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In a radiation-dominated background with comoving cold dark matter, neglect its contribution to gravity but retain the radiation perturbation. The regular adiabatic initial conditions select $F(x)={\int }_0^x(1-\cos t)dt/t+\sin x/x-(1-\cos x)/x^2-1/2$. It obeys $x^2{F}^{(4)}+5x{F}^{(3)}+x^2{F}^{\prime \prime }+x{F}^{\prime }=0$, with $F\sim x^2/8$ at zero and $F\sim \ln x+\gamma -1/2$ at infinity. The accompanying radiation contrast is $K[-2\cos x+4\sin x/x-4(1-\cos x)/x^2]/3$. This exact leading-radiation solution explains the transition from common superhorizon growth to acoustic radiation oscillations and logarithmic matter growth.

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

A cosmological entropy perturbation changes the relative composition of cosmic fluids while leaving the total curvature initially unchanged. For cold dark matter and photons, a common relative entropy variable is

The galactic angular-momentum catastrophe is the excessive loss or redistribution of baryonic specific angular momentum in early simulated galaxy formation, producing overly compact disks or large central concentrations. Premature cooling, gravitational torques, interactions, and resolution contributed. Improved stellar feedback and gas accretion histories can mitigate it; it is a historical simulation problem rather than a theorem against cold dark matter.

When baryons are a small fraction of the gravitating matter, their perturbation is forced by the cold dark matter potential. Comparing pressure frequency $c_s^2{k}^2$ with $4\pi G{a}^2{\bar{\rho }}_C$ yields the scale ${\lambda }_J=c_s\sqrt{\pi /(G{\bar{\rho }}_C)}$ for following the growing matter mode. It is not an absolute statement that every sub-Jeans baryon perturbation is zero or cannot be forced: a slowly varying response is ${\delta }_B\simeq (k_J^2/k^2){\delta }_C$. In the pressureless limit the difference from CDM is a constant plus a decaying mode during matter domination, so baryons catch up in relative amplitude.

The Lambda-CDM model combines a cosmological constant with cold dark matter and the standard baryonic and radiation components in an expanding Friedmann-Lemaître-Robertson-Walker metric.

Let ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

In deep matter domination, the scale factor is $a\propto {\tau }^2$, so $a^{\prime }/a=2/\tau$. Neglect radiation in the gravity source, including its backreaction, to obtainFor a power ${\tau }^p$ the Euler-Cauchy equation becomes $p(p-1)+2p-6=(p-2)(p+3)=0$. Its two independent matter-era growing and decaying density modes give the full approximate solutionThere is no $k$-dependent pressure term for cold dark matter. Thus this leading equation is valid both outside and inside the horizon in the stated comoving synchronous gauge, as long as the perturbations remain linear and the omitted radiation gravity is negligible. A different time slicing would give a different density variable.

For $k>0$, put $\omega =k/\sqrt{3}$, $C_g=A/{\tau }_i^2$ and $C_d=B{\tau }_i^3$. The remaining radiation equation is a forced harmonic oscillator:The matter-era forced radiation response can be written exactly within the matter-background approximation asThe last term is a Green's function particular solution; changing its lower limit only changes the two oscillatory constants. For the decaying source, a slowly varying particular solution obeys ${\delta }_{r,p}\simeq (4/3){\delta }_c^{\prime \prime }/{\omega }^2$, since a second time derivative is smaller than ${\omega }^2{\delta }_{r,p}$ by order $(k\tau {)}^{-2}$. Consequently the subhorizon asymptotic form isFor a pure growing mode, the constant offset and acoustic terms are an exact solution of this reduced radiation equation. In particular, the late radiation contrast does not follow the ever-growing ${\delta }_c$.

Before crossing, a regular growing adiabatic cosmological perturbation has both contrasts proportional to ${\tau }^2$, with ${\delta }_r=(4/3){\delta }_c$ to leading order. Afterwards cold dark matter keeps this quadratic growth, whereas radiation undergoes subhorizon radiation acoustic oscillations about a constant forced offset. To make the requested sketch concrete, one regular leading matter-era solution on both sides isIts small-$k\tau$ expansion gives ${\delta }_r=(4/3){\delta }_c+O((k\tau {)}^4)$, and its oscillation frequency is the radiation sound frequency. The sketch plots amplitudes in units of $\epsilon$; $\epsilon$ can be chosen small enough that the whole plotted interval remains linear. The line at ${\tau }_h$ uses the wavelength-to-particle-horizon convention in the question, rather than asserting that the smooth transition occurs at a sharp surface.

The perturbations are density contrasts, ${\delta }_c=\delta {\rho }_c/{\bar{\rho }}_c$ and ${\delta }_r=\delta {\rho }_r/{\bar{\rho }}_r$, evaluated in the common comoving synchronous gauge. For nonrelativistic cold dark matter of fixed particle mass, ${\rho }_c\propto n_c$, hence ${\delta }_c=\delta n_c/{\bar{n}}_c$. Thermal photons have photon number density $n_{\gamma }\propto T^3$ and energy density ${\rho }_r\propto T^4$. A small local temperature perturbation therefore givesFor adiabatic initial conditions, the ratio $n_{\gamma }/n_c$ is unperturbed, so $\delta n_{\gamma }/{\bar{n}}_{\gamma }=\delta n_c/{\bar{n}}_c$. Combining these equations provesEquivalently ${\delta }_r/(1+w_r)={\delta }_c/(1+w_c)$, with $w_r=1/3$ and $w_c=0$. The relation applies before causal pressure gradients can separate the two species on superhorizon scales; it need not remain true after cosmological horizon crossing.

Put $\mathcal{H}=a^{\prime }/a$ and choose the residual synchronous frame comoving with cold dark matter, so ${\mathbf{v}}_C=0$. Its Euler equation preserves this choice. The continuity equation then gives ${\delta }_C^{\prime }=h^{\prime }/2$. Neglect background baryon pressure and all unenhanced corrections of order $c_s^2$, but retain $c_s^2{k}^2$ because a short wavelength can make it important. The trace equation becomesUsing $h^{\prime }=2{\delta }_C^{\prime }$ provesFor baryons let ${\theta }_B=i\mathbf{k}\cdot {\mathbf{v}}_B$. The retained continuity and Euler terms are ${\delta }_B^{\prime }=-{\theta }_B+h^{\prime }/2$ and ${\theta }_B^{\prime }+\mathcal{H}{\theta }_B-c_s^2{k}^2{\delta }_B=0$. Differentiate the first and substitute the second:HenceThe pressure sign is stabilizing. The drag terms contain first derivatives of the density perturbations, as printed in the PDF; those primes are lost in the converted TeX. The approximation assumes slowly varying nonrelativistic background baryon pressure and the growing physical mode, not a residual synchronous gauge mode.

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.