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Match the cutoff exponent to get $2\beta =1/2$, so $\beta =1/4$. Independently, the low-mass exponent is then $\beta -2=-7/4$, as required. ConsequentlyMatching the coefficient of the power-law Press-Schechter stellar mass function givesThusAbout $5.1\%$ of the halo baryons form stars in this model, corresponding to $1.0\%$ of total halo mass. The independent stellar-density integral gives ${\rho }_{\ast }^{\mathrm{com}}/{\bar{\rho }}_{m,0}={ϵ}_{\ast }$, checking the mass-function normalization.

The index $-3/2$ is an effective slope of the processed matter spectrum on these mass scales, not the primordial index. The cosmological transfer function suppresses modes that enter the horizon during radiation domination, altering the approximately primordial $k$ law. Around and below galaxy scales the dimensional matter spectrum has a much more negative local slope, eventually tending toward the approximately $k^{-3}{\ln }^{2}k$ small-scale limit. A local power law with $n=-3/2$ is a useful idealization over a limited interval; there is no reason for it to equal the large-scale primordial value or to hold on every scale.

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.