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The Alcock-Paczyński parameter compares radial and transverse conversions between observed separations and comoving distances at one cosmological redshift. For a small intrinsically isotropic feature of comoving size $r$, the radial and transverse relations areHere $D_M$ is the transverse comoving distance, $H$ the Hubble parameter, and the angle $\Delta \theta$ is in radians. Eliminating the unknown ruler gives the dimensionless combinationwhere $D_A$ is the angular diameter distance. In spatially flat geometry $D_M=\chi$; the earlier expression $H\chi$ additionally uses units with $c=1$. For nonzero cosmological spatial curvature, the radial comoving distance cannot simply replace $D_M$.^[ref]

This ratio is useful because an absolute ruler calibration cancels. Its inference still assumes statistical isotropy of the selected feature. Peculiar velocities produce redshift-space distortions, so an observed anisotropy must be modeled jointly with these effects before being interpreted as an incorrect distance conversion.

Let $s$ be proper entropy density in a homogeneous FLRW metric and $u^{\mu }$ the comoving fluid four-velocity. Reversible adiabatic expansion has a conserved entropy current,Since a fixed comoving volume has physical volume proportional to $a^3$, the covariant divergence givesThus the conserved quantity is comoving entropy; the proper entropy density falls as $a^{-3}$. This dilution does not require a decrease of the total entropy in the selected material volume.

For equilibrium relativistic matter at negligible chemical potentials, $s$ is proportional to $g_{\ast s}{T}^3$, where $g_{\ast s}$ is the effective number of relativistic entropy degrees of freedom. Conservation gives $aT{g}_{\ast s}^{1/3}=\text{constant}$, so $T\propto a^{-1}$ only while $g_{\ast s}$ is constant. Annihilation into a still-coupled bath can change its temperature relative to a decoupled species while conserving the bath's comoving entropy.^[ref]

Irreversible entropy production replaces the zero divergence by a nonnegative source. Separately decoupled sectors need their own entropy balances; imposing one common temperature after decoupling would add an assumption.

In the Newtonian pressureless approximation, let $\mathbf{x}$ denote comoving positions, $\eta$ conformal time, $a$ the scale factor, and $\mathbf{v}=d\mathbf{x}/d\eta$ the physical peculiar velocity. With $\mathcal{H}=a^{\prime }/a$ and gravitational potential $\varphi$, the Euler equation isAt linear order, define the comoving-coordinate curl ${\mathbf{\omega }}_c={\nabla }_x\times \mathbf{v}$. The gradient force has zero curl, soThe physical spatial derivative is ${\nabla }_{\mathrm{phys}}=a^{-1}{\nabla }_x$. Therefore the physical vorticity is ${\mathbf{\omega }}_{\mathrm{phys}}=a^{-1}{\mathbf{\omega }}_c\propto a^{-2}$. Stating which derivative defines the vorticity prevents confusing these two dilution laws.

This smooth single-stream model omits pressure forces and velocity dispersion. Shell crossing or additional nongravitational forces can invalidate its simple description; the linear decay law is not a universal assertion about every rotating cosmic structure.^[ref]

At a specified Hubble parameter $H$, the critical energy density isThe Friedmann equation sets the spatial-curvature term to zero when the total energy density, including any cosmological constant contribution, equals this value. Dividing by $c^2$ gives the corresponding critical mass density $3H^2/(8\pi G)$. Confusing these two conventions loses a factor of $c^2$.^[ref]

The cosmological density parameter ${\Omega }_i={\rho }_i/{\rho }_{\mathrm{crit}}$ compares one component with this geometric normalization at the same epoch. Spatial flatness requires the sum of the component cosmological density parameters to be one, rather than each component separately. Since ${\rho }_{\mathrm{crit}}$ changes with $H$, a constant energy density need not have a constant cosmological density parameter.

The criterion concerns spatial geometry in an FLRW metric. It does not alone determine the future lifetime of an expanding universe: pressure and dark energy affect its subsequent dynamics. It also differs from critical densities defined by phase transitions or plasma processes.

Linear cosmological perturbation theory describes small departures from a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric. Introduce a bookkeeping parameter $ϵ$ and expand the metric tensor and stress-energy tensor asSubstituting into the Einstein field equations separates the background equations from linear equations for the perturbations; products of two first-order perturbations are discarded. Smallness of $h_{ab}$ is assessed relative to the background metric tensor, not by its determinant alone.

Spatial homogeneity lets one analyze each Fourier mode separately. The fields and particle species at a given wavevector can remain coupled, but different wavevectors do not exchange amplitude through the linear equations. Products retained in nonlinear cosmological perturbation theory instead generate convolutions between modes. The linear density equations help connect initially small fluctuations to the growth of cosmic structure.^[ref]

A cosmological gauge transformation changes how the perturbed and background spacetimes are identified. For example, under $\tilde{\eta }=\eta +T$, a density perturbation transforms aswhere the prime is a conformal time derivative. A nonzero coordinate-dependent density perturbation can therefore represent a time relabelling of unchanged matter. One fixes a gauge or forms invariant combinations before relating perturbations to observations.

Tensor cosmological perturbations describe gravitational-wave degrees of freedom, while density and velocity perturbations describe other responses. A weak gravitational potential does not guarantee a small density contrast; collapsed structures can require a nonlinear matter treatment even when their metric potentials remain small.

In a matter-dominated Einstein-de Sitter universe, the growing linear density contrast is proportional to the scale factor. When $|\delta |\sim 1$, linear evolution no longer describes collapse. An overdense region initially expands more slowly than the background, reaches cosmological turnaround, and then contracts under self-gravity.

The spherical top-hat collapse model supplies a useful idealization: pressureless shells obey $\ddot{r}=-GM/r^2$. At formal collapse, the linearly extrapolated growing mode isThis is a linear extrapolation, not the true finite nonlinear density at that instant. In the ideal energy-conserving spherical model, virial equilibrium gives a virial radius about half the turnaround radius; its mean density at the collapse time is $18{\pi }^2\simeq 178$ times the cosmic matter density.

Real perturbations are nonspherical: tides and anisotropic collapse produce sheets, filaments and haloes. Collisionless dark matter develops shell crossing and phase mixing rather than radiatively losing its orbital energy. Baryonic gas can shock, cool and collect within the resulting potential wells. Hierarchical merging then assembles larger systems where small-scale fluctuations have sufficient initial power. Press-Schechter formalism connects a collapse threshold with the statistics of the smoothed initial field, but halo abundances and internal structure need more than the single spherical model. These ideas apply after perturbations become nonlinear; one must not extend pressureless subhorizon equations through all radiation-era or horizon-scale conditions.^[ref]

Put $H=2/(3t)$ and let $f_B={\rho }_B/\bar{\rho }$, $f_D=1-f_B$ be constant background matter fractions. Define the total matter perturbation $S=f_B{\Delta }_B+f_D{\Delta }_D$ and the relative mode $D={\Delta }_D-{\Delta }_B$. Adding with the background weights and subtracting the two given equations respectively givesThe total and relative density modes of two pressureless matter fluids are consequentlyThese are the general consequences of the supplied equations; a single ratio additionally needs initial-mode assumptions.

For the usual dark-matter-dominated growing-mode approximation, take $f_B\ll 1$, neglect the decaying modes, and set the relative-velocity mode $E=0$. Then ${\Delta }_D\propto t^{2/3}\propto R$. With ${\Delta }_B(t_r)=0$ at recombination, $C={\Delta }_D(t_r)$, so the baryon catch-up approximation after recombination isAt redshifts $z,z_0\gg 1$, replacing $1+z$ by $z$ gives the printed ${\Delta }_B\simeq {\Delta }_D(1-z/z_0)$. The printed relation is a high-redshift approximation with a specified relative mode, not an exact identity for arbitrary solutions of the two-fluid equations.

For example, retaining a finite baryon fraction with a growing total mode and constant relative mode gives, with $x=R/R(t_r)$,Alternatively, if initially both ${\Delta }_B=0$ and ${\dot{\Delta }}_B=0$ while the dark-matter growing mode is already present, then in the dark-matter-dominated limit the relative transient is not zero andThese different solutions show why the initial-velocity convention matters. The quoted catch-up law is the dark-matter-dominated branch with the relative decaying mode neglected, together with the high-redshift approximation stated above. The general solution retains the transient and a finite baryon fraction, so the additional assumptions needed for that special case remain explicit.

The physical conclusion is robust: after decoupling, baryonic gas can fall into pre-existing dark-matter potential wells and its fractional density perturbation catches up. The absolute constant offset need not vanish; it becomes small relative to the growing common mode. The pressureless approximation applies on scales above the baryonic Jeans scale and while perturbations remain linear; subsequent collapse requires nonlinear treatment.^[ref]

Normalize the present scale factor to $a_0=1$. Conservation of the number of electrons associated with the baryons makes their physical number density scale as $a^{-3}$. At this early epoch they are free rather than bound in neutral atoms, soHere the specified present density is interpreted as the conserved total electron abundance, including electrons that might be bound today. A present free-electron density would require an ionization-fraction correction.

Density alone does not determine whether a gas is relativistic; its random kinetic energy must be compared with the rest energy. In the usual thermal Hot Big Bang model, use the measured cosmic microwave background temperature $T_0\simeq 2.7\mathrm{K}$ and the temperature scaling of thermal photon number and energy densities, $T_{\gamma }\propto a^{-1}$. Tight coupling then givesSince $m_e{c}^2\simeq 5.1\times 10^5\mathrm{eV}$,Their thermal speed is of order $\sqrt{3k_B{T}_e/m_e}\simeq 0.04c$. The gas is fully ionized because its temperature is far above the recombination temperature, but ionization does not imply relativistic electrons. Thermal electron-positron pairs are exponentially suppressed at this temperature.^[ref]

The given age and scale-factor ratio should be treated as independent supplied estimates in the following rate comparison. They do not describe the same epoch in standard radiation-era expansion: $a/a_0=10^{-6}$ normally corresponds to an age of order a year, not $10^4$ years. The density and temperature estimates above depend on the scale-factor ratio, not on assigning it the stated age.

A photon travels at $c$, so its typical scattering interval isabout four minutes. The supplied universe age isThusA photon scatters extremely frequently compared with the expansion timescale, of order the cosmic age. The plasma is opaque and its photons are tightly coupled to the electrons rather than freely streaming over a cosmological distance. In rate language ${\Gamma }_{\gamma }=n_e{\sigma }_{e}c\gg H$, where $H$ is the Hubble parameter.^[ref]

This is a statement about transport and momentum coupling. Elastic Thomson scattering alone does not create or destroy photons and hence is not, by itself, proof that an arbitrary photon spectrum reaches complete blackbody thermal equilibrium. The interaction interval calculated here is the interval experienced by a photon; the interval per electron differs because there are many more photons than electrons. Even replacing the inconsistent supplied age by the standard age of order a year at this scale factor would leave scattering much faster than expansion.

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.

There are two first-order effects: an intrinsic temperature change from adiabatic compression, and the gravitational redshift of the emerging photons. Evaluate the excess Newtonian gravitational potential at the boundary of the spherical fluctuation, with the exterior reference potential subtracted:In the weak-field approximation, gravitational time dilation gives $d{\tau }_{\mathrm{local}}/dt\simeq 1+\delta \Phi /c^2$. A local emitter's successive wave crests therefore arrive farther apart at a reference observer outside the potential well. Their frequency changes by $\delta \nu /\nu =\delta \Phi /c^2<0$. Since a redshifted black-body radiation spectrum has its temperature scaled by the same factor as its frequency, the gravitational contribution is

For the intrinsic effect, compare the local scale factor $R+\Delta R$ with the unperturbed value $R$ at the reference epoch. An equilibrium photon gas has energy density $u=a_{\mathrm{rad}}{T}^4$ and pressure $p=u/3$. For adiabatic compression, the first law of thermodynamics applied to a comoving volume $V\propto R^3$ givesThus $TR$ is constant along this adiabatic process, and the initial intrinsic contribution is $\Delta T/T=-\Delta R/R$. The smaller local expansion associated with the fluctuation leaves its radiation hotter; this is the compression or local expansion-history contribution, distinct from the clock-rate effect on the escaping radiation.

Multiplying the two first-order factors, and cancelling the common subsequent background redshift, givesFor the overdense region produced by adiabatic compression, $\Delta R<0$, so the final term is positive. It competes with the negative gravitational redshift term. Their sum is the intrinsic-plus-endpoint part of the Sachs-Wolfe combination; computing a full observed Cosmic microwave background anisotropy can also require Doppler CMB anisotropy and the Integrated Sachs-Wolfe effect.^[ref]