 Solution

Use primes for conformal-time derivatives. The cold-dark-matter Cosmological Euler equation in synchronous gauge is $v_C^{\prime }+\mathcal{H}{v}_C=0$, so multiplying by $a$ gives $(a{v}_C{)}^{\prime }=0$. ThereforeIn an expanding universe $a$ increases, damping any initial peculiar velocity as $a^{-1}$. During radiation domination $a\propto \tau$, so the velocity scales as ${\tau }^{-1}$; during matter domination $a\propto {\tau }^2$, it scales as ${\tau }^{-2}$. This is the nonrelativistic limit of the physical-momentum redshift derived in Q1. It concerns the bulk velocity in the synchronous coordinate frame, not a growing peculiar velocity sourced in another gauge. A cold-matter rest-frame choice also removes the residual synchronous time displacement relevant to that component. The decay statement assumes expanding evolution and the pressureless, linear approximation used by the equation.