Observations show that the universe is fairly homogeneous and isotropic
at scales
larger than about 150 *h*^{-1} Mpc where 1 Mpc
3
× 10^{24}
cm and *h* 0.7
is a parameter related to the expansion rate of
the universe. The conventional - and highly successful - approach to
cosmology separates the study of large scale (*l*
150
*h*^{-1} Mpc) dynamics
of the universe from the issue of structure formation at smaller
scales. The former is modeled by a homogeneous and isotropic
distribution of energy density; the latter issue is
addressed in terms of gravitational instability which will amplify the
small perturbations in the energy density, leading to the formation of
structures like galaxies.
In such an approach, the expansion of the background universe is
described by the metric (We shall use units with with *c* = 1
throughout, unless otherwise specified):

(1) |

with *S*_{k}()
= (sin,
,
sinh) for the three values
of the label *k* = (1,0,-1). The function *a*(*t*)
is governed by the equations:

(2) |

The first one relates expansion rate of the universe to the energy
density and
*k* = 0, ± 1 is a parameter which
characterizes the spatial curvature of the universe.
The second equation, when coupled with the equation of state
*p* =
*p*()
which relates the pressure *p* to the energy density,
determines the evolution of energy density
=
(*a*) in
terms of the expansion factor of the universe.
In particular if *p* =
*w* with
(at least, approximately) constant *w*
then,
*a*^{-3(1+w)} and (if we further assume *k* =
0, which is strongly favoured by observations) the first equation in
Eq.(2) gives
*a*
*t*^{2/[3(1+w)]}. We will also often use the
redshift *z*(*t*),
defined as (1 + *z*) = *a*_{0} / *a*(*t*)
where the subscript zero denotes quantities evaluated at the present
moment. in a *k*=0 universe,
we can set *a*_{0} = 1 by rescaling the spatial coordinates.

It is convenient to measure
the energy densities of different components in terms of a *critical
energy density*
(_{c})
required to make *k* = 0 at the present epoch. (Of course, since
*k* is a constant,
it will remain zero at all epochs if it is zero at any given moment of
time.) From Eq.(2), it is clear that
_{c} =
3*H*_{0}^{2} /
8 *G*
where *H*_{0}
( /
*a*)_{0} - called the Hubble constant -
is the rate of expansion of the universe at present. Numerically

(3) |

The variables
_{i}
_{i} /
_{c}
will give the fractional contribution of different components of the
universe (*i* denoting baryons, dark matter, radiation, etc.) to the
critical density. Observations then lead to the following results:

(1) Our universe has 0.98
_{tot}
1.08. The value
of _{tot}
can be determined from the angular anisotropy spectrum
of the cosmic microwave background radiation (CMBR; see
Section 6) and these observations (combined
with the reasonable
assumption that *h* > 0.5) show
[1]
that we live in a universe with critical density, so that *k* = 0.

(2) Observations of primordial deuterium produced in big bang
nucleosynthesis (which took place when the universe
was about few minutes in age) as well as the CMBR observations show
[2]
that the *total* amount of baryons in the
universe contributes about
_{B} =
(0.024 ± 0.0012) *h*^{-2}. Given the
independent observations
[3]
which fix *h* = 0.72 ± 0.07, we conclude
that _{B}
0.04-0.06. These
observations take into account all
baryons which exist in the universe today irrespective of whether they
are luminous or not. *Combined with previous item we conclude
that most of the universe is non-baryonic.*

(3) Host of observations related to large scale structure and dynamics
(rotation curves of galaxies, estimate of cluster masses, gravitational
lensing, galaxy surveys ..) all suggest
[4]
that the universe is
populated by a non-luminous component of matter (dark matter; DM
hereafter) made of weakly interacting massive particles which
*does* cluster at galactic scales. This component contributes
about _{DM}
0.20-0.35 and has the
simple equation of state
*p*_{DM}
0. The second equation in Eq.(2), then gives
_{DM}
*a*^{-3}
as the universe expands which arises from the
evolution of number density of particles:
=
*nmc*^{2}
*n*
*a*^{-3}.

(4) Combining the last observation with the first we conclude that there
must be (at least) one more component
to the energy density of the universe contributing about 70% of
critical density. Early analysis of several observations
[5]
indicated that this component is unclustered and has negative
pressure. This is confirmed dramatically by the supernova observations
(see Ref. [6];
for a critical look at the current data, see
Ref. [7]).
The observations suggest that the missing
component has *w* = *p* /
-0.78
and contributes
_{DE}
0.60-0.75. The simplest
choice for
such *dark energy* with negative pressure is the cosmological
constant which is a term that can be added to Einstein's equations. This
term acts like a fluid with an equation of state *p*_{DE} =
-_{DE};
the second equation in Eq.(2), then gives
_{DE} =
constant as universe expands.

(5) The universe also contains radiation contributing an energy density
_{R}
*h*^{2} = 2.56× 10^{-5} today most of which is
due to photons in the CMBR.
The equation of state is *p*_{R} = (1/3)
_{R};
the second equation in Eq.(2), then gives
_{R}
*a*^{-4}. Combining it with the result
_{R}
*T*^{4}
for thermal radiation, it follows that
*T*
*a*^{-1}.
Radiation is dynamically irrelevant today but since
(_{R} /
_{DM})
*a*^{-1}
it would have been the dominant component when
the universe was smaller by a factor larger than
_{DM} /
_{R}
4× 10^{4}
_{DM}
*h*^{2}.

(6) Taking all the above observations together, we conclude that our
universe has (approximately)
_{DE}
0.7,
_{DM}
0.26,
_{B}
0.04,
_{R}
5×
10^{-5}. All known observations are consistent with such an -
admittedly weird - composition for the universe.

Using _{NR}
*a*^{-3},
_{R}
*a*^{-4} and
_{DE} =
constant we can write Eq.(2) in a convenient dimensionless form as

(4) |

where =
*H*_{0} *t*, *a* = *a*_{0}
*q*(),
_{NR} =
_{B} +
_{DM} and

(5) |

This equation has the structure of the first integral for
motion of a particle with energy *E* in a potential
*V*(*q*).
For models with =
_{NR} +
_{DE} = 1,
we can take *E* = 0 so that (*dq* /
*d*) =
(*V*(*q*))^{1/2}. Based on the
observed composition of the universe, we can identify three distinct
phases in the evolution of the universe when the temperature is less
than about 100 GeV. At high redshifts (small *q*)
the universe is radiation dominated and
Img src="../../New_Gifs/qdot.gif" alt="dot{q}"> is independent
of the other cosmological parameters. Then Eq.(4) can be
easily integrated to give *a*(*t*)
*t*^{1/2} and the temperature of the universe
decreases as *T*
*t*^{-1/2}. As the universe expands, a time will come
when (*t* = *t*_{eq}, *a* = *a*_{eq}
and *z* = *z*_{eq}, say)
the matter energy density will be comparable to radiation energy
density. For the parameters described above,
(1 + *z*_{eq}) =
_{NR} /
_{R}
4× 10^{4}
_{DM}
*h*^{2}. At
lower redshifts, matter will dominate over radiation and we will
have *a*
*t*^{2/3} until fairly late when the
dark energy density will dominate over
non relativistic matter.
This occurs at a redshift of *z*_{DE} where (1 +
*z*_{DE}) =
(_{DE} /
_{NR})^{1/3}.
For _{DE}
0.7,
_{NR}
0.3, this
occurs at *z*_{DE}
0.33. In this phase,
the velocity
changes
from being a decreasing function to an increasing function leading to
an accelerating universe. In addition to these, we believe that the
universe probably went through a rapidly expanding, inflationary, phase
very early when
*T*
10^{14} GeV;
we will say more about this in Section 7. (For a
textbook description of these and related issues, see e.g.
Ref. [8].)

Before we conclude this section, we will briefly mention some key aspects of the background cosmology described by a Friedmann model.

(a) The metric in Eq.(1) can be rewritten using the
expansion parameter *a* or the redshift *z* =
(*a*_{0} / *a*)^{-1} -1 as the
time coordinate in the form

(6) |

This form clearly shows that the only dynamical content of the metric is
encoded in the function *H*(*a*) =
( / *a*). An
immediate consequence is
that any observation which is capable of determining the geometry of the
universe can only provide - at best - information about this function.

(b) Since cosmological observations usually use radiation received from distant sources, it is worth reviewing briefly the propagation of radiation in the universe. The radial light rays follow a trajectory given by

(7) |

if the photon is emitted at *r*_{em} at the redshift
*z* and received here today.
Two other quantities closely related to *r*_{em}(*z*)
are the luminosity distance, *d*_{L}, and the angular
diameter distance *d*_{A}. If
we receive a flux *F* from a source of luminosity *L*, then the
luminosity distance is defined via the relation *F*
*L* /
4
*d*_{L}^{2}(*z*) . If an object of transverse
length *l* subtends a small
angle , the angular
diameter distance is defined via (*l* =
*d*_{A}).
Simple calculation shows that:

(8) |

(c) As an example of determining the spacetime geometry of the universe
from observations, let us consider how one can determine
*a*(*t*) from the
observations of the luminosity distance. It is clear from the
first equation in Eq. (8) that

(9) |

where the last form is valid for a *k* = 0 universe. If we
determine the form of *d*_{L}(*z*) from observations -
which can be done if we can
measure the flux *F* from a class of sources with known value for
luminosity *L* - then we can use
this relation to determine the evolutionary history of the universe and
thus the dynamics.