Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 67 / 2

Normalize the region's reference scale factor to one. Pressureless matter has $\rho ={\rho }_0{a}^{-3}$, so the closed Friedmann equation isThe expanding branch can reach only $a\le a_m$, where $a_m={\Omega }_m/({\Omega }_m-1)$. Substitute $a=a_m(1-\cos \theta )/2$. Since $da/d\theta =a_m\sin \theta /2$ and $\dot{a}=H_0\sqrt{({\Omega }_m-1)(a_m/a-1)}$, cancellation on $0<\theta <\pi$ givesIntegration from the Big Bang establishesThe same parametrization continues through spherical-collapse turnaround at $\theta =\pi$ onto the contracting branch up to $\theta =2\pi$. Its maximum-expansion time is $t_m=\pi {\Omega }_m/[2H_0({\Omega }_m-1{)}^{3/2}]$.

The outer factor is explicitly $a_{\mathrm{bg}}/a_m=\frac{1}{4}(6\pi t/t_m{)}^{2/3}$, the normalized scale factor of an Einstein-de Sitter universe, with the same enclosed mass and initial bang time as the region. The correction describes a fractional reduction of the region's radius relative to this background:For fixed mass, density varies as inverse radius cubed, so ${\rho }_{\mathrm{region}}/{\rho }_{\mathrm{bg}}=(1-ϵ{)}^{-3}=1+3ϵ+O({ϵ}^2)$. The growing linear density contrast is thereforeThe full spherical collapse reverses its expansion at $t_m$ and reaches zero radius at $t=2t_m$, corresponding to $\theta =2\pi$. In the ideal cold, pressureless model this makes the nonlinear density contrast infinite. Extrapolating the linear growing mode to that same time instead gives the finite linear spherical-collapse thresholdThis is an extrapolation of a linear perturbation, not an accurate finite-radius approximation at the moment of collapse. Virial equilibrium requires relaxation of the cold coherent motion and replaces the ideal singular endpoint.

The regional mass density is $3H_0^2{\Omega }_m/(8\pi G{a}^3)$, while the Einstein-de Sitter universe has ${\rho }_{\mathrm{bg}}=1/(6\pi G{t}^2)$. Substituting the parametric spherical collapse from part (a) yieldsAt spherical-collapse turnaround, $\theta =\pi$, and henceFor an isolated uniform sphere, the Newtonian gravitational potential energy is $U=-3G{M}^2/(5R)$. At spherical-collapse turnaround the coherent kinetic energy is zero, so conservation of energy fixes $E=U_m$. At virial equilibrium, the virial theorem gives $2T_v+U_v=0$, hence $E=U_v/2$. Equating these energies proves the virial radius from turnaround energy:This uses the same mass and gravitational structure coefficient before and after collapse and neglects boundary pressure, mass loss and external energy exchange.

If virial equilibrium is assigned to $2t_m$, the sphere's density is eight times its turnaround density, whereas the background mass density has decreased by four. ConsequentlyThe often quoted value $18{\pi }^2$ is a density ratio; the density contrast defined as $(\rho -{\rho }_{\mathrm{bg}})/{\rho }_{\mathrm{bg}}$ is smaller by one. This finite virialized estimate and the infinite cold-collapse contrast in part (b) describe different physical endpoints.

First fix a genuine normalization error in the original PDF: the Gaussian exponent uses ${\sigma }^2$, so $\sigma$ must be the standard deviation, the variance is ${\sigma }^2$, and the normalized probability density function isThe printed prefactor has only $\sigma$ under its square root; its integral is $\sqrt{\sigma }$ rather than one. Use the normalized normal distribution above.

The one-scale probability of exceeding the linear spherical-collapse threshold is $P_M=\frac{1}{2}\mathrm{erfc}[{\delta }_c/(\sqrt{2}\sigma )]$. In the Press-Schechter formalism, the mass fraction above $M$ is assigned toThe factor two is necessary to recover all the mass in the hierarchical limit $\sigma \to \infty$: the uncorrected one-point upper tail tends to only one half. A precise justification is available in the excursion-set description of halo formation with a sharp-k smoothing filter. The smoothed Gaussian random field then becomes Brownian motion in its variance $S$. Reflecting a trajectory after its first crossing of ${\delta }_c$ pairs trajectories which crossed but end below the threshold with those which end above it. The Brownian reflection principle therefore gives $\mathrm{Pr}({\max }_{s\le S}\delta (s)>{\delta }_c)=2\mathrm{Pr}(\delta (S)>{\delta }_c)$.

This multiscale mass assignment is an extra modelling prescription: a Gaussian one-point probability and a collapse threshold alone do not mathematically determine a halo mass function. The “all matter” normalization additionally assumes arbitrarily large small-scale variance; a finite variance cutoff leaves an uncollapsed fraction.

Differentiate at fixed $z$ and fixed ${\delta }_c$:For decreasing $\sigma (M,z)$, a mass interval contains the positive fraction $-F^{\prime }(M)dM$. Its comoving mass is ${\rho }_{m,0}[-F^{\prime }(M)]dM$, so division by halo mass gives the Press-Schechter halo mass function:The absolute value, or equivalently a minus sign multiplying ${\partial }_M\sigma$, is essential for a positive abundance. The original PDF correctly places absolute values around the abundance and the full derivative expression; these bars are lost in the converted TeX. The density is comoving, so the prefactor uses today's matter mass density, rather than the physical density at redshift $z$.