Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 346 (Formation of Galaxies)

Write physical position as $\mathbf{r}=a(t)\mathbf{x}$ and physical velocity asSubtract the homogeneous expanding background from the pressureless Euler equation. To linear order in peculiar velocity and density contrast, the convective term is negligible andorHere $\nabla$ differentiates with respect to comoving position and $\Phi$ is the peculiar gravitational potential.

The linear continuity and Poisson equations areFor the growing mode $\delta (\mathbf{x},t)=D(a){\delta }_i(\mathbf{x})$, matter conservation gives ${\bar{\rho }}_m\propto a^{-3}$, so the potential scales asafter absorbing the fiducial normalization into ${\Phi }_i$. Integrating the linear Euler equation with a negligible decaying mode gives

Taking the divergence of Euler and using continuity and Poisson yields the linear growth equationIn the fiducial normalization used in the question this can be rearranged asThe cosmological Lagrangian displacement obeys $\dot{\mathbf{\psi }}=\mathbf{v}/a$. Combining the last two relations and choosing the growing displacement gives the Zeldovich approximation

DefineThen $\mathbf{x}={\mathbf{x}}_i-b\nabla {\Phi }_i$ and $\mathbf{v}=-a\dot{b}\nabla {\Phi }_i$. Keeping the lowest nonvanishing order in displacement in the halo angular momentum givesThe angular momentum vanishes for a spherical Lagrangian patch, and also whenever the patch's inertia principal axes align with the local tidal-field principal axes. It likewise vanishes in a locally isotropic tidal field.

Taylor-expandThe constant-gradient term vanishes by the barycentre definition. Withone obtains the tidal torque theory result$I$ is the protohalo inertia tensor and $T$ is the local tidal, or gravitational Hessian, tensor. Their eigenframe misalignment produces the torque.

In an Einstein-de Sitter universe, $D\propto a\propto t^{2/3}$. Hence $\dot{b}\propto \dot{a}\propto t^{-1/3}$ andThis linear estimate is normally stopped near turnaround. Simulated haloes subsequently gain angular momentum through nonlinear torques, anisotropic accretion, and mergers, including the orbital angular momentum of infalling subhaloes, so the early linear estimate underpredicts the final value.

The Schechter function has logarithmic slope $\alpha \simeq -1$ for $M\ll M_{\ast }$ and an exponential cutoff for $M\gg M_{\ast }$. Thus a log-log plot is approximately a straight line of slope $-1$ at low mass and bends sharply downward near $M_{\ast }$.

For a collisionless self-gravitating system, define the inertia tensorTwo time derivatives, the collisionless Boltzmann equation, and integration by parts give the tensor virial theoremwhereFor an isolated steady state, or a long-time average of a bounded system, ${\ddot{I}}_{ij}=0$ and surface terms vanish, soTaking the trace gives $2K+W=0$. Therefore

Introduce the gravitational radius by $W=-G{M}^2/R$. Virial equilibrium givesAssume dissipationless accretion, negligible escaping mass, no external work, and final re-virialization. The accreted systems bringWith $\eta =M_A/M_I$ and $ϵ=⟨v_A^2⟩/⟨v_I^2⟩$,Since $M_F=M_I(1+\eta )$ and $E_F=-M_F⟨v_F^2⟩/2$,Using $E=-G{M}^2/(2R)$ also gives

Many minor galaxy mergers have dynamically cold accreted material, $ϵ\ll 1$. If they double the mass, $\eta =1$, then $R_F/R_I\simeq 4$, andThe steep low-mass side of the Schechter function supplies many small satellites. Repeated dry galaxy mergers can therefore add stars mainly at large radius, increasing size much faster than mass and lowering mean density. This explains how compact, dense redshift-two galaxies can evolve into larger present-day elliptical galaxies without requiring comparable in-situ star formation.

For a spherical orbit,so the galaxy rotation curve is flat. The isotropic spherical Jeans equation with constant dispersion isSince $\rho \propto r^{-2}$,

WriteFor $X\ll 1$,so Chandrasekhar dynamical friction is linear in $\mathbf{v}$ at low speed. At $X\gg 1$, $\mathcal{F}(X)\to 1$, so its acceleration magnitude decays as $v^{-2}$. At zero speed the wake is symmetric and the drag vanishes; at high speed the subhalo spends too little time deflecting each background particle efficiently.

For a circular orbit $v=v_0$, so $X=1$ and $\mathcal{F}(1)=0.428$. The tangential acceleration isThe specific angular momentum is $L=r{v}_0$, hence $v_0\dot{r}=r{a}_{\mathrm{df}}$ andIntegration from $r_i$ to zero gives

For a circular orbit of radius $r$,Solving for $r$ givesSince $\eta =L/L_{\mathrm{circ}}(E)$,

The friction acceleration is antiparallel to velocity, so locally $\dot{L}/L=(1/v)\dot{v}$ and $\dot{E}=v\dot{v}$. Applying the chain rule $\dot{e}=(de/d\eta )\dot{\eta }$ givesBecause $d\eta /de<0$ and $\dot{v}<0$, at pericentre $v>v_0$ gives $\dot{e}<0$: friction circularizes. At apocentre $v<v_0$ gives $\dot{e}>0$: it makes the orbit more eccentric. Orbit averaging produces substantial cancellation; the denser pericentre region generally gives modest net circularization, but the eccentricity changes much less dramatically than the orbital energy and radius.

For a statistically homogeneous and isotropic density contrast,Using the Fourier transform conventionand translational invariance giveswithAt one point,

For a spherical top-hat volume $V=4\pi R^3/3$,The overlap of two radius-$R$ balls separated by $r\le 2R$ isTherefore

For $\xi (r)=(r/r_0{)}^{-1.8}$,ThusThis calculation assumes that the observed galaxies trace the matter field with unit, scale-independent galaxy bias, and that the fitted power law remains valid throughout the top-hat separations.

The photon mean free path isIn time $t$ a photon takes about $ct/{\ell }_{\gamma }$ independent steps. Isotropy assigns one third of the mean-square displacement to any coordinate, giving the Silk damping diffusion scale

In the matter era, $a\propto t^{2/3}$ and $n_e=n_{\mathrm{rec}}(a_{\mathrm{rec}}/a{)}^3$. Accumulating the diffusion variance while converting each displacement to comoving length givesHenceThe order-one coefficient depends on the precise diffusion-length convention; the expression uses the convention stated in the question.

In a universe containing only baryons and radiation, photon diffusion erases coupled baryon-photon perturbations below this scale before recombination. Baryons begin subsequent matter-era growth from a strongly smoothed field, suppressing small-scale structure. With cold dark matter, collisionless dark-matter perturbations are not Silk damped and provide surviving gravitational wells. After recombination baryons fall into those wells, so baryonic acoustic structure is damped but small-scale total-matter structure can continue to grow.