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Suppose a collisionless particle has momentum $p=T_r\propto a^{-1}$ during pure radiation domination. Since $a\propto t^{1/2}$, integrating $\chi ={\int }_0^{t}v(t^{\prime })d{t}^{\prime }/a(t^{\prime })$ gives the displayed comoving distance. The physical distance at observation is $a\chi =H^{-1}\mathrm{arsinh}(y)/y$, $y=m/T_r$. The printed expression without $a^{-1}$ is consequently a physical distance, or a comoving distance in coordinates normalized to $a=1$ at observation. Its ratio tends to one in the relativistic limit and to $\log (2y)/y$ in the nonrelativistic limit. The accumulated scale differs from an instantaneous velocity-over-expansion-rate estimate.

The dimensional matter power spectrum contains a factor $k^{n_s}$, rather than $k^{n_s+3}$. The cosmological transfer function tends to one below the matter-radiation equality scale and to $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ above it. The smooth spectrum therefore scales as $k^{n_s}$ at small wavenumber and $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$ at large wavenumber. The logarithmic factor records slow cold-matter growth during radiation domination. Multiplying by $k^3$ instead produces dimensionless matter power.

Before cosmological recombination, photon-baryon sound speed controls the baryonic comoving Jeans length. The tightly coupled photon-baryon fluid has $c_s^2=c^2/[3(1+R_b)]$, where $R_b=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })\propto a$. Using total background density in the instantaneous dynamical Jeans estimate, the scale grows as $a$ during radiation domination, approaches a plateau in strongly baryon-loaded matter domination, and drops when photon decoupling removes photon pressure support. Subsequently, adiabatic monatomic-gas cooling gives $T_b\propto a^{-2}$ and a comoving Jeans length proportional to $a^{-1/2}$. Residual Compton scattering can delay that cooling law.

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.

During a decelerating expansion the comoving Hubble radius grows. A fixed comoving wavenumber then passes from a superhorizon scale to a shorter scale when $k\simeq aH$. In radiation domination, $aH=1/\tau$, so this occurs at $k\tau \simeq 1$. Here horizon entry refers to Hubble radius crossing; a true causal horizon instead depends on a conformal-time integral.

Compare a flat background Friedmann equation with a nearly homogeneous curved region on a slicing of equal local Hubble parameter. To first order in its curvature parameter $K$, subtraction gives $\delta \rho =3K/(8\pi G{a}^2)$, or $\delta =K/(a^2{H}^2)$. Conserved curvature therefore gives $\delta \propto a^2$ in radiation domination and $\delta \propto a$ in matter domination. This construction specifies the slicing: two patches at equal proper time generally do not have the same Hubble parameter. For growing dust perturbations, the continuity equation gives $H_s-H=-\dot{\delta }/3$ to first order. These different time comparisons can share growth powers without sharing the same curvature-to-density coefficient.

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.

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 $T(k)\propto k^{-2}\log (k/k_{\mathrm{eq}})$.

In a radiation-dominated toy model with $\Gamma ={\alpha }_Y^4{T}^5/m_Y^4$, setting the interaction rate equal to the Hubble parameter gives the displayed decoupling scale. The transition $T_D=m_Y$ defines $m_{\mathrm{nr}}$. For smaller masses, $x_D=m_Y/T_D=(m_{\mathrm{nr}}/m_Y{)}^{1/3}\gg 1$. An instantaneous chemically equilibrated cosmological particle freeze-out estimate gives $n_Y/s=[45g_Y/(2{\pi }^2{g}_{\ast s})(2\pi {)}^{-3/2}]x_D^{3/2}{e}^{-x_D}$. This estimate presupposes number-changing thermal equilibrium, radiation domination and the assumed rate law, rather than following from elastic thermalization alone.

Within the equal-expansion construction of a, subtract the flat-background Friedmann equation to obtainThe background density scales as ${\rho }_b\propto a^{-4}$ during radiation domination and ${\rho }_b\propto a^{-3}$ during matter domination. ThereforeThese are the regular adiabatic growing-mode scalings on scales outside the Hubble radius, with a specified time slicing. In a consistent comoving description the numerical relation between curvature and density contrast differs from the naive subtraction, but the powers agree. The result does not imply that every gauge's density contrast, every decaying mode, or subhorizon dark matter during radiation domination has these powers.

For $r_s\gg d_H$, a regular adiabatic perturbation follows the large-scale density contrast growth in b; this does not require causal communication across the perturbation. Since $d_H=c/H$, the Friedmann equation gives $d_H\propto a^2$ during radiation domination and $d_H\propto a^{3/2}$ during matter domination. A fixed comoving perturbation has physical size $r_s=aR$, so its ratio to $d_H$ decreases as $a^{-1}$ or $a^{-1/2}$ until cosmological horizon entry.

The numerical “exactly at equality” assertion is inconsistent with the supplied parameters and $d_H=c/H$. Normalize $a_0=1$ and use the supplied ${\Omega }_{\gamma }$ as the whole radiation component. ThenBoth terms must be retained at matter-radiation equality. The present comoving size corresponding to its Hubble radius isIts physical size then is $0.0355\mathrm{Mpc}$. A mode with $R=79\mathrm{Mpc}$ has physical size only $0.0263\mathrm{Mpc}$ at equality, already inside the Hubble radius. More explicitly, putting $q=R{H}_0/c$ in $aR=c/H(a)$ givesThis is a direct numerical counterexample using the stated data. Neglecting one component exactly at equality instead gives $150.5\mathrm{Mpc}$, which also does not produce $79\mathrm{Mpc}$. Late-time dark energy is negligible at these early times.

For conserved nonrelativistic dark matter and radiation, ${\rho }_{\mathrm{DM}}\propto a^{-3}$ and ${\rho }_{\gamma }\propto a^{-4}$, so $y={\rho }_{\mathrm{DM}}/{\rho }_{\gamma }=a/a_{\mathrm{eq}}$ obeys $\dot{y}=yH$. Denote differentiation with respect to $y$ by a prime. ThenThe supplied Friedmann equations implySubstitution in the linear growth equation and division by $y^2{H}^2$ give the Mészáros equationFor $D_1=1+3y/2$, $D_1^{\prime }=3/2$ and $D_1^{\prime \prime }=0$, making the two remaining terms cancel exactly. For reduction of order, put $\delta =f{D}_1$ and $P=(2+3y)/[2y(1+y)]=1/y+1/[2(1+y)]$. The equation reduces to $f^{\prime \prime }+(2D_1^{\prime }/D_1+P)f^{\prime }=0$, whenceUsing the supplied antiderivative and reversing an irrelevant overall sign gives the independent solutionIts Wronskian is $D_1{D}_2^{\prime }-D_1^{\prime }{D}_2=-1/[y\sqrt{1+y}]\ne 0$. The asymptotic expansions areThus subhorizon dark matter during radiation domination has constant and logarithmic modes, rather than the superhorizon $a^2$ growth of b. A combination with negative $A_2$ supplies positive logarithmic growth. During matter domination, the growing mode is proportional to $a$, agreeing with b, while the second mode decays as $a^{-3/2}$. These results neglect subhorizon radiation clustering; the Mészáros equation is the approximation stated in the question, not a full photon–matter perturbation system through horizon entry.

For a $1\mathrm{Mpc}$ mode, evolve the regular superhorizon density contrast as $\delta \propto a^2$ until cosmological horizon entry. In the radiation domination approximation its entry scale is $a_{\mathrm{ent}}\simeq (H_{0}R/c)\sqrt{{\Omega }_{\gamma }}=2.2\times 10^{-6}$, well before matter-radiation equality. At that point use the Mészáros equation basis of d, fixing both constants by continuity of $\delta$ and $\dot{\delta }$, equivalently $\delta$ and $d\delta /d\ln a$.

Explicitly, for entry data ${\delta }_e$ and ${\delta }_e^{\prime }=2{\delta }_e/y_e$, solveThe nonzero Wronskian guarantees a unique match. This produces slow logarithmic growth before equality, then $\delta \propto a$ with a decaying transient after equality. For example, at $y_e\ll 1$, $A_2\simeq -2{\delta }_e$ and $A_1\simeq {\delta }_e+2{\delta }_e[\ln (4/y_e)-3]$. An abrupt match estimates the growth history; a precise amplitude requires a common gauge and the coupled relativistic perturbation equations through entry. Linear evolution also ceases to apply once $|\delta |$ approaches unity, and late dark energy further suppresses growth.

cold dark matter is nonrelativistic and effectively collisionless on the scales of interest. Its small velocity dispersion limits free streaming, allowing small-scale gravitational wells to survive. baryons coupled to photons undergo acoustic motion and damping before recombination; dark matter does not share that photon pressure, and provides potential wells into which baryons can later fall.

For a subhorizon pressureless perturbation, once a Newtonian treatment is applicable,During radiation domination the matter contribution to the background gravity is small; neglecting it gives $\delta \simeq C_1+C_2\log t$. This slow subhorizon growth is the Mészáros effect. During matter domination,in an Einstein-de Sitter universe. Once $\delta$ approaches unity, linear theory fails and collapse, virialization and merging become important.

Modes entering the horizon during radiation domination therefore grow less before equality than longer modes entering later. This scale dependence is encoded in a transfer function, with a schematic linear matter power spectrum $P(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$. cold dark matter supports hierarchical assembly; a sufficiently large warm- or hot-dark-matter free-streaming length suppresses small-scale structure. Superhorizon density contrasts change under a gauge transformation in cosmological perturbation theory, so the subhorizon growth laws should not be applied indiscriminately outside the horizon. The original cold-dark-matter formation model explicitly connects weak free streaming to the spectrum that seeds galaxies and clusters.^[ref]

Growing and decaying modes. Neglect pressure and insert the Einstein-de Sitter background. A trial $\Delta \propto t^p$ givesThereforeThe quoted growth law selects the growing mode; arbitrary initial data can also contain the decaying one. In a static homogeneous Jeans model, $\ddot{\Delta }=(4\pi G\bar{\rho }-k^2{c}_s^2)\Delta$, so an unstable mode grows exponentially, $e^{\sqrt{4\pi G\bar{\rho }-k^2{c}_s^2}t}$. Expansion introduces the $2H\dot{\Delta }$ damping term and a falling background density, yielding power-law growth instead. If the wavevector in the supplied pressure term is comoving, that term would need $R^{-2}$; the pressureless result does not depend on this convention.

Amplitude problem. Using the stated recombination epoch, matter-era growth from then until today is only about $1501$. A baryonic density perturbation of order $10^{-5}$ would reach merely $1.5\times 10^{-2}$, far below the order-unity regime needed for nonlinear collapse. photon coupling also prevents baryons from freely collapsing before recombination. The observed microwave temperature contrast is not an exact direct measurement of baryon density contrast at every scale: acoustic transfer and gravitational redshift matter. The estimate expresses the insufficient growth of tiny photon-coupled baryonic seeds, not an exact conversion between the two observables.

Coupled gravitational response. Both pressureless components respond to the same perturbed potential. The equations areTheir difference obeys $\ddot{D}+2H\dot{D}=0$, where $D={\Delta }_D-{\Delta }_B$. Thus $\dot{D}\propto R^{-2}$ and, in Einstein-de Sitter, $D=C_0+C_1{t}^{-1/3}=C_0+{\tilde{C}}_1{R}^{-1/2}$.

Catch-up branch and its condition. With dark matter dominating the source and in its growing mode, write ${\Delta }_D=AR$. The full baryon solution isThe requested baryon catch-up approximation after recombination selects the branch with negligible relative decaying mode, ${\tilde{C}}_1=0$. Defining $C_0=AR(z_0)$ then givesThis branch applies at and after $z_0$ and has ${\Delta }_B(z_0)=0$ and matched baryon/dark-matter growth velocities at that epoch. It does not follow uniquely from vanishing initial baryon density contrast alone. If both ${\Delta }_B$ and its initial time derivative vanish at $R=R(z_0)$, the zero-initial-baryon-velocity catch-up transient instead givesBoth approach unity for $x\gg 1$. The displayed target therefore needs its stated no-relative-decaying-mode assumption; a finite baryon fraction modifies the transient further.

Physical consequence. cold dark matter is not held to the radiation by Thomson scattering, and can build gravitational wells before baryon decoupling, with appreciable matter-era growth after equality. Its growing perturbations then attract baryons once pressure and radiation drag become unimportant. The relative offset stays constant or decays while the common mode grows, so baryons catch up. Nonlinear structure need not grow solely from the tiny baryonic perturbation inferred at decoupling. cold dark matter growth during radiation domination is slower than the matter-era law, and the stated formula is neither a radiation-era nor a nonlinear-collapse solution.

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

As the temperature falls, electrons and protons form neutral hydrogen through cosmological recombination. The free electron fraction drops sharply, and so does the scattering rate ${\Gamma }_{\gamma }=n_e{\sigma }_{T}c$. When this rate becomes comparable to or smaller than the Hubble parameter, the plasma becomes transparent and photon decoupling occurs. The Thomson scattering cross-section itself does not abruptly vanish. The rapid change is in the abundance of free targets.

The Hydrogen binding energy is $13.6\mathrm{eV}$, but recombination occurs at a much smaller thermal energy, of order $0.3\mathrm{eV}$, or a few thousand kelvin. There are roughly a billion photons per baryon, so the high-energy tail can keep hydrogen ionized long after $k_{B}T$ drops below the binding energy. The falling exponential ionizing tail then makes the transition comparatively sharp. Last scattering follows when sufficiently few electrons remain free.^[ref]

To estimate a horizon, specify an expansion model as well as the redshift. The proper particle horizon at emission isAs a transparent analytic estimate, approximate a spatially flat matter-dominated universe by $a(t)=(t/t_{\mathrm{now}}{)}^{2/3}$ throughout. ThenTaking the illustrative present value $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, this model has $t_{\mathrm{now}}=2/(3H_0)\simeq 9.3\times 10^9\mathrm{yr}$. At $z_{\ast }=10^3$,This is a proper horizon radius at emission; its corresponding present comoving length is about $270\mathrm{Mpc}$.

For the same model the comoving distance to the Cosmic microwave background last-scattering surface and its angular diameter distance areTherefore the angular extent associated with one radial horizon length isThis is an order-one-degree estimate. Including the earlier radiation domination and late dark energy changes the coefficient; it does not make the horizon span the sky. A horizon patch diameter has twice this angle, whereas the acoustic scale uses a sound horizon and differs again. The redshift by itself, without an expansion model or normalization, is insufficient for a unique horizon distance.

The horizon problem is that the observed cosmic microwave background has nearly the same mean temperature across angular separations vastly greater than this scale. In the purely decelerating Hot Big Bang model, sufficiently separated last-scattering regions had neither time to exchange signals nor a common post-Big-Bang causal region that could establish their observed thermal uniformity. The puzzle concerns the missing causal explanation of those initial correlations; it is not an inconsistency of specifying uniform initial conditions mathematically. An earlier period of cosmic inflation can enlarge a formerly causally connected region beyond the later observable horizon.

After the decay products dominate the background, radiation domination gives ${\rho }_r\propto a^{-4}$. Neglecting curvature and vacuum terms, the Friedmann equation implies $\dot{a}/a=C/a^2$, so $a\propto (t-t_B{)}^{1/2}$ andThe displayed $1/(2t)$ form uses a cosmic-time origin with $t_B=0$. After a late transition the extrapolated radiation-era origin need not be the original Big Bang; this time shift does not change the growth law.

For the remaining pressureless matter, neglect its self-gravity to leading order in ${\rho }_m/{\rho }_r$. Its density contrast obeys $\ddot{\delta }+(t-t_B{)}^{-1}\dot{\delta }=0$. Using the conserved expansion-weighted derivative of a passive density perturbation givesThus logarithmic growth of matter perturbations during radiation domination replaces the static exponential growth and the matter-era $t^{2/3}$ growth. The expansion persists while its dominant radiation component supplies little sustained clustering in this pressureless test-component approximation. The residual matter can eventually become important again because ${\rho }_m/{\rho }_r\propto a$; the approximation is for the radiation-dominated interval. The equation is not a pressureless growth equation for the relativistic decay products themselves, whose pressure and, when a fluid description applies, radiation acoustic oscillations must be retained.

For $\Lambda =0$, use the instantaneous cosmological density parameterInsert the Friedmann equation from the preceding part and multiply numerator and denominator by $a^4$:This holds on an expanding or contracting branch with $H\ne 0$; in a recollapsing model the density parameter diverges at turnaround.

During radiation domination, while curvature is small, $|\Omega -1|\simeq |{\Omega }_0-1|a^2/{\Omega }_{R0}$. During matter domination the corresponding behavior is $|\Omega -1|\simeq |{\Omega }_0-1|a/{\Omega }_{M0}$. Thus the exactly flat solution $\Omega =1$ is unstable against relative curvature during decelerating expansion: a small departure grows rather than being dynamically driven to zero. The flatness problem is that a universe with modest present curvature must have had an extremely tiny initial curvature-to-density ratio at an early radiation epoch. The fact that every such solution tends to $\Omega =1$ when extrapolated to $a\to 0$ does not supply a mechanism selecting that small departure at a specified finite initial epoch. An early period of cosmic inflation can instead shrink the comoving Hubble radius and suppress $|\Omega -1|=|k|/(aH{)}^2$.