<0.1$ eV, only normal hierarchy is allowed, and thus a mass
determination can help disentangle the hierarchy. There is, however, another
effect: neutrinos of different masses become non-relativistic at slightly
different epochs; the free-streaming length is slightly different for the
different species, and thus the detailed shape of the small-scale power
suppression depends on the individual neutrino masses (not just on their sum).

As discussed in Ref. [21], in cosmology one can safely neglect the impact of
the solar mass splitting. Thus two masses characterize the neutrino mass
spectrum, the lighter one, $m$, and the heavier one, $M$. The mass splitting
can be parametrized by $\Delta=(M-m)/\Sigma$ for normal hierarchy and
$\Delta=(m-M)/\Sigma$ for inverted hierarchy. Cosmological data are very
sensitive to $|\Delta|$; the direction of the splitting (the sign of $\Delta$)
introduces a subdominant correction to the main effect. Planned surveys should
be able to measure $|\Delta|$ and may be able to help to determine the
hierarchy (i.e. the mass splitting and its sign) if far enough away from the
degenerate hierarchy (i.e. if $\Sigma<0.13$).

  4. 4.

“Are neutrinos their own antiparticle?”
If the answer is “yes”, then neutrinos are Majorana fermions; if not, they are
Dirac. If neutrinos and antineutrinos are identical, there could have been a
process in the early Universe that affected the balance between particles and
antiparticles, leading to the matter–antimatter asymmetry we need to exist.
This question can, in principle, be resolved if neutrinoless double-beta decay
is observed. But if such experiments lead to a negative result, the
implications for the nature of neutrinos depend on the hierarchy. As shown in
Ref. [21] (and references therein), in this case cosmology can offer
complementary information by helping to determine the hierarchy.

  5. 5.

Number of neutrino species.
After decoupling, neutrinos contribute to the relativistic energy density.
Cosmology is sensitive to the physical energy density in relativistic
particles in the early Universe $\omega_{\rm rel}$, as it affects the
expansion history; $\omega_{\rm rel}$ includes photons and neutrinos
$\omega_{\rm rel}=\omega_{\gamma}+N_{\rm eff}\omega_{\nu}$, where
$\omega_{\gamma}$ is the energy density in photons, $\omega_{\nu}$ the energy
density in one neutrino and $N_{\rm eff}$ is the effective number of neutrino
species. In the standard model with three neutrinos $N_{\rm eff}=3.046$ to
account for quantum chromodynamic effects and for neutrinos being incompletely
decoupled during electron–positron annihilation.

With $\omega_{\gamma}$ being extremely well constrained from the measurement
of the CMB temperature, constraints on $\omega_{\rm rel}$ can be interpreted
in terms of $N_{\rm eff}$, although a deviation from $N_{\rm eff}=3.046$ may
indicate not only extra neutrino species but also any process that affects the
expansion history. Any light particle that does not couple to electrons, ions
and photons will act as an additional relativistic species. The relativistic
energy density $\omega_{\rm rel}$ (and therefore $N_{\rm eff}$) impacts the
Big Bang nucleosynthesis through it effect on the expansion rate. The
relativistic energy density also affects the CMB via the expansion rate, which
alters the epoch of matter–radiation equality.

##  0.5 Inflation

The popular Big Bang model (combined with general relativity and the
cosmological principle) is extremely successful at predicting observations
such as Hubble’s law, the CMB and the abundance of light elements. However, it
leaves some major open problems: the flatness problem, the horizon problem and
the monopole problem (and the origin of the cosmological perturbations). In
other words, we still have the following puzzles:

  1. (a)

Why is the Universe so flat? The Friedmann equations extrapolated back in
time555See class slides. basically say that, in order for $|\Omega-1|<0.1$,
say, as observations indicate, very early on $|\Omega-1|<10^{-60}$.

  2. (b)

Why is the Universe so big? The Universe is homogeneous on large scales. Take,
for example, the CMB radiation: two antipodal points are separated by 30 000
Mpc and are both at 2.726 K (modulo the CMB temperature fluctuations, which
are, however, very very small), but the age of the Universe back then was only
380 000 years. Light could have travelled only 380 000 light years, i.e. only
points closer than that (i.e. closer than about 0.2 Mpc) could have been in
causal contact. How, then, can two antipodal points be at the same
temperature?

  3. (c)

If the Universe cooled from an extremely hot phase, in the very early Universe
there should have been phase transitions and they should have left
cosmological defects. Of all the types of cosmological defects, monopoles are
the most dangerous ones: more than one in the observable Universe would be
catastrophic, i.e. would violate all observations.

  4. (d)

Finally, where do the primordial perturbations we talked about before come
from?

The paradigm of inflation comes to the rescue. The concept of inflation was
started in 1981 by Alan Guth, but it is still a very active area of research
and is still a paradigm rather than a specific model or theory. Inflation
postulates a brief period of accelerated expansion in the very early Universe,
something not too dissimilar from a cosmological constant. In particular, one
postulates the existence of one (or more) field (the inflaton) which rolls
down a potential very slowly. Specific models have different shapes for the
inflaton potential.

This solves all the above puzzles in one go. (c) Monopoles are diluted away
very quickly so that there would be no more than one in the observable
Universe. (a) The flatness problem is solved, as a rapid expansion by many
e-foldings would make any non-flat geometry arbitrarily close to flat. (b) In
order to understand the solution of the horizon problem, we need to realize
that in cosmology there are different types of horizons and this happens
because light travels at finite speed (which gives a horizon) but at the same
time the Universe is expanding. Thus the particle horizon is the maximum
comoving distance light could have propagated in a given interval of time from
a specific $t_{1}$ to $t_{2}$, and takes into account the past expansion
history. Note that, for the same time interval, the particle horizon can be
very different if the expansion history changes! Then there is the Hubble
horizon, which is just the speed of light divided by the time interval. For
normal expansion histories, the Hubble horizon is not too different from the
particle horizon, but for weird expansion histories the two can be very
different. For example, in an accelerated expansion, the particle horizon
becomes much bigger than the Hubble horizon. So all regions of the Universe,
even if separated by many Hubble horizons, are so uniform because in the past
they were in causal contact – they are all within each other’s particle
horizon! Because of (b), inflation generated super-horizon perturbations (of
course, this refers to the Hubble horizon). (d) As an added bonus inflation
generates Gaussian perturbations: quantum fluctuations stretched to become
classical by the expansion. The mechanism is similar to Hawking radiation.
(The event horizon being the complementary concept to the particle horizon.)
It turns out that the power spectrum of these perturbations will be a power
law: different models of inflation predict slightly different power laws, but
all are close to “scale-invariant”.

One may ask, can this paradigm of inflation be tested? Inflation predicts a
flat Universe, and so far observation says that the Universe is consistent
with being flat. Inflation predicts a power-law primordial power spectrum
close to scale-invariant, and specific models yield slightly different slopes.
Observations are fully consistent with a power-law power spectrum close to
scale-invariant, and we hope to be able to distinguish different inflationary
models by determining the slope accurately. For example, current data give
$n_{s}=0.968\pm 0.012$ [13], which excludes the perfectly scale-invariant
spectrum at the 99.5% confidence level. Inflation predicts primordial
perturbations very close to Gaussian, and indeed this is confirmed by
observations. Inflation predicts super-horizon perturbations and a stochastic
background of gravity waves. These prediction can also be tested with
observations. To understand this, we first need to introduce CMB polarization.

###  0.5.1 CMB polarization

The CMB light was predicted to be polarized shortly after the CMB was
discovered. The CMB polarization signal was first detected on a patch of the
sky by the DASI experiment in 2002 [22, 23] and then mapped over the full sky
by WMAP.

Temperature quadrupole at the last scattering surface generates polarization
via Thomson scattering.666See class slides. On sub-horizon scales it is the
density perturbations that give the quadrupolar pattern. One should therefore
expect a radial pattern around a hot spot and a tangential one around cold
spots. This has been seen (albeit by stacking the pattern around many spots)
in 2010 (see Ref. [13]). For density fluctuations on large (super-horizon
scales) it is the velocity they source (from hot spots to cold spots) that
generates the quadrupolar pattern. Since the velocity is out of phase with the
density perturbation, and on super-horizon scales the temperature signal is
sourced by the density and the polarization by the velocity, one should expect
an anticorrelation between temperature and polarization on these scales [24,
25], which has been seen (see Ref. [26]). This is the signature of the
existence of super-horizon perturbations as predicted by inflation.

There is more. Density perturbations have no handedness and therefore the
polarization perturbations also do not have handedness (recall: radial or
tangential patterns around spots). Gravity waves stretch space-time and also
generate a quadrupolar pattern at the last scattering surface. But gravity
waves are tensor perturbations and have a handedness. The CMB polarization
pattern can be decomposed into two modes in analogy to electromagnetism: the E
mode (which has no handedness) and the B mode (with handedness). The B-mode
pattern cannot be generated by scalar perturbations but can be generated by
gravity waves. A detection of primordial B-mode polarization in the CMB would
be a detection of the inflationary stochastic gravity wave background. This is
the holy grail of CMB polarization.

The measurement of CMB primary polarization is, however, very difficult. In
fact, the signal is very small (orders of magnitude below the temperature
signal) and is contaminated by the emission of our own Galaxy.

In summary, quantum fluctuations get stretched to become classical and “super-
horizon” because of the accelerated expansion. The spacing of the fluctuations
(their power as a function of scale) depends on how fast they exited the
horizon (i.e. on the Hubble parameter during inflation), which in turn depends
on the inflaton potential. Thus the shape of the primordial power spectrum
encloses information on the shape of the inflaton potential! In general, the
observational constraints of $N_{\rm efold}>50$ requires the potential to be
flat (not every scalar field can be the inflaton). Detailed measurements of
the shape of the power spectrum can rule in or out different potentials. So
far, observations are consistent with the simplest inflationary models; from
current datasets, some specific inflationary models are being critically
tested.

For more information and an overview of references and possible forthcoming
data, see Refs. [27, 28] and references therein.

##  0.6 Dark energy

If we go back to look at Fig. 3 we quickly realize that in the standard
cosmological model 96% of the Universe is missing, i.e. it is supposed to be
in the form of dark matter or dark energy, about which we know very little.
While the dark matter at least is a form of matter and may soon be detected
directly by one of the underground ongoing experiments, there are two major
open questions of the model that can be observationally addressed exclusively
by looking at the sky: What created the primordial perturbations? What makes
the Universe accelerate? These two questions may not be unrelated. The first
question was examined in section 0.5, so here we concentrate on the second
question.

A major part of the observational effort in cosmology today is devoted to the
accelerating Universe and dark energy. It is important to stress a peculiarity
of cosmological observations: this sets a fundamental limit on how well a
property of the Universe or a parameter of the model can be measured, which is
called the cosmic variance limit. The cosmological principle of homogeneity
and uniformity implies that we can use statistical properties of the Universe
as observables and as a measurable quantity to make connection with
theoretical predictions. Inflation also tells us that the visible Universe is
only a small part of the Universe. It is only possible to observe part of the
Universe at one particular time, so it is difficult to make statistical
statements about cosmology on the scale of the entire Universe, as the number
of independent observations (sample size) is finite. In cosmology we are
interested in the properties of the underlying model, which is valid for the
entire Universe, not just the observable Universe. This sets an error floor,
and a fundamental limit to any measurement, the so-called cosmic variance
error.

Like every cloud has a silver lining, this means that in principle in
cosmology one can plan the perfect experiment: repeating the experiment or
improving it will not improve the statistical errors. In cosmology, therefore,
we can do what I call “ultimate experiments”. The interesting new development
is that planned and some ongoing surveys are ultimate experiments.

Back to dark energy. If we assume a model where the Universe is composed by
matter and a cosmological constant, and we impose spatial flatness, then CMB
observations alone indicate that $\Omega_{m}\sim 0.27$ and
$\Omega_{\Lambda}\sim 0.73$. If we do not impose flatness, then we need to
consider along with the CMB also supernovae data or large-scale structure data
or measurements of the Hubble parameter (or more than one of these, of
course). In this case, then, a flat model with the above values for matter and
cosmological constant is a very good fit to the observations.

In Einstein’s equations, the cosmological constant can be interpreted as
vacuum energy. However, when computing the expected vacuum energy density from
the quantum field theory approach, the theoretically predicted value exceeds
the measured one by 123 orders of magnitude. This has been dubbed the
“cosmological constant problem”. The vacuum energy density (which does not
dilute as the Universe expands) would have made a negligible contribution in
the early Universe but the future Universe would grow so quickly as to be
essentially empty of matter. This is the “Why now?” problem: Why is the brief
epoch when the cosmological constant is non-negligible but still comparable
with the matter density coincident with the present day? Why is the present
day so special?

To overcome these problems, one needs to look for different physical
mechanisms. A dynamical option is to suppose that a cosmic scalar field,
minimally coupled to gravity, called “quintessence” , changing with time and
slowly varying across space, is slowly approaching its ground state. In the
basic quintessence scenario, the dark energy enters only at late times in the
evolution of the Universe. More realistic scalar field models of quintessence
track the dominant component of the cosmological cosmic fluid until the
current epoch, when the quintessence energy overtakes the matter density. In
the case of quintessence, dark energy can be interpreted as a fluid with an
effective equation-of-state parameter $w=p/\rho$, where $p$ demotes the
pressure. For a cosmological constant $w\equiv-1$, for quintessence $w\neq-1$
and it can be a function of redshift. Current constraints on $w$ assumed to be
constant in redshift and assuming a flat Universe are shown in Fig. 5. Note
that, so far, observations are consistent with a cosmological constant.

Figure 5: Observational constraints on the dark energy equation-of-state
parameter assumed to be constant with redshift and assuming a flat Universe
geometry: black, WMAP, seven years only; red, WMAP \+ BAO + $H_{0}$; yellow,
WMAP \+ supernovae; cyan, WMAP \+ BAO + supernovae.

It is customary to parametrize a possible evolution of the equation-of-state
parameter as a linear function of the scale factor. This parametrization is
useful to quantify the performance of future surveys or to quantify how
different from a cosmological constant dark energy may be as long as
observations give back a $w$ consistent with $-1$, but it does not have any
physical motivation.

The conclusion that we live in such a strange Universe relies on the
assumption that general relativity holds on scales comparable to the horizon.
In fact the evidence for dark energy comes from observations of the Universe
on extremely large scales, comparable to the size of the observable Universe.
However, precision tests of gravity have been made only up to Solar System
scales. An enormous extrapolation (13 orders of magnitude) is needed to apply
it to horizon scales. It may be that gravity as we know it breaks down on
those scales, and we interpret that as an accelerated expansion. Is it
possible to distinguish between all these different possible explanations for
dark energy (i.e. cosmological constant, a slowly varying scalar field or
modifications of general relativity on large scales)?

Let us review the observational effects of dark energy. They can be divided
into two broad classes. The first is observations of the expansion history of
the Universe. These are the recession velocity versus brightness of standard
candles (e.g. supernovae), the position of the CMB acoustic peaks, which gives
the angular diameter distance to the last scattering surface, and the BAO,
which give angular diameter distance and Hubble parameter as a function of
redshift. The second is the growth rate of structures. The growth of
cosmological perturbations is driven by gravity but its rate is a result of
two competing effects: the pull of gravity and the expansion rate of space-
time. If gravity is well described by GR, then the growth rate of structure
can yield unequivocally the expansion rate and thus the properties of a
cosmological constant or quintessence. Galaxy (or gravitational lensing)
surveys can yield a measurement of the growth of structure because they can
measure the amplitude of the power spectrum as a function of redshift. The
clustering of galaxies, however, may not be a faithful tracer of the density
field, as galaxies may be biased tracers. In fact, the process of galaxy
formation is complex and may not be solely driven in a deterministic and
simple way by the dark matter overdensity.

On the other hand, galaxy surveys also probe the peculiar velocity field. In
fact, galaxy surveys use the observed galaxy redshift as a distance indicator;
this would be a faithful tracer of the distance in a perfectly uniform
Universe where everything moved with the Hubble flow. In a clumpy Universe,
the Hubble flow is disturbed by overdensities and so clustering in the line-
of-sight direction gets distorted. While this may seem a nuisance, if it can
be correctly modelled and interpreted, it is not. In fact, such distortions
directly probe the velocity field. The velocity field is sourced directly by
the potential, thus it is a direct probe of the density perturbations and it
is not as subject to bias as is the galaxy overdensity field.

On the other hand, if gravity on large scales deviates from general
relativity, this would affect both the expansion history and the growth rate
of structure. Any expansion history (due to quintessence or due to
modifications of gravity) can be interpreted as a quintessence model with a
suitable $w(z)$. But if gravity is modified, then there would be a mismatch
between expansion history and growth. This is the reason why forthcoming
surveys are poised to measure both expansion history and growth of structures
– see e.g. Refs. [29, 30] and references therein.

##  0.7 Conclusions

The standard cosmological model is extremely successful: with a handful of
parameters it can describe observations of the Universe on large scales from
380 000 years after the Big Bang until today (13.7 billion years after the Big
Bang). The parameters of the model are constrained observationally with per
cent precision. However, we have no real understanding of what these
parameters mean. Cosmology is now in a similar stage in its intellectual
development to particle physics four decades ago when particle physicists
converged on the current Standard Model. The Standard Model of particle
physics fits a wide range of data, but does not answer many fundamental
questions, for example: What is the origin of mass? Why is there more than one
family? Similarly, the standard cosmological model has many deep open
questions: What is the dark energy? What is the dark matter? What is the
physical model behind inflation (or something like inflation)? The questions
address an emerging model of the Universe that connects physics at the most
microscopic scales to the properties of the Universe and its contents on the
largest physical scales.

It is interesting to note that, while dark matter may soon be detected
directly by underground experiments, the open questions of dark energy and
inflation can be addressed almost exclusively by observing the cosmos. Dark
energy and inflation both rely on an accelerated expansion of the Universe.

Despite inflation happening 13.7 billion years ago and dark energy happening
today, we seem to know much less about dark energy. In fact, we can test about
10 efoldings of inflation by looking at cosmological structures, but we cannot
see dark energy perturbations and we can only see about two efoldings. But we
can follow the (recent) expansion history and the growth of cosmological
structures and this is what forthcoming surveys are poised to do. The wealth
of forthcoming cosmological data will provide ever more rigorous tests of the
cosmological standard model and search for new physics beyond the Standard
Model.

## Acknowledgements

We wish to thank the organizers of the 2011 CERN–Latin-American School of
High-Energy Physics in Natal, Brazil, and all the participants and lecturers
for a memorable week. LV acknowledges support of FP7 IDEAS-Phys.LSS 240117.

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