pp 107-108

On these pages  we discuss the fact that the CDM density contrast grows logarithmically during radiation domination, starting at the epoch of horizon entry. The growth leads to the logarithmic factor in the transfer function Eq. (5.13), which is crucial for small-scale perturbations such as those responsible for reionization (Section 11.6). Contrary to what we state in the text, this logarithmic growth does not depend on the validity of the Meszaros equation. Instead, it arises simply  because the cold dark matter density perturbation is evolving in a practically unperturbed spacetime, corresponding to the equation

\delta_c''+ (aH) \delta_c' =0

where a prime denotes differentiation with respect to conformal time. The independent solutions of this equation are (constant) and (log a), and imposing the initial condition mentioned at the top of page 108 leads to logarithmic growth starting at horizon entry.

To derive the above equation one should start with the relativistic equations Eqs. (15.1) and (15.2), in which the space-time perturbation is described by the potentials \Phi and \Psi. At horizon entry both potentials are of order \delta_c. Well after horizon entry the neutrino perturbation has free-streamed away. Then the two potentials are equal and they fall like (aH)^2 (using Eq. (4.121) and bearing in mind that the total density contrast coming primarily from the baryon-photon fluid oscillates with roughly constant amplitude). With the potentials ignored, Eqs. (15.1) and (15.2) are obviously equivalent to the above equation.

None of this invokes the Meszaros equation, which is equivalent to keeping only the matter contribution to the potentials, or in other words to neglecting the radiation contribution. We end this note by discussing the validity and use of the Meszaros equation, following the treatment of Weinberg (astro-ph/0207375). As we point out in the text, the neglect of the radiation contribution is obviously justified on scales below the Silk scale where photon diffusion damps the photon/baryon contribution. The comoving
scale falls below the Silk scale during radiation domination only for very small comoving scales, enclosing mass less than about a million solar masses, and the  Meszaros equation is not literally true on larger comoving scales. However,  averaging over a few oscillations effectively damps the radiation even on these scales, justifying in practice the use of the Meszaros equation.

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Last updated:  January 2003
Andrew Liddle