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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.

During matter domination, the spatially flat background has $a\propto {\tau }^2$ and $\mathcal{H}=2/\tau$. Let ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$ denote the scalar velocity divergence. Linear stress-energy conservation in synchronous gauge givesThe first equation follows from the perturbed volume element $a^3(1+h/2)$; the second states that peculiar dust velocity redshifts as $a^{-1}$. Differentiating the first and substituting the trace of the Einstein field equations, with $w_m=0$ and ${\Omega }_m=1$, yieldsThereforeTrying ${\delta }_m\propto {\tau }^p$ gives $p(p-1)+2p-6=(p-2)(p+3)=0$. The complete solution of this density equation isThese are the matter-era growing and decaying density modes, proportional to $a$ and $a^{-3/2}$. In a dust-comoving synchronous gauge, ${\theta }_m=0$ and $h^{\prime }=-2{\delta }_m^{\prime }$. That rest-frame choice distinguishes physical mode coefficients from changes of density induced by residual synchronous-gauge freedom.

During matter domination, ${\rho }_C\propto a^{-3}$. The flat Friedmann equation in conformal time implies $a^{\prime 2}\propto a$, and henceWith radiation neglected, the CDM density equation in a matter-radiation universe reduces toFor this Euler-Cauchy equation, ${\delta }_C\propto {\tau }^p$ requires $p(p-1)+2p-6=(p-2)(p+3)=0$. Thus the matter-era growing and decaying density modes are ${\tau }^2$ and ${\tau }^{-3}$, respectively, andThere is no explicit $k$ in this pressureless equation, so the result holds on all scales within the stated CDM-comoving synchronous gauge description. The amplitude can still depend on $k$ through its earlier evolution; the result does not equate density contrasts on different time slicings.

During matter domination, the radiation source can be neglected irrespective of whether the mode is inside or outside the horizon in the supplied linear cosmological density perturbation equation. The Friedmann equation gives $\mathcal{H}=2/\tau$ and $4\pi G{a}^2{\rho }_C=6/{\tau }^2$, just as in part (a). ThereforeSubstituting ${\tau }^q$ gives $(q-2)(q+3)=0$, so for every Fourier modeThus the matter-era growing and decaying density modes have growing behaviour ${\delta }_{C,\mathbf{k}}\propto {\tau }^2\propto a$ for all $k$. There is no pressure-gradient term introducing a $k$-dependent growth rate for cold dark matter. The coefficients retain their different earlier histories; common growth does not erase the spectral shape. A specially chosen pure decaying solution is of course possible, so the growing-mode statement does not assert that every initial condition grows.

In matter domination, the Einstein field equations with constant $\Phi$ give the Newtonian-gauge matter density from a constant gravitational potentialWell inside the Hubble radius, the first term is negligible. Since ${\mathcal{H}}^2\propto a^{-1}$ and $\Phi$ is constant,Equivalently, the matter-era growing and decaying density modes follow from ${\delta }_c^{\prime \prime }+\mathcal{H}{\delta }_c^{\prime }-(3/2){\mathcal{H}}^2{\delta }_c=0$: they are ${\tau }^2\propto a$ and ${\tau }^{-3}\propto a^{-3/2}$.

The lower panel of the preceding figure gives the three requested density contrast sketches in Newtonian gauge in cosmology. Their early superhorizon density is nearly constant, rather than proportional to $a^2$ in this gauge. A large-$k$ mode enters first and grows only approximately logarithmically during radiation domination. Once the rapid radiation forcing has subsided, ${\delta }_c^{\prime \prime }+\mathcal{H}{\delta }_c^{\prime }\simeq 0$ with $\mathcal{H}\simeq 1/\tau$, so ${\delta }_c\simeq C+D\ln \tau \simeq C+D\ln a$. This is the Mészáros effect; forcing around entry determines the coefficients. After equality its growing part becomes proportional to $a$.

A mode near $k_{\mathrm{eq}}$ begins substantial growth around equality. A small-$k$ mode keeps its superhorizon constant term until entering during matter domination, then follows the same $a$ growth law. Their entry scale factors satisfy $a_{\mathrm{ent}}\propto k^{-1}$ during radiation domination and $a_{\mathrm{ent}}\propto k^{-2}$ during matter domination. At a common late time, the scaled amplitudes are therefore of orderup to common dimensional constants and order-one matching terms. Thus the three late growth curves have the same logarithmic slope one as functions of $a$, but different amplitudes. These amplitude statements require the modes to have entered the Hubble radius; the full Newtonian-gauge formula above remains available for modes that have not.

In matter domination, the radiation source is negligible and cold matter has negligible pressure. The comoving density equation consequently contains no Jeans-pressure term proportional to $k^2{\Delta }_c$. On every linear scale, including superhorizon and subhorizon modes, it reduces toThe matter-era growing and decaying density modes are thereforeThe absence of a $k$-dependent pressure term makes the time dependence scale-independent; the coefficients still depend on wavenumber and on the earlier radiation-era evolution. This is linear growth in the stated cold-matter model, before nonlinear collapse or any additional late-time component changes the evolution.

The linearized cosmological continuity equation, the divergence of the cosmological Euler equation, and the cosmological Poisson equation giveEliminating $\theta =-{\delta }^{\prime }$ yields the equation for a linear cosmological density perturbation:In an Einstein-de Sitter universe, ${\Omega }_m=1$ and $\mathcal{H}=2/\tau$, soSubstitution of the power-law ansatz $\delta \propto {\tau }^p$ gives $(p-2)(p+3)=0$. HenceThe leading one of the matter-era growing and decaying density modes is $\delta \propto a\propto {\tau }^2$, as direct substitution confirms, and its velocity divergence is