Spherical-collapse model

The spherical-collapse model applies a spherical top-hat filter to an overdense region and predicts collapse when its linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.

For ideal isolated spherical collapse, energy conservation and the virial theorem give $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$ and ${\rho }_{\mathrm{vir}}=8{\rho }_{\mathrm{ta}}$ if the density-profile shape is unchanged. The cycloid gives $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$ and ${\rho }_{\mathrm{ta}}/\bar{\rho }(t_{\mathrm{ta}})=9{\pi }^2/16$. Since the Einstein-de Sitter universe has $\bar{\rho }\propto t^{-2}$, the overdensity at the nominal collapse time is $32(9{\pi }^2/16)=18{\pi }^2$. Hence ${\rho }_{\mathrm{vir}}=18{\pi }^2{\rho }_0(1+z_{\mathrm{coll}}{)}^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.

For spherical collapse in an Einstein-de Sitter universe, take $y=R/R_{\mathrm{ta}}$ and $\tau =H_{\mathrm{ta}}t$, with ${\Delta }_{\mathrm{ta}}=(3\pi /4{)}^2$. The expanding branch has $\tau ={\Delta }_{\mathrm{ta}}^{-1/2}{\int }_0^y[u/(1-u){]}^{1/2}du$. Its small-radius expansion is $\tau =(8/9\pi )y^{3/2}(1+3y/10+O(y^2))$, and its turnaround time is $2/3$. These constants are linked: using another leading coefficient without changing the time normalization no longer gives a density contrast approaching zero at early times. The collapsing branch is $4/3-{\tau }_{\mathrm{expand}}(y)$.

For a homologous, isolated collapsing sphere, write its Newtonian gravitational potential energy as $U=-aG{M}^2/R$. At spherical-collapse turnaround the coherent kinetic energy vanishes. Conserving total energy and applying the virial theorem to the final equilibrium gives $U_{\mathrm{ta}}=U_{\mathrm{vir}}/2$ and hence $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$. If the gravitational structure coefficient changes, the ratio is $a_{\mathrm{vir}}/(2a_{\mathrm{ta}})$ instead. Mass loss, surface pressure, external tides and energy exchange also change this estimate.

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.

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.

A fixed-mass spherical shell has acceleration $\ddot{R}=-GM/R^2+\Lambda R/3$, in units $c=1$. At a finite maximum radius $R_m$, inward acceleration requires $\eta =\Lambda R_m^3/(3GM)<1$. Equality is the unstable static radius, approached only in the limiting separatrix case rather than a finite expansion-to-collapse turnaround. The effective potential reaches its maximum there, so an initially expanding bound inner-branch shell also needs energy below that maximum.

For an isolated uniform sphere with fixed positive cosmological constant, energy conservation and the virial theorem give $2\eta x^3-(2+\eta )x+1=0$, $x=R_v/R_m$. The physical root starts at $x=1/2$ when $\eta =0$ and decreases for $0<\eta <1$. At the limiting value $\eta =1$, the cubic factors as $(x-1)(2x^2+2x-1)$; the collapsing branch approaches $(\sqrt{3}-1)/2$. Finite turnaround therefore gives a strict lower bound, whose infimum is approached rather than attained. Compared with zero vacuum energy at the same turnaround data, the negative vacuum potential and its virial contribution require stronger final gravitational binding.

In an Einstein-de Sitter universe, extrapolating the linear growing mode to the time when the spherical top-hat solution collapses gives

With linear growth normalized to unity today, a collapse threshold ${\delta }_{\mathrm{sc}}$ at epoch $z$ corresponds to a present-extrapolated initial barrier ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$. Comparing this barrier to present variance is equivalent to comparing ${\delta }_{\mathrm{sc}}$ to variance evolved to the collapse epoch. Mixing the two conventions counts growth twice.

The peak height of a halo of mass $M$ is $\nu ={\delta }_c/\sigma (M)$, where ${\sigma }^2(M)$ is the variance of the linear density field smoothed on the corresponding mass scale.

The Press-Schechter formalism estimates the halo abundance from the Gaussian probability that a smoothed linear overdensity crosses the spherical-collapse threshold, with a factor of two enforcing total mass conservation.

A halo is associated with the first upcrossing of the linear spherical-collapse threshold by the smoothed density contrast as its variance increases. With a sharp-k smoothing filter, the Brownian reflection principle makes the crossing probability twice the one-scale Gaussian upper tail. This handles trajectories that crossed earlier but finish below the barrier and leads to the Press-Schechter halo mass function.

A small-scale spectral cutoff bounds the smoothing variance by $S_{\max }$. A sharp-k excursion remains Brownian up to this terminal variance, with total crossed fraction $\mathrm{erfc}({\delta }_c/\sqrt{2S_{\max }})<1$. The ordinary first-crossing density remains valid before the cutoff. Below the sharp-filter cutoff radius the variance is constant, so there are no new crossings and the idealized differential halo abundance vanishes. The missing fraction need not be artificially renormalized to one.

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.

The Press-Schechter differential abundance is

The virial theorem and hydrostatic equilibrium relate a cluster's gas temperature to $GM/R$, while $M=(4\pi /3){\Delta }_v{\rho }_c{R}^3$. A calibrated mass-temperature relation converts the Press-Schechter halo mass function into $dn/dT=(dn/dM)|dM/dT|$. Its exponential rare-cluster tail is sensitive to the density contrast variance on cluster scales, hence to the power-spectrum amplitude, often parameterized by ${\sigma }_8$. Mass calibration, temperature scatter, nonthermal pressure, sample selection and the matter density must be included rather than treating temperature counts as an independent measurement of amplitude alone.

If the linearly extrapolated rms amplitude scales as $\sigma (M)\propto M^{-s}$, a fixed peak-height cutoff in the Press-Schechter halo mass function has $M_{\ast }\propto D^{1/s}$, where $D$ is the linear growth factor. The differential prefactor scales as $M^{s-2}$ and the exponential argument as $M^{2s}$. The cutoff convention must be stated when fixing its absolute normalization: $\nu =1$ and ${\nu }^2/2=1$ select different masses, although their redshift scaling is the same.

For comoving differential number density $n(M)=dn/dM$, the abundance in an order-unity logarithmic mass bin is approximately $Mn(M)$. A characteristic mean spacing is therefore $[Mn(M){]}^{-1/3}$. A cumulative mass fraction cannot replace this number directly; cumulative halo counts require integrating $n(M)$ over mass.

For a fixed narrow mass bin with time-independent smoothing slope, the Press-Schechter halo mass function has time dependence $n\propto \nu e^{-{\nu }^2/2}$, with halo peak height $\nu \propto D^{-1}$. Differentiating gives the displayed rate. It is positive only for $\nu >1$; the bin can lose objects to higher masses later. Cumulative counts require integrating the differential abundance, and a moving mass boundary adds a selection term.

At fixed mass, the Press-Schechter halo mass function has abundance per logarithmic interval proportional to $\nu e^{-{\nu }^2/2}$. This increases as $\nu$ falls from infinity to one and then decreases as $\nu$ falls toward zero, reflecting incorporation of small dark-matter haloes into larger ones. A target below the maximum therefore has two mathematical crossings if the linear growth factor can increase without limit. The large-$\nu$ crossing describes first appearance; the small-$\nu$ crossing is late depletion. Late vacuum domination limits growth, so a formal future crossing need not be reached physically.

Defining characteristic mass by an exponential factor $\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$ fixes its peak height to $\nu =\sqrt{2}$, rather than one. Hence $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and the variance at other masses follows the same power-law scaling. The precise characteristic-mass convention determines this factor.

For $\sigma (M,0)=S{M}^{-q}$ with $q=(n+3)/6>0$, matching the Press-Schechter halo mass function to $A{M}^{q-2}\exp [-(M/M_{\ast }{)}^{2q}]$ requires $\sigma (M_{\ast },0)={\delta }_c(z)/\sqrt{2}$. The comoving normalization is $A=2q{\bar{\rho }}_{m,0}{M}_{\ast }^{-q}/\sqrt{\pi }$. Consequently both $A$ and $M_{\ast }$ evolve with the linear growth factor; $M_{\ast }\propto D^{1/q}$.

For a power-law matter spectrum, smoothed variance scales as $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. Differentiating the Press-Schechter collapsed fraction gives a halo mass function proportional to $M^{\beta -2}\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$. A constant stellar-to-halo mass ratio preserves both exponents and rescales characteristic mass and normalization.