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The baryon conversion efficiency of a halo is the fraction of its baryonic mass incorporated into stars. If the halo retains the cosmic baryon fraction $f_b$, its stellar-to-total-mass ratio is $f_b{f}_{\ast }$. This efficiency depends on halo mass, epoch, gas supply and feedback and must not be confused with a star-formation efficiency per free-fall time.

If a measured cluster baryon fraction represents the cosmic baryon fraction ${\Omega }_b/{\Omega }_m$, combining it with the cosmological baryon density inferred from Big Bang nucleosynthesis gives the displayed estimate. Gas and stellar baryons must both be counted, and all distance or Hubble constant dependence must be consistent. Baryon depletion or feedback modifies the identification: with $f_{b,\mathrm{cluster}}=b{\Omega }_b/{\Omega }_m$, the corrected answer is ${\Omega }_m=b{\Omega }_b/f_{b,\mathrm{cluster}}$. The method measures total matter rather than vacuum or radiation density.

The cosmic baryon fraction is the ratio of mean baryonic matter density to total matter density. Assigning this fraction to a halo is a simplifying assumption; cooling, ejection and other baryonic processes can change the halo's retained gas and stellar fractions. A baryon conversion efficiency becomes a stellar-to-total-halo mass ratio only after multiplying by the applicable baryon fraction.

The integrated halo star-formation efficiency is the fraction of a dark matter halo's reference baryonic mass converted into stars: ${ϵ}_{\ast }=M_{\ast }/(f_b{M}_h)$, with cosmic baryon fraction $f_b$. It is distinct from star-formation efficiency per free-fall time. Cooling, gas supply and retention, stellar feedback and active-galactic-nucleus feedback make it strongly dependent on dark matter halo mass. With stellar mass-to-light ratio ${{\rm Y}}_{\ast }$, a simple central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. Changing this efficiency changes both the mapping and the logarithmic Jacobian between the dark matter halo mass function and the luminosity function.

Measure the cluster's baryonic mass, dominated by its intracluster medium with a smaller stellar contribution, and divide it by an independently inferred total mass. If this cluster baryon fraction $f_b$ is representative of the cosmic baryon fraction, then $f_b={\Omega }_b/{\Omega }_m$. Thus the cluster baryon-fraction estimate of matter density isBig Bang nucleosynthesis constrains the baryon-to-photon ratio and hence ${\Omega }_b{h}^2$ through primordial light-element abundances. A specified $h$ converts this to ${\Omega }_b$; the cluster mass and gas-mass estimates also have distance-dependent $h$ factors, which must be used consistently. This method estimates the total nonrelativistic matter density, including baryonic and dark matter. It does not add radiation or vacuum energy to that inferred ${\Omega }_m$; in a matter-dominated interpretation the question's $\Omega$ is this matter parameter.

Primordial deuterium as a baryon-density indicator works because increasing baryon density makes the destruction of deuterium into heavier nuclei more efficient during Big Bang nucleosynthesis. Smaller inferred primordial D/H therefore implies larger ${\Omega }_b$, with the other nucleosynthesis assumptions fixed. For a fixed measured cluster baryon fraction, a smaller primordial deuterium abundance raises the inferred matter density. A low abundance caused instead by stellar processing is not a lower primordial abundance; depletion, gas ejection and the representativeness of clusters must also be assessed before identifying the cluster fraction with the cosmic one.

A genuine nondegenerate bounce has ${\dot{a}}_{\ast }=0$ and ${\ddot{a}}_{\ast }>0$; using ${\ddot{a}}_{\ast }\ge 0$ also includes the limiting stationary case. Put $s=a_{\ast }^{-1}=1+z_{\ast }$, and define the present matter density parameter ${\Omega }_m=8\pi G{\rho }_0/(3H_0^2)$, ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)$ and ${\Omega }_k=-k/H_0^2$. The dust-plus-cosmological-constant Friedmann equation and its acceleration equation becomeAt the bounce these giveFor a past bounce with $s>1$, eliminate ${\Omega }_k$ from the first equation:Subtracting ${\Omega }_m{s}^3/2$ and multiplying by the positive denominator givesThus the matter bound from a Friedmann bounce isStrict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.

An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_{\ast }\ge 6$. The denominator of the bound increases with positive $z_{\ast }$, soThe observational lower bound is much larger. Cluster mass-to-light ratios, inferred independently from motions, gas temperatures and gravitational lensing, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the cosmic baryon fraction from representative clusters, combined with the baryon cosmological density parameter inferred from Big Bang nucleosynthesis, provide another estimate: ${\Omega }_m\simeq {\Omega }_b/f_b$. Large-scale clustering and peculiar velocities provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for ${\Omega }_m\gtrsim 0.2$.

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.

Objects near the derived cutoff are too faint for their absence in ordinary star counts alone to exclude a massive dark matter halo. The decisive test is gravitational microlensing: compact objects passing in front of background stars produce transient magnification even if they emit no detectable light. For a lens at distance $D_l$ and a source at $D_s$, the physical Einstein radius and event timescale areFor $D_l=10\mathrm{kpc}$, $D_s=50\mathrm{kpc}$ and $v_{\perp }=200\mathrm{km}{\mathrm{s}}^{-1}$, a $10^{-3}{M}_{\odot }$ lens gives $t_E\simeq 2.2$ days, within the regime tested by suitably sampled gravitational microlensing surveys. The microlensing optical depth isso at fixed compact-object mass density it is independent of the individual lens mass: reducing individual mass changes the event durations rather than hiding the total lensing probability.

EROS observations of the Magellanic Clouds found far fewer candidate events than a halo dominated by compact objects would predict and excluded such objects as the principal Galactic halo component over $6\times 10^{-8}<m/M_{\odot }<15$, under their tested halo models. Their often-quoted eight-percent bound is specifically near $0.4M_{\odot }$, not a uniform bound at every lens mass. A halo dominated by the approximately Jupiter-mass objects needed here is incompatible with those microlensing constraints. Deep optical and infrared counts also restrict a large population of luminous low-mass stars and brown dwarfs, while the cosmic baryon fraction provides a separate constraint on explaining all cosmological dark matter with baryonic objects.

Take the retained disc mass to be the cosmic baryon fraction of the total dark matter halo mass: $M_d=f_b{M}_h$, where $f_b=0.05/0.25=0.2$. Adopt $v_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$ and describe the disc gravity by $v_d^2=C_{d}G{M}_d/R_d$. The structure coefficient $C_d$ must be stated: the scale-free Mestel disc gives $C_d=1$ when its flat rotation speed and enclosed mass are used, while a finite disc requires boundary and thickness information. The equality of edge specific angular momentum gives $R_d{v}_d=q{r}_{\mathrm{vir}}{v}_{\mathrm{vir}}$, with $q=0.1$. Combining these relations gives the angular-momentum-conserving self-gravitating disc radius:With the commonly intended $C_d\simeq 1$ approximation,The corresponding masses are $M_h\simeq 2.49\times 10^{11}{M}_{\odot }$ and $M_d\simeq 4.97\times 10^{10}{M}_{\odot }$, where $M_{\odot }$ is the solar mass.

For a definite formation estimate, assume the dark matter halo acquired this mass and radius at virial equilibrium, retained them afterward, and had mean density ${\Delta }_v{\bar{\rho }}_m(z_f)$ with ${\Delta }_v=18{\pi }^2$. This uses the matter-dominated spherical-collapse model as an approximation to the specified matter-plus-vacuum cosmology. Since ${\bar{\rho }}_m(z_f)=3H_0^2{\Omega }_{m0}(1+z_f{)}^3/(8\pi G)$,The approximate distance therefore gives formation modestly before observation. Using the numerical rather than approximate distance integral gives $z_f\simeq 1.22$, within the accuracy of these structural assumptions. The calculation also assumes inclination-corrected spectroscopy, negligible disc pressure support, conserved edge specific angular momentum, no substantial later accretion or mergers, and no significant baryon loss.

This formation redshift is model-dependent, not uniquely fixed by the stated disc data. Keeping $C_d$ explicit gives $1+z_f\simeq 2.209C_d^{-4/3}({\Delta }_v/18{\pi }^2{)}^{-1/3}$. In particular, replacing the mean-density convention by $200$ times the critical density gives $H(z_f)=v_{\mathrm{vir}}/(10r_{\mathrm{vir}})$ and $z_f\simeq 0.87$, later than the observation. That inconsistent chronological result cannot be silently used as the formation epoch.

There is a further idealization in applying $C_d=1$ exactly at the edge: an abruptly truncated razor-thin disc with nonzero edge surface density of a disk has a logarithmically divergent in-plane force at that edge. Locally the available mass occupies a half-plane, and its radial contribution is proportional to $G{\Sigma }_d{\int }_{ϵ}^{\ell }ds/s$, with no opposite exterior half-plane to cancel it. Finite thickness or a smooth taper removes that sharp-edge force singularity of a thin disk. Thus $C_d=1$ is an explicitly qualified Mestel disc or monopole-scale approximation, not an exact finite-disc calculation; enclosed mass does not determine a disc rotation curve.

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ a comoving smoothing radius and $W$ the top-hat filter in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and dark-matter halo mergers. This statistical growth from smaller bound systems to larger ones is hierarchical galaxy formation. It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.

For linear modes the linear growth factor gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with galaxy bias; it should not be equated directly with an unbiased matter measurement.

The broad linear shape was set by the cosmological transfer function before and around matter-radiation equality, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The cold-dark-matter transfer function behaves approximately as $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$:The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for hierarchical galaxy formation.

Cold dark matter has negligible primordial thermal velocities and a very short collisionless free streaming length on galactic scales. Warm dark matter has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its cosmological transfer function is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless dark matter develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound dark matter halo. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.

The unheaded request about baryon conversion efficiency of a halo is also answered here. Define $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ using the cosmic baryon fraction $f_b$. In small halos, shallow potentials let stellar feedback drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\mathrm{vir}}^2$. Photoheating during reionization also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher virial temperature and lower cooling efficiency let a substantial hot atmosphere persist. Active-galactic-nucleus feedback can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass. Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.

Use the supplied cosmological cosmic baryon fraction $f_b={\Omega }_{\mathrm{bar}}/{\Omega }_{\mathrm{mat}}=0.2$, and assume the halo initially has that fraction. Its total baryon mass is then $M_{b,\mathrm{tot}}=6\times 10^{11}{M}_{\odot }$. The selected baryons are the lowest-angular-momentum fractionHere the given cumulative specific angular momentum distribution must be normalized separately for the baryon component: equal baryon and dark-matter distributions mean equal normalized fractions, not equal absolute masses.

The virial velocity and the largest specific angular momentum in the selected inner baryon population areEstimate the outer radius of the settled low-angular-momentum component using circular rotational support and an enclosed-mass approximation to its self-gravity. With negligible dark matter inside the component, $v_d^2\simeq G{M}_b/r_d$ and $j_d=r_d{v}_d$, soThe given rounded gravitational constant produces the same estimate. The very small radius follows from selecting a small low-$j$ fraction, rather than assigning all central baryons the halo-edge angular momentum.

These estimates assume that gas radiates energy, preserves each parcel's specific angular momentum, and settles with negligible pressure support and no strong redistribution or cancellation of its angular-momentum vectors. They also assume that the phrase “innermost baryons” selects the lowest-$j$ material, the original cosmic baryon supply is retained, and the central gas supplies the dominant gravity. A possible central black hole or an exact flattened disk potential changes the numerical coefficient; the stated baryonic mass is used for this estimate.

The low-angular-momentum baryonic disk estimate uses a cumulative distribution uniform in $j$ from zero to $j_d$, so its mean is $j_d/2$. The quoted $r_d$ is the outer radius based on the cutoff angular momentum, not a one-zone radius based on the mean. Within the same enclosed-mass approximation, $M(<j)=M_{b}j/j_d$ and circular balance give $r(j)=j{j}_d/(G{M}_b)$: the rotation curve is approximately flat and the half-mass radius is $r_d/2$. Treating every baryon as one shell with the mean $j$ would instead give $r_d/4$ and twice the velocity, a different radius convention rather than the outer edge of the supplied distribution.

Let $M_h,R_h$ describe the halo and let $v=v_{\mathrm{vir}}$. The cosmic baryon fraction is $f_b={\Omega }_b/{\Omega }_m=0.20$, so the specified settled fraction gives $M_d=0.10M_h$. The specific angular momentum constraint at the disk edge isTo infer an actual circular speed one must specify the disk's radial mass distribution: enclosed mass does not determine a disc rotation curve. Introduce a finite geometry coefficient $\kappa$ by $v_d^2=\kappa G{M}_d/R_d$. A rounded, centrally concentrated disk permits a monopole estimate $\kappa \simeq 1$ at its outer edge; it is an explicit approximation, not the spherical shell theorem applied exactly to a razor-thin disk. Ignore the force of the unsettled baryons as well as the dark matter within the disk. Using $v^2=G{M}_h/R_h$, the angular-momentum estimate of a self-gravitating galactic disk givesAn exact disk answer cannot be fixed by total disk mass alone; the dependence on $\kappa$ records that missing input.

For the numerical virial radius of a dark-matter halo, adopt mean density ${\Delta }_c=200$ times the critical density at the formation epoch. This conventional definition givesThe supplied expansion law gives $H(3)=286.49\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, and hence $R_h=v/(10H)=97.74\mathrm{kpc}$. Taking $\kappa =1$ yieldsThe corresponding virial mass of a dark-matter halo is $M_h\simeq 1.78\times 10^{12}{M}_{\odot }$, and $M_d\simeq 1.78\times 10^{11}{M}_{\odot }$. Other overdensity conventions give $R_d\propto {\Delta }_c^{-1/2}$ at fixed $H,v,\kappa$.

Interpret the wavelength separation as an observed-frame local mean near the redshift in question. Since Lyman-alpha absorption appears at ${\lambda }_{\mathrm{obs}}={\lambda }_{\alpha }(1+z)$, the incidence isAssume one counted absorption system per intercepted disk, no unrelated forest systems or missed absorbers, unity neutral covering fraction, and a locally slowly varying population. Let ${\sigma }_{\mathrm{abs}}$ be the proper interception cross-section and $n_c$ the comoving number density. The proper density is $n_p=n_c(1+z{)}^3$, and the proper line element along the light path is $d{l}_p=cdz/[H(z)(1+z)]$. Thus the absorber incidence and comoving number density relation isFor a thin circular disk, the projected area is $\pi R_d^2|\cos i|$. Isotropically oriented normals have $⟨|\cos i|⟩=1/2$, so the random-orientation absorbing-disk cross-section is ${\sigma }_{\mathrm{abs}}=\pi R_d^2/2$. ConsequentlyIf all disks are taken face-on instead, $n_c\simeq 0.398{\mathrm{Mpc}}^{-3}$. Generally the random-orientation result scales as ${\kappa }^2({\Delta }_c/200)$ and is divided by the neutral covering fraction. The finite mean redshift spacing is sizable, so a precision inference would integrate the incidence over the actual redshift interval rather than identify it with one local value.

There is also a halo abundance mass-budget bound on this formal result. The present mean matter density for these parameters is ${\bar{\rho }}_{m,0}={\Omega }_{m}3H_0^2/(8\pi G)\simeq 3.40\times 10^{10}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Distinct haloes of this mass cannot have $n_c{M}_h>{\bar{\rho }}_{m,0}$, even if every matter particle belonged to them. Yet the inferred random-orientation population hasEven the face-on estimate exceeds this bound by about twenty-one. The numerical incidence result is conditional; the supplied population assumptions are not cosmologically consistent under this standard virial and compact-disk estimate. A larger neutral-gas absorption radius, a different absorber population, or different physical inputs are needed. For example, random orientations would require an absorbing radius at least about $63\mathrm{kpc}$ merely to reach the all-matter upper bound, much larger than the calculated centrifugal radius. This check does not change the algebraic answer, but prevents treating it as a realizable population.

Treat the features in the QSO spectrum as Lyman-alpha absorption and use the cosmological wavelength-redshift relation to convert their spacing into $\Delta z=304/1216=0.25$. In the specified flat matter-plus-cosmological-constant model,A local radial interval has comoving length $d\chi =cdz/H(z)$ and proper length $dl=d\chi /(1+z)$. Evaluating the mean interval locally at $z=3$ gives the proper absorber separation from a wavelength intervalThe corresponding comoving interval is about $262\mathrm{Mpc}$. We neglect peculiar-velocity shifts and treat $H$ as approximately constant over one mean interval.

The proper number density is $n_p=(1+z{)}^3{n}_{\mathrm{gal}}=179.2\mathrm{Mp}{\mathrm{c}}^{-3}$. To compute the mean projected area of a randomly oriented thin disc, note that a disc whose normal makes angle $i$ to the sightline has projected area $\pi R_d^2|\cos i|$. Isotropic normals haveFor independent intersections with fully covering thin discs, the mean free path through randomly oriented disc absorbers is ${\ell }^{-1}=n_p\pi R_d^2/2$. ThusAll radii here are proper radii at the absorber epoch. The projected-area average is over the underlying orientation distribution; it already accounts for the larger intersection probability of face-on discs.

The cosmic baryon fraction is $f_b={\Omega }_b/{\Omega }_m=0.2$, so the disc mass is $M_d=f_d{M}_h$ with $f_d=f_b/4=0.05$. Define $v_{\mathrm{vir}}^2=G{M}_h/R_{\mathrm{vir}}$. Disc gravity depends on its radial mass profile, which has not been specified. Parameterize its edge circular speed by $v_d^2=C_{d}G{M}_d/R_d$ and use $C_d=1$ for a centrally concentrated disc in a monopole approximation at its outer edge. The angular-momentum condition givesCombining the equations yields the angular-momentum-conserving self-gravitating disc radiusFor the stated $C_d=1$ choice, $R_{\mathrm{vir}}\simeq 36.8\mathrm{kpc}$ and $v_d=0.5v_{\mathrm{vir}}$. The factor $C_d$ makes explicit the otherwise undetermined disc-profile dependence; an enclosed-mass formula is not exact for an arbitrary flattened disc.

To determine the halo speed, a virial-density convention is also needed. Choose a spherical overdensity ${\Delta }_v=200$ relative to the critical density at $z=3$:The virial velocity of a spherical-overdensity halo then obeysThe nearly matter-dominated spherical-collapse choice ${\Delta }_v\simeq 18{\pi }^2$ instead gives about $99.5\mathrm{km}{\mathrm{s}}^{-1}$. More generally the numerical radius and velocity scale respectively as $C_d$ and $C_d\sqrt{{\Delta }_v/200}$. With $G\simeq 4.30\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$, the adopted model has $M_h\simeq 9.5\times 10^{10}{M}_{\odot }$ and $M_d\simeq 4.8\times 10^9{M}_{\odot }$; these reproduce the edge speed and the specified angular momentum.

The calculation assumes homogeneous, unclustered intersections, one detectable absorption feature per disc crossing, unit neutral-gas covering factor, isotropic orientations, universal halo baryon fraction and conservation of the specified edge specific angular momentum. It assumes the halo is virialized at the absorber epoch and neglects its inner gravity as instructed. The extra profile and overdensity choices are necessary to turn the geometric and angular-momentum constraints into unique numerical halo properties.

The gravitational lensing mass includes dark matter, hot intracluster gas, stars, and any other material in the projected aperture. Most is not stellar: in the literal convention above the stellar fraction is only $2.22\times 10^{12}/(5.89\times 10^{14})\simeq 0.00376$.

Assume the cluster shares the cosmic cosmic baryon fraction, $f_b={\Omega }_b/{\Omega }_m=1/6$, and compare the same aperture for all components. ThenNeglecting stars entirely would give $9.82\times 10^{13}{M}_{\odot }$. Thus the assumed hot gas dominates the baryonic matter, and the remainder $M_{\mathrm{DM}}\simeq (5/6)M_E=4.91\times 10^{14}{M}_{\odot }$ is dark matter. The budget is consistent: stars are about $2.26\%$ of the inferred baryonic matter, and all components sum to $M_E$.

For a thermal scale, take gas extent $R_g\simeq 2R_E=465\mathrm{kpc}$ and mean particle mass $\mu m_p$ with $\mu \simeq 0.61$ for ionized gas. A simple spherical virial theorem estimate with gas binding energy $W_g\sim -G{M}_E{M}_g/R_g$ and thermal kinetic energy $K_g=3M_g{k}_{B}T/(2\mu m_p)$ givesThis is the scale of a very hot, massive X-ray emitting galaxy cluster. The numerical prefactor depends on the radial mass density profile, gas distribution and boundary pressure; for example an isothermal-sphere estimate $k_{B}T\sim \mu m_{p}GM/(2R_g)$ gives about $2.0\times 10^8\mathrm{K}$. Moreover $M_E$ is a projected aperture mass at $R_E$, whereas the estimate needs a spherical enclosed mass at $2R_E$; the given data do not determine that conversion. The quoted temperature is therefore an order-of-magnitude estimate, not a uniquely measured value. Nor do the data determine the total gas mass outside the Einstein ring without a profile. The gas budget above pertains to the matched aperture under the cosmic-fraction assumption.