Having discussed the evolution of the background universe, we now turn
to the study of structure formation.
Before discussing the details, let us briefly summarise the broad picture
and give references to some of the topics that we will *not* discuss.
The key idea is that if there existed small fluctuations in the energy
density in the early universe, then gravitational instability can
amplify them in a well-understood manner leading to structures like
galaxies etc. today. The most popular model for generating these
fluctuations is based on the idea that if the very early universe went
through an inflationary phase
[9],
then the quantum fluctuations of the field driving the inflation can
lead to energy density fluctuations
[10
, 11].
It is possible to construct models of inflation such that these
fluctuations are described by a Gaussian random field and are
characterized by a power spectrum of the form *P*(*k*) =
*A* *k*^{n} with *n*
1 (see
Sec. 7). The models cannot predict the value
of the amplitude *A* in an unambiguous manner but it can be
determined from CMBR observations. The CMBR observations are consistent
with the inflationary model for the generation of perturbations and
gives *A*
(28.3
*h*^{-1} *Mpc*)^{4} and *n* = 0.97
± 0.023 (The first results were from COBE
[12] and
WMAP has reconfirmed them with far greater accuracy).
When the perturbation is small, one can use well defined linear
perturbation theory to study its growth. But when
(
/
) is comparable
to unity the perturbation theory breaks down. Since there is more power
at small scales, smaller scales go non-linear first and structure forms
hierarchically. The non linear evolution of the *dark matter halos*
(which is an example of statistical mechanics of self gravitating
systems; see e.g.
[13])
can be understood by simulations as well as theoretical models based on
approximate ansatz
[14]
and nonlinear scaling relations
[15].
The baryons in the halo will cool and undergo collapse
in a fairly complex manner because of gas dynamical processes.
It seems unlikely that the baryonic collapse and galaxy formation can be
understood by analytic approximations; one needs to do high resolution
computer simulations to make any progress
[16].
All these results are broadly consistent with observations.

As long as these fluctuations are small, one can study their evolution
by linear perturbation theory, which is what we will start with
[17].
The basic idea of linear perturbation theory is well defined and
simple. We perturb the background FRW metric by
*g*_{ik}^{FRW}
*g*_{ik}^{FRW} + *h*_{ik} and also
perturb the source energy momentum tensor by
*T*_{ik}^{FRW}
*T*_{ik}^{FRW} +
*T*_{ik}. Linearising the Einstein's equations, one can
relate the perturbed quantities by a relation of the form
(*g*_{ik}^{FRW}) *h*_{ik} =
*T*_{ik}
where is second order
linear differential operator depending on the back ground metric
*g*_{ik}^{FRW} .
Since the background is maximally symmetric, one can separate out time
and space; for e.g, if *k* = 0, simple Fourier modes can be used
for this purpose and we can write down the equation for any given mode,
labelled by a wave vector **k** as:

(42) |

To every mode we can associate a wavelength normalized to today's value:
(*t*) =
(2 / *k*)(1 +
*z*)^{-1} and a corresponding mass scale
which is invariant under expansion:

(43) |

The behaviour of the mode depends on the relative value of
(*t*) as compared
to the Hubble radius *d*_{H}(*t*)
( /
*a*)^{-1}. Since the Hubble radius:
*d*_{H}(*t*)
*t* while the
wavelength of the mode:
(*t*)
*a*(*t*)
(*t*^{1/2},
*t*^{2/3}) in the radiation dominated and matter dominated
phases it follows that
(*t*) >
*d*_{H}(*t*) at sufficiently early times. When
(*t*) =
*d*_{H}(*t*), we say that the mode is entering the
Hubble radius. Since the Hubble radius at *z* =
*z*_{eq} is

(44) |

it follows that modes with
_{0} >
_{eq} enter
Hubble radius in MD phase while the more relevant modes with
<
_{eq} enter
in the RD phase. Thus, for a given mode we can identify three distinct
phases: First, very early on, when
>
*d*_{H}, *z* > *z*_{eq} the dynamics
is described by general relativity. In this stage, the universe is
radiation dominated, gravity is the only relevant force and the
perturbations are linear. Next, when
<
*d*_{H} and *z* > *z*_{eq} one can
describe the dynamics by Newtonian considerations. The perturbations are
still linear and the universe is radiation dominated. Finally, when
<
*d*_{H}, *z* < *z*_{eq} we have a
matter dominated universe in which we can use the Newtonian formalism;
but at this stage - when most astrophysical structures form - we need to
grapple with nonlinear astrophysical processes.

Let us now consider the metric perturbation in greater detail. When the
metric is perturbed to the form: *g*_{ab}
*g*_{ab} + *h*_{ab} the perturbation can be
split as *h*_{ab} = (*h*_{00},
*h*_{0 }
*w*_{},
*h*_{}). We also
know that any 3-vector **w**(**x**) can be split as **w** =
**w**^{} +
**w**^{||} in which **w**^{||} =
^{||} is
curl-free (and carries one degree of freedom) while
**w**^{} is
divergence-free (and has 2 degrees of freedom). This result is obvious
in **k**-space since we can write any vector
**w**(**k**) as a sum of two terms, one along **k** and one
transverse to **k**:

(45) |

Fourier transforming back, we can split **w** into a curl-free and
divergence-free parts. Similar decomposition works for
*h*_{} by
essentially repeating the above analysis on each index. We can write:

(46) |

The *u*_{}^{} is divergence free and
*h*_{}
_{} is traceless and
divergence free. Thus
the most general perturbation *h*_{ab} (ten degrees of
freedom) can be built out of

(47) |

We now use the freedom available in the choice of
four coordinate transformations to set four conditions:
^{||} =
_{1} = 0 and
*u*_{}^{} = 0 thereby
leaving six degrees of freedom in (*h*_{00}
2,
,
**w**^{},
*h*_{}
_{}) as nonzero.
Then the perturbed line element takes the form:

(48) |

To make further simplification we need to use two facts from Einstein's
equations. It turns out that the Einstein's equations for
**w**^{} and
*h*_{}
_{} decouple from those
for (,
). Further, in the absence
of anisotropic stress, one of the equations give
=
. If we use these two
facts, we can simplify the structure of perturbed metric drastically. As
far as the growth of matter perturbations are concerned, we can ignore
*w*_{}^{} and
*h*_{}
_{} and work with a
simple metric:

(49) |

with just one perturbed scalar degree of freedom in . This is what we will study.

Having decided on the gauge, let us consider the evolution equations for
the perturbations. While one can directly work with the Einstein's
equations, it turns out to be convenient to use the equations of motion
for matter variables, since we are eventually interested in the matter
perturbations. In what follows, we will use the over-dot to denote
(*d* /
*d*)
so that the standard Hubble parameter is *H* = (1 /
*a*)(*da* / *dt*) =
/
*a*^{2}. With this notation, the continuity equation
becomes:

(50) |

Since the momentum flux in the relativistic case is
( + *p*)
*v*^{}, all
the terms in the above equation are intuitively obvious, except probably
the term. To
see the physical origin of this term, note that the perturbation in
Eq. (49) changes the factor in front of the spatial metric from
*a*^{2} to *a*^{2}(1 -
2) so that
ln*a* ln*a*
- ; hence the effective
Hubble parameter from
( / *a*) to
( / *a*) -
which explains the
extra term.
This is, of course, the exact equation for matter variables in the
perturbed metric given by Eq. (49); but we only need terms which are of
linear order. Writing the curl-free velocity part as
*v*^{}
= ^{} *v*, the
*linearised* equations, for dark matter (with *p*=0) and
radiation (with *p* = (1/3)
) perturbations
are given by:

(51) |

where *n*_{m} and *n*_{R} are the
number densities of dark matter particles and radiation.
The same equations in Fourier space [using the same symbols for, say,
(*t*,**x**) or
(*t*,**k**)] are
simpler to handle:

(52) |

Note that these equations imply

(53) |

For long wavelength perturbations (in the limit of *k*
0), this will
lead to the conservation of perturbation
(*s* /
*n*_{m}) in the entropy per particle.

Let us next consider the Euler equation which has the general form:

(54) |

Once again each of the terms is simple to interpret. The
( + *p*)
arises because the pressure also contributes to inertia in a
relativistic theory and the factor 4 in the last term on the right hand
side arises because the term
*v*^{}
_{}( + *p*) on the
left hand side needs to be compensated. Taking the linearised limit of
this equation, for dark matter and radiation, we get:

(55) |

Thus we now have four equations in Eqs. (52), (55) for the five
variables (_{m},
_{R},
*v*_{m}, *v*_{R},
). All we need to do is
to pick one more from Einstein's equations to complete the set. The
Einstein's equations for our perturbed metric are:

(56) | |||

(57) | |||

(58) |

where *A* denotes different components like dark matter, radiation
etc. Using Eq. (57) in Eq. (56) we can get a modified Poisson
equation which is purely algebraic:

(59) |

which once again emphasizes the fact that in the relativistic theory, both pressure and density act as source of gravity.

To get a feel for the solutions let us consider a flat universe
dominated by a single component of matter with the equation of state
*p* = *w*
. (A
purely radiation dominated universe, for example, will have *w* =
1/3.) In this case the Friedmann background equation gives
*a*^{-3 (1 +
w)} and

(60) |

The equation for the potential can be reduced to the form:

(61) |

The second term is the damping due to the expansion while last term is
the pressure support that will lead to oscillations. Clearly, the factor
*k*
determines which of these two terms dominates. When the pressure term
dominates (*k*
>> 1),
we expect oscillatory behaviour while
when the background expansion dominates (*k*
<< 1),
we expect the growth to be suppressed. This is precisely what happens.
The exact solution is given in terms of the Bessel functions

(62) |

From the theory of Bessel functions, we know that:

(63) |

This shows that if we want a finite value for
as
0, we
can set *C*_{2} = 0. This gives the gravitational potential
to be

(64) |

The corresponding density perturbation will be:

(65) |

To understand the nature of the solution, note that
*d*_{H} = (
/ *a*)^{-1}
and
*kd*_{H}
*d*_{H} /
*k*. So
the argument of the Bessel function is just the ratio
(*d*_{H} /
).
From the theory of Bessel functions, we know that for small values of
the argument
*J*_{}(*x*)
*x*^{} is a power law while for large values of the
argument it oscillates with a decaying amplitude:

(66) |

Hence, for modes which are still outside the Hubble radius (*k*
<<
^{-1}),
we have a constant amplitude for the potential and density contrast:

(67) |

That is, the perturbation is frozen (except for a decaying mode) at a
constant value. On the other hand,
for modes which are inside the Hubble radius (*k* >>
^{-1}),
the perturbation is rapidly oscillatory (if *w*
0). That is the pressure
is effective at small scales and leads to acoustic oscillations in the
medium.

A special case of the above is the flat,
matter-dominated universe with *w*=0. In this case, we need to take the
*w* 0 limit and
the general solution is indeed
a constant =
_{i}(**k**)
(plus a decaying mode
_{decay}
^{-5}
which diverges as
0).
The corresponding density perturbations is:

(68) |

which shows that density perturbation is "frozen" at large scales but grows at small scales:

(69) |

We will use these results later on.