Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 55 / 2

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.