Does the FLRW metric account for the decrease in density for an expanding universe?

For what I remember I would say that FRW metric certainly account for that; in fact a constant energy density in an expanding universe should mean that there is energy production, but from where exactly? I can cite the Hubble parameter equation begin{equation*} left(H(t)right)^2 = left(H(t_r)right)^2 left( frac{Omega_{text{mat}}(t_r)}{s^3(t,t_r)} +frac{Omega_{text{rad}}(t_r)}{s^4(t,t_r)} +Omega_Lambda(t_r) +frac{Omega_alpha(t_r)}{s^2(t,t_r)} right) end{equation*} where you have an emission time $t$, a receiving time $t_r$ and a normalized scale factor $s(t,t_r)doteqtilde{S}(t)/tilde{S}(t_r)$ (where $tilde{S}$ represents $S/sqrt{|K|}$). You can recognize the presence of

• Matter density parameter, decreasing with $s^3$, representing exactly the tridimensional volume change
• Radiation density parameter, that goes with $s^4$, why? because the expansion redshifts the radiation!
• Vacuum density parameter, just constant because the vacuum has an unaltered nature
• Curvature density parameter, a really esotic and wild animal!

Considering also that $H(t)approxpartial_t tilde{S}(t)/tilde{S}(t)$ this means that the universe expands changing scale factor due to the fact that energy density changes.

I hope it was helpful and without big errors.