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Normalize the linear growth factor to $D(0)=1$. The collapse overdensity at the collapse epoch is ${\delta }_{\mathrm{sc}}\simeq 1.686$ for the matter-dominated spherical-collapse model. The present-extrapolated spherical-collapse barrier isIt is the initial linear overdensity, extrapolated to today, required to collapse by $t$; it is not the nonlinear density contrast of a virialized halo. The smoothed matter density variance ${\sigma }^2(M,z)$ is the variance of the linear density contrast smoothed on a Lagrangian comoving scale containing mass $M$. For a spherical top-hat filter,Consequently $\sigma (M,z)=D(z)\sigma (M,0)$ for scale-independent linear growth. The equivalent halo peak height conventions are $\nu ={\delta }_c(t)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$; using both an evolved barrier and an evolved variance would count growth twice.

To calculate a number-density growth time, use a narrow fixed-mass bin, not the collapsed mass fraction itself. Differentiate the Press-Schechter formalism mass fraction and divide the mass density in the resulting interval by $M$. This gives the Press-Schechter halo mass functionAt fixed $M$, the mass factor and logarithmic slope do not depend on time. Since $\dot{\nu }/\nu =-\dot{D}/D$, the Press-Schechter abundance growth at fixed mass isThe prefactor $\nu$ matters here: the exponential-only rare-peak approximation drops the minus one and is accurate only for $\nu \gg 1$.

Read the supplied variance relation as a mass-scale calibration using the numerical velocity label given for the selected population. It gives $\sigma (M,3)=0.85\sqrt{250/10}=4.25$. Matter domination between the two high-redshift epochs gives $D(19)/D(3)\simeq 4/20$, so $\sigma (M,19)\simeq 0.85$ and $\nu \simeq 1.984$. At the epoch in question,Thus the requested fixed-mass-bin growth estimate with that calibration isAn exponential-only approximation gives about $8.1\times 10^7\mathrm{yr}$, which is somewhat shorter because this is only a roughly two-sigma population.

The wording leaves two sample conventions worth distinguishing. First, an actual virial velocity of a spherical-overdensity halo is epoch-dependent at fixed mass. If the variance fit is instead calibrated using physical virial velocities at redshift three, the same mass whose velocity is $10\mathrm{km}{\mathrm{s}}^{-1}$ at redshift nineteen has $v(3)\simeq 10\sqrt{4/20}=4.47\mathrm{km}{\mathrm{s}}^{-1}$. The halo virial-velocity conversion between epochs then gives $\sigma (M,19)\simeq 1.271$, $\nu \simeq 1.327$ and $t_d\simeq 4.2\times 10^8\mathrm{yr}$ for a fixed-mass bin. The two numerical answers reflect the velocity-label convention in the supplied fit, not two ways of differentiating one fixed fit.

Second, a cumulative number density is $n(>M)={\int }_M^{\infty }(dn/d{M}^{\prime })d{M}^{\prime }$, not simply ${\bar{\rho }}_{m,0}f(>M)/M$: the latter is a mass fraction divided by a threshold mass, not the number of objects. The cumulative derivative is an abundance-weighted average of $(\nu (M^{\prime }{)}^2-1)\dot{D}/D$ over that integral. If the supplied power-law variance fit is extended over all larger masses, with the same velocity-label convention, direct integration gives a cumulative-number growth time of about $8.4\times 10^7\mathrm{yr}$ instead. A sample maintained at fixed physical velocity at successive epochs additionally moves its mass boundary and needs a selection convention. The numerical estimate above explicitly uses the ordinary fixed-mass differential interpretation. These distinctions are important when an exact growth time rather than a rare-tail estimate is intended.

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.