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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.

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.

The cosmological perfect-fluid continuity equation gives ${\rho }_r={\rho }_{r,0}{a}^{-4}$ and ${\rho }_m={\rho }_{m,0}{a}^{-3}$. In conformal time, $\dot{a}=a^{\prime }/a$ and $H=a^{\prime }/a^2$. Thus the flat Friedmann equation isChoose the expanding branch and put the Big Bang at $\tau =0$. Integration givesInverting it supplies the radiation-matter scale factor in conformal time:The early linear term gives radiation domination, while the late quadratic term gives matter domination. The solution presumes the stated radiation-plus-matter model, without a cosmological constant.

At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$. Since ${\Omega }_{r,0}+{\Omega }_{m,0}=1$,In units $c=1$, the comoving particle horizon at equality is ${\tau }_{\mathrm{eq}}$; its physical size is $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$. The angular diameter distance to the equality surface is $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$. Therefore the horizon angle at matter-radiation equality corresponding to one horizon length isFor ${\Omega }_{r,0}=10^{-4}$, this is $4.20\times 10^{-3}$ radians, or $0.241^{\circ }$, about $14.4$ arcminutes. If quoting the full diameter of a horizon-sized patch, the result is twice this, $0.481^{\circ }$. The small-angle approximation is excellent. Expanding in $\sqrt{{\Omega }_{r,0}}\ll 1$ would give $4.14\times 10^{-3}$ radians; retaining both cosmic components near equality avoids an inaccurate pure-matter extrapolation there. The scale factor cancels between physical size and angular diameter distance, and $H_0$ cancels from the ratio.

If cosmological recombination occurs only a modest expansion after equality, its causal patch still subtends a small part of the sky. Widely separated regions of the observed cosmic microwave background have nearly the same temperature even though their past particle horizons did not overlap in the ordinary decelerating history. Local thermalization after the Big Bang therefore cannot explain this large-angle uniformity. This is the horizon problem; inflation supplies an earlier connected region that can grow to encompass the observed sky.

Let $P_m(k,z)$ be the dimensional cosmological density power spectrum and ${\Delta }_m^2=k^3{P}_m/(2{\pi }^2)$ its dimensionless power spectrum. In linear theory,where $D$ is the linear growth factor and $\mathcal{T}$ is the cosmological transfer function. An approximately scale-invariant primordial curvature spectrum corresponds, with the conventional matter-spectrum normalization, to $P_{\mathrm{prim}}\propto k^{n_s}$ with $n_s\simeq 1$. This does not say that the dimensional density spectrum is white noise or that the early curvature and density perturbations are the same variable.

The crucial scale dependence develops around horizon entry and matter-radiation equality. Modes entering during radiation domination undergo only logarithmic cold-matter growth, whereas modes entering later avoid that suppression. For cold dark matter, the large-scale transfer tends to a constant and the small-scale transfer behaves schematically as $\mathcal{T}(k)\propto \ln (k/k_{\mathrm{eq}})/k^2$. ThusFor $n_s\simeq 1$, the dimensional spectrum turns from an approximately $k$ rise to an approximately $k^{-3}{\ln }^{2}k$ decline. Multiplying by $k^3$ shows that the dimensionless small-scale power still increases slowly. The resulting variance of density fluctuations smoothed on mass $M$ generally increases as $M$ decreases, over the scales relevant to galaxy assembly.

After equality, and over scales where linear growth is scale-independent, the transfer shape is approximately preserved while the amplitude grows. During matter domination, $D\propto a$; late accelerated expansion slows that growth. A region collapses when its linearly evolved density contrast reaches the linear spherical-collapse threshold. Smaller mass scales, having larger variance, typically reach this threshold earlier. Small haloes form first, then accrete matter and merge into larger haloes; galaxies form from cooling baryons in those wells and themselves merge. This small-to-large assembly is hierarchical galaxy formation. It is a statistical trend: rare high peaks can produce unusually massive early objects, and gas cooling and feedback prevent a one-to-one identification of halo collapse with star formation.

For cold dark matter, random particle velocities are small enough that collisionless free streaming erases little power on galactic scales. Warm dark matter retains a larger early velocity dispersion and a larger free-streaming length, suppressing power below a finite scale. There are consequently fewer low-mass haloes, delayed formation of the smallest galaxies, and a lower limit to the hierarchy that can develop. Above the cutoff, warm-matter structure can still assemble hierarchically. The distinction concerns the particles' velocity history and transfer function, not the thermal temperature of gas in a galaxy today. The broad late-time shape was largely established by early horizon-entry physics, with subsequent nonlinear collapse and merging altering the spectrum at high wavenumber.

Normalize the scale factor so that $a_0=1$. With separately conserved pressureless matter and radiation in cosmology, their energy densities scale as $a^{-3}$ and $a^{-4}$. Hence the flat Friedmann equation isEquality of the two densities defines matter-radiation equality, with $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$. Define the dimensionless variableSubstituting $a=a_{\mathrm{eq}}y$ givesBoth reference density fractions are positive. In the two-component flat model their sum at the reference epoch is one; no dark-energy component is being included in this background calculation.

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.

For an adiabatic cosmological perturbation, the baryon and photon fractional density perturbations obey ${\delta }_b=3{\delta }_{\gamma }/4$. Since the pressure of the photon-baryon fluid is supplied by the photons,The photon-baryon sound speed is therefore

Use ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$ in the comoving Jeans lengthWell before matter-radiation equality, photon inertia dominates, $c_s\simeq c/\sqrt{3}$, andOnce baryon loading dominates while tight coupling still holds, $c_s\propto a^{-1/2}$, so ${\lambda }_{J,\mathrm{com}}$ is approximately constant. At cosmological recombination, photon pressure support disappears and the baryonic Jeans scale drops sharply. For a subsequently adiabatic monatomic gas, $T_b\propto a^{-2}$ and $c_s\propto a^{-1}$, givingin an Einstein-de Sitter universe. The requested graph therefore rises as $t^{1/4}$, flattens before recombination, jumps downward there, and then decreases as $t^{-1/3}$.

For collisionless matter, the same instantaneous estimate uses its one-dimensional velocity dispersion ${\sigma }_v$ instead of $c_s$; more precisely, suppression is described by collisionless free streaming. ThusWhile the particles are relativistic, ${\sigma }_v\simeq c$ and ${\lambda }_{J,\mathrm{com}}\propto a^{1/2}$. Once nonrelativistic but thermally coupled to radiation, $T_{\mathrm{cdm}}\propto a^{-1}$, so ${\sigma }_v\propto a^{-1/2}$ and the scale is constant. After kinetic decoupling, momentum redshifts as $a^{-1}$, so ${\sigma }_v\propto a^{-1}$ and ${\lambda }_{J,\mathrm{com}}\propto a^{-1/2}$. The second graph joins these three power laws at $t_{\mathrm{NR}}$ and $t_{\mathrm{dec}}$; unlike the baryonic graph, its final decline begins at dark-matter kinetic decoupling rather than recombination.

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.