Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 3

For radiation in cosmology, ${\bar{P}}_r={\bar{\rho }}_r/3$ and $\delta P_r=\delta {\rho }_r/3$, so both terms proportional to the equation-of-state difference vanish. During matter domination the gravitational potential is constant, ${\Phi }^{\prime }=0$. Writing ${\theta }_r=\nabla \cdot {\mathbf{v}}_r$, the continuity and Euler equations reduce toDifferentiate the first equation and substitute the second:Thus the radiation perturbation obeys the forced acoustic equation

After a spatial Fourier transform, the preceding equation isThe constant particular solution is ${\delta }_r=-4\Phi$. Hence the Sachs-Wolfe combinationcontains only the homogeneous harmonic oscillator modes and satisfiesFor adiabatic initial conditions on large scales, $S=\Phi /3=\mathcal{R}/5$ and $S^{\prime }=0$ at the start of matter domination. Up to an irrelevant choice of the phase origin,Thus the radiation acoustic transfer function is flat at $1/5$ as $k\to 0$ and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the Cosmic microwave background power spectrum.

The baryon transfer function should also oscillate soon after cosmological recombination: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the photon-baryon fluid, and recombination does not instantly erase the resulting density pattern.

Let $k_{\mathrm{eq}}$ be the horizon scale at matter-radiation equality. At a fixed time soon after recombination, the cold-dark-matter transfer function defined as $T_c={\Delta }_c/\mathcal{R}$ has the schematic behaviorwith a smooth turnover near $k_{\mathrm{eq}}$. Large-scale modes enter the horizon during matter domination and the Poisson equation converts an almost scale-independent primordial potential into a density contrast proportional to $k^2$. Small-scale modes enter during radiation domination; radiation controls the potential and pressure prevents it from clustering, so cold-dark-matter growth is only logarithmic until equality. Subsequent matter-era growth multiplies all these modes by the same linear growth factor.

Equivalently, if the conventional matter transfer function is normalized to one as $k\to 0$, it is constant for $k\ll k_{\mathrm{eq}}$ and falls approximately as $\log (k/k_{\mathrm{eq}})/k^2$ for $k\gg k_{\mathrm{eq}}$. Multiplication by the Poisson factor $k^2$ gives the behavior of the definition used here.

Subtracting the baryon equation from the cold-dark-matter equation cancels their common gravitational source and gives the baryon--cold-dark-matter relative density modeBecause the background fractions ${\bar{\rho }}_c/{\bar{\rho }}_m$ and ${\bar{\rho }}_b/{\bar{\rho }}_m$ are constant after recombination, their weighted sum givesDuring matter domination, $a\propto t^{2/3}$. The first equation gives $a^2\dot{D}=\text{constant}$ and henceThe cosmic-time matter density modes giveSolving the definitions for the individual contrasts,soThus $\boxed{f=1}$. Both species feel the same gravitational potential after baryon pressure becomes negligible; the growing total mode overtakes the constant relative mode, so baryons catch up with cold dark matter.

Yes. The baryons carry the oscillatory baryon transfer function inherited from the pre-recombination photon-baryon fluid. Their later infall toward cold-dark-matter overdensities suppresses the relative mode and drives ${\delta }_b/{\delta }_c\to 1$, but gravitational evolution does not remove every scale-dependent acoustic phase. The total cosmological density power spectrum therefore contains the small wiggles known as a baryon acoustic oscillation in the matter power spectrum.