Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 3

Use the comoving cold dark matter choice ${\mathbf{v}}_C=0$ in synchronous gauge. Its continuity equation gives ${\delta }_C^{\prime }=-h^{\prime }/2$. Substituting $h^{\prime }=-2{\delta }_C^{\prime }$ and $h^{\prime \prime }=-2{\delta }_C^{\prime \prime }$ into the metric tensor equation yieldsFor radiation, put ${\vartheta }_R=i\mathbf{k}\cdot {\mathbf{v}}_R$. The continuity and Euler equations become ${\delta }_R^{\prime }=-(4/3){\vartheta }_R-(2/3)h^{\prime }$ and ${\vartheta }_R^{\prime }=k^2{\delta }_R/4$. Differentiating the first and eliminating the other two variables givesThe radiation sound speed is $1/\sqrt{3}$.

During radiation domination, $a\propto \tau$, $\mathcal{H}=1/\tau$ and ${\Omega }_R\simeq 1$. Outside the horizon, $k\tau \ll 1$, the gradient term is negligible. The regular initially adiabatic mode has ${\delta }_R=(4/3){\delta }_C$ to leading order. Consequentlywith powers ${\tau }^2$ and ${\tau }^{-2}$. Keeping the regular growing branch givesThe superhorizon density contrasts here are in the specified comoving synchronous gauge; density contrast is not a gauge-independent superhorizon observable.

Deep inside the horizon, still before equality, radiation pressure produces bounded acoustic oscillations at frequency $\omega =k/\sqrt{3}$. The leading form isTo see the logarithmic growth, average the rapidly oscillating radiation forcing in the Cold dark matter equation and neglect the small matter self-gravity term; the remaining equation is $(\tau {\delta }_C^{\prime }{)}^{\prime }\simeq 0$. Radiation does not have a secular growing acoustic amplitude in this approximation. Its source $(4/3){\delta }_C^{\prime \prime }$ supplies only a smaller slowly varying response, asymptotically $-4C_2/(k^2{\tau }^2)$, while the radiation oscillations induce suppressed oscillatory corrections to Cold dark matter.

The matching can be made explicit in the leading ${\Omega }_R=1$ system. Put $x=k\tau /\sqrt{3}$ and defineEliminating radiation gives $x^2{F}^{(4)}+5x{F}^{(3)}+x^2{F}^{\prime \prime }+x{F}^{\prime }=0$. Direct differentiation verifies that the regular adiabatic solution isIndeed $F(x)\sim x^2/8$ at zero, and the radiation bracket is $x^2/2+O(x^4)$, recovering the early amplitudes. At infinity, $F(x)=\log x+{\gamma }_E-1/2+O(x^{-2})$ and ${\delta }_R=-(2K/3)\cos x+O(x^{-1})$. This adiabatic radiation-era cold-dark-matter transfer solution fixes the logarithmic coefficient and leading acoustic phase rather than leaving matching arbitrary.

The constants and acoustic phase are fixed by matching through horizon entry, not by imposing the early adiabatic relation at every later time. Since entry occurs at ${\tau }_h\sim k^{-1}$, their amplitude scale is set by $A(k)/k^2$. This is the Mészáros effect: logarithmic subhorizon Cold dark matter growth accompanies radiation oscillations until matter becomes dynamically important. Factors such as $2\pi$ in defining a wavelength do not change the two asymptotic regimes.

Continue to choose the CDM-comoving synchronous frame. For nonrelativistic baryons, neglect $O(c_s^2)$ corrections to gravitational density weights and Hubble friction, but retain $c_s^2{k}^2$, since gradients may compensate its small coefficient. ThenTogether with the metric tensor equation these giveThe friction terms contain first derivatives, as the original PDF confirms; they are missing in the TeX's transcription of these equations.

When ${\Omega }_B\ll {\Omega }_C$, Cold dark matter sets the gravitational forcing. Its matter-era growing mode is ${\delta }_C\propto a\propto {\tau }^2$. Compare pressure with the gravitational timescale:Converting the comoving wavenumber to a physical wavelength givesFor $k\ll k_J$, pressure is negligible and baryons can follow the growing Cold dark matter mode. In that limit subtracting the two equations gives $D^{\prime \prime }+2D^{\prime }/\tau =0$ for $D={\delta }_B-{\delta }_C$, so $D=D_0+D_1/\tau$. Relative to ${\delta }_C\propto {\tau }^2$, this offset vanishes: baryons catch up.

The phrase “only grow” needs this gravitational-growth interpretation. For a small baryon fraction the baryon equation is forced, not an autonomous Jeans-instability equation. At $k\gg k_J$, its slowly varying response isplus acoustic transients. It is suppressed rather than identically absent. For example, with constant sound speed, $\mathcal{H}=2/\tau$ and ${\delta }_C=A{\tau }^2$, the equation is ${\delta }_B^{\prime \prime }+2{\delta }_B^{\prime }/\tau +c_s^2{k}^2{\delta }_B=6A$. Its particular solution $6A/(c_s^2{k}^2)$ is constant, and its oscillatory homogeneous part decays as $1/\tau$. This demonstrates Jeans support versus forced baryon response; the scale above is not a theorem forbidding every forced or transient sub-Jeans change.

Before decoupling, tight photon coupling makes the effective sound speed large, about $1/\sqrt{3}$. The small-$c_s$ approximation just derived is not quantitative in that regime; the coupled photon-baryon fluid must be used. Its large pressure supports acoustic oscillations, and baryon clustering below the corresponding large Jeans/sound scale is inhibited while Cold dark matter wells grow. After decoupling the supplied sound speed is tiny and decreases further as the gas cools. The Jeans length drops sharply, so many previously supported modes fall into Cold dark matter wells and approach its growing density contrast. Modes still below the instantaneous pressure-support scale remain suppressed. The subsequent detailed thermal scaling depends on the gas's temperature evolution; the given sound-speed history suffices for this qualitative conclusion.