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For constant barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation gives $\bar{\rho }\propto a^{-3(1+w)}$. In conformal time the flat Friedmann equation becomes ${\mathcal{H}}^2=(8\pi G/3)a^2\bar{\rho }$, where $\mathcal{H}=a^{\prime }/a$. Integration, for $w\ne -1/3$, givesChoose the conformal-time origin so that ${\eta }_B=0$ and work on the appropriate expansion branch. Write $\beta =2/(1+3w)$, so $\mathcal{H}=\beta /\eta$ and $2{\mathcal{H}}^{\prime }+(1+3w){\mathcal{H}}^2=0$.

Combining the supplied energy and momentum constraints gives the comoving-gauge density contrast Poisson relationFor this barotropic fluid $\delta P=w\bar{\rho }\delta$. Substitute the energy constraint into the pressure equation to obtainso the bracket vanishes and the coefficient of ${\varphi }^{\prime }/\eta$ is $A=6(1+w)/(1+3w)$.

Because $a^2\bar{\rho }=C{\eta }^{-2}$, the Poisson relation gives $\Delta =({\eta }^2/(4\pi GC)){\nabla }^2\varphi$. Apply the same potential equation to $F={\eta }^2{\nabla }^2\varphi$: differentiating ${\eta }^{-2}F$ givesSince $F$ is a constant multiple of $\Delta$, the constant-equation-of-state comoving density equation isThe primes here denote conformal-time derivatives, corresponding to the overdots in the question. A pure vacuum fluid $w=-1$ has zero enthalpy and no independent fluid velocity or nonzero-wave-number density perturbation; its formal equation should be interpreted with that degeneracy in mind.

For pressureless matter, $w=0$, the constant-equation-of-state comoving density equation reduces to ${\Delta }^{\prime \prime }+2{\Delta }^{\prime }/\eta -6\Delta /{\eta }^2=0$. Substitution of $\Delta \propto {\eta }^s$ gives $s^2+s-6=(s-2)(s+3)=0$. Since $a\propto {\eta }^2$,In the growing mode, $a^2\bar{\rho }\Delta$ is constant, so the Poisson constraint makes $\varphi$ constant for a nonzero Fourier mode. The momentum constraint and the flat Friedmann equation giveThis is the growing-mode velocity potential in matter domination, in the question's convention where the peculiar velocity is $\nabla v$.