 Solution

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.