Cosmological transfer function

A cosmological transfer function relates a late perturbation to a specified primordial perturbation mode by mode. It records the scale-dependent evolution caused by horizon entry, pressure, free streaming, and changes in the dominant cosmic component.

During matter domination with nearly constant gravitational potential, a radiation density perturbation obeys a forced acoustic equation. The observable Sachs-Wolfe combination is the homogeneous oscillatory part.

At fixed time after equality, the comoving cold-dark-matter density contrast per unit primordial curvature grows as $k^2$ for $k\ll k_{\mathrm{eq}}$ and only logarithmically for $k\gg k_{\mathrm{eq}}$. Small-scale modes entered during radiation domination and could grow only logarithmically until equality.

The baryon transfer function retains acoustic oscillations generated while baryons belonged to the photon-baryon fluid. After recombination, baryons fall into cold-dark-matter potential wells, reducing the relative perturbation while leaving a small oscillatory imprint.

After recombination and when pressure is negligible, cold dark matter and baryons feel the same gravitational acceleration. Their density-contrast difference therefore obeysDuring matter domination $D=D_0+D_1{a}^{-1/2}$, whereas the growing total-matter mode is proportional to $a$, so ${\delta }_c/{\delta }_b\to 1$.