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2.2. Radiation-dominated universe

At early enough times, the universe was radiation dominated (cs = c / 31/2) 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:

Equation 26 (26)

in Eulerian units. The main change from the previous treatment come from factors of 2 and 4/3 due to this (rho +3p / c2) term, and other contributions of the pressure to the relativistic equation of motion. The resulting evolution equation for delta is

Equation 27 (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 Omega = 1, we have rho0 propto 1/t2:

Equation 28 (28)

Every term in the equation for delta is thus the product of derivatives of delta and powers of t, and a power-law solution is obviously possible. If we try delta propto tn, 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 eta ident integ dt / a:

Equation 29 (29)

The quantity eta 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 Omega = 1). Poisson's equation tells us that - k2 Phi / a2 propto rho delta; since rho propto a-3 for matter domination or a-4 for radiation, that gives Phi propto delta / a or delta / a2 respectively, so that Phi 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.

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