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In a matter-dominated Einstein-de Sitter universe, the growing linear density contrast is proportional to the scale factor. When $|\delta |\sim 1$, linear evolution no longer describes collapse. An overdense region initially expands more slowly than the background, reaches cosmological turnaround, and then contracts under self-gravity.

The spherical top-hat collapse model supplies a useful idealization: pressureless shells obey $\ddot{r}=-GM/r^2$. At formal collapse, the linearly extrapolated growing mode isThis is a linear extrapolation, not the true finite nonlinear density at that instant. In the ideal energy-conserving spherical model, virial equilibrium gives a virial radius about half the turnaround radius; its mean density at the collapse time is $18{\pi }^2\simeq 178$ times the cosmic matter density.

Real perturbations are nonspherical: tides and anisotropic collapse produce sheets, filaments and haloes. Collisionless dark matter develops shell crossing and phase mixing rather than radiatively losing its orbital energy. Baryonic gas can shock, cool and collect within the resulting potential wells. Hierarchical merging then assembles larger systems where small-scale fluctuations have sufficient initial power. Press-Schechter formalism connects a collapse threshold with the statistics of the smoothed initial field, but halo abundances and internal structure need more than the single spherical model. These ideas apply after perturbations become nonlinear; one must not extend pressureless subhorizon equations through all radiation-era or horizon-scale conditions.^[ref]

Galaxy formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics. Primordial density fluctuations supply a statistical seed distribution. In linear theory, each Fourier mode evolves through a growth factor and a scale-dependent transfer function; the power spectrum describes the variance by scale. Radiation pressure, free streaming and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.

When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through accretion and mergers. Cold dark matter permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the Press-Schechter formalism, organize an approximate halo abundance; departures from spherical collapse and correlated environments limit its literal interpretation.

Gas falling into a halo is compressed and may be shock-heated toward a virial scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere the denominator specifies a chosen optically thin astrophysical cooling function convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, ionization state, density and metal abundance matter. A stable virial shock is not obligatory for all accretion: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.

The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.^[ref]

Angular momentum inherited from tidal torques and later accretion helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms stars. Stellar radiation, winds and supernovae then heat gas, drive turbulence, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.

Galaxy growth is usefully expressed as a gas-budget competition:Here $\mathcal{R}$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.

Accreting central black holes can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and ram pressure in a surrounding medium provide additional environmental routes. Quiescent galaxy assembly can still add stellar mass through dry mergers, so cessation of star formation is not identical to cessation of all mass assembly.

Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth accretion can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between galaxy luminosity and total halo mass.

The Press-Schechter formalism starts from a statistically homogeneous Gaussian initial density contrast with variance specified by its linear matter power spectrum. Average it on mass scale $M=(4\pi /3){\bar{\rho }}_{m0}{R}^3$, usually with a spherical top-hat. Let $\sigma (M,t)=b(t){\sigma }_0(M)$, normalizing $b$ consistently. Assume that a region becomes a collapsed halo when its linearly extrapolated smoothed density exceeds a mass-independent linear spherical-collapse threshold ${\delta }_{msc}\simeq 1.686$ in the Einstein-de Sitter approximation. This extrapolates a linear Gaussian statistic to a nonlinear collapse criterion, and ignores the detailed nonspherical dynamics.

The one-scale Gaussian upper tail only assigns half the mass to overdense regions when the variance grows without limit. The conventional factor of two is inserted to account for material in larger collapsed regions even if it falls below the threshold on a smaller scale, the cloud-in-cloud issue. A first-crossing excursion-set argument can justify that factor for an ideal sharp-k filter; it is not a theorem that an arbitrary top-hat one-scale probability alone supplies it. With this prescription, the mass fraction in objects above $M$ isConvert mass fraction to comoving number density by assigning mass $M$ to each halo, using $dn/dM=({\bar{\rho }}_{m0}/M)|dF/dM|$. This yields the Press-Schechter halo mass functionEquivalently one can use a present-extrapolated threshold ${\delta }_c(t)={\delta }_{msc}/b(t)$ and present variance; using both that threshold and an already grown variance would double-count growth. The model assumes approximate mass conservation, a smoothing-to-halo-mass assignment and a spherical universal barrier. It is an approximation to gravitational halo formation, not an exact partition of the real cosmic web.

In hierarchical formation, ${\sigma }_0(M)$ decreases with mass: smaller regions become nonlinear first and are subsequently incorporated into larger haloes. Define a characteristic mass by $b(t){\sigma }_0(M_{\ast })={\delta }_{msc}$. It increases as growth proceeds, moving the exponential cutoff towards larger masses. For a scale-free spectrum with well-defined top-hat variance, $-3<n<1$, ${\sigma }_0\propto M^{-(n+3)/6}$ and hence $M_{\ast }\propto b^{6/(n+3)}$. At a fixed mass, the derivative of $\ln (dn/dM)$ with respect to $\ln b$ is ${\nu }^2-1$: rare high-mass abundances grow, while the small-mass abundance can decline as objects merge into larger systems. Thus one must not claim that every fixed-mass bin increases forever. The growth of $M_{\ast }$ is the scale-free characteristic mass in the Press-Schechter formalism.

For the temperature test, virial equilibrium and gas hydrostatic equilibrium give $k_{B}T\sim \mu m_{p}GM/R$. Defining the halo radius by $M=(4\pi /3){\Delta }_v{\rho }_c(z)R^3$ yieldsA calibrated monotone mass-temperature relation converts the mass function to $dn/dT=(dn/dM)|dM/dT|$, with temperature scatter treated by convolution when necessary. The cluster temperature function normalization is strongly sensitive to the linear fluctuation amplitude: at fixed mass and spectral shape, $\partial \ln n/\partial \ln {\sigma }_8={\nu }^2-1$. Rare hot clusters occupy the exponential tail, so even a modest amplitude change can produce a large abundance change. Here sigma eight is the rms linear density contrast today in an $8h^{-1}\mathrm{Mpc}$ top-hat, not the full spectrum.

Compare observed cluster-temperature counts and their redshift evolution with these predictions to constrain that normalization jointly with matter density and growth. The mass-temperature calibration, selection volume, temperature scatter, nonthermal support and deviations from the ideal halo mass function limit a temperature-only inference. The abundance of rare hot clusters is a sensitive amplitude probe once their mass calibration and cosmology are specified.

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 $z$.

Let ${\delta }_{\mathrm{sc}}\simeq 1.686$ be the linear spherical-collapse threshold and let $D(t)$ be the linear growth factor, normalized to one at the epoch to which the random density field has been extrapolated. The time-dependent barrier is ${\delta }_c(t)={\delta }_{\mathrm{sc}}/D(t)$. The smoothed matter density variance ${\sigma }^2(M)$ is the variance of the linear density contrast smoothed with a spherical top-hat window function whose comoving mass is $M$. Both quantities must refer to the same linear normalization; equivalently use $\nu ={\delta }_{\mathrm{sc}}/\sigma (M,t)$.

Let $N$ denote the comoving number density and write $\mathcal{N}(M,t)=dN/dM$ for the positive differential Press-Schechter halo mass function. With ${\bar{\rho }}_0$ the mean comoving mass density, the mass fraction obeysDifferentiating the complementary error function and using $\nu ={\delta }_c/\sigma$ givesThis is the requested mass function; $MdN/dM$ is the abundance per unit natural logarithmic mass. For physical rather than comoving number density, multiply by $(1+z{)}^3$. The normalization in the Press-Schechter formalism includes its factor of two relative to the positive Gaussian tail, so $f(>M)$ tends to one as the smoothing mass tends to zero when $\sigma$ diverges. It is a collapse prescription, not the probability that an arbitrary already nonlinear density field stays Gaussian.

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.

Let $D(z)$ be the linear growth factor normalized to $D(0)=1$. The present-extrapolated spherical-collapse barrier is ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$, where ${\delta }_{\mathrm{sc}}\simeq 1.686$ is the linearly evolved overdensity required for spherical collapse at that epoch. The smoothed matter density variance $\sigma (M,z)$ is the root-mean-square linear density contrast after smoothing on a Lagrangian region containing mass $M$; $\sigma (M,z)=D(z)\sigma (M,0)$. Thus the barrier-to-variance ratio is equivalently ${\delta }_c(z)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$.

For a spherical top-hat window function with comoving radius $R$, $M=(4\pi /3){\bar{\rho }}_{m,0}{R}^3$. A scale-free spectrum givesChanging variable to $kR$ gives the scale-free smoothed density variance scaling $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. For a top-hat the scale-free integral converges for $-3<n<1$; the inferred index below lies in this interval.

Differentiate the Press-Schechter formalism mass fraction and divide the resulting mass density by halo mass. Writing $\nu ={\delta }_{\mathrm{sc}}/\sigma (M,z)$,This is a comoving halo abundance. Introduce $M_{h,\mathrm{ch}}$ by ${\nu }^2/2=(M/M_{h,\mathrm{ch}}{)}^{2\beta }$; then $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and

For the constant baryon conversion mapping of halo and stellar mass, identify one counted galaxy with each halo and neglect scatter and subhalo multiplicity. If $f_{\ast }$ is the fraction of the halo's baryons incorporated into stars, its stellar-to-halo mass ratio is ${ϵ}_{\ast }=f_{\ast }{f}_b$, with $f_b=0.2$, and $M_{\ast }={ϵ}_{\ast }M$. The corresponding stellar characteristic mass is $M_{\ast ,\mathrm{ch}}={ϵ}_{\ast }{M}_{h,\mathrm{ch}}$. Transforming $d{n}_h/dM$ with $dM/d{M}_{\ast }=1/{ϵ}_{\ast }$ gives the power-law Press-Schechter stellar mass function:It has the required power-law low-mass behavior and stretched exponential cutoff. Matching both exponents and the normalization determines the effective spectral index and stellar conversion efficiency.

On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since ${\delta }_h(M)=b(M){\delta }_m$, mass conservation requires

For the Press-Schechter formalism, the mass-fraction measure becomesIt is normalized and is a half-normal distribution, soUsing the linear Eulerian halo biastherefore giveswhich verifies the consistency relation.

If each collapsed baryon supplies $N_{\gamma }$ effective ionizing photons and $N_{\mathrm{req}}$ photons per cosmic baryon are required, the threshold collapsed fraction is $f_{\mathrm{coll}}=N_{\mathrm{req}}/N_{\gamma }$. The Press-Schechter formalism gives $\mathrm{erfc}[{\delta }_{\mathrm{sc}}/(\sqrt{2}\sigma (M_{\min },z))]=f_{\mathrm{coll}}$. Escape losses and recombinations must either be incorporated in the effective yield/requirement or counted separately, not twice.