Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 312 / 2

Throughout this question dots denote conformal time derivatives, and $\mathcal{H}=\dot{a}/a$ is the conformal Hubble parameter. The monopole of the photon Boltzmann hierarchy gives ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$. With the specified velocity convention, the photon continuity equation is thereforeThe dipole equation is ${\dot{\Theta }}_1+k(2{\Theta }_2/5-{\Theta }_0)=\dot{\tau }({\Theta }_1+v_b)+k\psi$. Thus the photon Euler equation isIn particular, $\dot{\tau }<0$ makes the scattering term damp the photon-baryon velocity slip; it does not amplify it.

The linear momentum density of the photon-baryon fluid is weighted by enthalpy, not merely energy density:Elastic Thomson scattering exchanges momentum internally. Its photon force density is $(4{\bar{\rho }}_{\gamma }/3)\dot{\tau }(v_{\gamma }-v_b)$, so the baryon force density is its negative. Dividing by ${\bar{\rho }}_b$ gives the baryon Euler equation with Thomson dragThis $R$ is the baryon loading parameter, the baryon-to-photon inertia ratio. The two collision terms cancel exactly when the equations are weighted by their enthalpies. Background conservation gives ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, so $\dot{R}=\mathcal{H}R$.

The tight-coupling approximation requires the positive scattering rate $\kappa =-\dot{\tau }$ to exceed both $k$ and the conformal Hubble parameter. The photon-baryon velocity slip and photon quadrupole are then small, and the leading fluid motion has $v_b=v_{\gamma }=v$. One must combine the equations before setting the slip to zero: a small slip times a large collision rate can exert a finite force.

Add the photon Euler equation to $R$ times the baryon Euler equation with Thomson drag. The collisions cancel and, neglecting ${\Theta }_2$ at this order, the result isSince $\dot{R}=\mathcal{H}R$, the common velocity obeysThe inertia of the baryons reduces the photon-baryon sound speed to $c_s^2=1/[3(1+R)]$.

Differentiate the photon continuity equation and substitute this velocity equation:Thus the photon-baryon acoustic oscillator isThe terms on the right drive the oscillation gravitationally; the first-derivative term comes from changing baryon inertia. Diffusion damping enters only beyond this leading tight-coupling approximation.

Expand the fractional CMB temperature fluctuation as $\Theta (\mathbf{n})={\sum }_{\ell m}{a}_{\ell m}{Y}_{\ell m}(\mathbf{n})$. Under statistical isotropy, its CMB angular power spectrum is defined byThe normalization is for dimensionless temperature, rather than temperature in kelvin. The spherical harmonics separate angular scales, with large $\ell$ corresponding to small angles.

For approximately constant $R,\varphi ,\psi$, a useful solution of the photon-baryon acoustic oscillator isHere $r_s$ is the comoving sound horizon. With slowly varying $R$, the oscillatory phase is still $k{r}_s$ and its homogeneous amplitude varies approximately as $(1+R{)}^{-1/4}$ under the short-period approximation. Evolving potentials add gravitational driving rather than erase this phase coherence.

The regular growing adiabatic initial conditions produce an initial density displacement and negligible initial acoustic velocity outside the horizon. They therefore select the cosine phase, with $B$ approximately zero, for all modes. At last scattering, extrema of compression and rarefaction occur at $k{r}_s({\eta }_{\ast })\simeq n\pi$. The observed monopole-plus-potential Sachs-Wolfe combination is ${\Theta }_0+\psi ={\delta }_{\gamma }/4+\psi$, so baryon loading shifts its equilibrium and makes alternating peaks unequal. The velocity produces a phase-shifted Doppler CMB anisotropy.

Projection from the Cosmic microwave background last-scattering surface places a mode near $\ell \simeq k{\chi }_{\ast }$, where ${\chi }_{\ast }={\eta }_0-{\eta }_{\ast }$ is its comoving radial distance in a flat universe. The resulting adiabatic acoustic-peak spacing isCoherent acoustic phases turn successive compressions and rarefactions into the CMB peak series. Finite-width last scattering, Doppler projection, evolving potentials and diffusion shift or broaden actual peaks, so this is the requested approximate spacing, not an exact prediction for every peak.