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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.