= Solution
Let $D(z)$ be the <linear growth factor> normalized to $D(0)=1$. The <present-extrapolated spherical-collapse barrier> is $\delta_c(z)=\delta_{\rm sc}/D(z)$, where $\delta_{\rm sc}\simeq1.686$ 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 $\delta_c(z)/\sigma(M,0)=\delta_{\rm sc}/\sigma(M,z)$.
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 gives
$$
\sigma^2(M,z)\propto D^2(z)\int_0^\infty k^{n+2}|W(kR)|^2\,dk\propto D^2(z)R^{-(n+3)}.
$$
Changing 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 $\nu=\delta_{\rm sc}/\sigma(M,z)$,
$$
\frac{dn_h}{dM}=\frac{\bar\rho_{m,0}}{M}\left|\frac{df(>M)}{dM}\right|=\sqrt{\frac2\pi}\frac{\bar\rho_{m,0}}{M^2}\,\beta\nu\,e^{-\nu^2/2}.
$$
This is a comoving halo abundance. Introduce $M_{h,\rm ch}$ by $\nu^2/2=(M/M_{h,\rm ch})^{2\beta}$; then $\sigma(M_{h,\rm ch},z)=\delta_{\rm sc}/\sqrt2$ and
$$
\frac{dn_h}{dM}=\frac{2\beta}{\sqrt\pi}\frac{\bar\rho_{m,0}}{M_{h,\rm ch}^2}\left(\frac{M}{M_{h,\rm ch}}\right)^{\beta-2}e^{-(M/M_{h,\rm ch})^{2\beta}}.
$$
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 $f_*$ is the fraction of the halo's baryons incorporated into stars, its <stellar-to-halo mass ratio> is $\epsilon_*=f_*f_b$, with $f_b=0.2$, and $M_*=\epsilon_*M$. The corresponding stellar characteristic mass is $M_{*,\rm ch}=\epsilon_*M_{h,\rm ch}$. Transforming $dn_h/dM$ with $dM/dM_*=1/\epsilon_*$ gives the <power-law Press-Schechter stellar mass function>:
$$
\boxed{\frac{dn_*}{dM_*}=\frac{2\beta}{\sqrt\pi}\frac{\bar\rho_{m,0}\epsilon_*}{M_{*,\rm ch}^2}\left(\frac{M_*}{M_{*,\rm ch}}\right)^{\beta-2}\exp\left[-\left(\frac{M_*}{M_{*,\rm ch}}\right)^{2\beta}\right].}
$$
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.
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