4.2. Isothermal density fluctuations
An orthogonal mode to the adiabatic fluctuations discussed in the previous section has the solution,
for the fractional density perturbations in the matter and radiation. The solution corresponds to a primordial perturbation in the entropy-per-baryon (see Peebles, 1980, Section 94, or Wilson and Silk, 1981). The solution is sometimes referred to as the "isothermal" mode because the radiation dominated era T / T ~ r << m. The matter density stays at a nearly constant amplitude prior to decoupling and while the Universe is radiation dominated.
Although the amplitude of the radiation brightness perturbation is small in the radiation era, it becomes comparable to that of the original fluctuation once the Universe is matter-dominated. The fluctuations in the radiation brightness generated by primordial entropy perturbations are of great importance in estimating the resulting radiation anisotropy and will be discussed in detail later.