Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 64 / 2 / i / Solution

Let $\mathcal{H}=a^{\prime }/a$ and ${\theta }_N=i\mathbf{k}\cdot {\mathbf{v}}_N$. Choose the synchronous gauge rest frame of cold dark matter, so ${\theta }_C=0$. Its continuity equation then gives $h^{\prime }=-2{\delta }_C^{\prime }$. Substitution in the trace Einstein equation givesFor radiation in cosmology the continuity and Euler equations are ${\delta }_R^{\prime }+4{\theta }_R/3+2h^{\prime }/3=0$ and ${\theta }_R^{\prime }=k^2{\delta }_R/4$. Differentiate the first, eliminate ${\theta }_R^{\prime }$ and $h^{\prime \prime }$, and obtainradiation in cosmology pressure supplies the acoustic restoring term while its energy density contributes twice as strongly to the trace gravitational source through $1+3w_R=2$.

Deep in radiation domination, $a\propto \tau$, $\mathcal{H}=1/\tau$, ${\Omega }_R\simeq 1$ and ${\Omega }_C\simeq 0$. At long wavelengths $k\tau \ll 1$, the adiabatic initial conditions require ${\delta }_C=3{\delta }_R/4$. The cold-matter equation reduces toA power ${\tau }^p$ gives $p^2-4=0$. The regular growing solution therefore hasThe other adiabatic solution diverges as ${\tau }^{-2}$ and is discarded for the regular primordial growing mode. The relation between the density contrasts fixes their relative entropy to zero, not their subsequent equality inside the acoustic horizon.

One can also solve the leading radiation-era equations explicitly, which proves that the same regular mode develops logarithmic cold-matter growth after horizon entry. Put $x=k\tau /\sqrt{3}$. The first equation gives ${\delta }_R=(x^2{\delta }_{C,xx}+x{\delta }_{C,x})/3$. Substituting in the second givesDefineIts useful derivative is $F^{\prime }=1/x-2\sin x/x^2+2(1-\cos x)/x^3$. Direct differentiation verifies the fourth-order equation and gives the associated radiation in cosmology solution:At zero, $F=x^2/8+O(x^4)$ and ${\delta }_R=K{x}^2/6+O(x^4)$, so this is precisely the regular adiabatic radiation-era cold-dark-matter transfer solution with $K=24A/k^2$. The other indicial behaviors of the fourth-order equation are $1,x,x^{-2}$; they are either nonadiabatic at leading order or singular and do not add another regular growing adiabatic mode.

For $x\gg 1$, the integral is $\gamma +\ln x-\mathrm{Ci}(x)$, where $\gamma$ is the Euler's constant and $\mathrm{Ci}$ the cosine integral. Its oscillatory tail cancels the explicit leading $\sin x/x$ term, givingThus the requested late sound-horizon behavior isAn additive constant in ${\delta }_C$ is absorbed into the reference ${\tau }_{\ast }$; the regular adiabatic solution fixes the acoustic phase, with $D=0$ in the explicit time-origin convention above. A general acoustic mixture has both terms. The physical expansion parameter is $k\tau /\sqrt{3}$; comparisons with $2\pi /k$ describe the same long- and short-wavelength regimes up to fixed numerical factors.

The contrast grows much more slowly after entry during radiation domination: radiation in cosmology undergoes pressure-supported acoustic oscillations and cold matter only the logarithmic Mészáros effect. Modes entering well before matter-radiation equality consequently have suppressed late amplitudes relative to modes that stay outside the horizon until near equality. After matter domination begins, pressureless growing modes can grow as $a$. This scale-dependent history produces the turnover and small-scale suppression of the matter transfer function and affects the initial conditions for large-scale structure of the universe. The early subhorizon limit requires both $k\tau \gg 1$ and $\tau \ll {\tau }_{\mathrm{eq}}$; it is not applicable to a mode that enters only in the matter domination.