**2.2. Radiation-dominated universe**

At early enough times, the universe was radiation dominated
(*c*_{s} = *c* / 3^{1/2}) and the analysis so
far does not apply.
It is common to resort to general relativity perturbation theory
at this point. However, the fields are still weak, and
so it is possible to generate the results we need by
using special relativity fluid mechanics and Newtonian gravity
with a relativistic source term:

(26) |

in Eulerian units.
The main change from the previous treatment come from factors
of 2 and 4/3 due to this
( +3*p* /
*c*^{2}) term, and other contributions
of the pressure to the relativistic equation of motion.
The resulting evolution equation for
is

(27) |

so the net result of all the relativistic corrections is a driving term on the rhs that is a factor 8/3 higher than in the matter-dominated case (see e.g. Section 15.2 of Peacock 1999 for the details).

In both matter- and radiation-dominated universes with
= 1, we have
_{0}
1/*t*^{2}:

(28) |

Every term in the equation for
is thus the product of
derivatives of and
powers of *t*, and a power-law
solution is obviously possible. If we try
*t*^{n},
then the result is *n* = 2/3 or -1 for matter domination;
for radiation domination, this becomes *n* = ± 1.
For the growing mode, these can be combined rather conveniently
using the conformal time
*dt*
/ *a*:

(29) |

The quantity is proportional to the comoving size of the cosmological particle horizon.

One further way of stating this result is that gravitational
potential perturbations are independent of time (at least while
= 1). Poisson's
equation tells us that
- *k*^{2}
/ *a*^{2}
;
since
*a*^{-3}
for matter domination or *a*^{-4} for radiation, that gives
/ *a* or
/ *a*^{2}
respectively, so that
is independent of *a* in either case. In other words,
the metric fluctuations resulting from potential perturbations are
frozen, at least for perturbations with wavelengths greater than the
horizon size.