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The Harrison-Peebles-Zeldovich spectrum is the scale-invariant primordial growing-mode spectrum: in a fixed conformal time slice before cosmological horizon crossing, the matter power spectrum is proportional to ${\tau }^{4}k$ in the usual growing-mode density convention. At horizon entry ${\tau }_H\sim k^{-1}$, its dimensionless mass variance $k^{3}P(k,{\tau }_H)$ is independent of $k$. This statement fixes a spectral shape, not its amplitude. Radiation domination subsequently suppresses small-scale density growth through the Mészáros effect, producing the asymptotic matter power spectrum $P\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$ after matter-radiation equality.

After cosmological horizon entry during radiation domination, neglect averaged radiation forcing and the small matter self-gravity term. The CDM density equation in a matter-radiation universe becomes $(\tau {\delta }_C^{\prime }{)}^{\prime }=0$, giving the displayed logarithmic solution. Entry at ${\tau }_h\sim k^{-1}$ matches a superhorizon amplitude of order $A_R/k^2$, fixing the coefficient scale. Detailed radiation forcing near entry determines its order-one numerical factor. This is the Mészáros effect for the initially adiabatic mode.

In deep radiation domination, $a\propto \tau$ and $a^{\prime }/a=1/\tau$. Neglect ${\Omega }_c{\delta }_c$ against the radiation gravity source, so the supplied CDM density equation in a matter-radiation universe becomesOn superhorizon scales omit the $k^2{\delta }_r/3$ pressure-gradient term. Under adiabatic initial conditions, ${\delta }_r=(4/3){\delta }_c$ then givesThe Euler-Cauchy equation characteristic polynomial is $p^2-4$, soThese are the general growing and decaying adiabatic solutions in this approximation. A regular primordial growing mode discards the divergent ${\tau }^{-2}$ solution. Although its conformal-time growth is again quadratic, now ${\delta }_c\propto a^2$, rather than ${\delta }_c\propto a$ in matter domination.

If “general” is taken without imposing adiabaticity, both independent relative-entropy integration constants must also be retained. The gradient-free radiation equation givesSubstitution into the matter equation producesThis is the general solution of the leading supplied coupled system, described by the superhorizon radiation-era entropy integration constants. Setting both $S_0$ and $S_1$ to zero recovers the adiabatic result. Regularity and the full initial Einstein field equations constraints can further restrict which of these leading solutions represents a prescribed physical initial mode.

On scales well inside the horizon, radiation pressure drives oscillations at frequency approximately $k/\sqrt{3}$. Their amplitude is bounded at leading order, and their gravitational forcing of a slowly varying matter contrast can be averaged away. With this approximation, as well as negligible matter self-gravity,hence the logarithmic CDM growth after radiation-era horizon entry isThis is the Mészáros effect. It does not mean that ${\delta }_r$ itself vanishes. For a rapidly oscillating source $R_{\gamma }\cos (\omega \tau +\phi )$, the matter equation has a particular responseIts leading second derivative reproduces $3{\delta }_r/{\tau }^2$; the remaining terms are smaller by powers of $(k\tau {)}^{-1}$. Thus the suppressed cold-matter forcing by rapid radiation oscillations is only order ${\delta }_r/(k\tau {)}^2$ and does not change the secular logarithm. Radiation forcing near horizon entry determines $C_0,C_{\log }$ and cannot be averaged away there. Neither this smoothing approximation nor the neglect of matter gravity remains uniformly valid near equality.

Neglecting the rapidly oscillating radiation source on subhorizon scales, stress-energy conservation and the trace of the Einstein field equations giveEquivalently in cosmic time, ${\delta }_{m,tt}+2H{\delta }_{m,t}-4\pi G{\rho }_m{\delta }_m=0$. Radiation still contributes to the background expansion, even when its perturbation source is averaged away.

Put $\eta =a/a_{\mathrm{eq}}$ and let ${\rho }_{\mathrm{eq}}$ be the density of either component at matter-radiation equality. Then ${\rho }_m={\rho }_{\mathrm{eq}}{\eta }^{-3}$ and ${\rho }_r={\rho }_{\mathrm{eq}}{\eta }^{-4}$. With $C=(8\pi G/3){\rho }_{\mathrm{eq}}{a}_{\mathrm{eq}}^2$, the Friedmann equation givesApply the chain rule to the density equation:Thus the Mészáros equation isSubstituting $D_{+}=1+3\eta /2$ gives $(2+3\eta )(3/2)-3(1+3\eta /2)=0$, sois indeed a solution. It is approximately constant during radiation domination and grows as $a$ during matter domination. The second independent solution iswhich behaves as $\log (4/\eta )-3$ at small $\eta$. Thus smoothing radiation does not eliminate every logarithmic solution; matching through actual horizon entry fixes the combination excited.

Modes with $k\ll k_{\mathrm{eq}}$ enter after equality and retain the $P(k)\propto k$ shape. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination: their horizon-entry spectrum is $P_H\propto k^{-3}$ and they gain little growth until equality. The late matter power spectrum therefore turns over near $k_{\mathrm{eq}}$ and has a large-scale $k$ branch and a small-scale $k^{-3}$ branch, with the more accurate radiation-era matching givingEquivalently the cold-dark-matter transfer function falls approximately as $(k_{\mathrm{eq}}/k{)}^2$, up to the logarithm. The turnover is the consequence of the Mészáros effect; it is not a change of the primordial scale-invariance condition.

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.

There is a conformal-time normalization error in the printed density parameter. For physical background density, $H=\mathcal{H}/a$, so the cosmological density parameter isThe printed definition lacks the numerator's $a^2$. The scale factor in a conformal-time density parameter is essential for ${\Omega }_c\simeq 1$ in the intended flat matter-dominated approximation.

During matter domination, pressureless matter supplies both the expansion and the gravitational source: $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$ and ${\Omega }_c\simeq 1$ when CDM is taken to constitute that matter. ThusTry ${\delta }_c\propto {\tau }^p$. Its characteristic polynomial is $p(p-1)+2p-6=(p-2)(p+3)$, givingRadiation-era photon pressure and radiation perturbations affect the metric source, and matter alone does not dominate the expansion, so this matter-only closure and its power laws are inadequate before equality. At late acceleration the growth also ceases to follow $a\propto {\tau }^2$ and slows. The question's claim that the differential form itself is valid only in matter domination is too strong: for subhorizon pressureless matter with smooth dark energy it still applies with the correct time-dependent $\mathcal{H}$ and ${\Omega }_m$. Indeed, the pressureless growth equation in proper time is $\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$ for smooth dark energy; substituting $d/d{t}_{\mathrm{proper}}=a^{-1}d/d\tau$ gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }-4\pi G{a}^2{\bar{\rho }}_m\delta =0$, the same conformal differential form. It is the matter-dominated coefficients and simple solutions that are restricted to the stated interval.

Outside the radiation-era horizon, the regular growing adiabatic mode has nearly constant curvature, while its comoving-synchronous density perturbation begins at gradient order $k^2{\tau }^2$. Equivalently its metric trace grows as ${\tau }^2$ and the volume relation in (i) gives ${\delta }_c\propto {\tau }^2$. This is a relativistic superhorizon solution, not instantaneous Newtonian communication between causally disconnected regions. Inside the horizon, pressure makes the dominant radiation oscillate and its gravitational potential decay, while CDM self-gravity is still subdominant. CDM then grows only weakly, logarithmically in the more accurate Mészáros effect; the stated freezing approximation replaces that slow growth by a constant.

Assume all modes being compared start outside the horizon at common ${\tau }_i$, and define ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and $G_0=({\tau }_0/{\tau }_i{)}^2$. Modes with $k\le k_{\mathrm{eq}}$ never undergo the radiation-era frozen interval, so their growth is $G_0$. For $k>k_{\mathrm{eq}}$, multiply growth before horizon entry, freezing until equality, and subsequent matter growth:Thus the frozen-radiation approximation to the matter transfer function isThis $T$ includes total growth because the question defines it directly between two density contrasts. The usual transfer normalized to one on large scales is $T/G_0$.

Power is the variance of a Fourier amplitude, so $P_0=|T{|}^2{P}_i$. The prescribed primordial density spectrum givesIt turns over at the comoving equality-horizon scale $2\pi /k_{\mathrm{eq}}={\tau }_{\mathrm{eq}}$. The turnover and reduced small-scale growth help organize the broad spectrum of galaxy clustering, clusters and superclusters; they do not select one unique object size. The dimensionless power $k^3{P}_0/(2{\pi }^2)$ is approximately flat above equality in this crude approximation, so a decreasing dimensional spectrum does not imply no small-scale structures. Restoring radiation-era logarithmic growth adds logarithmic corrections to the small-scale tail. Baryon acoustic features, a smooth equality transition, late growth suppression and nonlinear halo formation are absent from this simple linear broken-power approximation.

For subhorizon pressureless matter during radiation domination, expansion is too rapid for significant self-gravitating growth. Neglecting the small matter-gravity term gives ${\ddot{\delta }}_m+t^{-1}{\dot{\delta }}_m\simeq 0$, hence a constant plus a logarithm of time; the exact coupled radiation problem supplies the detailed amplitudes. This is the Mészáros effect. Modes entering the horizon during radiation domination are suppressed relative to modes entering later, while after equality ordinary matter growth resumes.

The matter power spectrum consequently has a turnover near the equality horizon scale $k_{\mathrm{eq}}\sim a_{\mathrm{eq}}{H}_{\mathrm{eq}}$ in units with $c=1$. Shorter-scale modes retain the radiation-era suppression. This statement concerns cold pressureless matter; baryons coupled to photons also show acoustic effects before recombination.

The continuity and Cosmological Euler equation in synchronous gauge combine particularly simply. Differentiate ${\delta }_C^{\prime }=-\nabla \cdot v_C-h^{\prime }/2$ and add $\mathcal{H}{\delta }_C^{\prime }$:The velocity terms cancel by the Cosmological Euler equation in synchronous gauge, so this step does not require setting the initial velocity exactly to zero.

For subhorizon modes deep in radiation domination, take the rapidly oscillating radiation perturbation to be negligible in the averaged gravitational forcing. Also neglect the subdominant matter self-gravity to leading order in ${\rho }_C/{\rho }_r$. With $a\propto \tau$ and $\mathcal{H}=1/\tau$, the resulting equation isIntegrating twice giveswhere the arbitrary reference time ${\tau }_{\ast }$ makes the logarithm dimensionless and can be absorbed into $B$. This is logarithmic growth of matter perturbations during radiation domination.

The neglect of matter self-gravity is needed for the displayed form to be exact within the leading radiation-background approximation. Uniform radiation alone leaves a nonzero term $4\pi G{a}^2{\rho }_C{\delta }_C$ if the cold-matter density is retained. Relative to ${\mathcal{H}}^2$, its coefficient is $3{\Omega }_C(a)/2$, small deep in the radiation era but not near equality. The Mészáros equation retains that effect in a matter-plus-radiation background; its solution basis tends to a constant and a logarithm at $a/a_{\mathrm{eq}}\ll 1$, recovering the present leading result. It connects smoothly to matter-era growth rather than allowing the logarithmic approximation to be extrapolated indefinitely.

Galaxy-scale modes enter the Hubble radius before matter-radiation equality. The earlier radiation forcing and horizon-entry matching ordinarily generate a nonzero logarithmic coefficient, even though subsequent oscillatory radiation forcing is small. A Fourier mode with entry time ${\tau }_h\sim k^{-1}$ accumulates $\log ({\tau }_{\mathrm{eq}}/{\tau }_h)\sim \log (k{\tau }_{\mathrm{eq}})$ before equality. Across a large scale-factor range this factor need not be small, so treating the density contrast as exactly frozen would lose an important part of the galaxy-scale seed amplitude and its scale dependence. Growth is nevertheless much slower than ${\delta }_C\propto a$ in matter domination. This is the Mészáros effect, and the logarithmic matching contributes to the small-scale matter cold-dark-matter transfer function.

After cosmological horizon entry in radiation domination, the oscillating radiation source averages away. Neglecting both that source and CDM self-gravity in the CDM density equation in a matter-radiation universe givesso ${\delta }_C=C_1(k)+C_2(k)\log \tau$. The superhorizon adiabatic CDM mode in radiation domination has amplitude $A_R(k){\tau }_h^2\sim A_R(k)k^{-2}$ at entry ${\tau }_h\sim k^{-1}$, fixing this scale for the subsequent logarithmic CDM growth after radiation-era horizon entry.

More explicitly, in a sharp-transition approximation with ${\tau }_h=1/k$, continuity of ${\delta }_C$ and ${\delta }_C^{\prime }$ givesThe matching the CDM growing mode at horizon entry therefore yields the late subhorizon scalingHere $\simeq$ suppresses order-one coefficients and the nonlogarithmic term, as in the question. A smooth treatment of the radiation forcing through entry changes those coefficients; the factor two is exact only for the sharp matching model just used. This logarithmic growth, rather than growth proportional to $a$, is the Mészáros effect.

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.

After cosmological horizon crossing, the rapidly oscillating radiation source can be averaged away as specified. Well inside radiation domination, the remaining cold dark matter self-gravity is also subleading because ${\rho }_C/{\rho }_R\ll 1$. To leading order the linear cosmological density perturbation equation is thereforeTwo integrations giveThe reference time makes the logarithm dimensionless. A growing superhorizon mode reaches entry with a nonzero derivative, so matching generally excites $B\ne 0$. For example, an abrupt leading-order match from $\delta \propto {\tau }^2$ gives $B={\tau }_H{\delta }_H^{\prime }\simeq 2{\delta }_H$, whereas its exact numerical coefficient depends on the transition and the radiation forcing.

This logarithmic growth is the Mészáros effect. It is an asymptotic radiation-era result: after dropping the oscillating radiation perturbation, the original equation still contains $4\pi G{a}^2{\rho }_C\delta$. Retaining that term and the changing background leads to the Mészáros equation, which provides the interpolation near matter-radiation equality. It would be incorrect to claim that a pure logarithm solves the complete mixed-background equation exactly.

First consider $k{\tau }_{\mathrm{eq}}<1$. Such a mode remains outside the horizon through matter-radiation equality, grows as ${\tau }^2$ on both sides, and keeps the same matter-era growth after entry. Squaring this growth preserves the shape of the HPZ spectrum:up to a scale-independent matching coefficient.

For $k{\tau }_{\mathrm{eq}}\gg 1$, entry occurs in radiation domination at ${\tau }_H\sim k^{-1}$, with $P_H\sim \mathcal{C}{k}^{-3}$. The density amplitude then grows only logarithmically until equality. ThusDuring matter domination, the matter-era linear growth factor of the density is $(\tau /{\tau }_{\mathrm{eq}}{)}^2$, so its square multiplies the matter power spectrum:The turnover at $k\sim {\tau }_{\mathrm{eq}}^{-1}$ records whether a mode entered during radiation domination. Earlier-entering small-scale modes miss the rapid ${\tau }^2$ growth and receive only the logarithmic Mészáros effect. This is the origin of the logarithmic small-scale matter transfer. The formula is a large-$k$ asymptote: the additive constant in the logarithmic solution and smooth equality matching must be retained near $k{\tau }_{\mathrm{eq}}=1$, where the spectrum does not vanish.

Multiplying by $k^3$ gives the small-scale mass-fluctuation varianceThis gives the requested estimate for modes entering before equality, with the logarithm replaced by a smooth order-one factor in the transition region. On comoving scale $L\sim k^{-1}$, unit variance today requiresFor a numerical illustration faithful to this question's flat, matter-plus-radiation background, take $h=0.7$ and ${\Omega }_r=4.2\times 10^{-5}{h}^{-2}$, hence ${\Omega }_C=1-{\Omega }_r\simeq 1$ rather than the $0.3$ appropriate to a vacuum-dominated model. Part 1(b), now with ${\Omega }_m={\Omega }_C$, gives ${\tau }_{\mathrm{eq}}\simeq 33\mathrm{Mpc}$. In this two-component model the same integral is valid all the way to today, givingWith $L=3\mathrm{Mpc}$, ${\tau }_0/{\tau }_{\mathrm{eq}}\simeq 258$ and $\ln ({\tau }_{\mathrm{eq}}/L)\simeq 2.39$. Therefore $\mathcal{C}\sim [258^4(2.39{)}^2{]}^{-1}\simeq 4\times 10^{-11}$, corresponding to a horizon-entry root-mean-square fluctuation of roughly $6\times 10^{-6}$. A very small primordial amplitude, of order $10^{-10}$ in this schematic normalization, can reach unit small-scale variance by today. Factors from the Fourier convention, smoothing window and equality matching preclude a precise normalization here. The calculation marks the onset of nonlinear structure formation, where extrapolating the linear solution ceases to be valid. For a universe with late vacuum domination, one must replace the matter-only growth law by its actual growth factor before making a numerical inference.

For a fixed coordinate cell, the spatial volume element isThus its perturbation relative to the background expansion, or its perturbed comoving volume, is $\Delta V_c/V_c=h/2$. In the cold-matter rest frame each such cell contains a fixed amount of matter. Since density is mass divided by physical volume,Equivalently, stress-energy conservation gives ${\delta }_c^{\prime }=-h^{\prime }/2$; the stated relation fixes the spatially dependent integration constant by the initial mass-coordinate convention. A contraction of comoving volume gives a positive density contrast.

In the matter era $a\propto {\tau }^2$, so $\mathcal{H}=2/\tau$ and the matter density fraction is approximately one. The evolution equation reduces toFor ${\delta }_c\propto {\tau }^p$, the indicial equation is $p(p-1)+2p-6=(p-2)(p+3)=0$. ThereforeThe growing mode is proportional to $a$, and the decaying mode to $a^{-3/2}$. These are the density solutions after the comoving coordinate choice; the residual time mode found in part (i) explains why a similar power in an unfixed density variable cannot be interpreted in isolation.

There is a normalization omission in the printed density parameter if ${\bar{\rho }}_c$ denotes physical density. Its usual definition in conformal time isThe printed expression lacks $a^2$, unless its density is already rescaled by that factor. The matter-era statement ${\Omega }_c\simeq 1$ and the growth equation use the physical density fraction above.

On superhorizon scales during radiation domination, the regular adiabatic cosmological perturbation has approximately constant primordial curvature. The comoving density perturbation is gradient-suppressed, of order $(k/\mathcal{H}{)}^2$ times that curvature. Since $\mathcal{H}\propto {\tau }^{-1}$, it grows as ${\delta }_c\propto k^2{\tau }^2$ for fixed mode amplitude. This is the regular superhorizon solution in the specified slicing, not causal collapse driven by communication across the entire wavelength.

After radiation-era horizon entry, the dominant radiation develops pressure-supported acoustic oscillations. Its potential no longer provides a steady growing gravitational source, while cold matter is too small a fraction of the total density to produce rapid self-gravitating growth. The result is the Mészáros effect: growth is strongly suppressed until equality. The supplied stagnation approximation replaces that slow evolution by a constant. More accurately, ignoring the rapidly oscillating source at leading order gives ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau \simeq 0$, with a constant and a logarithmic solution. Thus radiation-era subhorizon growth is at most slow logarithmic growth, rather than the matter-era power law; treating it as constant is the explicit approximation used for the simple transfer estimate below.

The broad turnover near $k\sim 10^{-2}{\mathrm{Mpc}}^{-1}$ marks the matter-radiation equality scale. Modes entering the Hubble radius after matter-radiation equality grow efficiently during matter domination. Modes entering earlier have their cold-matter growth delayed by radiation domination: their subhorizon growth is logarithmic rather than proportional to $a$. This is the Mészáros effect. The smaller undulations on the declining side are baryon acoustic oscillations inherited from the photon-baryon fluid, with an oscillatory wavenumber spacing set approximately by $\pi$ divided by the sound horizon. They are damped at sufficiently short scales by photon diffusion and the transfer of the baryon perturbation into the total matter field. The plotted spectrum is a linear prediction; nonlinear evolution changes the actual sufficiently small-scale spectrum at redshift zero.

For an adiabatic growing mode, the comoving curvature perturbation transfers to the matter overdensity asup to an irrelevant overall sign convention for $\mathcal{R}$. Normalize the cosmological transfer function by $T\to 1$ at small $k$ and the linear growth factor by $D\to a$ in matter domination. The relation distinguishes a dimensional matter power spectrum from a dimensionless cosmological power spectrum: the dimensional curvature spectrum is proportional to $k^{-3}{\mathcal{P}}_{\mathcal{R}}(k)$. ThereforeFor locally constant scalar spectral index $n_s$, the asymptotic cold-dark-matter power spectrum isbecause the cold-dark-matter transfer function tends to one at small $k$ and scales as $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. The asymptotes refer to the smooth envelope, not the individual baryonic wiggles. The large-$k$ local logarithmic slope is $n_s-4+2/\ln (k/k_{\mathrm{eq}})$. For the quadratic potential at $N\simeq 60$, derived below, $n_s\simeq 0.967$, so the limiting powers are approximately $k^{0.967}$ and $k^{-3.033}{\ln }^{2}k$. Multiplying by $k^3$ to form dimensionless matter power would change both exponents by three; that is not the plotted quantity.

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.

Once a mode is well inside the horizon, radiation pressure opposes gravitational collapse. For an ideal radiation fluid its cosmological sound speed is $c_s=1/\sqrt{3}$, and its density perturbation undergoes subhorizon radiation acoustic oscillations with frequency approximately $k/\sqrt{3}$. Unlike the superhorizon growing mode, its amplitude does not acquire sustained gravitational growth. Free-streaming relativistic components can additionally phase-mix the perturbation; the acoustic-fluid description is the appropriate idealization for the stated system.

During deep radiation domination, the radiation source in the dark-matter equation oscillates rapidly compared with a Hubble time. Its accumulated late-time effect averages away. More quantitatively, an oscillatory source of magnitude ${\Delta }_r/{\tau }^2$ gives a rapidly varying particular solution of order ${\Delta }_r/(k\tau {)}^2$ when $k\tau \gg 1$. It can therefore be neglected in computing the slowly varying homogeneous dark-matter growth, although its forcing near horizon entry fixes the integration constants.

The dark-matter self-gravity term is also subleading while $a/a_{\mathrm{eq}}\ll 1$. Keeping the leading expansion drag and averaging the radiation source yieldsConsequentlyThis logarithmic growth of matter perturbations during radiation domination is the Mészáros effect. It is the leading behavior well after horizon entry but before equality; close to equality the neglected self-gravity matters. The radiation perturbation is not small simply because it is radiation: its rapid oscillations, rather than a small radiation background density, justify dropping its source for this slow mode.

The Mészáros effect suppresses the growth of modes which enter the horizon during radiation domination. Modes with comoving wavenumber $k\gg k_{\mathrm{eq}}$ enter earlier, but gain only a logarithmic growth factor until matter-radiation equality; modes entering during matter domination grow as $a$ after entry. Consequently small-scale amplitudes are reduced relative to an extrapolation of the unsuppressed large-scale cosmological density power spectrum, creating a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}={\mathcal{H}}_{\mathrm{eq}}$ in units $c=1$.

More quantitatively, for adiabatic primordial fluctuations with scalar spectral index $n_s$, the late-time linear spectrum can be written, up to a time-dependent normalization,The cold-dark-matter transfer function approaches a constant at $k\ll k_{\mathrm{eq}}$, whereas logarithmic small-scale matter transfer gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\log (k/k_{\mathrm{eq}})$ at large $k$. ThusFor approximately scale-invariant initial curvature fluctuations, $n_s\simeq 1$, the large-$k$ scaling is $k^{-3}{\log }^{2}k$, rather than $k$. This discussion ignores baryonic acoustic structure and nonlinear clustering. The Mészáros equation solution basis includes both a constant and a logarithmic branch during radiation domination; the physical logarithmic growth is selected by matching the earlier radiation forcing, so not every mathematical solution grows logarithmically.

During matter domination, $4\pi G{\bar{\rho }}_m=3H^2/2$ and $a\propto t^{2/3}$. Substitution of ${\delta }_m\propto t^p$ gives $p=2/3,-1$, henceDuring radiation domination, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leavesThereforeThis logarithmic behavior is the Mészáros effect for subhorizon cold dark matter.