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Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is $n_p=a^{-3}{n}_c$. A conserved population moving with homogeneous expansion has constant $n_c$ despite declining proper density. Formation, mergers or changing sample selection can make $n_c$ evolve.

If relic particle number and reference-gas entropy are separately conserved in a comoving volume, their ratio $Y=n/s$ is constant. A decay or entropy injection can change it. The reference entropy is that of the specified thermal sector, not necessarily the entropy of the decoupled relic itself.

The cosmic stellar mass density is the mass moment of a galaxy stellar mass function in a stated volume convention. With a comoving mass function it is mass per comoving volume, while proper density is larger by $(1+z{)}^3$. Comparing it to a critical or matter density requires keeping the epoch and volume convention explicit.

The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

A population of distinct, nonoverlapping dark-matter halos cannot contain more total mass per comoving volume than the cosmic mean matter density. A narrow mass population therefore obeys $n_c{M}_h\le {\bar{\rho }}_{m,0}$. Do not sum subhalo masses and their inclusive host masses in the same bound, because that double-counts material. This check can reveal incompatible inferred absorption incidence, object size and halo-mass assumptions.

Integrate galactic stellar mass, rather than galaxy count, against the galaxy stellar mass function. With $x=M_{\ast }/M_{\ast ,\mathrm{ch}}$, the comoving cosmic stellar mass density, evaluated by the stellar mass density from a square-root exponential cutoff, isPut $u=\sqrt{x}$, so $dx=2udu$ and $x^{-3/4}=u^{-3/2}$. The integral is $2\Gamma (1/2)=2\sqrt{\pi }$, givingThe supplied present mean matter density and ${\Omega }_m=0.25$ imply ${\rho }_{\mathrm{crit},0}=3.5\times 10^{10}/0.25=1.4\times 10^{11}{M}_{\odot }\mathrm{Mp}{\mathrm{c}}^{-3}$. ThereforeThis is about $0.25\%$ of today's critical density, or $1\%$ of today's mean matter density, expressed per present comoving volume. The proper stellar density at $z=3$ is $64{\rho }_{\ast }^{\mathrm{com}}$; if comparing that proper density directly with today's critical density its ratio is $0.162$. It must not be confused with the usual comoving-density comparison. The low-mass number integral diverges if extrapolated to zero mass, but the mass integral converges; real populations require a low-mass cutoff.

Use units $c=\hslash =k_B=1$, and write $H=\dot{a}/a$ for the Hubble parameter, with dots denoting cosmic time. Differentiating the Friedmann equation givesOn an interval where $H\ne 0$, use the Hubble parameter identity $\dot{H}=\ddot{a}/a-H^2$ and the Friedmann acceleration equation to findSubstitution cancels the curvature term and gives the continuity equationThe cosmological perfect-fluid continuity equation extends by continuity through a regular isolated turning point. Physically it states that the change of energy in a comoving volume is the negative of the pressure work: $d(\rho a^3)=-Pd(a^3)$.

For separately conserved cosmological fluids, each component obeys this equation individually. A constant equation-of-state parameter $w_i=P_i/{\rho }_i$ therefore giveswhere the present scale factor is normalized to $a_0=1$. Thus the component density laws areThe extra factor for radiation in cosmology is the cosmological redshift of each photon's energy; pressureless matter has only number dilution, while the specified dark energy is a cosmological constant.

The critical density at a given expansion rate is the total density that makes the spatial curvature vanish:These cosmological density parameters use the critical density at that same time. Put $E(a)=H(a)/H_0$ and ${\Omega }_{K,0}=-K/H_0^2$. The Friedmann equation yieldsConsequently the fractional-density evolution, rather than just the component-density evolution, isIn particular ${\sum }_i{\Omega }_i=1+K/(a^2{H}^2)$. The simple powers of $a$ alone apply to ${\rho }_i$, or to ${\rho }_i/{\rho }_{\mathrm{crit},0}$, not to the instantaneous cosmological density parameters. At a recollapse turning point $H=0$, those instantaneous ratios are undefined even though the component densities remain finite.

Consider a species in thermal equilibrium, with energy density and pressure depending on temperature and with vanishing chemical potential, as in the supplied thermodynamic relation. Put $U=\rho (T)V$. The first law of thermodynamics givesSince $dP/dT=(\rho +P)/T$,The differential is therefore exact:After fixing the irrelevant additive entropy zero, the entropy density at zero chemical potential isMore precisely the first law leaves an additive constant in $S$; extensivity fixes that constant for the usual entropy-density normalization. A nonzero chemical potential would instead require $Ts=\rho +P-\mu n$, so the stated formula is not a universal identity for a decoupled massive species with conserved particle number.

For a reversible thermodynamic process that is an adiabatic process in a comoving volume, $\dot{V}=3HV$ and the cosmological perfect-fluid continuity equation gives $\dot{\rho }=-3H(\rho +P)$. Differentiating the extensive expression explicitly, or using the exact differential above, givesThis is cosmological entropy conservation for the closed equilibrium gas with no entropy-producing energy injection.

For radiation in cosmology, $P=\rho /3$, so $s=4\rho /(3T)$. Comparing $s=c_1{g}_{\star }{T}^3$ and $\rho =c_2{g}_{\star }{T}^4$ yields the coefficient ratioWhen all relativistic species share the same temperature and count, the familiar normalizations are $c_1=2{\pi }^2/45$ and $c_2={\pi }^2/30$. More generally the energy and entropy effective counts can differ; the common $g_{\star }$ here uses the assumptions stated in the question.