 Cosmological continuity equation

In natural units, conservation of the stress-energy tensor for a homogeneous perfect fluid in an FLRW metric gives the displayed equation for its energy density and pressure. It is equivalent to $d(\rho a^3)=-Pd(a^3)$. For a constant equation-of-state parameter $w=P/\rho$, separation gives $\rho \propto a^{-3(1+w)}$. If $\rho$ instead denotes mass density, the pressure term is $P/c^2$.

This is local stress-energy conservation; it does not in general define a globally conserved total energy for an expanding universe. The adiabatic work in a comoving volume makes this distinction explicit.