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For a relaxed cluster, galaxy motions act as tracers of its gravitational potential. The virial theorem gives a dynamical mass of orderwith a profile-dependent coefficient and a projection correction because the observed velocity dispersion is usually only the line-of-sight component. For an isotropic uniform-sphere estimate using its one-dimensional dispersion, $T=3M{\sigma }_v^2/2$ and $W=-3G{M}^2/(5R)$ give $M=5R{\sigma }_v^2/G$. Measured dispersions require much more gravitating mass than the luminous galaxies provide, hence large cluster mass-to-light ratios. Interlopers, orbital anisotropy and departures from equilibrium are systematic uncertainties, so the independent gas and lensing measurements matter.

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.

Interpret the barred velocities as second moments of a steady nonstreaming collisionless stellar system, and write ${\sigma }_r^2=\overline{v_r^2}$. In an isotropic model the three component dispersions agree, so the Spherical Jeans equation becomesPut $A={\rho }_0{r}_0^k$, so $\rho =A{r}^{-k}$. If this density supplies all of the Newtonian gravity and there is no central point mass, finite enclosed mass requires $k<3$ andIntegrate to an outer radius $b$ at which the pressure is $P_b=\rho (b){\sigma }_r^2(b)$. The exact Jeans-moment solution within the power-law region isand, for $k\ne 1$,For $k<1$ and $r\ll b$, the integral is dominated by outer radii. ThereforeThe coefficient cannot be determined from the local density cusp alone. Extending the power law to infinity gives a divergent pressure integral when $k<1$, so an outer scale or boundary condition is essential. For the usual shallow cusp $0<k<1$, the central velocity dispersion tends to zero even though the density diverges.

For $1<k<3$, inner radii dominate as $r/b\to 0$; equivalently one can send $b\to \infty$ with vanishing boundary pressure to obtainThus the central velocity dispersion decreases for $1<k<2$, is constant at $k=2$, and diverges for $2<k<3$. At the omitted threshold $k=1$, ${\sigma }_r^2=(P_b/A)r+2\pi GAr\ln (b/r)$. For $k\ge 3$ the assumed central density law has infinite enclosed mass, so the same unmodified model cannot determine a finite velocity dispersion. These boundary and domain restrictions are the boundary dependence of an isotropic self-gravitating cusp; specifying density without them does not specify a physical finite galaxy.

Let $p(r)=\rho (r){\sigma }_r^2(r)$ and use the velocity-anisotropy parameter $\beta =1-{\sigma }_{\theta }^2/{\sigma }_r^2$, with ${\sigma }_{\theta }^2={\sigma }_{\varphi }^2$. Here the tangential variance is per component; a definition using the sum of both tangential variances instead has an extra factor of two. The original PDF has this correct per-component ratio, which the TeX corrupts.

At a point along the line of sight, $r^2=R^2+z^2$. The radial direction makes an angle with the sightline satisfying ${\sin }^2\chi =R^2/r^2$. Spherical symmetry eliminates mixed moments, and projection givesUsing both sides of the galaxy and $dz=rdr/\sqrt{r^2-R^2}$,For this formula use $I$ as the projected density of the same tracer represented by $\rho$. If $\rho$ is a mass density and $I$ an observed surface brightness, a constant mass-to-light ratio rescales $I$ by that factor; otherwise replace $\rho$ by the luminosity density throughout the tracer equation. The gravitating $M(r)$ remains the total mass, not necessarily the tracer's mass.

The Spherical Jeans equation is $p^{\prime }+2\beta p/r=-\rho GM/r^2$. Eliminate $\beta p=-r[p^{\prime }+\rho GM/r^2]/2$ from the projected integrand:Rearranging provesNo constancy of $\beta$ was needed. Convergence of the projected moments and the steady spherical assumptions are required. This is the integral form of the projected spherical Jeans residual.

Although $\beta$ has disappeared explicitly, the unknown radial pressure $p$ still carries the orbital information. Thus the identity does not remove the mass--anisotropy--density degeneracy: projected brightness and one projected second moment do not uniquely determine both $M(r)$ and the radial/tangential partition. A central rise in velocity dispersion can reflect a compact central mass, radial orbital bias, or a changing mass-to-light ratio; flattening and departures from equilibrium add further ambiguities.

For example, within a point-mass-dominated region with tracer $\rho \propto r^{-k}$ and constant $\beta$, direct solution gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$. For $k>1$ the projected velocity dispersion of a Keplerian cusp isThe beta function factors come from setting $r=R\sec \chi$ in the integrals along the line of sight. Different allowed $\beta$ generally change the mass needed to fit the same normalization; $k=2$ is a special cancellation, not the general case. Resolving the central black-hole sphere of influence and constraining orbital structure with higher velocity moments or additional kinematic directions strengthens a compact-mass inference. Brightness and velocity dispersion profiles alone do not establish a unique central black-hole mass, still less an event horizon.

The primordial light-element abundance determination uses minimally processed material and estimates its residual processing rather than directly observing the universe at a few minutes old.

For deuterium, high-redshift, low-metallicity quasar absorption systems resolve isotope-shifted D I lines from the H I Lyman series. The deuterium feature is displaced by about $82{\mathrm{km}\mathrm{s}}^{-1}$ from hydrogen; fitting several transitions and the velocity components distinguishes genuine deuterium from an unrelated hydrogen blend. The ratio of the inferred columns estimates D/H after the ionization and component structure are checked. Low metallicity minimizes stellar destruction, giving a representative primordial ratio of roughly $(2.5\text{–}3)\times 10^{-5}$.

For helium-4, helium and hydrogen recombination-emission lines in metal-poor H II regions of nearby dwarf galaxies give a mass fraction. Several lines constrain temperature, electron number density and corrections for collisional excitation, ionization structure, radiative transfer and underlying stellar absorption. Extrapolation of helium enrichment against metallicity towards zero metallicity estimates $Y_p$, approximately $0.24$–$0.25$, corresponding to helium/hydrogen by number near $0.08$.

For helium-3, the approximately $8.665$ GHz hyperfine line of ${}^3{\mathrm{He}}^{+}$ in Galactic H II regions probes the abundance, with ionization and nebular structure corrections. Stellar production and destruction make its primordial value hard to extract; a characteristic abundance near $10^{-5}$ relative to hydrogen is compatible with standard nucleosynthesis but is not a comparably clean primordial measurement.

For lithium-7, the approximately $6708$ Å Li I absorption doublet in warm, old, metal-poor halo stars gives a photospheric abundance after stellar-atmosphere and effective-temperature corrections. The Spite plateau suggests an initial number ratio of order $(1\text{–}2)\times 10^{-10}$ if depletion is small; transport below the photosphere, nuclear destruction and atmosphere systematics can lower the measured value relative to the birth abundance. Standard Big Bang nucleosynthesis at the deuterium- and microwave-background-favored baryon density instead predicts several $10^{-10}$, roughly $(4\text{–}5)\times 10^{-10}$ in exam-era calculations. This discrepancy is the cosmological lithium problem, not a measurement of an independently adjustable primordial lithium parameter.

The approximate values deliberately represent exam-era orders of magnitude rather than a claim of current precision; the contemporary abundance assessment discusses the inference and tensions.

Separate the observables before combining their parameter constraints. The baryon fraction is included within total matter: for the benchmark, ${\Omega }_{\mathrm{CDM}}={\Omega }_m-{\Omega }_b=0.255$, rather than adding ${\Omega }_b$ to ${\Omega }_m$ a second time. With $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})=0.7$, the physical density combinations areFor baryons, Big Bang nucleosynthesis predicts primordial light-element abundances as a function of the baryon-to-photon ratio; deuterium is especially sensitive. Independently, baryon inertia changes the relative heights of compressional and rarefaction peaks in the cosmic microwave background through the baryon loading parameter. Agreement of these two physical-density estimates supports a baryon abundance near the quoted value. For example, the three-year WMAP cosmological analysis found ${\Omega }_b{h}^2=0.02229\pm 0.00073$ in its fitted model.

For total matter, matter-radiation equality affects the microwave-background peak pattern and fixes a characteristic turnover in the matter power spectrum. galaxy clustering therefore constrains matter density, with corrections for galaxy bias and peculiar-velocity distortions. Cluster baryon fractions combine observed gas and stars with total masses inferred dynamically or by gravitational lensing; comparison with the cosmic baryon fraction also constrains ${\Omega }_m$. Galactic rotation, cluster dynamics and gravitational lensing give independent evidence that visible baryons alone cannot supply the gravitating mass. The precise ${\Omega }_m$ estimate comes from quantitative joint fits, rather than from a rotation curve by itself. Different 2006 data combinations gave somewhat different best fits, so $0.3$ is a rounded benchmark rather than a universal exact fit.

For a cosmological constant, standardized high-redshift Type Ia supernovae measure the shape of the luminosity-distance relation. In a flat matter-vacuum model,The observed supernova dimming relative to simple decelerating matter models favours late accelerated expansion. The original evidence is given by Riess and collaborators' 1998 supernova analysis. Combining the distance curve with matter constraints selects a positive vacuum fraction close to the quoted benchmark; it gives $q_0={\Omega }_m/2-{\Omega }_{\Lambda }=-0.55$.

For curvature, the angular scale of the Cosmic microwave background acoustic peaks compares the sound horizon with the angular diameter distance to last scattering. This strongly constrains spatial geometry, but curvature and late-time expansion parameters can partly compensate one another. Independent distances, notably baryon acoustic oscillations and supernova distances, help break that degeneracy. The 2005 SDSS acoustic-peak detection supplies a low-redshift standard ruler complementary to the microwave-background ruler. In combination, these observations support near-flat geometry and the closure relation ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k\simeq 1$.

The benchmark is supported by mutually constraining abundance, clustering, angular-scale and distance measurements. Its uncertainties are correlated and depend on model assumptions. An exactly zero curvature parameter has no meaningful percentage error; observational evidence establishes consistency with zero and an absolute uncertainty, not a literal fractional accuracy better than ten per cent on zero.

The cosmological principle assumes statistical spatial homogeneity and cosmological isotropy on sufficiently large scales. Spatial homogeneity means that coarse-grained statistical properties do not depend on location; cosmological isotropy means that they do not depend on direction. The claim concerns a large-scale description, allowing individual galaxies, clusters and voids. It motivates the FRW metric and makes cosmic expansion describable by a single scale factor and a small number of matter variables.

Its adoption combines empirical large-scale regularity with the Copernican principle, the assumption that our location is typical. The resulting models provide a coherent account of cosmic expansion, the cosmic microwave background and primordial abundances. Neither the Einstein field equations alone nor cosmological isotropy around just one observer proves global spatial homogeneity: a spherically symmetric universe specially centred on that observer would also look isotropic. Observations of a finite past light cone support the principle over sampled scales without establishing it beyond the observed region.

Three supporting observations probe complementary aspects. First, after the observer-motion dipole and local foregrounds are removed, the cosmic microwave background is extremely isotropic, with intrinsic temperature fluctuations of order $10^{-5}$ and a nearly identical thermal spectrum across the sky. Second, suitably corrected all-sky counts of distant radio sources and extragalactic background intensities show no large preferred direction; Galactic obscuration and survey selection must be accounted for. Third, three-dimensional galaxy redshift surveys show that clustering and voids average toward statistically similar distributions in sufficiently large regions: large-volume number counts approach volume scaling and correlations weaken on large separation scales. The first two primarily test cosmological isotropy, while the third adds evidence for large-scale spatial homogeneity. The principle is a well-supported large-scale working hypothesis whose observational and typicality assumptions remain explicit.

Use the stellar position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$. This is the $I$ entering the tensor virial theorem, rather than the engineering moment-of-inertia tensor $\mathrm{tr}(I)1-I$. For a density pattern rotating steadily about its third principal axis, choose $t=0$ when its in-plane axes are principal axes. Then $I(0)=\mathrm{diag}(I_1,I_2,I_3)$ andIn particular, $I_{11}=I_1{\cos }^2\Omega t+I_2{\sin }^2\Omega t$, $I_{22}=I_1{\sin }^2\Omega t+I_2{\cos }^2\Omega t$, and $I_{12}=(I_1-I_2)\sin \Omega t\cos \Omega t$, while $I_{33}=I_3$ and $I_{13}=I_{23}=0$. Two differentiations at $t=0$ prove the inertia-tensor acceleration of a steadily rotating stellar pattern:Both the off-diagonal acceleration and the in-plane trace acceleration vanish at this instant.

Write $\mathbf{u}=\overline{\mathbf{v}}$ for the local mean streaming. The stellar kinetic-energy tensor, stellar pressure-energy tensor, and stellar potential-energy tensor have the conventionsFor an isolated self-gravitating system, $\frac{1}{2}{\ddot{I}}_{ij}=2T_{ij}+{\Pi }_{ij}+W_{ij}$. Subtracting its second diagonal component from the first givessoThis quotient requires $I_{11}\ne I_{22}$. In an axisymmetric density distribution it becomes $0/0$: rotating the pattern about its symmetry axis does not change the density, and this equation cannot determine a pattern speed. Stellar streaming need not be rigid rotation with angular frequency $\Omega$.

Next add the first two diagonal virial equations and use the third. Their acceleration terms vanish, soFor $T_{33}=0$, ${\Pi }_{33}=-W_{33}$. With the planar virial trace ${\Pi }_{\perp }={\Pi }_{11}+{\Pi }_{22}$ and $W_{\perp }=W_{11}+W_{22}$, the prescribed mass-weighted quantities satisfy ${\Pi }_{\perp }=2M{\sigma }_0^2$, $2(T_{11}+T_{22})=M{V}_0^2$, and $1-\delta =2{\Pi }_{33}/{\Pi }_{\perp }$. Therefore $W_{33}=-(1-\delta )M{\sigma }_0^2$ and the planar equation becomesEliminating $M{\sigma }_0^2$ givesThe ${\nu }_0$ in the numerator of the PDF's displayed relation is a notation slip: the velocity subsequently defined is $V_0$. The relation follows from the traces, not from replacing a galaxy by a solid rotating body.

For the axisymmetric oblate spheroid, rotational symmetry gives ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$, ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$ and zero integrated off-diagonal random stresses. A line-of-sight unit vector can be chosen as $\mathbf{n}=(\sin i,0,\cos i)$, with $i=0$ face-on. Contracting the random stress tensor with this vector gives the inclination projection of an axisymmetric stellar velocity tensor:This is a mass-weighted random velocity dispersion after subtracting streaming, with consistent aperture weights. Along the projected major axis, azimuthal streaming lies along the direction whose line-of-sight projection is $\sin i$; a rotation amplitude $V_0$ consequently projects asThere is a necessary observational convention here. The globally defined $V_0^2=2(T_{11}+T_{22})/M$ is a streaming mean square. The displayed amplitude relation holds if the rotation curve is constant, or if the measured rotation-amplitude statistic is weighted consistently to represent this intrinsic quantity. Axisymmetry alone does not equate an arbitrary observed peak rotation to the global $V_0$. If $V_r$ instead means the full-aperture line-of-sight streaming root-mean-square, azimuthal averaging yieldsand the corresponding root-mean-square is $V_0\sin i/\sqrt{2}$. Keeping amplitude and root-mean-square distinct prevents a spurious factor of two in the kinetic energy.

Finally, let the intrinsic equatorial and polar semiaxes be $a$ and $c$, and put $q_t=c/a=1-{ϵ}_t$. An intrinsic boundary has $(x^2+y^2)/a^2+z^2/c^2=1$. Its sky minor-axis coordinate is $Y=y\cos i-z\sin i$. Eliminating the line-of-sight coordinate, or maximizing $Y$ on this ellipse, gives the projected minor semiaxis $\sqrt{a^2{\cos }^{2}i+c^2{\sin }^{2}i}$; the major semiaxis remains $a$. Thus the projected axis ratio of an oblate spheroid isSubtracting from unity provesThe face-on limit is round with zero projected rotation amplitude; the edge-on limit reveals the intrinsic flattening and in-plane random velocity dispersion. These limits check both the geometry and the convention for inclination.

The Lyman-alpha forest is the multitude of Lyman-alpha absorption lines blueward of the broad emission line in a distant quasar spectrum. Neutral hydrogen along the line of sight absorbs at its rest wavelength ${\lambda }_{\alpha }\simeq 1216\mathring{\mathrm{A}}$, so each absorber appears at ${\lambda }_{\mathrm{obs}}=(1+z_c){\lambda }_{\alpha }$. Much of the ordinary forest traces modest overdensities in the mostly ionized intergalactic medium, rather than compact objects made primarily of neutral gas. Strong self-shielded systems form a distinct absorption regime. Line strengths constrain neutral-hydrogen column density, while line widths contain thermal and velocity information.

The redshift evolution of the Lyman-alpha forest is normally described, for a fixed line selection, by an incidence $d\mathcal{N}/dz=A(1+z{)}^{\gamma }$. At redshifts of a few the line incidence and mean absorption increase rapidly with redshift; typical selected populations have $\gamma$ of order $2$–$3$, not a universal exponent. Low-redshift evolution is appreciably flatter. Geometry alone cannot explain the full change: for a fixed comoving abundance and fixed proper cross section one obtains $d\mathcal{N}/dz\propto (1+z{)}^2/H(z)$, only $(1+z{)}^{1/2}$ in matter domination. Density evolution, thermal evolution and the cosmic ionizing background change how many structures exceed the chosen absorption threshold. In optically thin photoionization equilibrium,so for roughly fixed overdensity the rising cosmic density alone makes $n_{\mathrm{HI}}$ scale approximately as $\alpha (T)(1+z{)}^6/{\Gamma }_{\mathrm{HI}}$. At low redshift, the declining ionizing radiation partly offsets the decrease in gas density. Near the end of reionization the forest becomes very opaque and the simple individual-line description ceases to suffice.

The quasar proximity effect is the statistically reduced absorption near the quasar's own redshift, contrary to the general increase of absorption towards higher redshift. It results from additional ionizing radiation from the nearby quasar. At fixed density and temperature the local photoionization rate becomes ${\Gamma }_{\mathrm{bg}}+{\Gamma }_Q$. Accordinglywhere $N_{\mathrm{HI}}^{(0)}$ is the value with the background alone. Lines weaken and some fall below the detection threshold; the absorbers need not have been physically removed. This gives the proximity-effect estimator of the ionizing background.

For an approximately isotropic quasar and a nearby cloud at proper distance $d$, calculate its flux from $F_{\nu ,Q}\simeq L_{\nu ,Q}/(4\pi d^2)$, allowing for absorption and relative redshift if necessary. The quasar and isotropic background ionization rates arewhere ${\nu }_0$ is the hydrogen ionization threshold, $h_P$ is the Planck constant and $J_{\nu }$ is the background specific intensity. Establish the ordinary absorption statistics far from quasars, then fit their reduction as a function of distance and known quasar luminosity. The scale at which $\omega \sim 1$ gives the background rate; an assumed spectral shape converts this to an intensity near the ionization threshold. For matching spectral shapes, $\omega \simeq L_{{\nu }_0,Q}/(16{\pi }^2{d}^2{J}_{{\nu }_0})$. A column-density distribution proportional to $N^{-p}$ gives a line-incidence suppression $(1+\omega {)}^{1-p}$ above a fixed threshold, by shifting that threshold to $(1+\omega )N_{\min }$ in the unperturbed distribution. This is the physical measurement principle established in adsabs.harvard.edu/pdf/1988ApJ...327..570B .

The inference is statistical. Quasars live in overdense environments, which enhance absorption and can mask extra ionization. Emission-redshift errors smear the effect, as part (iii) demonstrates. Finite source lifetimes, anisotropic emission, variability and departures from ionization equilibrium can also alter the radial profile. These effects must be modeled rather than attributed entirely to the cosmic ionizing background.

The background is produced mainly by ionizing radiation from accretion-powered quasars and hot massive stars in star-forming galaxies. Their relative importance changes with redshift. Quasars have hard spectra and can ionize helium as well as hydrogen; stellar photons depend strongly on their ionizing photon escape fraction from galaxies. Integrating source luminosities over the luminosity function and propagating radiation through absorbing gas gives the background. At high redshift the observed quasar population alone need not supply the required hydrogen ionization, so star-forming galaxies can be essential, as discussed in arxiv.org/abs/0807.4177 . Diffuse recombination radiation can contribute, but a residual primordial radiation field is not an adequate ionizing source.

The rotation-anisotropy diagram below uses the oblate-rotator flattening proxy for $\beta =0,0.2,0.4$. Each curve starts at $e=\beta$ and rises as $e$ increases. The plotted domains cover typical spheroid values, roughly $0\le e\lesssim 0.7$ and $0\le v/\sigma \lesssim 1.5$, with the upper axis extended to show the reference curve's rise. The shaded regions are schematic overlapping populations, not measurements or sharp classification boundaries.

Giant elliptical galaxies commonly have low ordered rotation for their observed flattening, lying below the oblate isotropic rotator reference. Anisotropic velocity dispersions and, in many cases, triaxial structure provide support. Lower-luminosity elliptical galaxies are more commonly rotating, disky systems near the isotropic reference. Classical galactic bulges of spiral galaxies often overlap this rotating-spheroid locus; disk-like bulges can be more strongly rotation supported. These are population trends with substantial overlap, not universal rules for individual galaxies.

Observations usually use projected ellipticity and a line-of-sight rotation measure such as peak speed divided by a central velocity dispersion. The intrinsic $e$ and global dispersions used in the derivation are different quantities. Inclination, apertures, and unresolved streaming can change both coordinates. Consequently a point below the reference suggests extra random-motion support, but does not uniquely measure $\beta$.

The stated kinematics do not, by themselves, fix a galaxy's intrinsic axis ratio. The requested curve is the usual simplified oblate-rotator flattening proxy; its additional geometric and streaming conventions should be explicit. Put $q=b/a=1-e$ and adopt the approximate moment prescriptionHere $v_p^2$ is the ordered support assigned to one in-plane component. Using ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$ and solving givesIn this prescription, isotropic random motions give the oblate isotropic rotator curve; positive $\beta$ allows flattening with less rotation. A real solution requires $0\le e<1$, $\beta <1$, and $e\ge \beta$. Equality $e=\beta$ gives no rotation. This is a useful diagnostic approximation, not an exact identity for arbitrary density distributions.

To see what the tensor virial theorem actually requires, take a finite stationary axisymmetric system with mass-weighted constant velocity dispersions and azimuthal streaming speed $v_{\mathrm{az}}$. Since $u_x=-v_{\mathrm{az}}\sin \varphi$, the azimuthal average has $⟨u_x^2⟩=v_{\mathrm{az}}^2/2$. The $xx$ and $zz$ components giveand therefore, with the oblate tensor-virial shape factor $Q=W_{xx}/W_{zz}$,For similar oblate self-gravitating density surfaces, write $s=\sqrt{1-q^2}$. The ellipsoidal force coefficients, obtained by integrating the Newtonian force over those surfaces, areThe common radial integral in the stellar potential-energy tensor cancels, but these shape coefficients remain. At $q=1/2$, $Q\simeq 1.79$, rather than $1/q=2$. With $\beta =0$, the intrinsic azimuthal ratio is about $1.26$, whereas the printed simplified ratio is $1$. Recovering the latter uses both $v_p=v_{\mathrm{az}}/\sqrt{2}$ and $Q\simeq 1/q$. Thus, if the printed $v$ literally means intrinsic azimuthal speed, its claimed exact relation omits necessary model assumptions and a streaming normalization. The next sketch displays the requested approximate curve with that qualification.

The relevant comparison is continuing hierarchical galaxy formation in the Lambda-CDM model against an idealized monolithic galaxy formation model in which both formation and assembly were essentially complete early. Each observation below concerns a population or a fossil record, rather than a currently merging pair. None individually proves that every galaxy formed through the same route.

Fossil substructure in stellar halos. Old stars in the Milky Way's stellar halo include coherent groups in orbital-integral space, and external halos contain extended debris and shells. Phase mixing can erase spatial concentration while retaining correlations of orbital energy or angular momentum. Such stellar streams, when linked to a disrupted dwarf progenitor, record accretion of an already formed stellar population. A smooth single collapse followed by quiet passive evolution has no equally direct mechanism for creating several dynamically and chemically distinct accreted populations. This supports a composite assembly history, without requiring that every halo star was accreted or counting a current merging pair as evidence.

Growth of the quiescent population. The galaxy red sequence contains old stellar populations, yet its inferred stellar mass density grows between approximately redshift one and the present. Pure fading of an already fixed early population changes luminosities and colours, but cannot increase the mass in that population. Quenching adds newly passive systems, while subsequent assembly combines older stellar populations into larger galaxies. The observed evolution therefore requires demographic evolution after the putative completed collapse epoch. It is consistent with hierarchical galaxy formation plus quenching, although red-sequence growth alone does not distinguish assembly from all possible histories of in-situ star formation; stellar-mass estimates and survey selection must be treated consistently.

Evolution of galaxy morphology. Distant galaxy samples contain a larger prevalence of irregular and peculiar structures than a simple non-evolving extrapolation of the local population. Settled discs and smooth spheroids become common outcomes as time proceeds. Continued gas supply, dynamical interactions, and internal rearrangement in hierarchical galaxy formation naturally allow this galaxy morphological evolution. A model in which structure became fixed during one early collapse and then only stellar light faded has difficulty producing the population change. Rest-frame wavelength, resolution, and surface-brightness selection can affect the observed classifications, and clumpiness is not unique evidence for a galaxy merger: unstable gas-rich discs can also be irregular.

Evolution of sizes at fixed stellar mass. Massive high-redshift galaxies, including concentrated systems, are often more compact than comparable populations at low redshift. Galaxy size evolution implies structural growth beyond passive luminosity fading. In particular, deposition of accreted stars at large radii in minor galaxy mergers can enlarge the outer stellar body while preserving an old central stellar population; gas accretion can grow discs from the inside outward. A completed early collapse with a fixed stellar structure does not naturally supply those new outskirts. Changes in sample membership and expansion associated with stellar mass loss also contribute, so this is evidence requiring continued structural evolution rather than a unique quantitative measurement of a merger rate.

Diffuse intracluster stars. Intracluster light and extended outer envelopes show that some stars no longer belong to individual cluster galaxies. Their irregular distribution and substructure retain evidence of redistribution within an assembling cluster. Tidal stripping and disruption of member systems supply stars to the common environment, as expected when smaller bound systems are incorporated into a growing hierarchy. Passive evolution of independent galaxies formed in isolated early collapses cannot itself move their stars into the space between galaxies. Early stellar formation followed by later stripping remains possible, but admitting that later dynamical assembly already departs from the strict completed-collapse picture. This evidence concerns the cumulative stellar distribution, not merely the existence of an occasional present-day merging pair.

The neutral-hydrogen column density is $N_{\mathrm{HI}}=\int n_{\mathrm{HI}}d\ell$, with units ${\mathrm{cm}}^{-2}$. Define the neutral-hydrogen column-density distribution byHere $\mathcal{N}$ counts absorption systems along a quasar sightline. A histogram per logarithmic column interval instead measures $N\ln (10)f(N,z)$; forgetting this factor changes the inferred slope. Another common convention uses the absorption distance $X$, with $dX/dz=H_0(1+z{)}^2/H(z)$, and $f_X=f/(dX/dz)$.

At a fixed epoch, weak Lyman-alpha forest lines have an approximately falling power-law distribution, $f(N,z)\simeq A(z)N^{-\beta }$ with $\beta \simeq 1.4$–$1.5$ over representative low-column ranges. Stronger absorbers deviate from one universal slope, and the normalization and fitted slopes depend on epoch and sample. The low-column fit is measured, for example, in Kim and collaborators' forest survey. The sketch distinguishes that empirical low-column trend from an illustrative higher-column continuation.

Ordinary Lyman-alpha forest lines cover roughly $10^{12}\lesssim N_{\mathrm{HI}}\lesssim 10^{17.2}{\mathrm{cm}}^{-2}$. They are numerous resonant Lyman-alpha absorption features from intervening gas at different cosmological redshifts, collectively resembling a forest in a quasar spectrum. Many are optically thin at the ionization edge even when their line centers are saturated.

A Lyman limit system becomes optically thick at the $91.2\mathrm{nm}$ hydrogen ionization edge. With ${\sigma }_{912}\simeq 6.3\times 10^{-18}{\mathrm{cm}}^2$, ${\tau }_{912}=N_{\mathrm{HI}}{\sigma }_{912}\gtrsim 1$ requiresIts spectrum consequently has a strong continuum decrement below the redshifted Lyman edge. In a classification separating the highest columns, Lyman limit systems extend up to the damped-system boundary.

A damped Lyman-alpha system conventionally has $N_{\mathrm{HI}}\gtrsim 2\times 10^{20}{\mathrm{cm}}^{-2}$. Its Lyman-alpha absorption is sufficiently strong that the Lorentzian wings of the Voigt profile are prominent. “Damped” refers to the finite radiative lifetime of the atomic excited state, not an inference that the gas has a high temperature. These systems trace substantial neutral reservoirs, usually associated with galaxies.

A typical globular cluster has undergone appreciable two-body relaxation, since its stellar relaxation time is $t_{\mathrm{rel}}\sim 0.1(N/\ln N)t_{\mathrm{cross}}$: for $N\sim 10^5$ and stellar crossing time $t_{\mathrm{cross}}\sim 10^6\mathrm{yr}$ this is about $10^9\mathrm{yr}$, shorter than its age, allowing a relatively relaxed, tidally limited structure. A galaxy with $N\sim 10^{11}$ and $t_{\mathrm{cross}}\sim 10^8\mathrm{yr}$ instead has $t_{\mathrm{rel}}\sim 4\times 10^{16}\mathrm{yr}$ and behaves as a collisionless stellar system, retaining differences from formation, galaxy mergers, and star formation. Relaxation erases more memory in clusters than in galaxies, although real globular clusters also differ in concentration and evolutionary state, so structural similarity is approximate.

A flux-limited galaxy selection samples a much larger volume for bright galaxies: in the nearby Euclidean limit $d_{\max }\propto \sqrt{L}$, hence $V_{\max }\propto L^{3/2}$, so very numerous faint dwarf galaxies are usually detectable only nearby. With a Schechter function $\varphi (L)\propto L^{{\alpha }_S}{e}^{-L/L_{\ast }}$, detected counts per logarithmic luminosity interval scale as $L^{{\alpha }_S+5/2}{e}^{-L/L_{\ast }}$ and, for ${\alpha }_S\sim -1$ to $-1.3$, peak near $L\sim (1.2\text{–}1.5)L_{\ast }$ rather than at the numerous faint end. The competition between increasing survey volume and the bright-end cutoff concentrates detected luminosities near $L_{\ast }$, with the precise width depending on the selection and survey depth.

Using the given constants, the mass is $2\times 10^{41}\mathrm{kg}$ and the velocity dispersion is $2\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The nominal galactic dynamical time isThus the collapse clock is a few tens of millions of years, of order $10^8$ years, with $R/\sigma \simeq 3\times 10^7$ years in the interior singular isothermal sphere convention. The nominal radius $GM/{\sigma }^2\simeq 3.5\times 10^{20}\mathrm{m}\simeq 12\mathrm{kpc}$ is a plausible galactic scale. A full stellar orbit takes several crossing times, giving order $10^8$ years near a galactic disc radius, whereas the age of the Milky Way is of order $10^{10}$ years. Many orbital passages therefore fit into the lifetime of the galaxy, as expected; a galactic dynamical time need not equal either its full orbital period or its age.