Nonrelativistic density equation in synchronous gauge (${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }+(c_s^2{k}^2-4\pi G{a}^2\bar{\rho })\delta \simeq 0$)

For a matter-dominated fluid with constant $c_s^2=w\ll 1$, combine stress-energy conservation in synchronous gauge with the scalar trace Einstein equation. Discard pressure corrections to the background gravitational and velocity-damping terms while retaining the potentially large pressure-gradient term $w{k}^2\delta$. This produces the displayed leading nonrelativistic equation. It is not an exact equation at finite $w$. At negligible pressure the growing and decaying matter-era solutions are ${\tau }^2$ and ${\tau }^{-3}$.

// Target: cosmology.bigb

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

During radiation domination, $a\propto \tau$ and $8\pi G{\rho }_R{a}^2=3/{\tau }^2$. The adiabatic initial conditions reduce the CDM density equation in a matter-radiation universe to ${\delta }_C^{\prime \prime }+{\delta }_C^{\prime }/\tau -4{\delta }_C/{\tau }^2=0$. Its Euler-Cauchy equation powers are two and minus two. The growing solution is the displayed one, in the stated CDM-comoving synchronous gauge.

In the leading radiation domination limit, neglect $k^2{\delta }_r$ and the matter contribution to gravity. The radiation equation gives $S^{\prime \prime }=0$. Substitution into ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau =3{\delta }_r/{\tau }^2$ givesThus adiabatic initial conditions select both $S_0=0$ and $S_1=0$, leaving the growing and decaying power-law modes. The four constants describe the leading supplied coupled system; regularity and additional initial Einstein field equations constraints can further restrict their physical interpretation.