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In an Einstein-de Sitter universe, $H(z)=H_0(1+z{)}^{3/2}$ and the radial comoving distance is $2H_0^{-1}[1-(1+z{)}^{-1/2}]$. A physical Hubble radius $H(z{)}^{-1}$ has comoving size $(1+z)/H(z)$. Dividing by the radial distance gives the displayed small angle, in radians. A Hubble diameter has twice this angle; the matter-era particle-horizon radius also equals twice the Hubble radius. These distinctions matter when comparing approximate CMB angular scales.

For spherical collapse in an Einstein-de Sitter universe, take $y=R/R_{\mathrm{ta}}$ and $\tau =H_{\mathrm{ta}}t$, with ${\Delta }_{\mathrm{ta}}=(3\pi /4{)}^2$. The expanding branch has $\tau ={\Delta }_{\mathrm{ta}}^{-1/2}{\int }_0^y[u/(1-u){]}^{1/2}du$. Its small-radius expansion is $\tau =(8/9\pi )y^{3/2}(1+3y/10+O(y^2))$, and its turnaround time is $2/3$. These constants are linked: using another leading coefficient without changing the time normalization no longer gives a density contrast approaching zero at early times. The collapsing branch is $4/3-{\tau }_{\mathrm{expand}}(y)$.

For pressureless linear cosmological density perturbations in an Einstein-de Sitter universe, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. The growth equation is an Euler differential equation whose indicial equation is $(p-2/3)(p+1)=0$. Hence the independent modes are ${\delta }_{+}\propto t^{2/3}\propto a$ and ${\delta }_{-}\propto t^{-1}\propto a^{-3/2}$. The first is the growing linear growth factor, while the second is a decaying transient.

In an Einstein-de Sitter universe normalized by $a_0=1$, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Integrating the radial null geodesic gives $r(z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })=2(c/H_0)[1-(1+z{)}^{-1/2}]$. The luminosity distance is $(1+z)r(z)$, yielding the displayed expression. Near the observer it is $(c/H_0)[z+z^2/4+O(z^3)]$; the limiting radial comoving distance is finite even though the luminosity distance diverges.

For nonevolving comoving number density $n_0$, identical spectra $L_{\nu }\propto {\nu }^{\alpha }$, and no attenuation, the cosmological background intensity from comoving emissivity in an Einstein-de Sitter universe is proportional to ${\int }_0^{\infty }(1+z{)}^{\alpha -5/2}dz$. The improper integral is finite exactly for $\alpha <3/2$, with value $1/(3/2-\alpha )$. Equality gives logarithmic divergence; larger $\alpha$ gives a power-law divergence. A formation epoch or a spectral break can regularize this idealized model.

For ideal isolated spherical collapse, energy conservation and the virial theorem give $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$ and ${\rho }_{\mathrm{vir}}=8{\rho }_{\mathrm{ta}}$ if the density-profile shape is unchanged. The cycloid gives $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$ and ${\rho }_{\mathrm{ta}}/\bar{\rho }(t_{\mathrm{ta}})=9{\pi }^2/16$. Since the Einstein-de Sitter universe has $\bar{\rho }\propto t^{-2}$, the overdensity at the nominal collapse time is $32(9{\pi }^2/16)=18{\pi }^2$. Hence ${\rho }_{\mathrm{vir}}=18{\pi }^2{\rho }_0(1+z_{\mathrm{coll}}{)}^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.

For dust, curvature and a constant cosmological constant, write $H^2=A{a}^{-3}+B{a}^{-2}+C$. Then $\dot{H}=-3A{a}^{-3}/2-B{a}^{-2}$ and $\ddot{H}=H(9A{a}^{-3}/2+2B{a}^{-2})$, which imply $\ddot{H}+2H\dot{H}-3A{a}^{-3}H/2=0$. Thus $H$ is a solution of the pressureless linear matter perturbation growth equation. In an Einstein-de Sitter universe, $H\propto t^{-1}$ is the decaying mode and the independent growing mode is $t^{2/3}$. Additional dynamical components need not preserve this identity.

For plane-parallel pressureless matter in an Einstein-de Sitter universe, before shell crossing, the symmetrized growing-mode standard perturbation theory density kernel isIn particular $F_{2,s}=(q_1+q_2{)}^2/(2q_1{q}_2)$. An unsymmetrized representation of the one-dimensional beta mode-coupling kernel is $(q_1+q_2)/q_1$; its symmetric part is the expression above.

For an open matter-dominated Friedmann solution, put $u=H_0\sqrt{1-{\Omega }_0}\tau$, $a={\Omega }_0(\cosh u-1)/[2(1-{\Omega }_0)]$ and $\cosh u_0=2/{\Omega }_0-1$. The cosmological bolometric background from conserved sources has $J(u)=\sinh (2u)/4-2\sinh u+3u/2$, since $J^{\prime }=(\cosh u-1{)}^2$ and $dt=adu/[H_0\sqrt{1-{\Omega }_0}]$. At fixed $H_0$, the limit ${\Omega }_0\to 0$ is $cL{n}_0/(2H_0)$, whereas the Einstein-de Sitter universe gives $2cL{n}_0/(5H_0)$. Keeping the cosmic age fixed instead reverses this ordering: the two values are respectively $cL{n}_0{t}_0/2$ and $3cL{n}_0{t}_0/5$.

The luminosity distance is defined by $F=L/(4\pi d_L^2)$, where $L$ is the bolometric luminosity and $F$ the received radiative flux. In the FRW metric a sphere through the observer around a comoving source has area $4\pi a_0^2{r}_e^2$. The cosmological redshift reduces each photon's energy by $(1+z{)}^{-1}$, and cosmological time dilation reduces their arrival rate by another such factor. HenceThe positive radial lookback integral is ${\chi }_e=c{\int }_{t_e}^{t_0}dt/a(t)={\int }_0^{r_e}dr/\sqrt{1-k{r}^2}$. The printed time limits are reversed; also the factor $c$ is required unless units $c=1$ are adopted. The areal coordinate $r_e$ and the radial comoving distance ${\chi }_e$ agree only when $k=0$.

For an Einstein-de Sitter universe, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Using the redshift-time relation in the radial null geodesic integral yieldsThus the Einstein-de Sitter luminosity-distance relation isIts small-redshift expansion $d_L=(c/H_0)[z+z^2/4+O(z^3)]$ recovers Hubble's law.

Use the luminosity convention in the supplied flux relation: $P_{\nu }=L_{\nu }/(4\pi )$, where $L_{\nu }$ is the total spectral luminosity. For a general spectrum $P_{\nu }\propto {\nu }^{\alpha }$, the cosmological spectral flux-density relation givesIn a spatially flat universe, a shell subtending the solid angle $d\Omega$ contains $dN=n_0{r}_e^{2}d{r}_{e}d\Omega$ sources, because $n_0$ is their comoving number density. The cosmological spectral background intensity per unit solid angle is consequentlyThis is the cosmological background intensity from comoving emissivity specialized to identical, nonevolving sources in an Einstein-de Sitter universe. For $\alpha =1$ the integrand reduces to $(1+z{)}^{-3/2}$, or directly $d{I}_{{\nu }_0}=n_0{P}_{{\nu }_0}d{r}_e$. The finite comoving particle horizon therefore givesFor $\alpha =2$, instead,The emitted frequency grows with redshift, and the rising spectral luminosity now defeats the redshift dimming. The Einstein-de Sitter spectral-background convergence criterion is $\alpha <3/2$; equality produces a logarithmic divergence. This divergence describes the unbounded power-law, eternal-population model. A finite formation epoch, a high-frequency spectral break or absorption changes that model and can make its background finite.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$ and $H=2/(3t)$. The Friedmann equation givesThis is an Euler differential equation. Substitution of $\delta =t^p$ gives $p(p-1)+(4/3)p-2/3=0$, namely $(p-2/3)(p+1)=0$. The Einstein-de Sitter density-growth modes are thereforeThe linear growth factor for the growing mode is proportional to $a$, rather than an exponential in cosmic time. These expressions apply while the density contrast is small enough for linear cosmological density perturbations.

An isotropic source's bolometric luminosity is distributed over a wavefront of physical area $4\pi a_0^2{r}^2$. Each photon loses an energy factor $(1+z{)}^{-1}$ through cosmological redshift, and successive photons arrive with a further rate factor $(1+z{)}^{-1}$ through cosmological time dilation. Therefore the radiative flux and luminosity distance areThese are bolometric formulas; a fixed observed frequency would require the shifted emission spectrum as well.

Normalize the scale factor by $a_0=1$. Summing the radiative flux over the source counts on an FLRW past light cone makes the transverse shell-area factors cancel:Thus ${\mathcal{F}}_0$ is power per effective detector area, summed over the whole sky; an all-sky detector of effective area $A_{\det }$ receives $A_{\det }{\mathcal{F}}_0$. The corresponding intensity per solid angle is ${\mathcal{F}}_0/(4\pi )$. Restoring the speed of light multiplies these flux integrals by $c$.

Spatially uniform $n(t)$ by itself does not specify its time evolution. For a persistent population comoving with the expansion, with neither creation nor destruction, the cosmological continuity equation for source number is $\dot{n}+3Hn=0$. Its conserved comoving number density is $n_0=a^{3}n$, so the cosmological bolometric background from conserved sources becomes $L{n}_0\int adt$. For the Einstein-de Sitter universe,The last equality uses $H_0=2/(3t_0)$. The early-time improper integral converges despite the diverging physical source density. If instead the proper number density were maintained at a constant value $n_0$ by continuous source creation, the same geometry would give $3L{n}_0{t}_0/11$. For a completely unspecified source history the general integral above is the determined answer; constancy of individual luminosity alone does not imply number conservation.

For the growing density mode, write $\delta (\mathbf{x},t)=b(t){\delta }_{\ast }(\mathbf{x})$. The new Poisson equation gives ${\nabla }_x^2\psi ={\delta }_{\ast }$ with no time dependence. With periodic zero-mean or fixed decaying boundary conditions, this determines a time-independent $\psi$ up to a spatially constant, dynamically irrelevant function. ThusIn the linear continuity equation, ${\partial }_b\delta ={\delta }_{\ast }$, so ${\nabla }_x\cdot \mathbf{u}=-{\delta }_{\ast }=-{\nabla }_x^2\psi$. For the pure longitudinal growing mode, this gives $\mathbf{u}=-{\nabla }_x\psi$ and hence a time-independent scaled velocity. Equivalently, neglect the quadratic advection term in the motion equation and put $\mathbf{h}=\mathbf{u}+{\nabla }_x\psi$. It obeysThe second expression follows directly from $\ddot{b}+2H\dot{b}=Cb/a^3$: its logarithmic derivative with respect to $b$ is the negative coefficient in the first equation. Excluding the homogeneous decaying-velocity component, $\mathbf{C}=0$, yields the pure growing-mode scaled velocityThe same holds along a trajectory to first order, since $\mathbf{u}\cdot \nabla \mathbf{u}$ is second order.

The pure growing-velocity assumption matters. A growing density alone does not exclude a divergence-free decaying velocity. For example, in an Einstein-de Sitter universe with $b=a$, one may add ${\mathbf{u}}_h=a^{-3/2}\sin (kx){\mathbf{e}}_y$. Its divergence is zero, it solves the homogeneous linear motion equation, and it changes neither $\delta =b{\delta }_{\ast }$ nor $\psi$, yet it is not constant. Thus the printed constant-velocity statement is correct for the intended pure scalar growing mode; without that condition the displayed field is a counterexample. This extra mode is removed by the usual irrotational, no-decaying-mode initial conditions.

In an Einstein-de Sitter universe, a slightly overdense spherical region initially expands nearly with the Hubble flow, with its early physical radius proportional to $t^{2/3}$. Its extra self-gravity gradually slows its expansion relative to the background. The radius reaches a maximum at spherical-collapse turnaround, where the coherent radial velocity is zero; it then contracts. The ideal pressureless top-hat would collapse to a point, but real collisionless matter undergoes crossing and relaxation into a bound system with random motions and virial equilibrium.

The bound-shell solution illustrates the stages:At $\theta =\pi$, $R_m=2A$ and $t_m=\pi B$; the formal cold collapse occurs at $\theta =2\pi$, $t_c=2t_m$. For a uniform isolated sphere, let $U=-3G{M}^2/(5R)$. At turnaround $K_m=0$, whereas the final virial theorem gives $2K_v+U_v=0$. Conservation of energy gives $U_m=K_v+U_v=U_v/2$, henceThe usual spherical-collapse prescription associates virialization with roughly the formal collapse epoch, not with a literal instantaneous halt of the cold shell. The sketch distinguishes the formal point collapse from the schematic settling at the virial radius:

The plateau onset is illustrative. A real cluster's relaxation, ongoing accretion and nonuniform density profile replace the cold solution, and surface pressure or energy exchange can change the half-radius estimate. The radius here is physical, not comoving.

Interpret the requested scale-dependent rms as the contribution from a logarithmic band of wavelengths, or equivalently a band centered at $k\sim R^{-1}$. A global uncut rms is not finite for this scale-free spectrum. The dimensionless density power per logarithmic wavenumber isThe cosmological Poisson equation gives ${\phi }_{\mathbf{k}}=-(4\pi G{a}^2\bar{\rho }){\delta }_{\mathbf{k}}/k^2$. ThereforeThis is the potential fluctuations per logarithmic wavenumber scaling. A simple dimensional argument gives the same answer: the density amplitude on scale $R$ is proportional to $R^{-(n+3)/2}$, and its potential amplitude is of order $G\bar{\rho }(aR{)}^2{\delta }_R$, supplying two additional powers of $R$.

For $n=1$, the Harrison-Zeldovich spectrum has scale-independent potential amplitude per logarithmic interval. If the density-spectrum amplitude grows as $A(t)=b^2(t)A_{\mathrm{in}}$, then $4\pi G{a}^2\bar{\rho }=C/a$, so the potential amplitude evolves as $b/a$. It is constant in the pure growing matter mode of an Einstein-de Sitter universe, where $b=a$.

A literal total variance would be proportional to ${\int }_0^{\infty }{k}^{n-2}dk$. Infrared convergence requires $n>1$ and ultraviolet convergence requires $n<1$, so no index makes both ends finite. At $n=1$ both divergences are logarithmic. Thus cutoffs or a band specification are essential for a numerical rms; the boxed statement describes the intended scale-local fluctuations rather than claiming a finite unsmoothed potential variance.

Use natural units and conformal time $dt=ad\tau$. The conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH$. For the barotropic equation of state $P=(\gamma -1)\rho$, the Friedmann acceleration equation givesThe Friedmann equation gives $8\pi G{a}^2\rho /3={\mathcal{H}}^2+k$. Hence the constant-equation-of-state conformal Riccati equation isFor $\alpha \ne 0$, separation and integration of this Riccati equation yield, after choosing the origin of conformal time,The open positive-energy density branch uses the hyperbolic cotangent, rather than the cotangent. Integrating $a^{\prime }/a=\mathcal{H}$ giveswhere the positive constants are fixed by the energy density normalization. These formulas apply on branches where the bases are positive. For $\alpha >0$, take the Big Bang at $\tau =0$. The cosmic time is thenThe integrals also give the general answer after selecting a suitable reference time when a finite Big Bang endpoint does not exist. The exceptional coasting fluid $\gamma =2/3$ has $\alpha =0$, so $\mathcal{H}=h$ is constant: $a=A{e}^{h\tau }$ and $t=(A/h)e^{h\tau }+t_{\ast }$. If $h=0$, instead $a=A$ and $t=A\tau +t_{\ast }$. A flat fluid with $\gamma =0$ is the constant-density de Sitter spacetime case, for which the flat time integral is logarithmic.

During matter domination, $\gamma =1$ and $\alpha =1/2$. Absorbing numerical factors into separate positive constants gives the closed matter-dominated Friedmann solution, the flat Einstein-de Sitter universe and the open matter-dominated Friedmann solution:To compare the age of a dust universe without a cosmological constant, specify the normalization. At fixed present Hubble parameter $H_0$, with $x=a/a_0$ and present matter cosmological density parameter ${\Omega }_m>0$, the Friedmann equation givesThe integrand decreases strictly with ${\Omega }_m$ for $0<x<1$. Therefore age ordering of dust universes at fixed Hubble parameter gives, on the expanding branches,The final bound is approached by the empty Milne universe. This compares models with the same $H_0$; fixing instead the present scale factor and energy density gives a different comparison.

For pressureless matter with curvature and a possible constant cosmological constant, write the background Friedmann equation asDifferentiating on the expanding branch givesConsequentlyThis proves that the Hubble parameter as a solution of the dust growth equation is valid, including constant-$\Lambda$ and curvature contributions to the background. It does not extend without change to a radiation background or arbitrary dynamical dark energy.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar{\rho }=2/(3t^2)$. The linear matter perturbation growth equation becomesSubstitute $D=t^p$. The indicial equation is $p(p-1)+4p/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe Hubble-parameter solution is the decaying mode in the Einstein-de Sitter universe.

An initially overdense spherical region first expands approximately with the background, but its self-gravity slows it more strongly. It reaches turnaround of spherical collapse at its maximum radius, recollapses, and forms a bound approximately stationary object when infall energy is redistributed into random motions. An exactly cold, uniform pressureless solution would instead collapse to zero radius; virialization replaces that formal endpoint.

For fixed mass $M$ and no cosmological constant, integrate $\ddot{R}=-GM/R^2$ for a bound shell. Its specific energy can be written $\epsilon =-GM/(2A)$, and the cycloidal parametrization isSubstitution in ${\dot{R}}^2/2-GM/R=\epsilon$ verifies this parametrization. At $\theta =\pi$, $R_{\mathrm{ta}}=2A$ and $t_{\mathrm{ta}}=\pi B$; formal collapse occurs at $\theta =2\pi$, so $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$.

For a uniform sphere, $W=-3G{M}^2/(5R)$. At turnaround the coherent kinetic energy vanishes, so $E=W_{\mathrm{ta}}$. At the final equilibrium, the virial theorem gives $2T_{\mathrm{vir}}+W_{\mathrm{vir}}=0$, whence $E=W_{\mathrm{vir}}/2$. Conservation of energy therefore impliesThis virial radius from turnaround energy assumes conserved mass and energy, negligible surface-pressure/tidal terms, and the same gravitational profile coefficient. The sketch's settling segment is qualitative; the cycloid is the exact pre-virialization cold model, not a dissipative relaxation solution.

To compare with the cosmological density, use $\bar{\rho }(t)=1/(6\pi G{t}^2)$ in an Einstein-de Sitter universe. The turnaround sphere hasSince the background density falls by four between $t_{\mathrm{ta}}$ and $t_{\mathrm{coll}}$, the Einstein-de Sitter virial overdensity isFinally $\bar{\rho }(z)={\rho }_0(1+z{)}^3$, so assigning the nominal collapse redshift givesThe density ratio eight compares the same object's physical densities at turnaround and virialization; $18{\pi }^2$ compares its final density with the background at the later collapse epoch.

The outer factor is explicitly $a_{\mathrm{bg}}/a_m=\frac{1}{4}(6\pi t/t_m{)}^{2/3}$, the normalized scale factor of an Einstein-de Sitter universe, with the same enclosed mass and initial bang time as the region. The correction describes a fractional reduction of the region's radius relative to this background:For fixed mass, density varies as inverse radius cubed, so ${\rho }_{\mathrm{region}}/{\rho }_{\mathrm{bg}}=(1-ϵ{)}^{-3}=1+3ϵ+O({ϵ}^2)$. The growing linear density contrast is thereforeThe full spherical collapse reverses its expansion at $t_m$ and reaches zero radius at $t=2t_m$, corresponding to $\theta =2\pi$. In the ideal cold, pressureless model this makes the nonlinear density contrast infinite. Extrapolating the linear growing mode to that same time instead gives the finite linear spherical-collapse thresholdThis is an extrapolation of a linear perturbation, not an accurate finite-radius approximation at the moment of collapse. Virial equilibrium requires relaxation of the cold coherent motion and replaces the ideal singular endpoint.