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Combining the degenerate neutrino and antineutrino energy density with a critical density bound constrains the neutrino degeneracy parameter through the fourth root of the available energy budget. The numerical bound depends on $T_{\nu ,0}/T_{\gamma ,0}$ and the number of degenerate flavors. The standard ratio $(4/11{)}^{1/3}$ assumes neutrino decoupling before electron-positron annihilation in cosmology; it should be stated as a thermal-history assumption if the degeneracy is very large. The massless formula is inappropriate for a species that is nonrelativistic today.

A standard thermal Cosmic neutrino background with one populated neutrino and one antineutrino helicity per species has $n_{\nu }+n_{\bar{\nu }}=3n_{\gamma }/11$ after electron-positron annihilation in cosmology. If this species is nonrelativistic today and dominates a spatially flat matter universe, equate $m_{\nu }(n_{\nu }+n_{\bar{\nu }})$ to the critical density $3H_0^2{M}_P^2/(8\pi )$, using the unreduced Planck mass. The photon number density gives the displayed conditional mass. Extra populated states or several massive species change the abundance count; vacuum-dominated cosmology changes the density available for matter.

At maximum expansion of a closed matter-dominated Friedmann solution, the Hubble parameter vanishes while the physical mass density stays positive and finite. Its critical density therefore vanishes and the cosmological density parameter diverges. This is a singularity of that ratio rather than a curvature singularity. In the cycloid variable $\theta$, ${\Omega }_M={\sec }^2(\theta /2)$, diverging at $\theta =\pi$ and approaching one at the Big Bang and Big Crunch ends.

The mass-density convention for the cosmological critical density. If $u$ denotes energy per unit volume, $\rho =u/c^2$ and $u_{\mathrm{crit}}=3c^2{H}^2/(8\pi G)$. Both conventions give the same dimensionless density parameter when numerator and denominator use consistent units. This distinction avoids inserting an extra $c^2$ into a Friedmann equation written in terms of mass-equivalent density.

The matter density parameter is the ratio of total nonrelativistic matter density, including baryonic and dark matter, to the critical density ${\rho }_{\mathrm{crit}}=3H^2/(8\pi G)$ at the same epoch. A present-epoch value uses $H_0$ and ${\rho }_{m,0}$. It is not the total cosmological density parameter when radiation or vacuum energy also contributes.

Let $f_X={\partial }^2\mathcal{N}/(\partial N\partial X)$ count neutral hydrogen absorbers per column density and absorption distance. The summed column per unit proper light-path length is the mean neutral number density. Since $d\ell =cdz/[(1+z)H(z)]$ and $dX/dz=H_0(1+z{)}^2/H(z)$, that density is $(H_0/c)(1+z{)}^3\int N{f}_{X}dN$. Dividing the proper mass density by $(1+z{)}^3$ gives the comoving density in the displayed formula. The denominator is today's critical density; including neutral helium requires a stated mass correction. For a distribution per redshift, replace $H_0{f}_X$ by $H(z)f_z/(1+z{)}^2$.

For an optically thin collisionally excited line, the volume emissivity is approximately $j_{ul}=n_{e}n(X^i)q_{lu}(T_e)h{\nu }_{ul}$ below the transition's critical density. The excitation coefficient includes a factor $T_e^{-1/2}\exp [-\Delta E/(k_B{T}_e)]$, so temperature is needed to convert luminosity into an ionic abundance. Density is needed both to assess collisional quenching of a metastable line and to convert an emission measure into a gas mass. For a uniform recombining ion, for example, $M_i=m_i{L}_{\mathrm{line}}/[n_e{\alpha }_{\mathrm{eff}}h\nu ]$. Temperature structure, density clumping, extinction and unobserved ionization stages introduce further corrections.

A useful density doublet has closely spaced upper energies but different spontaneous decay rates and hence different critical densities. Nearly equal excitation energies reduce temperature dependence of the ratio; similar wavelengths reduce differential extinction and relative calibration errors. Metastable upper levels have sufficiently small Einstein coefficients for collisional de-excitation to compete at nebular densities.

Use statistical weights $g_2=6$, $g_3=4$. The source's O II ground-state term has a typographical error: it is $2s^{2}2p^3{}^4{S}_{3/2}^{\mathrm{o}}$, not ${}^2{S}_{3/2}$. A ${}^{2}S$ term has $J=1/2$, so the printed term is internally inconsistent. The tabulated $J$ values, wavelengths and rates can still be used.^[ref]

At high density, collisions establish a Boltzmann distribution between the upper states. Their energy splitting is small compared with $k_B{T}_e$, so $n_2/n_3\simeq g_2/g_3$. If both lines escape without absorption,The exact ratio includes $\exp [-(E_2-E_3)/(k_B{T}_e)]$, an approximately $0.2\%$ correction at $10^4\mathrm{K}$.

At low density, assume excitation from the common ground state and radiative decay back to it dominate, with no substantial collisional transfer between upper states. Statistical equilibrium gives $n_j{A}_{j1}=n_1{n}_e{q}_{1j}$. Detailed balance relates excitation to the supplied de-excitation coefficient, yieldingEach excitation then produces one line photon, so the spontaneous rate cancels:At high density the upper populations are collisionally thermalized and the ratio retains the radiative lifetimes; at low density radiative escape is nearly certain and the ratio is set by the excitation probabilities instead.

At $T_e=10^4\mathrm{K}$, the two de-excitation coefficients are $C_{21}=1.161\times 10^{-14}$ and $C_{31}=1.1825\times 10^{-14}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. ThusThe useful density range lies between and around these scales. In the independent-upper-level approximation,The density sensitivity of a metastable doublet depends on the error measure. The fractional response $|d\ln R/d\ln n_e|$ is maximal atIf absolute ratio errors are constant, maximizing $|dR/d\ln n_e|$ instead gives $n_e=n_{c,2}$. An unspecified experimental noise model therefore does not define a unique sharper optimum than this transition-density range.

For Ne IV, the same calculation gives critical densities $4.23\times 10^{10}$ and $3.93\times 10^{11}{\mathrm{m}}^{-3}$, with fractional sensitivity peaking near $1.29\times 10^{11}{\mathrm{m}}^{-3}$. This is a useful higher-density diagnostic when O II is saturated. However, Ne${}^{3+}$ and O${}^{+}$ need not occupy the same ionization zone, so their inferred densities need not describe the same gas. The ultraviolet Ne IV pair also needs suitable observations and extinction correction. The proposal is useful with these qualifications, not a guaranteed replacement for the density of the O II emitting zone.

Take the initially expanding branch ${\dot{r}}_i=H_i{r}_i$; the printed squared velocity alone also permits an initially contracting branch, which would not reach a future maximum radius. Multiply the equation of motion by $\dot{r}$ and integrate. The conserved outer-shell specific orbital energy isThe initial mass follows from the background critical density:ConsequentlyIf $E$ means the energy of a test shell of mass $m$, $E=mϵ$. If it means the total energy of the entire uniform, homologously moving perturbation, direct integration instead gives $K=3M{\dot{r}}^2/10$, $U=-3G{M}^2/(5r)$ andThe common convention of writing the first integral as $E$ is therefore an energy per unit shell mass, not the total energy of the sphere.

At spherical-collapse turnaround, $\dot{r}=0$, so $ϵ=-GM/r_m$. HenceAny positive initial density contrast binds this pressureless perturbation in the stated Einstein-de Sitter universe.

The cosmological principle permits a Friedmann-Lemaître-Robertson-Walker metric. The Friedmann equation, with vanishing cosmological constant, isFor separately conserved pressureless matter, the cosmological continuity equation gives ${\bar{\rho }}_m={\bar{\rho }}_{m0}{a}^{-3}$: its rest energy is conserved while its physical volume grows as $a^3$. Define the present critical density and matter density parameter byEvaluating the Friedmann equation today gives $-k=H_0^2(1-\Omega )>0$. Thus the model has negative spatial curvature of an FLRW universe; the second contribution to $H^2$ represents that spatial curvature of an FLRW universe, rather than another material fluid. Taking the expanding branch givesHere $0<\Omega <1$. At early times, $a\ll \Omega /(1-\Omega )$, matter domination gives $\dot{a}\simeq H_0\sqrt{\Omega }{a}^{-1/2}$, hence, measuring cosmic time from the Big Bang,At late times the $a^{-2}$ term dominates the Friedmann equation. This curvature-dominated universe has $\dot{a}\to H_0\sqrt{1-\Omega }$ and thereforeThe late expansion approaches the coasting Milne model. The early limit belongs to this idealized matter-only model; including radiation in cosmology would alter the sufficiently early evolution.

For dust alone, write ${\dot{a}}^2=c^2(A/a-k)$, whereSet $\theta =\sqrt{k}c\eta$ and $a=(A/k){\sin }^2(\theta /2)$. Differentiation gives $\dot{a}=c\sqrt{k}\cot (\theta /2)$ once $dt=ad\eta$ is used, verifying the Friedmann equation. Integrating the time relation from the Big Bang yields the closed matter-dominated Friedmann solutionThe Hubble parameter and the cosmological density parameter areThe first decreases from positive infinity through zero at maximum expansion to negative infinity at the Big Crunch. The second approaches one near either singular endpoint and diverges at maximum expansion. This is a density-parameter divergence at a cosmological turnaround: $H=0$ makes the critical density vanish even though the physical dust density there is finite.

Let $t_0$ be the initial epoch, let $r(t)$ be the physical radius of the outer matter shell, and put $r(t_0)=r_0$. The background Einstein-de Sitter universe has critical density ${\rho }_b=3H_i^2/(8\pi G)$ at that epoch. Its mass inside $r_0$ is $M_b=M-\delta M$, where $M$ is the actual conserved mass. Hence the background Friedmann equation impliesFor spherical collapse of pressureless matter, with no shell crossing, the shell theorem gives $\ddot{r}=-GM/r^2$. Its conservation of energy first integral isConsequentlyFor $\delta M>0$, the initially expanding branch reaches cosmological turnaround at $r_{\mathrm{ta}}=A/B=M{r}_0/\delta M$. Since $dt=dr/\dot{r}$ on that branch, the elapsed cosmic time isThis is spherical turnaround from an unperturbed Hubble velocity. The integral measures the comoving matter observers' time interval in the pressureless model. The integration variable is the physical radius, not the radius divided by the background scale factor. If an absolute cosmic time is wanted, add $t_0$ to the displayed interval.

Normalize the scale factor to $a(t_0)=1$, use $c=1$, and define the instantaneous critical density by ${\rho }_{\mathrm{cr}}(t)=3H(t{)}^2/(8\pi G)$. Separately conserved pressureless matter and radiation in cosmology obey the cosmological continuity equation,The present cosmological density parameters areThe cosmological constant has equivalent energy density ${\rho }_{\Lambda }=\Lambda /(8\pi G)$. Substituting into the Friedmann equation yieldsAt the present epoch this gives $k=H_0^2({\Omega }_0-1)$, or curvature contribution ${\Omega }_{K0}=-k/H_0^2=1-{\Omega }_0$. Dividing each component by $H_0^2$ provesSpatial curvature is not included in the total physical-density parameter ${\Omega }_0$. In particular, positive vacuum density does not have the radiation or matter scaling.

Take the transition temperature $T_i\sim \eta$ and radiation-dominated universe. With $g_{\ast }\simeq 100$ relativistic degrees of freedom, ${\rho }_{r,i}=({\pi }^2/30)g_{\ast }{T}_i^4$ and the Friedmann equation givesA saturated causal correlation length is of order the radiation-era horizon $H_i^{-1}=2t_i$. Write $C_n\sim 1$ for the geometric/formation efficiency and $C_M=M_M/\eta$. One monopole per horizon volume gives the horizon-density monopole abundance bound estimateThe first quantity is number density and the second is defect energy density. Relative to the initial radiation critical density,This can initially be small even when the surviving monopole abundance is unacceptable today. Causality supplies an order-of-magnitude lower abundance estimate under the stipulated saturation; it is not a precise statistical count of defects.

In the observational setting of this paper, the cosmic microwave background favours nearly flat geometry, while cluster dynamics and large-scale structure favour a matter density near ${\Omega }_m\sim 0.3$. The missing contribution to the critical density is then roughly ${\Omega }_X\sim 0.7$. The supernova luminosity distance measurements require recent accelerated expansion. With present radiation negligible, $q_0<0$ requiresFor an X-dominated universe the threshold would instead be $w<-1/3$. Thus X cannot be ordinary pressureless matter or radiation. It must be comparatively smooth on cluster scales, so that it contributes to the background expansion without being counted as clustered dynamical matter. Its negative pressure makes its density dilute more slowly than matter, allowing late dominance. A cosmological constant, with $w=-1$, or sufficiently potential-dominated quintessence can have these properties.

The comparison with a perfect fluid in general relativity needs a distinction between background stress and perturbations. A homogeneous canonical scalar has canonical scalar stress as a perfect fluid:It has isotropic pressure and no background shear stress, just as a perfect fluid in general relativity does. Negative pressure therefore does not, by itself, prevent a perfect-fluid description. However, an adiabatic barotropic fluid with the constant closure $p=w\rho$ would have $c_s^2=dp/d\rho =w$. For negative $w$, a short-wavelength density mode then has ${\omega }^2\simeq w{k}_{\mathrm{phys}}^2<0$, the negative-w barotropic-fluid gradient instability. Such a closure cannot give a stable, smooth propagating X component.

A canonical scalar instead has rest-frame sound speed squared equal to one: its spatial and temporal kinetic terms have the same coefficient. Its pressure perturbation need not obey the background relation $\delta p=w\delta \rho$. This nonadiabatic response suppresses small-scale clustering without the negative-sound-speed instability. A cosmological constant is a separate limiting case with no independent propagating fluid density mode. Thus the issue is the microphysics and perturbation closure, not whether the background stress-energy tensor has perfect-fluid form.

Use units $c=1$ and take the initially expanding branch with ordinary positive matter or radiation density. In the Friedmann-Lemaître-Robertson-Walker metric, $k=+1,0,-1$ respectively describe positive, zero and negative spatial curvature. Separately conserved pressureless matter has ${\rho }_M\propto a^{-3}$ and radiation in cosmology has ${\rho }_R\propto a^{-4}$, so either dominates the curvature and vacuum terms sufficiently far back on a hot Big Bang branch.

With negligible cosmological constant, a spatially flat radiation universe has $a\propto t^{1/2}$ and a flat matter universe has $a\propto t^{2/3}$; both expand forever while decelerating. An open universe also expands forever and eventually becomes approximately curvature dominated, with $a\propto t$. A closed universe containing only ordinary matter/radiation reaches a maximum size and recollapses. For example, dust gives ${\dot{a}}^2=C/a-k$ with $C>0$, so only $k=+1$ admits the positive turning point $a=C$.

Curvature alone does not determine the fate when the cosmological constant is retained. Positive $\Lambda$ can cause late accelerated expansion and prevent even a closed universe from recollapsing; whether a closed expanding branch turns around depends on the relative matter and vacuum terms. Negative $\Lambda$ can cause recollapse even for flat or open spatial sections. These statements follow by inspecting ${\dot{a}}^2=C/a-k+\Lambda a^2/3$ for dust; the curvature sign is a geometrical classification, not by itself a universal destiny criterion.

Define the critical density ${\rho }_c=3H^2/(8\pi G)$ and cosmological density parameters ${\Omega }_M={\rho }_M/{\rho }_c$, ${\Omega }_R={\rho }_R/{\rho }_c$, ${\Omega }_{\Lambda }=\Lambda /(3H^2)$ and ${\Omega }_k=-k/(a^2{H}^2)$. The Friedmann equation gives ${\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }+{\Omega }_k=1$. Dividing the Friedmann acceleration equation by $-H^2$ givesConsequentlyThe deceleration parameter measures the sign and size of the acceleration relative to $a{H}^2$: $q>0$ means deceleration, $q<0$ accelerated expansion and $q=0$ instantaneously zero acceleration. In particular a falling Hubble parameter alone does not imply deceleration, since $q=-1-\dot{H}/H^2$.

For conformal time $d\eta =dt/a$, the conformal Hubble parameter is $\mathcal{H}=aH$. Using matter conservation and differentiating ${\Omega }_M\propto {\rho }_M/H^2$,This is the matter density flow with a cosmological constant. Neglecting ${\Omega }_{\Lambda }$ during matter domination reduces it to the requested relationThat last equation is not exact with nonzero $\Lambda$. For example, a flat dust-plus-vacuum state with $({\Omega }_M,{\Omega }_{\Lambda })=(0.9,0.1)$ gives ${\Omega }_M^{\prime }/\mathcal{H}=-0.27$, whereas the matter-only expression gives $-0.09$.

For the flatness problem, let ${\Omega }_{\mathrm{tot}}={\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }$. Curvature satisfies ${\Omega }_{\mathrm{tot}}-1=k/(aH{)}^2$, soIn decelerating near-flat radiation and matter eras, deviations therefore grow as $a^2$ and $a$ respectively. Maintaining small curvature at a late epoch requires very small early deviations unless a prior mechanism suppresses them. Cosmic inflation, with $q\simeq -1$, drives the deviation down approximately as $a^{-2}$. If vacuum energy is present, flatness means ${\Omega }_{\mathrm{tot}}=1$, not ${\Omega }_M=1$.

Use absorption distance $X$ and its distribution $f_X$. Summing hydrogen columns per unit $X$ gives $\int N{f}_{X}dN$. The physical light-path length per $X$ isThe mean neutral-hydrogen number density is therefore $(H_0/c)(1+z{)}^3\int N{f}_{X}dN$. Divide by $(1+z{)}^3$ to obtain its comoving density, multiply by hydrogen atomic mass, and divide by today's mass critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$. This derives the neutral gas density from an absorption distribution:If “neutral gas” includes helium, replace $m_H$ by ${\mu }_H{m}_H$, with a stated abundance correction approximately ${\mu }_H\simeq 1.3$. Equivalently, for the distribution per redshift the prefactor inside the integral is $m_{H}H(z)/[c{\rho }_{\mathrm{crit},0}(1+z{)}^2]$ multiplying $N{f}_z$. In the PDF's energy-density convention, both the mass density and critical density are multiplied by $c^2$, leaving their ratio unchanged. The usual quoted abundance uses comoving mass density relative to today's critical density; a ratio of physical density to the epoch's critical density instead multiplies this answer by $(1+z{)}^3{H}_0^2/H(z{)}^2$.

To infer total gas, one needs the neutral fraction $x_{\mathrm{HI}}=N_{\mathrm{HI}}/N_H$ for each absorber. The ionized gas density from an absorption distribution isThis is an ionization-corrected integral, not an inference available from the observed columns alone. For optically thin, almost fully ionized hydrogen, photoionization equilibrium gives $\Gamma n_{\mathrm{HI}}=\alpha (T)n_e{n}_{\mathrm{HII}}\simeq \alpha (T)n_H^2$. If the absorbing depth $L$ is specified, $N=\alpha (T)n_H^{2}L/\Gamma$ andThus photoionization rate, temperature and depth are all needed.

One useful physical closure sets $L$ to a local Jeans length, $L\propto (f_{g}T/n_H{)}^{1/2}$, with local gas fraction $f_g$. Solving $N\propto [\alpha (T)/\Gamma ](f_{g}T{)}^{1/2}{n}_H^{3/2}$ then gives the Jeans-length ionization correction for the Lyman-alpha forest:This illustrates why weak, highly ionized forest lines can dominate the total gas inventory even though damped Lyman-alpha systems dominate the neutral-column-weighted inventory. It is a model for locally confined optically thin gas; optically thick Lyman limit systems and damped Lyman-alpha systems require a separate radiative transfer treatment. The physical basis and observational application of this closure are developed in Schaye's absorption-based matter-density calculation.

No measured normalization of the neutral-hydrogen column-density distribution, ionizing field or gas temperature is supplied, so the paper's formulas do not determine numerical densities on their own. With observational data and suitable ionization corrections, the high-redshift Lyman-alpha forest gives a total gas density of order a few $10^{-2}$ relative to today's critical density, comparable to the cosmological baryon density ${\Omega }_b\sim 0.04$–$0.05$ for $H_0\sim 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. The conclusion is that a large fraction of baryons at $z\simeq 3$ reside in diffuse ionized gas, rather than that every baryon has been individually accounted for. Schaye's analysis and Kim and collaborators' survey support that interpretation under their stated gas and ionizing-background assumptions.

The neutral-gas reservoir traced predominantly by damped Lyman-alpha systems is much smaller, of order $10^{-3}$ in the same convention. It is broadly comparable in order of magnitude to today's luminous stellar matter, whose cosmic stellar mass density is a few $10^{-3}$, but exact comparisons depend on cosmology, helium correction and which stellar components are counted. Storrie-Lombardi, McMahon and Irwin's neutral-gas census finds especially close comparison with present stellar disks. Most high-redshift gas is ionized; the smaller neutral reservoir has a mass of roughly the same order as today's visible stellar population. This comparison does not imply that the measured neutral reservoir alone simply becomes all present-day stars: gas accretion, recycling, ionization changes and outflows continue during galaxy evolution.

The neutral gas density from an absorption distribution is comparatively robust for high columns where the hydrogen is mostly neutral, but saturated lines, dust-obscured quasars, incomplete surveys and small numbers of rare high-column systems affect the integral. The ionized gas density from an absorption distribution has additional systematic uncertainty in the cosmic ionizing background, temperature, absorber depth, helium treatment and small-scale density inhomogeneity. At low columns, line blending and incompleteness matter; at high columns, attenuation makes an optically thin ionization correction invalid. A single power law extrapolated outside the measured column range can also produce a misleading mass integral. Hot gas, molecular gas and gas missed by the selected absorption population are not included automatically. These are observational and modelling qualifications to a baryon census, not failures of the column-mass accounting itself.

Use natural units with $c=1$, write $H=\dot{a}/a$, and measure cosmic time from the Big Bang. The cosmological continuity equation follows by differentiating the Friedmann equation and using the Friedmann acceleration equation; for pressureless matter it gives $\dot{\rho }+3H\rho =0$, hence $\rho ={\rho }_0{a}^{-3}$ when the present scale factor is $a_0=1$. The present critical density and cosmological density parameter giveConsequently, with $C=H_0^2{\Omega }_0$, the Friedmann equation becomes ${\dot{a}}^2=C/a-k$.

Introduce conformal time through $dt=ad\tau$. A prime denotes a derivative with respect to $\tau$, so $a^{\prime }=a\dot{a}$ andSet $u=\sqrt{k}\tau$, with $u=0$ at the Big Bang. On the expanding branch the substitution $a=C(1-\cos u)/(2k)$ satisfies this equation. Integrating $dt=adu/\sqrt{k}$, with $t(0)=0$, gives the closed matter-dominated Friedmann solution:The parametrization continues smoothly through the maximum scale factor, reached at $u=\pi$, onto the contracting branch. Its curve in the $(t,a)$ plane is a rescaled cycloid.

For a presently expanding universe, $H_0>0$ selects $0<u_0<\pi$. The normalization $a(u_0)=1$ givesSubstitution yields the age of a closed dust universe:The first later zero of the scale factor occurs at $u=2\pi$, so the lifetime of a closed dust universe gives the Big Crunch timeThis is the total lifetime measured from the Big Bang; the remaining cosmic time is $t_{\mathrm{BC}}-t_0$. As a check, the limit ${\Omega }_0\downarrow 1$ gives $H_0{t}_0\to 2/3$, while the Big Crunch recedes to infinite time, as expected for a flat matter-dominated universe with zero cosmological constant.

Use natural units, assume negligible neutrino chemical potentials, and approximate neutrino decoupling as occurring before electron-positron annihilation in cosmology. At high temperature, weak interactions maintain a common temperature for neutrinos, antineutrinos, electrons, Positrons and photons. Since ${\Gamma }_{\nu }/H\propto T^3$, the reaction rate eventually loses against expansion. Decoupled relativistic neutrinos then free stream; their momenta redshift as $a^{-1}$, so their distribution retains a temperature parameter satisfying $a{T}_{\nu }=\mathrm{constant}$.

The still-coupled electromagnetic bath receives the entropy of annihilating electrons and Positrons. Its effective number of relativistic entropy degrees of freedom is initiallywhere the two photon polarizations count as bosonic states and the four electron-positron spin states have the fermion entropy weight $7/8$. After the pairs disappear, $g_{\ast s}^{\mathrm{EM}}=2$. Cosmological entropy conservation in this coupled bath gives $g_{\ast s}^{\mathrm{EM}}(a{T}_{\gamma }{)}^3=\mathrm{constant}$. The three already decoupled neutrino species are excluded from this bath's entropy count. At decoupling $T_{\nu }=T_{\gamma }$, henceSubsequent expansion preserves this ratio in the idealized history. Residual interactions during annihilation give small corrections to the instantaneous-decoupling approximation. If a relic species later becomes massive, $T_{\nu }$ here remains the redshifting momentum-distribution parameter; it need not be its nonrelativistic kinetic temperature.

A single ordinary neutrino species includes one populated neutrino helicity and one antineutrino helicity. For negligible chemical potential, integration of the Fermi-Dirac distribution givesTherefore the number density of a relativistic neutrino relative to photons, with the antiparticle included, isThe factor $3/4$ is the fermionic number-density weight, distinct from the energy and entropy weight $7/8$. This relation continues to hold after this species becomes nonrelativistic if its comoving particle number is conserved.

Under the proposed matter-dominated, spatially flat model, the Friedmann equation requires the present relic-neutrino energy density to be approximately the critical density. Using the unreduced Planck mass $M_P=G^{-1/2}$ givesThus the critical-density mass of a thermal relic neutrino isTaking the paper's rounded values, $T_{\gamma 0}=2\times 10^{-13}\mathrm{GeV}$, $H_0=h/(4.5\times 10^{41}{\mathrm{GeV}}^{-1})$ and $M_P\sim 10^{19}\mathrm{GeV}$, givesFor $0.5<h<1$, this is of order $30$--$100\mathrm{eV}$ with these approximate constants. More precise constants change the coefficient, not the requested mass scale. This mass is much larger than the present momentum scale $T_{\nu 0}\simeq 1.4\times 10^{-4}\mathrm{eV}$, validating the nonrelativistic energy density approximation. The result assumes that one thermal relic species dominates the matter density; it is conditional on that model rather than a measurement of a neutrino mass.

Specify the path convention before integrating the neutral-hydrogen column-density distribution. Write $f_z={\partial }^2\mathcal{N}/(\partial N\partial z)$ and $f_X={\partial }^2\mathcal{N}/(\partial N\partial X)$, with absorption distance satisfyingThe lower panel of the preceding sketch shows the approximate decrease of $f$ with $N=N_{\mathrm{HI}}$ near $z=3$. Ordinary forest columns are roughly described by $f\propto N^{-\beta }$, $\beta \sim 1.5$, but the slope changes through partially shielded Lyman limit systems and high-column damped Lyman-alpha systems. No single slope should be extrapolated through all these regimes. In logarithmic bins the number is $N\ln (10)f$, whereas their neutral mass contribution is proportional to $N^{2}f$; confusing these two weightings would give a wrong baryon inventory.

To derive the neutral gas density from an absorption distribution, a redshift interval $dz$ has proper light-path length $d\ell =cdz/[(1+z)H]$. The sum of neutral hydrogen columns per path length estimates the mean proper number density:The proper mass density is $m_H{\overline{n}}_{\mathrm{HI}}$; division by $(1+z{)}^3$ converts it to a comoving mass density. Thus, with today's critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$,A representative finite-sample estimator replaces the last integral by ${\sum }_j{N}_j/\Delta X$, with completeness and selection corrections. This defines a comoving inventory divided by today's critical density; a fraction of the instantaneous critical density has a different normalization. Neutral gas including accompanying helium is obtained by multiplying by a specified mass correction, approximately $1.3$; hydrogen-only ${\Omega }_{\mathrm{HI}}$ must not silently include it.

At $z\sim 3$ the observed neutral inventory is of order $10^{-3}$, far below the cosmological baryon density of order $0.04$–$0.05$ for $h\sim 0.7$. High-column damped Lyman-alpha systems contribute most of this neutral mass although weak forest lines dominate the counts; for $\beta =1.5$ the mass per logarithmic column interval increases as $N^{1/2}$ in the forest regime. The small neutral inventory therefore does not imply a small total gas inventory: the intergalactic medium contains a large ionized baryon reservoir invisible to a neutral-only count. Recovering that reservoir requires an ionization correction and a gas-density model. The survey normalization and high-column contribution are described in the neutral-gas inventory analysis.