Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 61 (The Origin and Evolution of Galaxies)

Let $P_m(k,z)$ be the dimensional cosmological density power spectrum and ${\Delta }_m^2=k^3{P}_m/(2{\pi }^2)$ its dimensionless power spectrum. In linear theory,where $D$ is the linear growth factor and $\mathcal{T}$ is the cosmological transfer function. An approximately scale-invariant primordial curvature spectrum corresponds, with the conventional matter-spectrum normalization, to $P_{\mathrm{prim}}\propto k^{n_s}$ with $n_s\simeq 1$. This does not say that the dimensional density spectrum is white noise or that the early curvature and density perturbations are the same variable.

The crucial scale dependence develops around horizon entry and matter-radiation equality. Modes entering during radiation domination undergo only logarithmic cold-matter growth, whereas modes entering later avoid that suppression. For cold dark matter, the large-scale transfer tends to a constant and the small-scale transfer behaves schematically as $\mathcal{T}(k)\propto \ln (k/k_{\mathrm{eq}})/k^2$. ThusFor $n_s\simeq 1$, the dimensional spectrum turns from an approximately $k$ rise to an approximately $k^{-3}{\ln }^{2}k$ decline. Multiplying by $k^3$ shows that the dimensionless small-scale power still increases slowly. The resulting variance of density fluctuations smoothed on mass $M$ generally increases as $M$ decreases, over the scales relevant to galaxy assembly.

After equality, and over scales where linear growth is scale-independent, the transfer shape is approximately preserved while the amplitude grows. During matter domination, $D\propto a$; late accelerated expansion slows that growth. A region collapses when its linearly evolved density contrast reaches the linear spherical-collapse threshold. Smaller mass scales, having larger variance, typically reach this threshold earlier. Small haloes form first, then accrete matter and merge into larger haloes; galaxies form from cooling baryons in those wells and themselves merge. This small-to-large assembly is hierarchical galaxy formation. It is a statistical trend: rare high peaks can produce unusually massive early objects, and gas cooling and feedback prevent a one-to-one identification of halo collapse with star formation.

For cold dark matter, random particle velocities are small enough that collisionless free streaming erases little power on galactic scales. Warm dark matter retains a larger early velocity dispersion and a larger free-streaming length, suppressing power below a finite scale. There are consequently fewer low-mass haloes, delayed formation of the smallest galaxies, and a lower limit to the hierarchy that can develop. Above the cutoff, warm-matter structure can still assemble hierarchically. The distinction concerns the particles' velocity history and transfer function, not the thermal temperature of gas in a galaxy today. The broad late-time shape was largely established by early horizon-entry physics, with subsequent nonlinear collapse and merging altering the spectrum at high wavenumber.

For an optically thin, low-density plasma, the astrophysical cooling function packages collisional radiation losses and temperature-dependent ion fractions into the coefficient multiplying $n_H^2$. Because the volume loss rate has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The energy-density unit printed in the numerical hint cannot be the unit of this coefficient. Interpret the quoted logarithmic values in the dimensionally consistent cooling-coefficient unit.

Assume collisional ionization equilibrium, a primordial hydrogen-helium mixture and no external photoheating. In the specified range, the primordial atomic cooling curve has the following features. Just above $10^4\mathrm{K}$, thermal electrons begin to excite neutral hydrogen efficiently; subsequent line emission, especially Lyman-alpha emission, causes a steep rise. The excitation rate contains a threshold factor of order $e^{-10.2\mathrm{eV}/(k_{B}T)}$. Hydrogen line cooling is strong near a few times $10^4\mathrm{K}$; the supplied value at $1.5\times 10^4\mathrm{K}$ provides a useful low-temperature label.

As hydrogen becomes ionized, neutral-hydrogen line cooling declines. Helium excitation and ionization produce a further shoulder or peak around $10^5\mathrm{K}$. Collisional excitation, collisional ionization and radiative recombination all contribute: excitation photons remove electron kinetic energy, ionization consumes it, and recombination produces free-bound radiation. Once hydrogen and helium are almost fully stripped, their bound-state cooling disappears and the curve falls into a relatively inefficient interval. At high temperature, thermal bremsstrahlung dominates, with an approximate $\Lambda \propto T^{1/2}$ tail and a weak Gaunt-factor correction.

The sketch uses the two supplied numerical labels and a qualitative hydrogen-helium interpolation; it is not a tabulated atomic-rate calculation.

Metal-line cooling raises the cooling coefficient markedly over much of $10^5$–$10^7\mathrm{K}$, because heavier elements supply many ions and excitation transitions after hydrogen and helium have lost their bound electrons. It also broadens and reshapes the line-cooling peaks. Fine-structure lines can permit cooling below the hydrogen atomic threshold; molecular hydrogen can likewise cool metal-free gas below that threshold, but lies outside the requested temperature range. At sufficiently high temperatures thermal free-free emission again dominates the continuum. The enhanced curve in the figure is a schematic comparison, not a numerical claim about a specified metallicity. Thus metals generally shorten the cooling time and extend the temperature range of efficient cooling; ionization state, abundance and radiation field determine the actual curve.

The optically thin gas cooling time is thermal energy density divided by the radiative loss rate. Write $\chi =n_{\mathrm{tot}}/n_H$ for the number of thermal particles per hydrogen nucleus, $X$ for the hydrogen mass fraction, and $f_g={\rho }_g/{\rho }_{\mathrm{grav}}$ for the gas fraction of the gravitating mass. ThenThe last expression is the free-fall time of a uniform sphere. Equating the two times gives the cooling-to-free-fall equality curveAt fixed temperature, $t_{\mathrm{cool}}/t_{\mathrm{ff}}=\sqrt{n_{H,\mathrm{crit}}/n_H}$: above the curve, $t_{\mathrm{cool}}<t_{\mathrm{ff}}$; below it, cooling is slower. The temperature dependence is approximately $n_{H,\mathrm{crit}}\propto T^2/{\Lambda }^2$. Strong line cooling therefore creates low-density troughs, whereas the fully ionized bremsstrahlung tail gives $n_{H,\mathrm{crit}}\propto T$.

For the diagram choose a self-gravitating uniform gas cloud, $f_g=1$, with $X=0.76$, proton mass $m_p\simeq 1.67\times 10^{-24}\mathrm{g}$, and the fully ionized particle ratio $\chi =2+3(1-X)/(4X)\simeq 2.24$. Using this fixed ratio is a convenient sketch normalization; the varying low-temperature ion fraction changes the coefficient by a factor of order unity. The supplied cooling values then give $n_{H,\mathrm{crit}}\sim 10^{-9}\mathrm{c}{\mathrm{m}}^{-3}$ at $1.5\times 10^4\mathrm{K}$ and $n_{H,\mathrm{crit}}\sim 0.1\mathrm{c}{\mathrm{m}}^{-3}$ at $10^7\mathrm{K}$. A dark-matter-dominated potential has $f_g<1$, shortening the free-fall time and moving the equality curve upward by $1/f_g$.

To relate the plane to mass, use the virial theorem for a uniform self-gravitating sphere. Its gravitational potential energy is $-3G{M}^2/(5R)$, so $k_B{T}_{\mathrm{vir}}=GM\mu m_p/(5R)$, where $\mu =1/(X\chi )$. With $M=(4\pi /3)\rho R^3$, the virial mass contours in a cooling diagram satisfyThe coefficient $5$ is for this uniform-sphere convention; other halo profiles change it. The horizontal mean-density label is $n_{H,0}\simeq 2\times 10^{-7}\mathrm{c}{\mathrm{m}}^{-3}$. Real collapsing clouds are overdense, and their characteristic densities rise at earlier epochs.

The cooling criterion for galaxy formation requires a cloud to lose shock-generated virial heat rapidly enough to contract and fragment. At higher virial masses and temperatures, hydrogen-helium line cooling ceases to be efficient, so a growing bound aggregate can remain a hot, pressure-supported atmosphere rather than condense as one giant luminous galaxy. Combined with virial mass contours and the densities at which clouds assemble, the diagram gives a characteristic upper galaxy-scale cooling mass, conventionally of order $10^{12}{M}_{\odot }$ in the simple baryonic-cloud argument. Larger aggregates are predominantly groups or clusters containing smaller galaxies and hot gas.

This upper galaxy mass from gas cooling is a physical scale, not an exact universal mass cutoff derivable from the two cooling labels alone. Composition, formation density, geometry, metals and a dark-matter potential shift it. Cooling that is slower than free fall but faster than the available cosmic time can still yield gradual central condensation. Mergers of existing stellar galaxies can also assemble a larger stellar system without rapidly cooling the entire gas mass of its host halo.

Treat the features in the QSO spectrum as Lyman-alpha absorption and use the cosmological wavelength-redshift relation to convert their spacing into $\Delta z=304/1216=0.25$. In the specified flat matter-plus-cosmological-constant model,A local radial interval has comoving length $d\chi =cdz/H(z)$ and proper length $dl=d\chi /(1+z)$. Evaluating the mean interval locally at $z=3$ gives the proper absorber separation from a wavelength intervalThe corresponding comoving interval is about $262\mathrm{Mpc}$. We neglect peculiar-velocity shifts and treat $H$ as approximately constant over one mean interval.

The proper number density is $n_p=(1+z{)}^3{n}_{\mathrm{gal}}=179.2\mathrm{Mp}{\mathrm{c}}^{-3}$. To compute the mean projected area of a randomly oriented thin disc, note that a disc whose normal makes angle $i$ to the sightline has projected area $\pi R_d^2|\cos i|$. Isotropic normals haveFor independent intersections with fully covering thin discs, the mean free path through randomly oriented disc absorbers is ${\ell }^{-1}=n_p\pi R_d^2/2$. ThusAll radii here are proper radii at the absorber epoch. The projected-area average is over the underlying orientation distribution; it already accounts for the larger intersection probability of face-on discs.

The cosmic baryon fraction is $f_b={\Omega }_b/{\Omega }_m=0.2$, so the disc mass is $M_d=f_d{M}_h$ with $f_d=f_b/4=0.05$. Define $v_{\mathrm{vir}}^2=G{M}_h/R_{\mathrm{vir}}$. Disc gravity depends on its radial mass profile, which has not been specified. Parameterize its edge circular speed by $v_d^2=C_{d}G{M}_d/R_d$ and use $C_d=1$ for a centrally concentrated disc in a monopole approximation at its outer edge. The angular-momentum condition givesCombining the equations yields the angular-momentum-conserving self-gravitating disc radiusFor the stated $C_d=1$ choice, $R_{\mathrm{vir}}\simeq 36.8\mathrm{kpc}$ and $v_d=0.5v_{\mathrm{vir}}$. The factor $C_d$ makes explicit the otherwise undetermined disc-profile dependence; an enclosed-mass formula is not exact for an arbitrary flattened disc.

To determine the halo speed, a virial-density convention is also needed. Choose a spherical overdensity ${\Delta }_v=200$ relative to the critical density at $z=3$:The virial velocity of a spherical-overdensity halo then obeysThe nearly matter-dominated spherical-collapse choice ${\Delta }_v\simeq 18{\pi }^2$ instead gives about $99.5\mathrm{km}{\mathrm{s}}^{-1}$. More generally the numerical radius and velocity scale respectively as $C_d$ and $C_d\sqrt{{\Delta }_v/200}$. With $G\simeq 4.30\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$, the adopted model has $M_h\simeq 9.5\times 10^{10}{M}_{\odot }$ and $M_d\simeq 4.8\times 10^9{M}_{\odot }$; these reproduce the edge speed and the specified angular momentum.

The calculation assumes homogeneous, unclustered intersections, one detectable absorption feature per disc crossing, unit neutral-gas covering factor, isotropic orientations, universal halo baryon fraction and conservation of the specified edge specific angular momentum. It assumes the halo is virialized at the absorber epoch and neglects its inner gravity as instructed. The extra profile and overdensity choices are necessary to turn the geometric and angular-momentum constraints into unique numerical halo properties.

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.