The spectrum of primordial fluctuations

In the description of the evolution of density inhomogeneities, one often make the simplifying assumption that the initial power-spectrum has the simple power-law expression $P(k)\propto k^n$. Although the scale-free Zeldovich spectrum, $P(k)\propto k$, is expected on the grounds of the classical inflationary scenario, nevertheless distortions of its shape should arise during the subsequent phases of cosmic expansion, and characteristic scales are imprinted on the form of $P(k)$ at the onset of the structure formation. Since the amount of such distortions and the scales at which they occur are strictly related to the nature of the fluctuations and to the matter content of the Universe, their knowledge becomes of crucial relevance in order to fix the initial conditions for the galaxy formation process.

Theoretical models for the determination of the power-spectrum, which is responsible for structure formation, starts from the assumption of a primordial $P_{pr}(k)$ at a sufficiently high redshift, $z_{pr} \gg z_{eq}$ [ $z_{eq}\simeq 4. 2\times 10^4 (\Omega h^2)$ is the redshift of the epoch of matter-radiation equality]. The usual choice $P(k)=Ak$ corresponds to the Harrison-Zeldovich spectrum. Due to the evolution of density perturbations, the slope of the power-spectrum is left unchanged at wavelengths $\lambda \gg \lambda_{eq}\sim ct_{eq}$, that exceed the horizon size at $t_{eq}$. On the contrary, the shape of the spectrum at $\lambda \ll \lambda_{eq}$ crucially depends on the nature of the matter which dominates the expansion. In order to account for these effects, the post-recombination spectrum is usually written as

$$P(k) = T^2(k) P_{pr}(k) ,$$ (21)

where the transfer function $T(k)$ conveys all the informations about the pre-recombination evolution and the nature of the matter content.