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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.