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For a homogeneous cosmic fluid, the physical volume bounded by fixed comoving coordinates obeys $V\propto a^3$. The cosmological continuity equation therefore gives $d(\rho c^{2}V)=-PdV$. This is the first law of thermodynamics with conserved entropy and no additional chemical work. The internal energy of that volume need not be constant: a component with negative pressure gains internal energy as the universe expands. Local stress-energy conservation does not imply a global conserved energy for an arbitrary expanding spacetime.

Comoving and proper distances are two cosmological distance conventions. For simultaneous radial separations in an FLRW model, proper distance equals the comoving radial distance multiplied by the scale factor at that cosmic time. A comoving coordinate remains fixed on an ideal comoving observer's worldline.

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving coordinate provides labels for the comoving and proper distances comparison; proper separation additionally contains the cosmological scale factor.

A comoving observer has constant comoving coordinates in a homogeneous expanding universe. Its physical separation from another comoving observer grows in proportion to the scale factor, giving relative recession governed by the Hubble parameter. Light exchanged between these observers undergoes cosmological redshift.

The comoving particle horizon is the particle horizon expressed in comoving coordinates, assuming a finite initial cosmic-time boundary at zero. In a spatially flat FLRW metric, it is the distance a photon could have travelled since that boundary. The corresponding proper distance in cosmology at time $t$ is $a(t)r_p(t)$. It differs from the future-directed cosmological event horizon and from the instantaneous comoving Hubble radius.

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.

For scale-independent linear growth $\delta (\mathbf{k},a)=D(a){\delta }_{\mathrm{ini}}(\mathbf{k})$, the logarithmic linear growth rate measures fractional amplification per logarithmic change of the scale factor. In an Einstein-de Sitter universe, $D\propto a$ and $f=1$. The cosmological Poisson equation for conserved pressureless matter gives $\psi \propto D/a$ at fixed comoving coordinate, and therefore ${\psi }^{\prime }=\mathcal{H}(f-1)\psi$. This relates suppression of matter growth to the Integrated Sachs-Wolfe effect.

At fixed positive redshift, intrinsic luminosity and present Hubble constant, a flat dust-vacuum model with $0<{\Omega }_{\Lambda 0}<1$ has luminosity distance between those of an Einstein-de Sitter universe and pure positive vacuum. The Hubble parameter of the dust-only model is largest at every positive integration redshift; that of pure vacuum is smallest. Therefore $c(1+z){\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$ is smallest for dust alone and largest for pure vacuum. Received radiative flux has the reverse ordering. This compares physical distances rather than arbitrarily normalized comoving coordinates.

Expand the density contrast in spatial Fourier modes using fixed comoving coordinates:The spatial Laplacian multiplies a mode by $-k^2$, while its comoving wavenumber $k=|\mathbf{k}|$ is time-independent. Writing $H=\dot{a}/a$, the linear cosmological density perturbation equation becomesThe proper wavenumber is $k^{(p)}=k/a$, since the proper wavelength is $2\pi a/k$. The proper Jeans wavenumber balances the pressure and gravity terms:

The four contributions have distinct physical origins. The intrinsic photon temperature perturbation is ${\delta }_{\gamma }/4$: a local blackbody radiation energy density satisfies ${\rho }_{\gamma }\propto T^4$, so its first-order fractional perturbation is four times the fractional temperature perturbation. The endpoint cosmological Newtonian potential term ${\psi }_{\ast }$ is the gravitational redshift between the emitting plasma and the observer; escaping a negative potential well lowers the measured photon energy. Together with the intrinsic contribution it produces the ordinary Sachs-Wolfe effect. The term $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_{\ast }$ is the Doppler CMB anisotropy, caused by the plasma's line-of-sight peculiar velocity at last scattering, with the source's direction convention.

Finally, $2\int d\chi {\psi }^{\prime }$ is the Integrated Sachs-Wolfe effect. A photon passing through a static potential well gains energy on entry and loses it on exit, with no net shift. When the potential evolves while the photon crosses, the two shifts fail to cancel. The factor two is the sum of the temporal and spatial scalar-potential contributions when anisotropic stress is negligible, so the two potentials are equal. The prime is a partial derivative at fixed comoving coordinate; it is not the total derivative along the light ray.

For a low-cosmological redshift galaxy sample, the intrinsic, endpoint-potential and last-scattering Doppler CMB anisotropy terms are localized near $z\simeq 1100$, whereas the galaxy density contrast is measured much nearer us. In the Limber approximation their radial kernels have no overlap, so their leading cross correlations vanish. The late Integrated Sachs-Wolfe effect samples the same nearby evolving potentials as the galaxies, giving the desired late-time CMB-matter cross correlation. The early Integrated Sachs-Wolfe effect near matter-radiation equality is likewise radially separated.

This is the leading linear, separated-kernel description intended here. The primary Cosmic microwave background anisotropy and late density contrast share primordial initial conditions, so their exact large-angle correlation need not be identically zero outside the Limber approximation. Additional secondary effects, such as the Sunyaev-Zeldovich effect, can also correlate with galaxies in real maps; they are absent from the stipulated linear expression.

Let $p(z)$ be the normalized galaxy probability density per unit cosmological redshift, $\int p(z)dz=1$, and let $b_g$ be its linear galaxy bias. In a spatially flat background with $c=1$, $d\chi =dz/H(z)$. Thus the projected galaxy density contrast isso its radial kernel is $q_g=Hbp$. The cosmological Poisson equation in Fourier transform variables givesHere ${\Omega }_m$ denotes today's matter density parameter, not its time-dependent value. For a growing linear cosmological density perturbation, write $\delta (\mathbf{k},a)=D(a){\delta }_{\mathrm{ini}}(\mathbf{k})$, where $D$ is the linear growth factor, and define the logarithmic linear growth rate $f=d\ln D/d\ln a$. At fixed comoving coordinate,ConsequentlyIn particular, $f=1$ during matter domination makes the linear potential constant and eliminates this Integrated Sachs-Wolfe effect.

To use the supplied Limber approximation without hiding its scale dependence in a radial kernel, take $U=\delta$, $V={\psi }^{\prime }$, $q_g=Hbp$, and $q_T=2$. The corresponding three-dimensional cross-power spectrum iswhere $P$ is the matter power spectrum. Substitution givesThe Limber approximation relation $k\chi =\ell +1/2$ cancels ${\chi }^{-2}{k}^{-2}$, yieldingBoth powers of $H$ are necessary: one converts the redshift selection to a distance kernel, and the other differentiates the potential in conformal time. The factor two in the Integrated Sachs-Wolfe effect cancels the factor $1/2$ from the cosmological Poisson equation. For positive galaxy bias and $f<1$, the predicted correlation is positive: an overdense region whose negative potential becomes shallower gives a positive integrated temperature shift. The result assumes linear, scale-independent growth and the Newtonian cosmological Poisson equation; the Limber approximation becomes less accurate at the very lowest angular multipoles.

On large scales the appropriate background is the spatially flat FLRW metric,Here $t$ is cosmic time, the proper time of a comoving observer; $R(t)$ is the scale factor; and $(r,\theta ,\varphi )$ are dimensionless comoving coordinates. With this convention $R$ has units of length. A fixed comoving radial interval has instantaneous proper length $R(t)dr$. The cosmological principle assumes spatial homogeneity and cosmological isotropy on the scales where this background description applies.

The general FLRW metric instead has $d{r}^2/(1-k{r}^2)$ in its radial part, with normalized $k=+1,0,-1$ describing positive, zero or negative spatial curvature of an FLRW universe. The best simple description takes $k=0$; observations constrain spatial curvature to be small rather than proving it identically zero. The late accelerated expansion is encoded by the time dependence of $R$, determined by the stress-energy tensor through the Friedmann equations, rather than by an additional term in the FLRW metric. Small-scale galaxies and other inhomogeneities require perturbations of this background.

For a perfect fluid in general relativity, let $\rho$ be rest-frame total energy density divided by $c^2$, $P$ its isotropic pressure, and $U^{\mu }$ its four-velocity, normalized by $U^{\mu }{U}_{\mu }=-c^2$. ThenThe contracted Bianchi identity and the Einstein field equations give stress-energy conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, for a constant cosmological constant. Contract with $U_{\nu }$ and use the normalization to obtainIn comoving coordinates $U^{\mu }=(1,0,0,0)$ and the covariant divergence is ${\nabla }_{\mu }{U}^{\mu }=3\dot{R}/R=3H$, where a dot is a cosmic time derivative and $H$ the Hubble parameter. ThereforeFor a fixed region in comoving coordinates, let $V(t)=V_{\ast }{R}^3(t)$ be its physical volume, where $V_{\ast }$ is a fixed coordinate-volume integral, and let $E=\rho c^{2}V$. Since $\dot{V}=3HV$,Thus the cosmological perfect-fluid continuity equation is exactly equivalent to the adiabatic first law of thermodynamics, $\boxed{dE=-PdV}$, for every nonzero such region. Conversely, expanding that first law of thermodynamics and dividing by $c^{2}V$ recovers the same equation. No heat or particle flux crosses the comoving boundary; interacting components separately need source terms, although the total still obeys this conservation equation.

There is a sign inconsistency in the supplied formula sheet worth isolating. Its Riemann curvature tensor and Ricci tensor conventions, together with its stated $-+++$ metric signature, give $G_{tt}=-3H^2$ for $R(t)=e^{Ht}$ with constant $H$. Its vacuum Einstein field equations with $+\Lambda g_{\mu \nu }$ then require $H^2=-\Lambda c^2/3$, whereas its printed vacuum Friedmann equation gives $H^2=+\Lambda c^2/3$. For example, $H>0$, $\rho =P=0$ and $\Lambda =3H^2/c^2$ satisfy the latter but leave the former's $tt$ component equal to $-6H^2$. Changing the geometric cosmological constant term to $-\Lambda g_{\mu \nu }$ repairs this mismatch. The conservation derivation is unaffected; below take the geometric cosmological constant to be zero and include any vacuum energy as a fluid, so no ambiguous sign enters.

The scale factor $a(t)>0$ specifies the common time dependence of proper distance in cosmology between comoving observers. Their comoving coordinates remain fixed, while their separation is multiplied by $a(t)$. Thus $\dot{a}>0$ means expansion and $\dot{a}<0$ means contraction. The coordinate $t$ is cosmic time, measured by the proper time of a comoving observer.

The parameter $k$ fixes the sign of the spatial curvature of an FLRW universe. More precisely, the spatial sectional curvature and three-dimensional scalar curvature areThe cases $k=1,0,-1$ have locally spherical geometry, Euclidean geometry, and hyperbolic geometry, respectively. For $k=\pm 1$, $a$ is the curvature radius of a constant-curvature space in the displayed normalization. Local curvature alone does not fix the global topology.

Integrating the given expansion equation gives $R^{3/2}=t/t_0$, with the integration constant fixed by $R(t_0)=1$. Thus the scale factor and Hubble parameter are $R=(t/t_0{)}^{2/3}$ and $H=2/(3t)$. The Newtonian derivation of the flat Friedmann equation gives $H^2=8\pi G\rho /3$, soUse the comoving coordinate $\mathbf{q}=\mathbf{x}/R$ and the density contrast $\delta ={\rho }_1/\rho$. Linearized mass conservation, the peculiar-velocity Euler equation for fluid motion and the Poisson equation for the peculiar gravitational potential areDifferentiate the first equation, substitute the second and use the third. A Fourier component of comoving wavenumber $k$ obeysSince $c^2{k}^2/R^2=k^2{c}^2(t_0)t_0^{2\gamma -2/3}{t}^{-2\gamma +2/3}$, the normalization is $\Lambda =kc(t_0)t_0^{\gamma -1/3}$, as printed in the PDF. The resulting linear cosmological density perturbation equation isTo reduce it to a Bessel equation, put $\nu =\gamma -4/3$, $z=\Lambda /(\nu t^{\nu })$ and $\epsilon =t^{a}y(z)$. As $\dot{z}=-\nu z/t$, multiplying the substituted equation by $t^{2-a}$ givesTaking $a=-1/6$ makes the first two coefficients proportional to those in the Bessel equation, and the remaining constant is $-25/36$. HenceFor noninteger $\lambda$, the Bessel density modes of a polytropic expanding universe areThere is an exceptional case within the allowed interval: if $\lambda =n$ is an integer, $J_{-n}=(-1{)}^n{J}_n$, so these are not two independent solutions. The full solution then uses $t^{-1/6}[A{Y}_n(z)+B{J}_n(z)]$, with the Bessel function of the second kind providing the missing branch. For example $\gamma =29/18$ gives $\lambda =3$. A basis using $J_{\lambda },Y_{\lambda }$ is valid for every allowed exponent.

For early times, $z\to \infty$. The large-argument asymptotic expansion of the Bessel function of the first kind has oscillatory phase $z-\pi a/2-\pi /4$ and envelope $(2/(\pi z){)}^{1/2}$ for order $a$; the Bessel function of the second kind has the same envelope. ThusThis decreases because $3\gamma <5$. At late times $z\to 0$, the leading powers are $J_{\pm \lambda }(z)\propto z^{\pm \lambda }$ for noninteger order, giving $t^{-1}$ and $t^{2/3}$. The $Y_n$ branch gives the same growing power in the integer case. Therefore a generic perturbation grows as $t^{2/3}$; an initial condition with zero growing-mode coefficient instead has the decaying $t^{-1}$ mode.

To find the growth scale including its dependence on $\nu$, set $\epsilon =t^{2/3}h(t)$. The equation becomesSubstitute $h=1+a{\Lambda }^2{t}^{-2\nu }+\cdots$. The leading coefficient equation is $-2\nu (5/3-2\nu )a+1=0$. The pressure correction to a growing polytropic cosmological mode is consequentlySince $ck/R=\Lambda t^{-\nu -1}$ and $t^{-2}=6\pi G\rho$, the correction becomes small when $(ck/R{)}^2{t}^2$ is small compared with $2\nu (5/3-2\nu )$. The factor $2(5/3-2\nu )$ is between two and $10/3$, giving the stated order-of-magnitude onset conditionFor a well-developed asymptotic growing mode, replace $\lesssim$ by $\ll$. This is a Jeans instability interpretation: the physical acoustic frequency must be sufficiently low compared with the gravitational expansion scale, or equivalently the physical wavelength must be sufficiently long for self-gravity to overcome pressure support. It is an approximate condition for the dust-like growing branch, not an exact initial-condition-independent instant at which any perturbation begins increasing. The instantaneous sign change of the coefficient in the original equation instead occurs at $c^2{k}^2/R^2=4\pi G\rho$.

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ be the density contrast, let $\mathbf{x}$ be a comoving coordinate, and use $a_0=1$. Define $C=3H_0^2{\Omega }_0/2$, so $4\pi G\bar{\rho }=C/a^3$ for the pressureless background. The perturbation Poisson equation isFor the comoving velocity field $\mathbf{w}=\dot{\mathbf{x}}$, local mass conservation givesOne obtains this by subtracting $\dot{\bar{\rho }}+3H\bar{\rho }=0$ from ${\partial }_t\rho +3H\rho +{\nabla }_x\cdot (\rho \mathbf{w})=0$. In the linear regime it reduces to ${\partial }_t\delta =-{\nabla }_x\cdot \mathbf{w}$, the linearized cosmological continuity equation. The spatial derivatives are with respect to $\mathbf{x}$, and the time derivative in the conservative continuity equation is at fixed $\mathbf{x}$. In contrast the acceleration of an individual fluid element is a material derivative. The physical peculiar velocity is $a\mathbf{w}$.

Write $H=\dot{R}/R$. A fluid element in the homologous spherical flow obeys $\dot{\mathbf{r}}=H\mathbf{r}$, so $\mathbf{r}/R$ remains constant. Also $\nabla \cdot {\mathbf{u}}_0=3H$. Along every element, the continuity equation therefore gives $D{\rho }_0/Dt=-3H{\rho }_0$. Every element starts with the same mass density, and hence has the same mass density at subsequent times:This proves preservation of uniformity, rather than assuming it when conserving the total mass.

By the shell theorem, only the mass interior to radius $r$ contributes to its gravitational acceleration. That mass is $4\pi {\rho }_0{r}^3/3$. The same result follows directly from the Poisson equation for Newtonian gravity: spherical symmetry gives $(r^2{\Phi }_r{)}_r=4\pi G{\rho }_0{r}^2$, and regularity at the centre sets the integration constant to zero. Hence ${\Phi }_r=4\pi G{\rho }_{0}r/3$ andThe material derivative of the background velocity is $(\dot{H}+H^2)\mathbf{r}=(\ddot{R}/R)\mathbf{r}$. There is no pressure force, and the momentum equation consequently reduces toMultiplication by $\dot{R}$ and integration, using collapse from rest, givesChoose the negative square root for contraction. Put $C^2=2K/R_0$ and $R=R_0{\cos }^2(\varphi /2)$. For $0<\varphi <\pi$, the energy integral then gives $\dot{R}=-C\tan (\varphi /2)$, while $dR/d\varphi =-R_0\sin \varphi /2$. Their ratio isIntegrating from the initial turning point proves the cycloid parametrization of pressureless homologous spherical collapse:It covers the entire contracting branch, with complete collapse at $\varphi =\pi$ and $t_c=\sqrt{3\pi /(32G{\rho }_{00})}$. At $R=0$ the uniform dust configuration is singular, so the smooth calculation is restricted to $t<t_c$.

Each Fourier mode among the comoving perturbations of a homologously collapsing cloud has its phase fixed on each fluid element. In fact, for $\mathbf{k}=\mathbf{q}/R$,Its physical wavelength contracts in proportion to $R$, while its wavelength in the comoving coordinate $\mathbf{r}/R$ stays fixed. This is the physical reason to use a time-dependent wavenumber; holding the physical wavenumber fixed would not follow the contracting pattern. Boundary effects are being excluded in this bulk Fourier mode calculation.

Let ${\rho }^{\prime }={\rho }_1(t)e^{i\mathbf{k}\cdot \mathbf{r}}$ and let the perturbed velocity and Newtonian gravitational potential have amplitudes ${\mathbf{u}}_1(t)$ and ${\Phi }_1(t)$. Linearizing the continuity equation, the uniform background eliminates ${\mathbf{u}}_1\cdot \nabla {\rho }_0$. The advected phase eliminates its own time derivative, leavingFor the density contrast $\delta ={\rho }_1/{\rho }_0$, use ${\dot{\rho }}_0=-3H{\rho }_0$ to obtain $\dot{\delta }=-i\mathbf{k}\cdot {\mathbf{u}}_1$. The linear momentum equation and Poisson equation for Newtonian gravity areDifferentiating the density contrast equation must include $\dot{\mathbf{k}}=-H\mathbf{k}$:Thus the compressive Fourier mode with $k\ne 0$ satisfiesThe pressureless approximation removes the usual acoustic restoring term of Jeans instability. A transverse velocity perturbation instead obeys ${\dot{\mathbf{u}}}_{1\perp }+H{\mathbf{u}}_{1\perp }=0$ and does not drive this density contrast.

Since $d/dt=(C/R)d/d\varphi$, the density contrast equation becomesFor a direct verification, put $s=\tan (\varphi /2)$. As $d/d\varphi =(1+s^2)d/(2ds)$, the first-derivative terms cancel, reducing the equation toThe function ${\delta }_g=s(1+s^2)/2$ has second derivative $3s$ and satisfies this equation. Transforming back gives the required growing branch:To establish what the other solution does, use reduction of order, not just the growth of this example. The coefficient of ${\delta }_{\varphi }$ is $d\log R/d\varphi$, so an independent solution is proportional to ${\delta }_g\int d\varphi /(R{\delta }_g^2)$. Choose the integration endpoint at complete collapse to select the decaying branch:For example, evaluating the integral with $a=\tan (s/2)$ gives integrand $4/[a^2(1+a^2{)}^2]$ and antiderivative $-4/a-6\arctan a-2a/(1+a^2)$. Its endpoint value gives the displayed expression. Its Wronskian with ${\delta }_g$ is $-(1+\cos \varphi {)}^{-1}$, so the two solutions are independent on $0<\varphi <\pi$.

For $ϵ=\pi -\varphi \to 0$, $R\sim R_0{ϵ}^2/4$, ${\delta }_g\sim 4{ϵ}^{-3}$ and the integral in ${\delta }_d$ is ${ϵ}^5/40+O({ϵ}^7)$. Consequently the two density modes of a collapsing pressureless sphere haveEquivalently, with $\tau =t_c-t$, the local powers are ${\tau }^{-1}$ and ${\tau }^{2/3}$. There is one independent growing branch: every solution is $A{\delta }_g+B{\delta }_d$, and only the branch with $A=0$ decays at collapse. The uniqueness is up to normalization and addition of the decaying solution; it does not mean that only one literal function can grow. A fixed nonzero initial perturbation will eventually violate $|\delta |\ll 1$, so the divergent linear result is not a nonlinear solution through collapse.

Use a radial null geodesic of the FRW metric in the form $d{s}^2=-d{t}^2+a(t{)}^{2}d{\chi }^2$ along a fixed spatial direction, with $c=1$. For an emitter and observer at fixed comoving coordinates, the comoving separation isApply this to two adjacent wave crests. Since their endpoint separation is the same, subtraction to first order gives $\delta t_o/a(t_o)=\delta t_e/a(t_e)$. These intervals are the local periods measured by the comoving observers, so the received wavelength is stretched by $a(t_o)/a(t_e)$. Consequently the cosmological redshift obeyswhen $a(t_o)=1$. This relation assumes comoving emitter and observer; additional peculiar velocities supply kinematic redshift factors.

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.