Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 61 / 3

Integrate galactic stellar mass, rather than galaxy count, against the galaxy stellar mass function. With $x=M_{\ast }/M_{\ast ,\mathrm{ch}}$, the comoving cosmic stellar mass density, evaluated by the stellar mass density from a square-root exponential cutoff, isPut $u=\sqrt{x}$, so $dx=2udu$ and $x^{-3/4}=u^{-3/2}$. The integral is $2\Gamma (1/2)=2\sqrt{\pi }$, givingThe supplied present mean matter density and ${\Omega }_m=0.25$ imply ${\rho }_{\mathrm{crit},0}=3.5\times 10^{10}/0.25=1.4\times 10^{11}{M}_{\odot }\mathrm{Mp}{\mathrm{c}}^{-3}$. ThereforeThis is about $0.25\%$ of today's critical density, or $1\%$ of today's mean matter density, expressed per present comoving volume. The proper stellar density at $z=3$ is $64{\rho }_{\ast }^{\mathrm{com}}$; if comparing that proper density directly with today's critical density its ratio is $0.162$. It must not be confused with the usual comoving-density comparison. The low-mass number integral diverges if extrapolated to zero mass, but the mass integral converges; real populations require a low-mass cutoff.

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.

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.

The characteristic halo mass inferred from the stellar mapping isThe peak-height calibration from an exponential mass-function cutoff and $\beta =1/4$ giveUsing ${\delta }_{\mathrm{sc}}\simeq 1.686$, appropriate to the nearly matter-dominated collapse at this redshift,The barrier in the original complementary-error-function expression is extrapolated to the present, but this answer is the variance at $z=3$. Rewriting the ratio with ${\delta }_{\mathrm{sc}}$ and $\sigma (M,3)$ consistently cancels the growth normalization; using the present-extrapolated barrier with the redshift-three variance would double-count growth.

The baryon conversion efficiency of a halo is not observationally constant. The stellar-to-halo mass ratio rises from low-mass haloes to a broad maximum near galactic halo masses, then decreases toward groups and clusters; it also varies with redshift and has intrinsic scatter. Photoheating and stellar-feedback-driven outflows suppress baryon retention and star formation in small haloes. Long cooling times, hot atmospheres and feedback from active nuclei limit efficient conversion in massive haloes. The peak commonly occurs around halo masses of order $10^{12}{M}_{\odot }$, with its position evolving with epoch.

Therefore a mass-independent $f_{\ast }\simeq 0.05$ is a simplified average, not a realistic stellar-to-halo relation. Observed ratios must be compared with ${ϵ}_{\ast }=f_b{f}_{\ast }$, not with $f_{\ast }$ alone. Scatter, satellite populations and the mass-dependent efficiency change the stellar mass-function shape and invalidate an exact constant-rescaling of the halo mass function.