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If a halo retains the mean density acquired at collapse, $3M/(4\pi R^3)={\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ estimates its formation redshift. In the non-dissipative spherical-collapse model during matter domination, ${\Delta }_{\mathrm{vir}}=18{\pi }^2$. Subsequent mass growth, profile evolution and the reference-density convention limit this inference.

Use conservation of energy for an isolated spherical perturbation of fixed mass $M$, with negligible initial random motion and no significant surface-pressure term. For a homologous density profile its Newtonian gravitational potential energy is $U=-\alpha G{M}^2/R$, with the same positive structure coefficient $\alpha$ at both stages; a uniform sphere has $\alpha =3/5$.

At spherical-collapse turnaround, coherent radial motion momentarily vanishes, so $E=U_{\max }=-\alpha G{M}^2/R_{\max }$. In the final virialized state, the virial theorem gives $2T_{\mathrm{vir}}+U_{\mathrm{vir}}=0$, hence $E=T_{\mathrm{vir}}+U_{\mathrm{vir}}=U_{\mathrm{vir}}/2$. ThereforeThis is the virial radius from turnaround energy in the spherical-collapse model. It is an idealized virial-radius estimate rather than the radius of the formal pressureless point collapse. If the profile coefficient changes, the same argument instead gives $R_{\mathrm{vir}}/R_{\max }={\alpha }_{\mathrm{vir}}/(2{\alpha }_{\max })$. Significant energy loss, mass loss or boundary work also changes the result.

Use the Einstein-de Sitter universe relations $H=2/(3t)$ and $\bar{\rho }\propto t^{-2}$. Write $H_{\mathrm{ta}}=H(t_{\mathrm{ta}})$ and $R_{\mathrm{ta}}=R(t_{\mathrm{ta}})$. Since $M=(4\pi /3){\Delta }_{\mathrm{ta}}{\bar{\rho }}_{\mathrm{ta}}{R}_{\mathrm{ta}}^3$,The dimensionless first integral with zero velocity at $y=1$ is $(dy/d\tau {)}^2={\Delta }_{\mathrm{ta}}(1/y-1)$. On the expanding branch this givesEvaluating the integral gives the inverse-trigonometric expression in the source. Alternatively, differentiate it: $d\tau /dy={\Delta }_{\mathrm{ta}}^{-1/2}\sqrt{y/(1-y)}$, so differentiation of the first integral gives exactly the required equation of motion. At $y=1$, the integral equals $\pi /2$. Thus ${\tau }_{\mathrm{ta}}=2/3$ fixes $\sqrt{{\Delta }_{\mathrm{ta}}}=3\pi /4$. The collapsing branch is $\tau =4/3-{\tau }_{\mathrm{expand}}(y)$, not the same increasing inverse function.

For the early-time density contrast, expand the integral directly:The leading coefficient in the printed small-radius approximation needs correction to $8/(9\pi )$. The printed coefficient is inconsistent with both the exact integral and approach to the homogeneous background. Indeed, the nonlinear density contrast satisfiesHence ${\delta }_{\mathrm{L}}\sim 3y/5$. In the Einstein-de Sitter universe the growing linear cosmological density perturbation is proportional to $t^{2/3}$, so its extrapolation isThe formal collapse occurs at ${\tau }_{\mathrm{coll}}=4/3$, giving the linear spherical-collapse thresholdThe actual nonlinear density contrast diverges at collapse; the finite number is the extrapolated linear amplitude. Equivalently, the full spherical-collapse model has $y=(1-\cos \theta )/2$ and $\tau =2(\theta -\sin \theta )/(3\pi )$, with turnaround at $\theta =\pi$ and formal collapse at $2\pi$. Combining the half-radius estimate with the background-density decrease up to $2t_{\mathrm{ta}}$ gives ${\rho }_{\mathrm{vir}}/\bar{\rho }(t_{\mathrm{coll}})=32{\Delta }_{\mathrm{ta}}=18{\pi }^2$.

In the spherical-collapse model, a bound overdense region initially expands with the background, but its additional Newtonian gravity slows that expansion. Its outer shell reaches a maximum turnaround radius, where its radial velocity is zero, and then falls inward. A perfectly spherical pressureless solution formally collapses to zero radius. In a realistic collisionless system, deviations from that idealization, shell crossing and violent relaxation convert coherent infall into random motions, producing a roughly stationary bound object.

Let the Newtonian gravitational potential energy of the homologous sphere be $U=-aG{M}^2/R$, where $a=3/5$ for a uniform sphere. At spherical-collapse turnaround, the kinetic energy of the bulk motion vanishes, so $E=U_{\mathrm{ta}}$. For the settled object, the virial theorem gives $2K_{\mathrm{vir}}+U_{\mathrm{vir}}=0$ and hence $E=U_{\mathrm{vir}}/2$. If energy and mass are conserved and the structure coefficient is unchanged,This is the virial radius from turnaround energy. It assumes negligible external pressure, mass loss and energy exchange. If the final structure coefficient differs, $R_{\mathrm{vir}}=(a_{\mathrm{vir}}/2a_{\mathrm{ta}})R_{\mathrm{ta}}$ instead. Half the turnaround radius is an idealized equilibrium size, not the radius at which each collisionless particle stops moving. The plotted settling time is schematic; the usual collapse-time convention is $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$.

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.

For a pressureless spherical mass shell with fixed enclosed mass and no shell crossing, the shell theorem gives $\ddot{R}=-GM/R^2$. A cosmological constant is absent in the Einstein-de Sitter universe. Parametric differentiation givesSince $R^2=A^2(1-\cos \theta {)}^2$, the equation of motion holds providedThe expanding branch runs from $\theta =0$ to the turnaround of spherical collapse $\theta =\pi$; the recollapsing branch runs from $\pi$ to $2\pi$. HenceThe common bang-time origin selects the growing overdensity, without an extra arbitrary shift in the time parameter.

In the Einstein-de Sitter universe, $H=2/(3t)$ and $\bar{\rho }=1/(6\pi G{t}^2)$. An unperturbed homogeneous sphere containing the same mass thus has $R_{\mathrm{bg}}^3=(9/2)GM{t}^2$. Comparing its volume with the perturbed sphere givesAt maximum expansion,For the non-dissipative, homologous spherical-collapse model, the potential energy of a uniform sphere is $W=-3G{M}^2/(5R)$. At turnaround the bulk kinetic energy is zero, so $E=W_{\mathrm{ta}}$. At virial equilibrium, $2K+W=0$ gives $E=W_{\mathrm{vir}}/2$. Conservation of energy implies $W_{\mathrm{vir}}=2W_{\mathrm{ta}}$, and therefore $R_{\mathrm{vir}}=R_{\max }/2$. This comparison assumes the same potential-energy structure coefficient; it is the usual top-hat model rather than an exact claim about every halo profile.

Halving the radius multiplies the density by eight. During the interval to $t_{\mathrm{coll}}=2t_{\max }$, the background density falls by four. ThusThe formal pressureless trajectory reaches zero radius; the physical virialized object instead has a finite radius. Its density is evaluated at the same collapse epoch, not at the instantaneous singular solution's density.

For a physical estimated virial radius and enclosed mass, ${\rho }_h=3M/(4\pi r_{\mathrm{vir}}^3)$. Equating this to ${\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ gives a halo density estimate of formation redshift:In the pure Einstein-de Sitter universe, ${\bar{\rho }}_{m,0}$ equals the present critical density. The inference assumes the measured density retains the collapse-epoch normalization. Later accretion, mergers and a changing virial-radius convention can change it, so it is a model-dependent formation estimate rather than a unique historical date.

The prescribed photon budget requires a collapsed mass fraction $f_{\mathrm{coll}}=3/3000=10^{-3}$, assuming baryons trace the collapsed mass fraction and using the supplied effective photon yield. No extra photon-escape or recombination correction should be counted on top of this stipulated budget.

For spectral slope $n=-2$, $q=1/6$. The mass-scale measurement givesBoth relevant epochs are matter-dominated, so the linear growth factor ratio is $D(z)/D(3)=4/(1+z)$, independent of any present-day normalization convention. Thus $\sigma (M_{\min },z)=4/(1+z)$. The Press-Schechter formalism givesUsing the supplied inverse value,This is the photon-budget reionization threshold in the specified toy model, not a measurement of the full astrophysical reionization history.

To estimate the characteristic mass, use the non-dissipative spherical-collapse model result ${\Delta }_{\mathrm{vir}}=18{\pi }^2$. At this high redshift the background matter density is practically the critical density. Circular virial velocity then obeysConsequentlyFor $v_{\mathrm{vir}}=9\mathrm{km}{\mathrm{s}}^{-1}$, this gives $H\simeq 7.87\times 10^2\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and $M_{\min }\simeq 2.3\times 10^7{M}_{\odot }$, with physical virial radius about $1.21\mathrm{kpc}$.

At the threshold mass the halo peak height is $\nu =\sqrt{2}(2.33)\simeq 3.30$, soThe present mean matter density is ${\bar{\rho }}_{m,0}=2.775\times 10^{11}{\Omega }_{\mathrm{mat}}{h}^2{M}_{\odot }{\mathrm{Mpc}}^{-3}\simeq 3.60\times 10^{10}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Hence the differential abundance per logarithmic mass interval is about $3.0{\mathrm{Mpc}}^{-3}$ andRounded inputs and the supplied approximate factor justify quoting about $0.7\mathrm{Mpc}$. The corresponding physical separation is about $0.088\mathrm{Mpc}$ at this epoch. The requested halo spacing from a differential mass function uses a logarithmic mass bin of order unity; it is not obtained by identifying $f_{\mathrm{coll}}\bar{\rho }/M$ with that differential abundance. A cumulative number density above the threshold requires a separate mass integral.

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.

A pressureless bound spherical shell first expands, reaches $\dot{R}=0$ at its maximum radius, and recollapses. In the cycloidal spherical-collapse model, turnaround occurs at $\theta =\pi$, with $R_{\max }=2A$ and $t_{\max }=\pi B$. Its mean density is $9{\pi }^2/16$ times the background density in an Einstein-de Sitter universe.