Baryon catch-up approximation after recombination

In a dark-matter-dominated growing mode with a constant relative offset and initially negligible baryon density contrast, ${\Delta }_B/{\Delta }_D=1-R_r/R=1-(1+z)/(1+z_r)$. Replacing this by $1-z/z_r$ requires high redshift. The initial relative velocity and finite baryon self-gravity alter transients, although the relative difference becomes small as the common growing mode increases.

Scale-factor ratios involve $(1+z)/(1+z_r)$ exactly. Replacing this by $z/z_r$ is accurate only when both redshifts are large; it cannot justify an exact zero-redshift identity.

In the dark-matter-dominated limit, imposing both ${\Delta }_B(t_r)=0$ and ${\dot{\Delta }}_B(t_r)=0$ requires a relative decaying mode. With $x=R/R_r$, the ratio becomes $1-3x^{-1}+2x^{-3/2}$. It has a different early-time approach from the constant-offset branch and the same late-time common growth.

With growing total matter mode, constant relative offset, initial ${\Delta }_B=0$, and $x=R/R_r$, the exact ratio is ${\Delta }_B/{\Delta }_D=f_D(x-1)/(f_{D}x+f_B)$. It reduces to $1-x^{-1}$ only in the dark-matter-dominated limit.