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Modes with $k\gg k_{\mathrm{eq}}$ enter the horizon during radiation domination. Their matter perturbations grow only logarithmically until matter-radiation equality, rather than as $a$. Matching the radiation-era logarithmic growth onto the matter-era growing mode gives the large-$k$ cosmological transfer function scaling shown, up to constants and the matching scale inside the logarithm. For a primordial scalar spectral index $n_s$, the late-time linear cosmological density power spectrum consequently changes from $P(k)\propto k^{n_s}$ at small $k$ to $P(k)\propto k^{n_s-4}{\log }^2(k/k_{\mathrm{eq}})$ at large $k$. These scalings omit baryonic acoustic structure and nonlinear evolution.

Here the cold-dark-matter transfer function is normalized so that ${\delta }_m(k,a)=A(a)k^{2}T(k)\mathcal{R}(k)$, where $\mathcal{R}$ is the primordial comoving curvature perturbation and $A(a)$ is independent of $k$. During radiation domination, horizon entry occurs at $a_{\mathrm{ent}}\propto k^{-1}$. An adiabatic entry amplitude of order $\mathcal{R}$ then acquires a factor $\log (a_{\mathrm{eq}}/a_{\mathrm{ent}})\sim \log (k/k_{\mathrm{eq}})$ by matter-radiation equality. Subsequent matter domination supplies a common growth factor; division by the conventional $k^2$ prefactor therefore gives $T(k)\propto k^{-2}\log (k/k_{\mathrm{eq}})$.

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.

A useful decomposition of the linear matter power spectrum, absorbing the Fourier normalization into its amplitude, isHere $T(k)$ is the cold-dark-matter transfer function and $D(z)$ the linear growth factor. On scales entering the horizon after matter-radiation equality, $T\simeq 1$, so $P\propto k^{n_s}$ with $n_s$ near one. Modes entering during radiation domination undergo suppressed growth relative to later entrants, producing a turnover near the equality scale and, well above it, $T\propto \ln (k/k_{\mathrm{eq}})/k^2$. Thus the linear spectrum asymptotes approximately to $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$. Baryon acoustic oscillations superpose a weak oscillatory pattern. At sufficiently short scales nonlinear gravitational instability invalidates this linear extrapolation.

The plotted curve illustrates the shape rather than a precision cosmological fit; the bands are overlapping approximate scale sensitivities, not sharp experiment boundaries. Cosmic microwave background anisotropies constrain primordial amplitudes and transfer physics on very large through acoustic scales; converting them to a late-time matter spectrum requires a cosmological model. Galaxy redshift surveys constrain approximately $k\sim 0.01$–$0.2h{\mathrm{Mpc}}^{-1}$, with galaxy bias and redshift-space distortions modeled. Peculiar velocities constrain large-scale mass fluctuations; cluster abundances constrain the fluctuation normalization near cluster scales. Weak gravitational lensing measures projected matter on intermediate and smaller scales. The Lyman-alpha forest extends constraints to roughly $k\sim 0.3$–a few $h{\mathrm{Mpc}}^{-1}$ at high redshift, requiring gas, temperature and ionizing-background modeling. These are complementary measurements of the underlying spectrum rather than interchangeable direct Fourier measurements.

The matter fraction controls the equality turnover. From $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$ and the Friedmann equation at equality,The measured background radiation fixes ${\Omega }_{r,0}{h}^2$, so the physical turnover scale measures approximately ${\Omega }_{m,0}{h}^2$. In units $h{\mathrm{Mpc}}^{-1}$ its location mainly constrains ${\Omega }_{m,0}h$; independent knowledge of $h$ is needed to extract ${\Omega }_{m,0}$. Standard radiation content gives the useful estimate $k_{\mathrm{eq}}\simeq 0.073{\Omega }_{m,0}{h}^2{\mathrm{Mpc}}^{-1}$, approximately $0.015h{\mathrm{Mpc}}^{-1}$ for ${\Omega }_{m,0}=0.3$, $h=0.7$.

The cosmological baryon density sets the inertia of the pre-decoupling photon–baryon fluid and changes its sound horizon and the oscillation pattern. The oscillation contrast and broadband suppression depend on the baryon fraction ${\Omega }_{b,0}/{\Omega }_{m,0}$; the acoustic positions depend on the sound horizon and the distance conversion. The cosmic microwave background peak-height pattern independently constrains ${\Omega }_{b,0}{h}^2$. Combining these with the equality scale and a distance or Hubble constant determination permits estimates of both requested density fractions. Spectral tilt, neutrino free streaming, galaxy bias and nonlinear evolution must be fitted rather than assigning every shape change to baryons. The acoustic signature was already observed in the 2005 galaxy correlation measurement; the first-year microwave-background parameter analysis demonstrates the complementary parameter constraints.

The Jeans length is the physical wavelength at which the restoring pressure term and self-gravity balance. In the elementary static-fluid approximation,Wavelengths longer than ${\lambda }_J$ are unstable to self-gravitating collapse, whereas shorter ones have restoring sound oscillations. Expansion and relativistic effects modify this elementary criterion. Ideal cold dark matter has $c_s=0$, hence no positive pressure-supported Jeans length in these equations. For the specified negative-pressure barotropic string fluid, $c_s^2=-1/3$; there is no real restoring-pressure Jeans threshold.

For $\eta \gg 1$, the matter growth equation has leading termsThe gravitational term is one inverse power smaller than the derivative coefficients. Dropping it at leading order gives $({\eta }^2{D}_{\eta }{)}_{\eta }=0$, hence a constant mode and a decaying ${\eta }^{-1}$ mode. A more controlled check is possible: direct substitution gives $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$ as an exact solution. Reduction of order then gives the growing mode, normalized by $D_{+}/\eta \to 1$ at early times,The integral grows as $\eta +O(\log \eta )$, while the prefactor is ${\eta }^{-1}[1+O({\eta }^{-1})]$. Thuswith corrections of order $\log \eta /\eta$. The perturbation freezes because the matter fraction tends to zero while the coasting fluid controls the expansion.

To describe the requested cold-dark-matter transfer function, its normalization must be specified. The early matter-era Hamiltonian constraint, in the negative-expansion convention for $K$, gives $-4k^2\Psi /a^2-4H\kappa =16\pi G\delta \rho$. For the growing comoving dust mode, the continuity equation gives $\kappa ={\dot{\delta }}_C=H{\delta }_C$ in proper time. Since $16\pi G\bar{\rho }=6H^2$, this yields ${\delta }_C=-2k^2\Psi /[5(aH{)}^2]$. On early superhorizon scales $\Psi$ is the conserved curvature amplitude, up to the chosen overall sign, and $(aH{)}^2\propto a^{-1}$. This establishes the $k^{2}a$ initial growing behavior rather than assuming identical horizon-entry amplitudes at all epochs. For regular adiabatic modes outside the horizon during early matter domination, $A_k=\mathcal{A}{k}^2{\mathcal{R}}_{\ast }(k)$, with a $k$-independent constant $\mathcal{A}$ and primordial curvature amplitude ${\mathcal{R}}_{\ast }$. The matter equation above has no $k$, so at fixed late timeConsequently the density-per-curvature transfer is proportional to $k^2{D}_{+}$ on scales entering after equality. The conventional shape transfer, which factors out $k^2$ and the common growth factor, is a plateau, $T(k)\simeq 1$ on these scales. Comparing with a matter-only universe that continues growing as $\eta$ instead gives a common amplitude suppression $D_{+}/\eta \simeq 5/(2\eta )$, still independent of $k$. A decline caused solely by later horizon entry would contradict the supplied scale-independent matter equation: the regular synchronous growing solution already evolves outside the Hubble radius. This is the scale-independent matter suppression by a coasting fluid.

There is also a limiting entry scale. Since $a^2{\rho }_S$ is constant, define $k_{\ast }^2=(8\pi G/3)a^2{\rho }_S$. ThenA mode enters the Hubble radius only if $k>k_{\ast }$; $k\gg k_{\ast }$ enters during matter domination, whereas $k\downarrow k_{\ast }$ enters arbitrarily late. Modes $k\le k_{\ast }$ never enter in this ideal coasting future. The requested post-equality entrants occupy $k_{\ast }<k<k_{\mathrm{eq}}$. The following original diagram distinguishes the two transfer conventions and marks this restricted interval; it does not attempt to model modes entering during radiation domination.

The adiabatic initial conditions for separately conserved components have equal ${\delta }_N/(1+w_N)$. Therefore the radiation relation is replaced here byTo check dynamically, put $S={\delta }_S-2{\delta }_C/3$. At $k=0$, subtracting two-thirds of the matter equation from the string equation gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$. Selecting $S=0$ and no independent relative-velocity mode preserves the relation. Thus the growing adiabatic string density contrast follows the matter mode and is also approximately frozen at late times to leading order in gradients. This is the adiabatic matter-string density relation.

At late times $\mathcal{H}\to k_{\ast }$, while ${\Omega }_C$ and ${\delta }_C^{\prime }$ vanish. The homogeneous string equation becomesFor $k\gg k_{\ast }$, one solution grows rapidly with exponent approximately $k/\sqrt{3}-k_{\ast }$, instead of undergoing radiation-like acoustic oscillations. This is a gradient instability of a negative-pressure perfect fluid. For $k\ll k_{\ast }$, the growing exponent is only $k^2/(6k_{\ast })$, while the other exponent is approximately $-2k_{\ast }$. The leading zero-gradient approximation is frozen, but at fixed nonzero superhorizon $k$ a slow instability can accumulate over sufficiently long conformal time. The strict gradient limit and infinite-future limit therefore need not commute.

These are the apparent behaviors of the equations given. They do not establish rapid collapse of a realistic cosmic string network: tension, anisotropic stress and the constitutive sound response were excluded in replacing the network by this barotropic perfect fluid. Once the modeled string perturbation becomes nonlinear, the linear calculation itself ceases to apply.

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.

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.

Assume matter-dominated growth over the two epochs, so $D(z)\propto (1+z{)}^{-1}$. A fixed definition of the characteristic mass selects fixed peak height ${\nu }_{\ast }$. Therefore $D(z)\sigma (M_{\ast })$ is constant. With $\sigma (M)\propto M^{-s}$, the observed ratios giveSince a spherical top-hat window function has $M\propto R^3$, the rms slope isFor a cosmological density power spectrum $P(k)\propto k^n$, the smoothed matter density variance isThe dimensionless integral converges for $-3<n<1$. Thus $m=-(n+3)/2$, andFor reference, substituting $s=1/4$ into the differential Press-Schechter halo mass function gives a low-mass prefactor $M^{-7/4}$ and an exponential argument proportional to $M^{1/2}$, so its displayed-form exponents are $\alpha =-7/4$, $\beta =1/2$. The prefactor normalization varies with epoch.

The effective galactic-scale index $-3/2$ need not equal the primordial nearly scale-invariant index one. The cosmological transfer function modifies perturbations between the primordial epoch and galaxy formation: modes entering the horizon in the radiation-dominated era grow differently from larger modes. Schematically $P_m(k,z)\propto D(z{)}^2{k}^{n_s}T(k{)}^2$, with $n_s\simeq 1$, while the cold-dark-matter transfer function decreases toward small scales. The result is a much redder late-time cosmological density power spectrum, with a scale-dependent slope. The inferred $-3/2$ is a local power-law approximation to the processed matter spectrum, not a measurement of the primordial index.

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ a comoving smoothing radius and $W$ the top-hat filter in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and dark-matter halo mergers. This statistical growth from smaller bound systems to larger ones is hierarchical galaxy formation. It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.

For linear modes the linear growth factor gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with galaxy bias; it should not be equated directly with an unbiased matter measurement.

The broad linear shape was set by the cosmological transfer function before and around matter-radiation equality, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The cold-dark-matter transfer function behaves approximately as $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$:The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for hierarchical galaxy formation.

Cold dark matter has negligible primordial thermal velocities and a very short collisionless free streaming length on galactic scales. Warm dark matter has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its cosmological transfer function is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless dark matter develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound dark matter halo. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.

The unheaded request about baryon conversion efficiency of a halo is also answered here. Define $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ using the cosmic baryon fraction $f_b$. In small halos, shallow potentials let stellar feedback drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\mathrm{vir}}^2$. Photoheating during reionization also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher virial temperature and lower cooling efficiency let a substantial hot atmosphere persist. Active-galactic-nucleus feedback can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass. Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.

A scale-invariant primordial potential has $P_{\Phi }(k)\propto k^{-3}$. On subhorizon linear scales, the cosmological Poisson equation and the cold-dark-matter transfer function give ${\delta }_c(k,a)\propto D(a)k^{2}T(k){\Phi }_{\mathrm{prim}}(k)$, where $D(a)\propto a$ during matter domination. ConsequentlyLarge scales enter only after equality and have $T(k)\simeq 1$. Small scales enter during radiation domination; their approximately logarithmic growth up to equality gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$. ThereforeIf the slow logarithm is suppressed in a rough sketch, the slopes are $+1$ and $-3$, with a turnover near $k_{\mathrm{eq}}$. The logarithmic correction is real and should not be mistaken for a different primordial spectral index. In the matter-only late-time approximation with $D(0)=1$, the cosmological redshift $z=1$ means $a=1/2$ and $D(1{)}^2=1/4$; it changes the amplitude, not these asymptotic shapes. Neither the initial normalization nor cosmological parameters needed for an absolute power are supplied.

This is a schematic smooth interpolation with the derived asymptotes, not a precision transfer-function fit. The linear ideal-fluid model excludes baryonic acoustic structure and small-scale nonlinear evolution.

There is also a gauge and horizon qualification. For a mode still outside the Hubble radius at $z=1$, the printed Newtonian-gauge density has ${\delta }_c\simeq -2\Phi$, and its formal dimensional spectrum is instead proportional to $k^{-3}$. The conventional large-scale $k$ branch describes modes that are large relative to the equality scale but already subhorizon at the observation time. Alternatively, the comoving matter density, ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ with ${\theta }_c=\nabla \cdot {\mathbf{v}}_c$, removes that constant gauge term and obeys ${\Delta }_c=-2k^2\Phi /(3{\mathcal{H}}^2)$ in the growing matter solution. The usual matter-spectrum sketch can be continued to small $k$ in this comoving-density convention. The dimensional spectrum requested here is $P$, rather than the dimensionless cosmological power spectrum $k^{3}P/(2{\pi }^2)$, whose slopes would differ by three.

Take regular adiabatic initial conditions and a common scale-invariant primordial amplitude, $k^{3/2}{\Phi }_{\mathrm{prim}}(k)=\text{constant}$. Let $k_{\mathrm{eq}}={\mathcal{H}}_{\mathrm{eq}}$. The three histories are distinguished by their time of cosmological horizon crossing:

The $9/10$ factor follows from conserving the large-scale adiabatic curvature: the constant potential is proportional to $3(1+w)/(5+3w)$, whose matter-to-radiation ratio is $(3/5)/(2/3)=9/10$. Define the cold-dark-matter transfer function by ${\Phi }_{\mathrm{late}}=(9/10)T(k){\Phi }_{\mathrm{prim}}$. Then $T\to 1$ on large scales, while $T\sim (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$ far below the equality length, as explained by the density growth below.

The upper panel sketches $k^{3/2}\Phi$ for the three modes. The lower panel provides the corresponding density contrast histories used in the next part. A negative common primordial potential was chosen so the growing density is positive; this arbitrary phase has no effect on a cosmological density power spectrum.

The curves integrate the ideal coupled radiation-fluid and pressureless-matter equations, rather than patching a pure-radiation solution onto a matter solution. They neglect baryons, free-streaming anisotropic stress, dark energy and nonlinear evolution, consistently with the stated mixture. Dots mark $k=\mathcal{H}$; equality is the dashed vertical line.

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.

Let $k_{\mathrm{eq}}$ be the horizon scale at matter-radiation equality. At a fixed time soon after recombination, the cold-dark-matter transfer function defined as $T_c={\Delta }_c/\mathcal{R}$ has the schematic behaviorwith a smooth turnover near $k_{\mathrm{eq}}$. Large-scale modes enter the horizon during matter domination and the Poisson equation converts an almost scale-independent primordial potential into a density contrast proportional to $k^2$. Small-scale modes enter during radiation domination; radiation controls the potential and pressure prevents it from clustering, so cold-dark-matter growth is only logarithmic until equality. Subsequent matter-era growth multiplies all these modes by the same linear growth factor.

Equivalently, if the conventional matter transfer function is normalized to one as $k\to 0$, it is constant for $k\ll k_{\mathrm{eq}}$ and falls approximately as $\log (k/k_{\mathrm{eq}})/k^2$ for $k\gg k_{\mathrm{eq}}$. Multiplication by the Poisson factor $k^2$ gives the behavior of the definition used here.