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For a pressureless overdensity initialized with the background Hubble flow in an Einstein-de Sitter universe, the cycloid $r=A(1-\cos \theta )$, $t+T=B(\theta -\sin \theta )$ has $A=r_i(1+{\delta }_i)/(2{\delta }_i)$, $B=(1+{\delta }_i)/(2H_i{\delta }_i^{3/2})$, and ${\theta }_i=2\arctan \sqrt{{\delta }_i}$. These follow from conservation of energy and matching the initial radius and velocity. With $t_i=2/(3H_i)$, $T=B({\theta }_i-\sin {\theta }_i)-t_i=-t_i{\delta }_i/5+O(t_i{\delta }_i^2)$. Thus an exactly common background and perturbation bang time is incompatible with these finite-time data. Linear mass conservation instead gives a growing amplitude $3{\delta }_i/5$ and a decaying amplitude $2{\delta }_i/5$, whose initial time derivatives cancel.

A system whose mutual gravity confines its constituents rather than letting them separate indefinitely. In an isolated two-body problem, the relative specific orbital energy $\epsilon =v^2/2-G(M_1+M_2)/r$ is negative exactly for bound, noncollisional Kepler orbits; $\epsilon =0$ is the marginal escape case. External forces, collisions and time-dependent potentials require additional qualifications. The separation of a gravitationally bound system is determined by its equations of motion and need not follow the cosmological Hubble flow.

Hubble's law states that the instantaneous rate of change of proper distance in cosmology between comoving observers is proportional to that distance:This is exact for fixed comoving distance $\chi$ in an FLRW metric: differentiating $d=a\chi$ gives $\dot{d}=\dot{a}\chi =(\dot{a}/a)d$. The dot uses cosmic time, and both distances are compared on spatial slices at the same cosmic time. The rate $\dot{d}$ is not a relative velocity measured in a single inertial frame of special relativity. Thus the instantaneous relation is not an exact observed redshift-distance formula. The associated Hubble flow describes the ideal expansion; peculiar velocities describe departures from it.

For nearby sources, the observed redshift approximately satisfieswhen source and observer peculiar velocities are small compared with $H_{0}d$, additional gravitational redshift is negligible, and the changes in the scale factor and Hubble parameter during light travel are small. Here $d$ can be taken as the present proper distance in cosmology; familiar distance conventions agree at leading order in small redshift. Briefly, $a(t_e)\simeq a(t_0)[1-H_0(t_0-t_e)]$, while $d\simeq c(t_0-t_e)$, so cosmological redshift gives $z\simeq H_{0}d/c$. At greater redshift, use an expansion history and an explicit distance convention instead.

Stipulate the illustrative Hubble constant $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and speed of light $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$. A $\mathrm{Mpc}$ is a million parsecs. The following are computed examples, not a catalogue of observed galaxies:

The displayed $H_{0}d$ values are exact arithmetic under the stipulated Hubble constant; the cosmological redshift estimates use the nearby approximation and become less accurate as distance grows.

Hubble's law turns a nearby observed redshift into an approximate distance indicator and makes the uniformity of cosmic expansion tangible: twice the proper distance in cosmology means twice the instantaneous separation rate. Recentring the ideal Hubble flow on another comoving observer preserves the same proportionality, as illustrated under scale factor, so recession in every direction does not select a central observer.

In his 1929 PNAS paper, Edwin Hubble presented a relation between the distances and radial velocities of extragalactic objects. His fits gave coefficients of $465$ and $513\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ for 24 individual objects and nine groups respectively, and he used about $500\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as a rounded representative value. Those historical fits used the paper's distance estimates; they are distinct from the stipulated modern-scale illustration above. ^[ref].

The Lyman-alpha forest is the multitude of foreground Lyman-alpha absorption features in a distant quasar's spectrum, at wavelengths shorter than its own redshifted Lyman-alpha emission. The quasar supplies a bright background; intervening neutral hydrogen atoms remove photons near their local Lyman-alpha line resonance. Many dips can therefore come from one atomic transition encountered at different redshifts, rather than from many different chemical elements. Some features are instead transitions of other ions, so a dip alone does not identify Lyman-alpha absorption. This makes faint foreground gas detectable through the light it removes, even when its own emission would be difficult to see. ^[ref].

Ignoring peculiar velocity for the moment, an absorber at cosmological redshift $z_a$ puts its Lyman-alpha line atThe rest wavelength is in the ultraviolet; at $z_a=2$ it appears at $3647.01\text{A˚}$ in the optical spectrum. In a toy sightline to a quasar at $z_Q=3$, absorbers at $z_a=2,2.35,2.7$ appear at $3647.01,4072.4945,4497.979\text{A˚}$, all shortward of the quasar's Lyman-alpha emission position $4862.68\text{A˚}$. The absorbing atoms see the selected photons at their own resonance; the observed dip's redshift belongs to the foreground gas. Peculiar velocity adds a Doppler shift, so the inferred $z_a$ is not a pure distance coordinate. The rest wavelength and resonance interpretation are given in Rauch (1998), section 1.

The diagram uses invented optical depths and 26 Doppler cores; three selected foreground regions are marked. Their drawn positions and sizes are schematic. The dashed comparison imposes a uniform optical depth $\tau =2$, giving transmission $e^{-2}\simeq 0.135$ throughout the chosen foreground band. It is an idealized smooth-absorption model, not a measured spectrum or a prediction of the actual intergalactic medium.

The smooth comparison explains the Gunn-Peterson effect: neutral hydrogen distributed throughout an expanding intergalactic medium can absorb continuously over a wavelength interval, because different parts of the path resonate with different observed wavelengths. The resolved Lyman-alpha forest instead has strong transmission fluctuations and distinguishable absorption lines. These need not represent isolated compact clouds: diffuse sheets and filaments in the cosmic web can generate peaks in optical depth along one line of sight. A continuously varying gas distribution can look like a collection of lines. ^[ref]. ^[ref].

An early observation makes the identification concrete. Lynds's 1971 paper analyzed six spectrograms of the quasar 4C 05.34, with emission redshift $2.877$, taken in spring 1970 using an image-tube spectrograph on Kitt Peak's 84-inch telescope. Five covered $3200$–$7500\text{A˚}$ with approximately $10\text{A˚}$ resolution; the most informative covered about $3500$–$6000\text{A˚}$ at $5\text{A˚}$ resolution. Its reproduced spectrogram shows the strong Lyman-alpha emission flanked mostly on the short-wavelength side by numerous absorption lines. Lynds tabulated their wavelengths and estimated strengths, identified five absorption redshift systems, and proposed that many remaining unidentified lines were the same Lyman-alpha line at additional redshifts. He explicitly warned about blending, an incomplete census of weak lines, and wavelength errors around $0.5$–$1\text{A˚}$. The result established a striking spectral pattern; the paper's proposed interpretation did not measure a separate physical cloud for every dip. ^[ref].

A contrasting observation demonstrates smooth, very strong absorption. Becker and collaborators (2001) used the Echelle Spectrograph and Imager on Keck II to observe the quasar SDSSp J103027.10+052455.0 at $z_Q=6.28$. Sky lines gave an instrumental full width at half maximum of about $1.8\text{A˚}$ near $8500\text{A˚}$, or spectral resolving power $R\simeq 4700$. After sky subtraction and flux calibration, the mean continuum-normalized transmission over $8450$–$8710\text{A˚}$ was $0.0038\pm 0.0026$, consistent with zero; this corresponds to absorption redshifts approximately $5.95$–$6.16$. They identified a Gunn-Peterson trough. Crucially, resonant Lyman-alpha absorption is so sensitive that even a small neutral fraction can make transmission undetectable: the trough alone does not establish a mostly neutral intergalactic medium or uniquely date reionization. ^[ref].

What a spectrum constrains is a radiative transfer problem. For an ideal foreground screen with negligible emission into the observed beam, the intrinsic continuum-normalized transmission isHere $F$ is spectral radiative flux, $F_{\mathrm{cont}}$ is its unabsorbed continuum, $\tau$ is optical depth, $n_{\mathrm{HI}}$ is neutral hydrogen number density, and $N_{\mathrm{HI}}$ is neutral hydrogen column density along proper path length $\ell$. A Voigt profile model constrains redshift, column density and a velocity width together. A nearly black line has lost much of its sensitivity to further increases in optical depth; blending and an uncertain continuum can also admit different fitted components. Finite spectral resolving power and noise limit which widths and weak features can be measured. ^[ref]. ^[ref].

For a purely thermal Doppler core, use the velocity convention $\tau (v)\propto \exp [-(v-v_0{)}^2/b^2]$. The Maxwell-Boltzmann velocity distribution then giveswhere $k_B$ is the Boltzmann constant, $T$ is temperature and $m_{\mathrm{H}}$ is a hydrogen atom's mass. This $b$ is $\sqrt{2}$ times the Gaussian distribution's standard deviation. Higher temperature spreads the same column density over more velocities. An instrument also broadens a line: locally, for a nearly flat continuum, the recorded transmission is approximately the convolution of $\mathcal{T}$ with the normalized instrumental response, not the exponential of a convolved optical depth. A Gaussian response with full width at half maximum $c/R$ has standard deviation $c/[R\sqrt{8\log 2}]$, where $c$ is the speed of light and $R$ the spectral resolving power.

The upper model fixes the integral $\int \tau (v)dv$, hence the neutral hydrogen column density for this transition, while changing temperature. A cooler line blurred by an instrument resembles a warmer intrinsic line. The lower model has two equal Doppler cores separated by $24\mathrm{km}{\mathrm{s}}^{-1}$: reducing spectral resolving power merges them into one apparent feature. Both panels use invented profiles, Gaussian instrumental responses, and no noise or damping wings. They illustrate an inverse problem, rather than measurements of temperature or a universal relation between a dip and a cloud.

Better observations provide the inputs to that inverse problem. Kim, Cristiani and D'Odorico (2001) analyzed three quasars with the VLT's UVES spectrograph. For HE2217–2818, Table 1 records $z_Q=2.413$ and $16200\mathrm{s}$ of exposure in the $3050$–$3860\text{A˚}$ setup in October 1999. Isolated thorium-argon lamp lines measured $R\simeq 45000$; the data were resampled into $0.05\text{A˚}$ bins, with signal-to-noise ratios about $45$–$50$ per pixel over $3550$–$4050\text{A˚}$. They fitted a local polynomial continuum, fitted Voigt profiles, and compared statistics of the transmitted flux. Their analysis excluded the region near the quasar to avoid the quasar proximity effect and used the interval between its Lyman-beta and Lyman-alpha emission to reduce confusion with other Lyman-series absorption. Features below $N_{\mathrm{HI}}=10^{12.5}\mathrm{c}{\mathrm{m}}^{-2}$ became confused with noise, setting their adopted lower column density limit. Figure 1 shows the measured spectrum, fitted profiles and residuals. This was a calibrated absorption measurement, with explicit selection limits. ^[ref].

Even after instrumental correction, a width need not be purely thermal Doppler broadening. Hubble flow across extended gas, peculiar velocity structure and unresolved blends can contribute. If an isolated line's intrinsic width obeys $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nt}}^2$ with an independent Gaussian nonthermal contribution $b_{\mathrm{nt}}^2\ge 0$, then $T\le m_{\mathrm{H}}{b}^2/(2k_B)$; $b=25\mathrm{km}{\mathrm{s}}^{-1}$ gives a computed upper bound about $3.79\times 10^4\mathrm{K}$. A complex blend need not satisfy this single-component model. Noise can also bias the fitted widths of weak lines. This is why the thermal cutoff of Lyman-alpha line widths, calibrated against gas models, is more informative than assigning a temperature to every dip. ^[ref].

Finally, neutral hydrogen column density is not total gas mass density. In a pure hydrogen approximation, with gas optically thin to ionizing radiation and nearly fully ionized in photoionization equilibrium, the Lyman-alpha forest photoionization scaling giveswhere $n_{\mathrm{H}}$ counts all hydrogen nuclei, $\Gamma$ is the photoionization rate, $\alpha (T)$ is the radiative recombination coefficient and $x_{\mathrm{HI}}=n_{\mathrm{HI}}/n_{\mathrm{H}}$ is the neutral fraction. At fixed temperature and $\Gamma$, doubling $n_{\mathrm{H}}$ quadruples $n_{\mathrm{HI}}$; doubling $\Gamma$ halves it. These computed examples explain both the sensitivity to density contrast and the ambiguity: stronger absorption can mean denser gas, weaker ionizing radiation, a different temperature, or more path length. The Jeans-length ionization correction for the Lyman-alpha forest adds a size model rather than an independently measured depth. Optically thick Lyman limit systems and damped Lyman-alpha systems require different treatment. ^[ref]. ^[ref].

The beauty of the Lyman-alpha forest is that a spectrum converts otherwise faint foreground gas into a sequence of markers through cosmic time. With a model of radiative transfer, velocities, ionizing radiation and the temperature-density relation of photoionized intergalactic gas, its transmission statistics probe the cosmic web and gas thermal history. A single line of sight remains one narrow sample, not a unique three-dimensional density map. ^[ref].

First retain the exact initial normalization. The scale factor obeys $a/a_i=(t/t_i{)}^{2/3}$, so the cosmological redshift isAlso $\rho =3M/(4\pi r^3)$ and ${\rho }_m=1/(6\pi G{t}^2)$. Using $A^3=GM{B}^2$ givesThese equations exactly accommodate the finite-time initial conditions in b.

Now neglect $T$ and retain leading order in ${\delta }_i\ll 1$. Since $t_i/B=4{\delta }_i^{3/2}/[3(1+{\delta }_i)]$, the cosmological redshift relation reduces toDefine ${\delta }_0=\frac{3}{5}{\delta }_i(1+z_i)$. ThenThe second equality is exact for the ideal common-bang growing-mode spherical-collapse model; it is approximate for the paper's finite-time initial Hubble-velocity data.

The factor $3/5$ has a physical origin. Linear mass conservation gives ${\dot{\delta }}_i=0$ when the initial local expansion equals the background Hubble flow. The general linear dust solution is ${\delta }_L=C_g(t/t_i{)}^{2/3}+C_d(t/t_i{)}^{-1}$. Imposing ${\delta }_L(t_i)={\delta }_i$ and ${\dot{\delta }}_L(t_i)=0$ yieldsThus ${\delta }_0$ is the present extrapolation of the growing component, after neglecting the decaying transient. It is not the exact present value of the full linear solution for finite $t_i$.

For the hypothetical Hubble flow, Hubble's law gives $v=H_{0}d$, where $d=4\times 10^5\mathrm{km}$ and $H_0$ is the Hubble constant. The accompanying constants sheet is not present in the supplied PDF, so retain its dependence explicitly. Taking $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as an illustrative input, and using one million parsecs equal to $3.08568\times 10^{19}\mathrm{km}$,A year of $365.25$ days contains $3.15576\times 10^7\mathrm{s}$. ThereforeEquivalently, $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ gives $v=4.09h\mathrm{cm}\mathrm{y}{\mathrm{r}}^{-1}$. This is a counterfactual recession rate for independently comoving bodies, rather than a prediction for the actual Earth-Moon orbit.

The Earth-Moon pair is a gravitationally bound system, so its separation does not obey $\dot{r}=H_{0}r$. The Hubble flow describes the motion of the approximately homogeneous cosmological matter distribution; it is not a compulsory extra velocity assigned to the members of every local orbit. Thus the numerical proximity of the counterfactual rate to $3.8\mathrm{cm}\mathrm{y}{\mathrm{r}}^{-1}$ is not evidence for that explanation.

The observed recession arises principally from tidal dissipation. The Earth rotates faster than the Moon orbits, so the displaced tidal response transfers angular momentum from the Earth's spin into the lunar orbit. For a nearly circular orbit, increasing orbital angular momentum increases its size. A model of this effect must include the appropriate tidal response, rather than add a universal Hubble flow recession.

A residual local cosmological tidal acceleration can exist, but its background scale is $|(\ddot{a}/a)r|\sim H_0^{2}r$, not an imposed first-order secular drift $H_{0}r$. At $r=4\times 10^8\mathrm{m}$,using $G(M_E+M_M)\simeq 4\times 10^{14}{\mathrm{m}}^3{\mathrm{s}}^{-2}$. The cosmological scale is about $10^{-24}$ of the mutual gravitational acceleration. Its detailed coefficient depends on how the local system is embedded in the cosmological background. Cosmological expansion can be neglected for the lunar recession calculation at the stated precision; this conclusion follows from binding and the tiny residual local cosmological tidal acceleration, rather than from comparing two recession speeds.

Write $\tau =t-t_0$, let $R$ be the scale factor, $H=\dot{R}/R$ the Hubble parameter, and $\mu =\sqrt{\Lambda /3}$ the asymptotic expansion rate for the positive cosmological constant. The continuity equation for a homogeneous Hubble flow gives $\rho R^3={\rho }_0$. Isotropy and the Poisson equation for Newtonian gravity give a relative acceleration proportional to displacement from any chosen origin:This uses the local relative gravitational acceleration, rather than trying to sum an absolutely convergent potential over an infinite universe. A fluid element with constant comoving coordinate $\mathbf{x}$ has $\mathbf{r}=R\mathbf{x}$. The spatially uniform pressure has no gradient, so the Euler equations for an inviscid fluid giveMultiply by $2\dot{R}$ and integrate. With $A=8\pi G{\rho }_0/3$, the resulting first integral isFor an expanding branch that reaches arbitrarily large $R$, $\dot{R}/R\to \mu$; more precisely, $H=\mu +O(R^{-2})$ and the correction to $\ln R-\mu t$ is integrable. Therefore $R\sim C{e}^{\mu t}$. Positive $\Lambda$ alone does not make every initially expanding branch escape: sufficiently large positive $\kappa$ can give a turning point or a limiting static radius. Under the stated vacuum-dominance inequalities at $R=1$, however, $A/R$ and $|\kappa |$ remain small compared with ${\mu }^2{R}^2$ for the future expanding branch, and $R\simeq e^{\mu \tau }$ is the required approximation.

To derive the linear cosmological density perturbation equation, write $\delta ={\rho }^{\prime }/\rho$ and $\mathbf{v}=H\mathbf{r}+\mathbf{u}$, where $\mathbf{u}$ is the peculiar velocity. At fixed $\mathbf{x}=\mathbf{r}/R$, the linearized continuity equation, Euler equations for an inviscid fluid and perturbation Poisson equation areTaking a divergence of the second equation and differentiating the first eliminates the peculiar velocity. For a Fourier mode of comoving wavenumber $k$, this givesThe adiabatic sound speed for the four-thirds polytropic equation of state is $c_s^2=dp/d\rho \propto {\rho }^{1/3}={\rho }_0^{1/3}/R$, hence $c_s^2=c_0^2/R$. Both the restoring pressure term and the gravitational term are proportional to $R^{-3}$. Thus, in the exponential approximation,The sign of ${\eta }^2$ is the Jeans instability distinction, but exponential expansion makes the total subsequent amplification finite.

Remove Hubble friction by setting $\epsilon =e^{-\mu \tau }y$. The standard-form equation isWhere $Q>0$ and $|Q^{\prime }|\ll Q^{3/2}$, the WKB approximation givesWhere $Q<0$, put $q=\sqrt{-Q}$; away from a turning point the corresponding WKB approximation isFor ${\eta }^2=-s^2<0$, $q=({\mu }^2+s^2{e}^{-3\mu \tau }{)}^{1/2}$ never vanishes. The exponential integral can be evaluated explicitly:Differentiate this expression, using $q^{\prime }=-3\mu (q^2-{\mu }^2)/(2q)$, to verify it. Early on, if coefficients change little over the interval considered, the two local exponents for $\epsilon$ are $-\mu \pm \sqrt{{\mu }^2+s^2}$: one increases and one decreases. At late times $q\to \mu$ and the displayed integral is $\mu \tau +O(1)$. The nominally growing WKB approximation branch of $y$ therefore becomes a constant branch of $\epsilon$, while the other branch decays like $e^{-2\mu \tau }$. The WKB approximation is not automatically accurate at all intermediate times; its stated derivative criterion must be checked.

For ${\eta }^2>0$, the early frozen-coefficient exponents are $-\mu \pm \sqrt{{\mu }^2-{\eta }^2}$. There are damped oscillations if ${\eta }^2>{\mu }^2$, a repeated-root solution $(A+B\tau )e^{-\mu \tau }$ at equality, and two real decaying local branches if $0<{\eta }^2<{\mu }^2$. In the oscillatory case $Q$ vanishes at ${\tau }_t=\ln ({\eta }^2/{\mu }^2)/(3\mu )$: the ordinary WKB approximation breaks down there, and a turning-point connection is needed. After that transition the solutions again approach a constant or decay. A positive pressure-minus-gravity restoring term does not imply indefinitely continuing oscillation; its physical frequency is redshifted away.

The limiting conclusion does not depend on the accuracy of WKB approximation. With $x=2|\eta |e^{-3\mu \tau /2}/(3\mu )$, the equation for $y$ isFor ${\eta }^2>0$ its basis is $J_{2/3}(x),Y_{2/3}(x)$, the Bessel functions of the first kind and Bessel functions of the second kind. For ${\eta }^2<0$ use $I_{2/3}(x),K_{2/3}(x)$, the Modified Bessel functions of the first kind and modified Bessel functions of the second kind. Their small-$x$ branches are proportional to $x^{2/3}$ and $x^{-2/3}$. Multiplication by $e^{-\mu \tau }$ gives the decaying and constant branches respectively. At ${\eta }^2=0$ the exact solution is $A+B{e}^{-2\mu \tau }$. Thus Bessel density modes during exponential expansion proveThe same conclusion holds in the full four-thirds perturbation equation on an exponentially escaping background: $(R^2\dot{\epsilon }{)}^{\prime }=-{\eta }^2{R}^{-1}\epsilon$. Its integrated equation has an absolutely integrable Volterra kernel, bounded by a constant times $e^{-3\mu \tau }$. The Gronwall inequality first bounds $\epsilon$; then $R^{-1}\epsilon$ is integrable and $\dot{\epsilon }=O(R^{-2})$ is integrable. A finite limiting amplitude follows without setting $H$ exactly constant. This argument requires positive $\Lambda$ and the escaping branch; it is not a claim about $\Lambda =0$ or recollapse.

The requested strict increase of that limit needs an initial-velocity qualification. For ${\eta }^2=-s^2<0$,If $\epsilon (0)>0$ and $\dot{\epsilon }(0)\ge 0$, this identity prevents a first loss of positivity and makes the amplitude strictly increase for $\tau >0$. Integrating it twice givesThis is monotone Jeans growth with positive cosmological constant. Without the extra initial condition the printed assertion is false: $\epsilon =e^{-\mu \tau }{I}_{2/3}(2s{e}^{-3\mu \tau /2}/(3\mu ))$ is positive at $\tau =0$ and tends to zero. An arbitrarily small multiple remains an infinitesimal linear cosmological density perturbation, so linearity does not remove the counterexample.

The cosmological principle asserts that, after averaging over sufficiently large scales, the universe is spatially homogeneous and isotropic on constant-cosmic time slices. Homogeneity means no distinguished location; isotropy means no distinguished spatial direction. These are statistical large-scale statements, not assertions that individual galaxies, clusters or the cosmic web are uniform, and they do not require the universe to be unchanging in time.

The nearly isotropic cosmic microwave background is strong evidence for large-scale isotropy about us. Its dominant dipole is explained by our peculiar velocity; the much smaller residual fluctuations provide the seeds and record of structure formation, rather than invalidating the principle. Large-area galaxy and radio-source distributions and deep galaxy surveys support statistical isotropy and a tendency towards statistical homogeneity after smoothing beyond the strongly clustered scales. An approximately isotropic Hubble flow and broadly uniform background radiation supply complementary checks.

There is an important logical distinction: isotropy seen by one observer alone does not prove homogeneity. A spherically symmetric inhomogeneous universe centered on that observer could look isotropic. Combining the observations with the assumption that our location is not privileged, so comparable observers would see comparable large-scale isotropy, motivates the homogeneous FLRW metric. Selection effects, foregrounds, finite survey volumes and the fact that observations lie on our past light cone must be accounted for. The evidence supports a large-scale statistical principle, not exact uniformity on every scale.

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.

A comoving coordinate labels a location in the expanding background and is constant for an object moving with the Hubble flow. For example, a radial comoving geodesic coordinate $\chi$ gives proper distance in cosmology $d(t)=a(t)\chi$ along the constant-cosmic-time slice. In a flat background one can write physical coordinates $\mathbf{r}=a(t)\mathbf{x}$ and split the physical velocity asThus proper distance in cosmology is measured using the spatial metric at a specified epoch, whereas a comoving distance factors out the overall expansion. A proper distance on a simultaneous spatial slice is not automatically the luminosity distance or a distance directly inferred from light received today. In a curved spatial geometry $a\chi$ is the radial proper distance, while the sphere's circumference involves the corresponding curved radial function.

Let ${\nu }_c$ be the photon frequency in the local comoving frame just after emission. The kinematic redshift gives ${\nu }_e/{\nu }_c=1+z_{\mathrm{kin}}$, and propagation in the FRW metric gives ${\nu }_c/{\nu }_0=1+z_{\mathrm{cosm}}$. Multiplication givesEquivalently, $z_{\mathrm{tot}}=z_{\mathrm{cosm}}+z_{\mathrm{kin}}+z_{\mathrm{cosm}}{z}_{\mathrm{kin}}$. Both redshifts being small is what permits their approximate addition. Here the observer's own peculiar velocity is assumed already corrected to the Hubble flow frame.

LetSubtracting the measured distance modulus from the measured apparent magnitude givesFor the Hubble flow, defineThe Hubble law, with the Hubble constant parametrized by $\theta =5{\log }_{10}h$, gives $r_i\sim N(M_0-\theta ,H_i)$. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function

In the homogeneous background, $\mathbf{v}=0$ and the physical velocity is the Hubble flowThe comoving coordinate $\mathbf{x}$ is held fixed by the time derivative in the fluid equation, soMoreover, spatial homogeneity gives $\nabla \bar{P}=0$. The Euler equation therefore reduces toTaking the divergence in three spatial dimensions givesConsistency with the Poisson equation,then requiresThis is the pressureless, zero-cosmological-constant Raychaudhuri equation, obtained by the Newtonian fluid derivation of the Raychaudhuri equation.

The elementary Newtonian derivation of the Friedmann equation follows a test particle on the boundary of a finite expanding sphere. It therefore introduces a distinguished centre and can misleadingly suggest that all matter expands away from one origin.

The fluid derivation is local: the Euler equation and Poisson equation must be compatible in every comoving neighbourhood. It consequently describes a genuinely homogeneous infinite universe, in which every comoving observer sees the same Hubble flow, without selecting a physical centre. This is the improvement captured by the Newtonian fluid derivation of the Raychaudhuri equation.

Write proper position as $\mathbf{r}=a(t)\mathbf{x}$ and proper velocity asSubstitute this decomposition into the proper-coordinate Euler equations for an inviscid fluid, use ${\nabla }_r=a^{-1}{\nabla }_x$, and subtract the homogeneous-background acceleration. With $\rho =\bar{\rho }(1+\delta )$, one obtains the comoving peculiar-velocity equationThe peculiar gravitational potential $\Phi$ is the total Newtonian potential with the potential of the exactly homogeneous expanding background subtracted. Its gradient therefore generates only accelerations relative to the Hubble flow.

For a pressureless fluid, linearization discards the quadratic advection term, leavingIn the early Einstein-de Sitter universe, the growing mode has $\Phi (\mathbf{x},t)=[D(a)/a]{\Phi }_i(\mathbf{x})$. Integration gives

The linear growth factor obeysSince $\bar{\rho }={\bar{\rho }}_i/a^3$ after choosing $a_i=1$, this is equivalent toIt follows that $a\mathbf{v}=-a^2\dot{D}\nabla {\Phi }_i/(4\pi G{\bar{\rho }}_i)$. Since $\dot{\mathbf{x}}=\mathbf{v}/a$, one final integration gives the Zeldovich approximationwhere an additive initial displacement has been absorbed into ${\mathbf{x}}_i$.

An Einstein-de Sitter universe has $a\propto t^{2/3}$, $H=2/(3t)$, and homogeneous density $\rho =1/(6\pi G{t}^2)$. HenceBefore the perturbation appreciably changes the motion, every shell follows the Hubble flow,

By Newton's shell theorem, only matter currently inside a shell contributes to its radial acceleration. Labelling matter by initial radius givesBefore shell crossing, the enclosed mass of a given shell is $M_i$. Its initial specific energy isAt the turnaround radius, $E_i=-G{M}_i/r_{\ast }$, soThe radial Kepler solution from the Big Bang to apocenter givesAfter turnaround a shell falls inward. Shell crossing makes its enclosed mass time dependent; shells then oscillate through the center, form caustics near successive apocenters, and phase mix into a halo.

For self-similar secondary infall, writeSubstitution in the shell equation givesUsing $G{t}_{\ast }^2/r_{\ast }^3={\pi }^2/(8M_i)$,

Now imposeThe case $ϵ=0$ is a scale-independent fractional overdensity, for which all shells turn around together in the ideal model. The case $ϵ=1$ has constant excess mass $\delta M_i=M_0$, corresponding to a central point-mass seed outside its core.

Since $r_{\ast }/r_i=(M_i/M_0{)}^{ϵ}$, while $r_i^3=(9/2)G{t}_i^2{M}_i$, the turnaround formulas giveThe shell turning at time $t$ therefore hasFor a shell with $\tau =t/t_{\ast }(M_i)$,Consequently its autonomous similarity equation isThe exponent $2/(3ϵ)$ follows directly from the preceding mass ratio; the printed $2ϵ/3$ in the final displayed equation is inconsistent with the supplied initial condition and preceding requested result. The corrected nonlinear equation is integrated numerically after turnaround.

The peculiar gravitational potential is the Newtonian potential after subtracting the potential of the homogeneous expanding background. In comoving coordinates it obeys a Poisson equation sourced by the density contrast, so its gradient accelerates matter relative to the Hubble flow.

For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term $H\mathbf{v}$ and force $-\nabla \Phi /a$ from the peculiar gravitational potential.

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.