Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

Often in physics we have to apply relatively
simple physical laws to extremely complicated situations. Although it
is straightforward to write down the equations necessary to calculate
what a given physical system will do, it may be difficult to solve
these equations unless the situation has a particular symmetry, or if
some aspects of the problem can be neglected. A good example lies in
the construction of **cosmological models** using **general
relativity**. The
**Einstein equations** are extremely complicated, and no general
solutions
are available. However, if we assume that the Universe is completely
homogeneous and isotropic (i.e. if we invoke the **cosmological
principle**), then the special symmetry implied by the **Robertson-Walker
metric** drastically simplifies the problem. We ends up with the
**Friedmann models**, and only the relatively simple *Friedman
equation* to solve.

But the Universe is not exactly homogeneous and isotropic now, even
if the **cosmic microwave background radiation** suggests that it must
have been so earlier on (see **large-scale structure**). One of the most
important questions asked by cosmologists is how this structure came
about. In order to make a theory of **structure formation**, surely we
need to solve the Einstein equations for the general case of an
inhomogeneous and anisotropic universe? To be precise, the answer to
this question is `yes', but to be reasonably accurate the answer is
`no'. Even though our Universe is not exactly homogeneous, it is
almost so. If we calculate the expected departures from the
Robertson-Walker metric for all the mass concentrations we know about,
we find them to be small - about one part in a hundred thousand. So we
need solutions of the Einstein equations that describe an almost but
not quite homogeneous universe. For this we need a model which is
almost a Friedmann model, but not quite.

The mathematical technique for generating solutions of equations
that are *almost* the same as solutions you already know is called
*perturbation theory*, and it is used in many branches of physics other
than cosmology. The basic idea can be illustrated as follows. Suppose
we have to calculate (1.0001)^{9} without using a calculator. This
problem can be thought of as being almost like calculating
1^{9}, because the quantity in brackets is not far from 1; and
1^{9} is just 1. Suppose
that we represent the extra 0.0001 we have to deal with by the symbol
. The problem now is to
calculate the product of nine terms 1 + : (1
+ ) (1 + )....(1 + ). Now imagine multiplying out
this expression
term by term. Since there are nine brackets each with two terms (a 1
and an ), there are
2^{9} = 512 combinations altogether - quite a
task. The first term would be a 1, which is obtained by multiplying
all the 1's in all the brackets. This would be the biggest term,
because there are no other terms bigger than 1 and all the terms
containing are much
smaller. If we multiplied the in the first
bracket by the 1's in all the others, we would get . By taking one
and eight 1's in every possible way from the nine brackets we would
get nine terms altogether, all of which are . Now, any other terms
made in more complicated ways that this, like five 1's and four 's,
would result in powers of
(in this case ^{4}). But because is
smaller than 1, all these terms are much smaller than itself, and
very much smaller than 1. It should therefore be a good approximation
just to keep the nine terms in which appears on its own, and ignore
terms that contain ^{2} or ^{3} or higher powers of . This suggests that
we can write, approximately,

^{9} 1 + 9

Going back to our original problem, we can put = 0.0001, from which
we find that the approximate answer to be 1.0009. In fact, the right
answer is 1.000 900 36. So our approximation of taking only the
lowest-order correction ()
to a known solution (1^{9} = 1) works very
well in this case.

The way to exploit this idea in cosmology is to begin with the
equations that describe a Friedmann model for which the
Robertson-Walker metric (which we denote here by *g*) holds. We know how
to handle these equations, and can solve them exactly. The we write
the equations again, not in terms of *g* but in terms of some other
quantity *g'* = *g* + *h* where *h* is a small
correction like in the above
example - in other words, a *perturbation*. If *h* is small, we can
neglect all the terms of order higher than *h* and obtain a relatively
simple equation for how *h* evolves. This is the approach used to study
the growth of small **primordial density fluctuations** in the expanding
Universe.

Of course, the approach breaks down when the small correction
becomes not so small. The method used above does not work at all well
for (1.1)^{9}, for example. In the study of structure formation
by means
of the **Jeans instability**, the fluctuations gradually grow with time
until they become large. We then have to abandon perturbation methods
and resort to another approach. In the example above, we have to reach
for a calculator. In cosmology, the final nonlinear stages have to be
handled in a similar brute-force way, by running ** N-body
simulations**.

FURTHER READING:

Coles, P. and Lucchin, F., Cosmology: *The Origin and Evolution of
Cosmic Structure* (John Wiley, Chichester, 1995), Part 3.