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Let $X$ be the hydrogen mass fraction, $f_g={\rho }_g/{\rho }_{\mathrm{tot}}$ the gravitating gas fraction, and $\chi =n_{\mathrm{tot}}/n_H$ the total-particle-to-hydrogen ratio. The optically thin gas cooling time and free-fall time of a uniform sphere arewhere ${\rho }_{\mathrm{tot}}=m_p{n}_H/(X{f}_g)$. Equating them gives the cooling-to-free-fall equality curve:At fixed composition, its shape is $n_{\mathrm{crit}}\propto T^2/{\Lambda }^2$. Gas above this curve cools faster than it falls; gas below it cools more slowly. Strong line cooling produces a deep low-density trough. In the high-temperature free-free regime, $\Lambda \propto T^{1/2}$ gives $n_{\mathrm{crit}}\propto T$.

The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu \simeq 0.59$ and $\chi \simeq 2.24$. Holding these ionized-gas factors fixed makes the diagram transparent, but near the atomic threshold their actual temperature dependence changes numerical normalizations. With these conventions the supplied cooling values imply $n_{\mathrm{crit}}(1.5\times 10^4\mathrm{K})\simeq 9.6\times 10^{-10}\mathrm{c}{\mathrm{m}}^{-3}$ and $n_{\mathrm{crit}}(10^7\mathrm{K})\simeq 0.107\mathrm{c}{\mathrm{m}}^{-3}$. For gas supported in a dark-matter-dominated dark-matter halo, take $f_g<1$: at fixed $n_H$ the free-fall time of a uniform sphere is shorter and the critical curve shifts upward by $1/f_g$.

To add virial mass contours in a cooling diagram, use the uniform self-gravitating sphere consistently. Its Newtonian gravitational potential energy is $-3G{M}^2/(5R)$ and its thermal kinetic energy is $(3/2)M{k}_{B}T/(\mu m_p)$. The virial theorem gives $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(5R)$, while $M=(4\pi /3)R^3{\rho }_{\mathrm{tot}}$. Eliminating $R$ givesThus constant total gravitating mass gives slope three in the logarithmic density-temperature plane, with larger mass contours lying at smaller density. A different density profile changes the structural coefficient; the customary halo temperature scale $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,\mathrm{vir}}\simeq {\Delta }_v(2\times 10^{-7})(1+z{)}^3\mathrm{c}{\mathrm{m}}^{-3}$, using ${\Delta }_v=18{\pi }^2$. They show why increasing formation cosmological redshift helps gas reach the rapid-cooling region.

At the low-mass end, a dark-matter halo's virial temperature may not reach the atomic threshold. The atomic and molecular cooling thresholds for galaxy formation therefore favor an atomic-cooling mass of order $10^8{M}_{\odot }$ at early formation epochs, with its precise value varying as approximately $(1+z{)}^{-3/2}$. Smaller primordial systems require molecules; photoheating and stellar feedback further suppress their luminous output.

At the high-mass end, the virial temperature at a fixed assembly density scales as $M^{2/3}$. Beyond the line-cooling interval the gas becomes increasingly hot, while thermal bremsstrahlung improves only as $T^{1/2}$. Hence at fixed assembly density $t_{\mathrm{cool}}/t_{\mathrm{ff}}\propto T^{1/2}\propto M^{1/3}$ in that regime. Large systems can remain pressure-supported hot atmospheres for many dynamical times, instead of promptly fragmenting into one stellar object. This is the upper galaxy mass from gas cooling: it associates efficient condensation with roughly galactic, rather than group or cluster, masses and motivates the broad $10^8$–$10^{12}{M}_{\odot }$ interval.

The diagram gives characteristic cooling scales, not exact universal mass bounds. Density profiles, gas fraction, enrichment and formation epoch move the boundaries; the two supplied values alone do not determine a complete numerical cutoff. Some gas can cool in dense cores of larger dark-matter haloes, early smaller progenitors can form stars before merging, and subsequent galaxy mergers can assemble more massive galaxies. Long-lived hot atmospheres also require heating to prevent excessive late-time cooling. These qualifications explain why the cooling criterion for galaxy formation is useful without equating each dark dark-matter halo to one galaxy.

For an optically thin low-density gas in collisional ionization equilibrium, the astrophysical cooling function collects losses per pair of hydrogen nuclei into $\mathcal{C}=n_H^2\Lambda (T)$. Its units are $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$; the time unit is missing from the printed guide to the vertical-axis labels. For the primordial atomic cooling curve, only hydrogen and helium provide bound-state coolants.

Near $10^4\mathrm{K}$, collisional excitation of hydrogen becomes effective and subsequent photon emission removes thermal energy. The excitation rate is exponentially suppressed below the atomic energy threshold. Atomic line cooling and collisional ionization generate a strong hydrogen feature around a few $10^4\mathrm{K}$; helium excitation and ionization produce further structure around $10^5\mathrm{K}$. Radiative recombination also contributes. Once the gas is highly ionized, these bound-state losses weaken. At $T\gtrsim 10^6$–$10^7\mathrm{K}$, thermal bremsstrahlung dominates, with approximately $\Lambda \propto T^{1/2}$ apart from slowly varying factors. The figure is an original qualitative sketch anchored to the two supplied values, not an atomic-rate calculation.

Increasing galactic metallicity adds many ionic transitions, producing metal-line cooling and substantially enhancing cooling, particularly in the $10^5$–$10^7\mathrm{K}$ range. Metals also provide low-energy fine-structure transitions below $10^4\mathrm{K}$. The change is not a uniform vertical shift: the positions and strengths of features depend on the elemental abundances and ionization state.

Photoionization removes bound electrons even where collisions alone would leave atoms neutral, often suppressing the hydrogen and helium line peaks and changing the metal-ion population. It also adds photoionization heating. The net thermal loss is then $\mathcal{C}-\mathcal{H}$, with possible heating-cooling equilibrium near $10^4\mathrm{K}$. A single density-independent $\Lambda (T)$ no longer describes every irradiated cloud: the answer also depends on radiation intensity, spectrum, shielding and density. The illustrative photoionized curve denotes altered cooling alone, not the net loss including heating.

Below $10^4\mathrm{K}$, primordial gas can use molecular line cooling, especially rotational and vibrational transitions of molecular hydrogen and, in suitable chemical conditions, hydrogen deuteride. Molecule formation and protection against photodissociation are essential. Enriched gas can additionally use metal-line cooling, other molecular species and dust thermal emission, with gas-dust energy exchange important at high density. At high cosmological redshift, residual free electrons can transfer energy to the Cosmic microwave background through Compton cooling by the cosmic microwave background if the gas is hotter than the radiation. Expansion can also cool gas adiabatically, but neither that process nor Compton cooling by the cosmic microwave background is an $n_H^2$ atomic loss law. The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.

A scale-invariant primordial potential has $P_{\Phi }(k)\propto k^{-3}$. On subhorizon linear scales, the cosmological Poisson equation and the cold-dark-matter transfer function give ${\delta }_c(k,a)\propto D(a)k^{2}T(k){\Phi }_{\mathrm{prim}}(k)$, where $D(a)\propto a$ during matter domination. ConsequentlyLarge scales enter only after equality and have $T(k)\simeq 1$. Small scales enter during radiation domination; their approximately logarithmic growth up to equality gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$. ThereforeIf the slow logarithm is suppressed in a rough sketch, the slopes are $+1$ and $-3$, with a turnover near $k_{\mathrm{eq}}$. The logarithmic correction is real and should not be mistaken for a different primordial spectral index. In the matter-only late-time approximation with $D(0)=1$, the cosmological redshift $z=1$ means $a=1/2$ and $D(1{)}^2=1/4$; it changes the amplitude, not these asymptotic shapes. Neither the initial normalization nor cosmological parameters needed for an absolute power are supplied.

This is a schematic smooth interpolation with the derived asymptotes, not a precision transfer-function fit. The linear ideal-fluid model excludes baryonic acoustic structure and small-scale nonlinear evolution.

There is also a gauge and horizon qualification. For a mode still outside the Hubble radius at $z=1$, the printed Newtonian-gauge density has ${\delta }_c\simeq -2\Phi$, and its formal dimensional spectrum is instead proportional to $k^{-3}$. The conventional large-scale $k$ branch describes modes that are large relative to the equality scale but already subhorizon at the observation time. Alternatively, the comoving matter density, ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ with ${\theta }_c=\nabla \cdot {\mathbf{v}}_c$, removes that constant gauge term and obeys ${\Delta }_c=-2k^2\Phi /(3{\mathcal{H}}^2)$ in the growing matter solution. The usual matter-spectrum sketch can be continued to small $k$ in this comoving-density convention. The dimensional spectrum requested here is $P$, rather than the dimensionless cosmological power spectrum $k^{3}P/(2{\pi }^2)$, whose slopes would differ by three.

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.

In the local orthonormal tetrad, the components of the photon number current areThey are respectively the photon number density and the photon number flux. Thus $J^{\mu }$ is the photon number four-current, not an energy current or an electric charge current. Photons carry no electric charge. For the isotropic background,Integrating the covariant Boltzmann equation over the invariant mass-shell measure, with vanishing momentum-boundary flux, giveswhere $C[f]$ is the collision operator in the covariant streaming equation. ThereforeCollisionless photon propagation satisfies this condition, as do Thomson scattering and ordinary Compton scattering when all participating photons are counted: each collision has one incoming and one outgoing photon. Photon emission, photon absorption, bremsstrahlung, double Compton scattering and electron-positron annihilation in cosmology or its inverse generally give a nonzero number source. A cancellation of net creation and destruction can nevertheless make the integrated source vanish even when individual number-changing reactions occur. Cosmological redshift itself changes photon energy, not photon number; in an expanding homogeneous universe a conserved photon number current gives ${\bar{n}}_{\gamma }\propto a^{-3}$.

For comoving galaxies their relative comoving vector $\mathbf{\chi }$ is constant, and the physical displacement is $\mathbf{r}(t)=a(t)\mathbf{\chi }$, where $a$ is the scale factor. Differentiating gives the Hubble lawPeculiar velocity relative to this comoving frame is excluded in this relation.

Transport a wave crest between neighbouring comoving observers separated by a small physical distance $d\ell$. Their recession speed is $Hd\ell$, while the light travel time is $dt=d\ell /c$ to first order. The local Doppler shift therefore changes its wavelength by $d\lambda /\lambda =Hdt$. Multiplying these infinitesimal shifts along the ray, or integrating their logarithms, yieldsThe use of the local first-order Doppler law is exact in the limit of infinitesimal steps: the accumulated second-order errors tend to zero. A single special-relativistic recession velocity assigned to widely separated galaxies would not justify this cosmological redshift relation.

For a component with constant barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation givesHence ${\rho }_i={\rho }_{i,0}{a}^{-3(1+w_i)}$. Choose the present scale factor to be $a_0=1$, so the cosmological redshift obeys $1+z=a^{-1}$, and define the present cosmological density parameter bySubstitution in the spatially flat Friedmann equation then yields

The Soltan argument compares the time-integrated luminosity density of the cosmological AGN population with the present comoving mass density in dormant supermassive black holes. If $U_{\mathrm{AGN}}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar{\eta }$ predictsIn practice $U_{\mathrm{AGN}}$ comes from integrating AGN luminosity functions over luminosity and cosmological redshift, with corrections for obscured sources and missed wavebands, while ${\rho }_{\mathrm{BH}}$ is inferred from local galaxy--black-hole scaling relations.

The inferred efficiency is of order the canonical thin-disk value, about ten per cent, so most cosmic black-hole mass was accumulated in radiatively efficient, optically thick accretion episodes. Radiatively inefficient flows can dominate low-luminosity activity or brief extreme phases, and mergers redistribute existing mass, but neither naturally accounts for the observed integrated AGN radiation while supplying most of the final mass.

The given formula is the integral of the Planck photon distribution. SetThenwhereEach photon of frequency $\nu$ has energy $h\nu$, so the energy density isThis is the temperature scaling of thermal photon number and energy densities.

After decoupling, cosmological redshift givesThe occupation number is conserved along the freely propagating photon trajectory. Its exponential argument at decoupling becomesThus the spectrum still has the Planck form ifFrequency and temperature redshift by the same factor, which is the redshift preservation of a thermal photon spectrum.

A suitable Spatially flat FLRW metric isBoth source and detector are comoving, so cosmological redshift stretches the wave period in proportion to the scale factor. If $f_e$ is the local source frequency at emission and $f_o$ is the frequency measured in Earth's local frame, thenThe same relation applies to gravitational waves propagating on the FLRW background in the geometric-optics regime.

A local redshift spacing between absorbers corresponds to proper path length $c\Delta z/[H(z)(1+z)]$. Using the same rest-frame transition for all lines gives $\Delta z=\Delta \lambda /{\lambda }_{\mathrm{rest}}$. This conversion assumes cosmological redshift dominates the line spacing and uses a local Hubble-rate approximation.