 Solution

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.