Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 2

Take $w$ constant and $w\ne -1$, with a barotropic perturbation $\delta P=w\delta \rho$. Define the density contrast $\delta =\delta \rho /\bar{\rho }$ and use the scalar velocity potential convention $v_i(\mathbf{k})=i{k}_i\theta (\mathbf{k})$, so ${\partial }_i{v}^i=-k^2\theta$. This $\theta$ is a potential, not the sometimes-used velocity-divergence variable. In synchronous gauge in cosmology, the inverse metric is $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}({\delta }^{ij}-h^{ij})$ to first order. Discarding products of perturbations in the perfect fluid in general relativity givesFor the linearization, each $h_{ij}$ must be small: the PDF's determinant condition alone is insufficient, since $h=\mathrm{diag}(L,0,0)$ has zero determinant even for large $L$.

Write $\mathcal{H}=a^{\prime }/a$. The time component of covariant conservation of stress-energy isHere ${\Gamma }_{\mu 0}^{\mu }=4\mathcal{H}+h^{\prime }/2$. To first order, ${\Gamma }_{ij}^0{T}^{ij}=a^{-2}w\bar{\rho }[3\mathcal{H}(1+\delta )+h^{\prime }/2]$: the two metric-$h$ contributions cancel. Multiplying the conservation equation by $a^2$, including ${\Gamma }_{00}^0{T}^{00}$, givesThe background cosmological perfect-fluid continuity equation, ${\bar{\rho }}^{\prime }=-3\mathcal{H}(1+w)\bar{\rho }$, cancels the first and third terms. ThusThe $h^{\prime }$ term accounts for perturbation of the spatial volume expansion in synchronous gauge in cosmology. The PDF's correct connection ${\Gamma }_{0j}^i=\mathcal{H}{\delta }_j^i+h^{\prime i}{}_j/2$ is garbled in the TeX aid.

In the decoupled perfect fluid in general relativity limit, photons have $w=1/3$ and no anisotropic stress, while cold baryons have $w\simeq 0$ with a small pressure correction $c_b^2{\delta }_b$. The density equations express particle and energy conservation; Thomson scattering transfers momentum rather than leading-order rest-frame energy. The photon Euler equation consequently gains a drag toward the baryon velocity potential, and the baryon Euler equation gains the equal-and-opposite drag weighted by their inertias.

Let ${\Gamma }_T=a{n}_e{\sigma }_T>0$. With the velocity potential used in part (i), these forces are $-{\Gamma }_T({\theta }_{\gamma }-{\theta }_b)$ and $-R{\Gamma }_T({\theta }_b-{\theta }_{\gamma })$, with $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$. Multiplication by the photon and baryon enthalpies, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$, makes their sum zero. Note that this paper's $R$ is the reciprocal of the usual baryon loading parameter. Thomson scattering also relaxes the photon quadrupole, giving the $-{\Gamma }_T{\sigma }_{\gamma }$ term in the stipulated simplified hierarchy. This quadrupole is an anisotropic stress, omitted by a perfect-fluid model; finite relaxation creates Silk damping.

The printed baryon density sign and collision normalization are inconsistent with part (i). The density equation must be ${\delta }_b^{\prime }-k^2{\theta }_b=-h^{\prime }/2$. As a direct check, with $h^{\prime }=0$, nonzero common velocity, and initially ${\delta }_{\gamma }=4{\delta }_b/3$, the printed plus sign gives $({\delta }_{\gamma }-4{\delta }_b/3{)}^{\prime }=8k^2\theta /3$, violating the tight-coupling adiabatic condition immediately. Likewise division of the collision forces by $k^2$ belongs to neither this potential convention nor the usual common velocity-divergence convention; the physical relaxation rate is ${\Gamma }_T$, independent of $k$. Both printed drag terms still cancel in the weighted sum used below, so that particular algebraic combination is unaffected by this normalization error.

For a nonrelativistic distribution, $T^{ij}=\int d^{3}p{p}^i{p}^{j}f/E$ is down by $v^2$ relative to the rest-mass energy density, and the corresponding anisotropic stress is also suppressed. The streaming terms coupling successive angular moments are proportional to $kv$, and stresses from the velocity distribution disappear in the cold limit. Thus, on scales where velocity dispersion is negligible, density and bulk velocity suffice; the photon Boltzmann hierarchy has no analogous small-speed suppression. This argument needs a cold distribution or a controlled small-dispersion approximation. Collisionless nonrelativistic matter with appreciable dispersion or multistreaming need not be an exact perfect fluid, and near photon decoupling the higher photon moments cannot all remain negligible.

Use the corrected baryon density equation from part (ii). Subtracting $4{\delta }_b^{\prime }/3$ from the photon equation gives $({\delta }_{\gamma }-4{\delta }_b/3{)}^{\prime }=(4/3)k^2({\theta }_{\gamma }-{\theta }_b)$. Tight coupling therefore preserves adiabatic initial conditions to leading order: ${\theta }_b={\theta }_{\gamma }=\theta$ and ${\delta }_b=3{\delta }_{\gamma }/4$.

Multiply the photon Euler equation by the paper's inertia ratio $R$ and add the baryon Euler equation before setting the velocities equal. The Thomson scattering terms cancel exactly. With $\mathcal{H}=a^{\prime }/a$ and $c_b^2$ denoting the baryon sound speed, this yieldsNo derivative of $R$ occurs in this instantaneous weighted addition; writing a derivative of total momentum instead would also differentiate the background inertia. Define the photon-baryon sound speedThen the coupled equations areThe sound speed is radiation pressure divided by the total photon-baryon inertia. Baryons add inertia with little pressure, reducing it below $1/\sqrt{3}$. The combined formula is meaningful despite the printed collision-factor error because the same erroneous factor cancels; maintenance of the assumed adiabatic relation does require correcting the baryon continuity sign.

For $R\gg 1$, $3{\tilde{c}}_s^2\simeq 1$, and the baryon-inertia corrections to the common Euler equation vanish. Differentiating the photon continuity equation givesSet aside the gravitational driving terms and balance the quadrupole streaming source against Thomson scattering. With ${\tau }_c={\Gamma }_T^{-1}$ and ${\sigma }_{\gamma }^{\prime }\simeq 0$,The positive friction is Silk damping; this particular truncation gives shear-only photon diffusion damping. Its coefficient follows the supplied simplified quadrupole equation; adding polarization and the full velocity-slip correction changes that coefficient, as in the general photon-baryon diffusion damping equation.

For completeness, put ${\omega }_0=k/\sqrt{3}$, $b=4k^2{\tau }_c/15$ and $D=\frac{1}{2}{\int }_{{\tau }_0}^{\tau }b(s)ds$. Substitution ${\delta }_{\gamma }=e^{-D}y$ gives the exact transformed equationFor constant ${\tau }_c$ the two independent solutions are $e^{-b(\tau -{\tau }_0)/2}\cos [\sqrt{{\omega }_0^2-b^2/4}(\tau -{\tau }_0)]$ and the corresponding sine. Their frequency correction begins at second order in ${\tau }_c$. For variable ${\tau }_c$, retaining first order alone does not automatically remove $b^{\prime }/2$: a general first-order representation isThe intended cosmological approximation additionally uses slow variation on an acoustic period, $\mathcal{H}/k\ll 1$, with tight coupling $k{\tau }_c\ll 1$. Then the $b^{\prime }$ correction to the frequency is subleading and the two independent photon-baryon acoustic oscillator solutions take the resummed damping formA shift of the time origin is absorbed in $A,B$. If the ionization fraction is constant, $n_e\propto a^{-3}$ gives ${\tau }_c\propto a^2\propto {\tau }^4$ during matter domination, and an initial origin at zero gives $k_D^{-2}=2\tau {\tau }_c/75$. Across recombination the ionization fraction varies, so the integral is the general answer. The PDF prints $O({\tau }_c^{-2})$, while the TeX aid prints $O({\tau }_c^2)$; the small-mean-free-time expansion requires the latter interpretation.

The intrinsic photon temperature fluctuation is $\delta T/T={\delta }_{\gamma }/4$. The cosine and sine produce the Cosmic microwave background acoustic peaks; the envelope suppresses their small-scale amplitudes. The corresponding contribution to the Cosmic microwave background power spectrum contains $e^{-2k^2/k_D^2}$, before angular projection and the other temperature sources are included. Thus Silk damping erases the high-multipole acoustic structure progressively rather than shifting every peak to a new frequency.