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The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat Friedmann-Lemaître-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

For an expanding Friedmann-Lemaître-Robertson-Walker metric with positive energy density, put $\Omega =8\pi G\rho /(3H^2)$ and $w=P/\rho$, in units $c=1$. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation giveFor constant $w$, linearizing at $\Omega =1$ gives $\Omega -1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp \int (1+3w(a))d\log a$. This describes the flatness problem for decelerating expansion, away from epochs at which $H=0$.

For a comoving observer in a spatially flat Friedmann-Lemaître-Robertson-Walker metric, a finite future conformal time interval gives an event horizon at the displayed comoving radius. Its physical radius at time $t$ is $a(t){\chi }_{\mathrm{eh}}(t)$.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with initial conformal time zero, two comoving worldlines at separation $d$ can exchange a one-way light signal by time $\eta$ only if $d\le \eta$. Their past light cones overlap if $d\le 2\eta$: a common event halfway between them can influence both. These are different statements. For opposite emission events observed at conformal time ${\eta }_0$, their separation is $2({\eta }_0-{\eta }_e)$, so the respective thresholds are ${\eta }_e\ge 2{\eta }_0/3$ and ${\eta }_e\ge {\eta }_0/2$.

The Lambda-CDM model combines a cosmological constant with cold dark matter and the standard baryonic and radiation components in an expanding Friedmann-Lemaître-Robertson-Walker metric.

This is the first-order sector of cosmological perturbation theory.

Linear cosmological perturbation theory expands the metric and matter fields to first order about a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, negative extrinsic curvature, and $K=-3H+\kappa$, the Hamiltonian constraint gives the displayed scalar perturbation equation. Indeed $K^2=9H^2-6H\kappa$ and $K^i{}_j{K}^j{}_i=3H^2-2H\kappa$ to first order. The spatial Ricci scalar is $4\Delta \Phi$; subtract the background equation $6H^2=16\pi G\bar{\rho }$ and divide by four.

Neighboring freely falling comoving particles in a homogeneous Friedmann-Lemaître-Robertson-Walker metric have separation $\mathbf{r}=a(t)\mathbf{\chi }$ and relative acceleration $\ddot{\mathbf{r}}=(\ddot{a}/a)\mathbf{r}$, where $t$ is cosmic time and $\mathbf{\chi }$ is constant. This is a local tidal force effect, of order $H^{2}r$ when the dimensionless background parameters are of order one. In a weak-field central-mass model retaining the homogeneous background tide, one writes $\ddot{r}=-GM/r^2+{\ell }^2/r^3+(\ddot{a}/a)r$. The detailed residual tide depends on the cosmological embedding; a bound orbit is not required to satisfy $\dot{r}=Hr$. Comparing $H^2$ with $GM/r^3$ estimates whether that background tide can affect its dynamics. A locally vacuum region with a cosmological constant has the corresponding outward term $\Lambda c^{2}r/3$.

For an open dust-only Friedmann-Lemaître-Robertson-Walker metric with zero cosmological constant and $0<\Omega <1$, $H(z)=H_0(1+z)\sqrt{1+\Omega z}$. The present radial cosmological proper distance is $d_S={\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$, and the areal radial coordinate satisfies $a_{0}r=\sinh [H_0\sqrt{1-\Omega }{d}_S]/[H_0\sqrt{1-\Omega }]$. Integrating with $v=\sqrt{1+\Omega z}$ gives the displayed relation. It has $r\sim z/(a_0{H}_0)$ near the observer and $r\to 2/(a_0{H}_0\Omega )$ as $z\to \infty$. The luminosity distance is $(1+z)a_{0}r$, not the radial cosmological proper distance.

For pressureless matter in a Friedmann-Lemaître-Robertson-Walker metric with negative spatial curvature of an FLRW universe and no cosmological constant, set $x=(1-{\Omega }_{m0})a/{\Omega }_{m0}>0$. A decaying linear growth factor is $D_{-}(x)=\sqrt{(1+x)/x^3}$. Reduction of order gives the regular growing linear growth factorIts Wronskian with $D_{-}$ is $x^{-3/2}(1+x{)}^{-1/2}$, so the modes are independent. The limits are $D_{+}\sim 2x/5$ during matter domination and $D_{+}\to 1$ in a curvature-dominated universe. Multiplying by $5/2$ normalizes the early mode to $x$. Negative spatial curvature of an FLRW universe thus stops late linear amplification even though the model contains no dark energy.

The density contrast is $\delta (\mathbf{x},t)=[\rho (\mathbf{x},t)-{\rho }_0(t)]/{\rho }_0(t)$, and ${\delta }_k(t)$ is one spatial Fourier mode with comoving wavevector $\mathbf{k}$. The scale factor is $a(t)$, $H=\dot{a}/a$ is the Hubble parameter, $c_s$ is the physical sound speed, ${\rho }_0$ the mean matter density, and $G$ the gravitational constant. For each fixed $k$ the displayed equation is an ordinary differential equation in cosmic time. It compares pressure restoration, self-gravity and expansion-induced damping:The pressure term resists compression; self-gravity enhances an overdensity; the $2H\dot{\delta }$ term slows the motion relative to a static background. The Newtonian treatment is appropriate to small perturbations on subhorizon scales within the specified fluid model, rather than an arbitrary gauge-independent treatment of superhorizon perturbations.

The physical Jeans length follows by equating pressure and gravity:The comoving Jeans wavenumber is $a{k}_{J,\mathrm{phys}}$. Perturbations well below the physical Jeans length are pressure-supported acoustic oscillations whose amplitude is modified by expansion; those above it admit gravitational growth when the background evolution permits. Pressureless matter in a matter-dominated universe is the example below. Merely having a large wavelength is not sufficient to apply this Newtonian equation outside the horizon.

For a gas of particle mass $m$ and temperature $T$, write $c_s^2=\Gamma k_{B}T/m$, with $\Gamma =1$ for the isothermal sound-speed convention or the appropriate adiabatic heat-capacity ratio otherwise. ThenAt fixed density it is larger for hotter or lighter particles, and it decreases with increasing density. Numerical coefficients differ between sound-crossing and self-gravitating-sphere conventions; the expression here uses the Fourier instability boundary of the stated equation.

In an Einstein-de Sitter universe, the spatially flat Friedmann-Lemaître-Robertson-Walker metric is $d{s}^2=-c^{2}d{t}^2+a^2(t)d{\mathbf{x}}^2$. Conservation of pressureless matter gives ${\rho }_0{a}^3=\mathrm{constant}$; the Friedmann equation then gives $a\propto t^{2/3}$, $H=2/(3t)$ and ${\rho }_0=1/(6\pi G{t}^2)$. With $c_s=0$,Trying $\delta =t^p$ gives $p^2+p/3-2/3=(p-2/3)(p+1)=0$. Hence the Einstein-de Sitter density-growth modes areThe growing mode increases only as a power law because the expansion dilutes and slows the collapsing matter.

Conformal time is defined by $d\eta =dt/a(t)$, so the metric becomes $d{s}^2=a^2(-c^{2}d{\eta }^2+d{\mathbf{x}}^2)$. Choosing its origin at the big bang gives $\eta \propto t^{1/3}$ and $a\propto {\eta }^2$. ThereforeEquivalently the equation is ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\eta -6\delta /{\eta }^2=0$, where primes denote conformal-time derivatives.

Before and through cosmological recombination, photons diffuse between hotter and cooler parts of the tightly coupled photon–baryon fluid. Radiative viscosity and heat transport erase short-wavelength acoustic perturbations. As free electrons disappear, the photon mean free path grows, enhancing this diffusion before decoupling. Silk damping can be represented by a factor approximately $e^{-k^2/k_D^2}$ in the linear acoustic amplitude. The comoving Silk wavelength may be defined as ${\lambda }_D=2\pi /k_D$; the associated diffusion-length convention is $r_D=1/k_D$.

A random-walk estimate is $r_D^2\sim \int c{\lambda }_{\gamma }dt/a^2$ up to transport and baryon-loading factors, with physical ${\lambda }_{\gamma }=(n_e{\sigma }_T{)}^{-1}$. Scales much shorter than the damping wavelength are strongly suppressed. This is photon diffusion, not the Jeans boundary and not erasure of every cold-dark-matter perturbation. After decoupling, photon free streaming is a distinct part of the evolution.

Use the dimensionally correct flat Friedmann-Lemaître-Robertson-Walker metricThe source's $-cd{t}^2$ is missing a factor $c$, and its angular term must mean the unit-sphere line element $d{\Omega }^2$, not the solid angle element $d\Omega =\sin \vartheta d\vartheta d\phi$. For example, $cd{t}^2$ has dimensions length times time, while $d{s}^2$ has dimensions length squared. The corrected convention above removes both ambiguities.

A radial light ray has $d{s}^2=0$ and $|d\chi |=c|dt|/a$. ThusA proper shell volume at emission is $d{V}_{\mathrm{proper}}=a^3{\chi }^{2}d\chi d\Omega$. Multiplying by $n(z)=n_0{a}^{-3}$ cancels the expansion factors, equivalently using the comoving volume element directly. HenceThis counts every galaxy in the shell per solid angle, rather than intersections weighted by individual galaxy areas. Near $z=0$ it reduces to $n_0(c/H_0{)}^3{z}^2$, as expected for a Euclidean volume shell.

The cosmological principle permits a Friedmann-Lemaître-Robertson-Walker metric. The Friedmann equation, with vanishing cosmological constant, isFor separately conserved pressureless matter, the cosmological continuity equation gives ${\bar{\rho }}_m={\bar{\rho }}_{m0}{a}^{-3}$: its rest energy is conserved while its physical volume grows as $a^3$. Define the present critical density and matter density parameter byEvaluating the Friedmann equation today gives $-k=H_0^2(1-\Omega )>0$. Thus the model has negative spatial curvature of an FLRW universe; the second contribution to $H^2$ represents that spatial curvature of an FLRW universe, rather than another material fluid. Taking the expanding branch givesHere $0<\Omega <1$. At early times, $a\ll \Omega /(1-\Omega )$, matter domination gives $\dot{a}\simeq H_0\sqrt{\Omega }{a}^{-1/2}$, hence, measuring cosmic time from the Big Bang,At late times the $a^{-2}$ term dominates the Friedmann equation. This curvature-dominated universe has $\dot{a}\to H_0\sqrt{1-\Omega }$ and thereforeThe late expansion approaches the coasting Milne model. The early limit belongs to this idealized matter-only model; including radiation in cosmology would alter the sufficiently early evolution.

The Friedmann-Lemaître-Robertson-Walker metric implements the cosmological principle: on the scales being modelled, the spatial slices have spatial homogeneity and cosmological isotropy. There is a family of comoving observers whose orthogonal slices share one cosmic time and one scale factor. Each spatial slice is a locally maximally symmetric space, with time-independent normalized $k$.

The matter is represented by a perfect fluid in general relativity, at rest relative to the comoving observers, with energy density $\varrho (t)$ and pressure $p(t)$ depending only on cosmic time. Its stress-energy tensor isThere is no preferred spatial direction, so there is no anisotropic stress or spatial heat flux in that rest frame. Gravity obeys the Einstein field equations; the equations in this paper contain no separate cosmological constant, although a component with $p=-\varrho$ can represent one. An equation of state relating $p$ and $\varrho$ closes the evolution equations. Spatial homogeneity and cosmological isotropy are idealizations of the cosmological background; they do not assert that individual local structures are uniform.

Each separately conserved component obeys the constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$. For the flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equations therefore giveWrite $H(z)/H_0=1+h_{1}z+O(z^2)$, where $h_1=[3{\Omega }_m+4{\Omega }_r+3(1+w){\Omega }_X]/2$. Expanding the reciprocal inside the luminosity distance integral givesMatching its quadratic coefficient gives the deceleration parameter $q_0=h_1-1$. Using flatness in this part,A negative deceleration parameter signifies accelerated expansion; negative pressure must overcome the positive matter and radiation terms. Restoring units replaces $H_0{d}_L$ by $H_0{d}_L/c$ in the expansion, leaving $q_0$ unchanged.

Varying the canonical scalar action gives $\square \varphi +V_{,\varphi }=0$. In a homogeneous flat Friedmann-Lemaître-Robertson-Walker metric written in conformal time, $\sqrt{-g}=a^4$ and $g^{00}=a^{-2}$, soThe quadratic potential therefore givesEquivalently $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$ in cosmic time. The intrinsic roll/oscillation time is $m^{-1}$ and the damping time is $H^{-1}$. If $m\gg H_0$, the scalar has long since begun rapid oscillations: averaging over a quadratic oscillation gives equal kinetic and potential energies, $⟨p⟩\simeq 0$, so it behaves as matter rather than accelerating dark energy. To remain nearly frozen or only begin rolling today requires the mass scale of quadratic quintessenceIf its energy is predominantly potential, equating $m^2{\varphi }_0^2/2$ to ${\rho }_{X0}=3M^2{H}_0^2{\Omega }_X$, with $M$ the reduced Planck mass, givesTaking the natural onset estimate $m\sim H_0$ and ${\Omega }_X\sim 0.7$ gives $|{\varphi }_0|\sim 2M\sim 5\times 10^{18}\mathrm{GeV}$, of order the Planck mass. More strongly frozen, negative-pressure evolution requires a smaller mass and correspondingly larger field. The density fixes the product $m|{\varphi }_0|$ at this accuracy, rather than a unique mass and field separately.

A horizon separates regions according to a specified causal, dynamical or geometrical criterion. The criterion must be stated: an event horizon, an apparent horizon, a Killing horizon and a Cauchy horizon answer different questions. Some horizons describe a black hole, others an observer's acceleration or the universe's expansion. A vanishing coordinate metric component alone does not establish any of these definitions.

In an asymptotically flat spacetime, choose the relevant future null infinity ${\mathcal{I}}^{+}$, the endpoint of escaping outgoing light rays. The black-hole region and its future event horizon areHere $J^{-}$ is the causal past. Events in $\mathcal{B}$ cannot send a causal signal to the chosen infinity. This is a global definition: identifying the horizon requires the full future development, so a local measurement need not reveal whether an event lies on it. Smooth portions of an event horizon are null hypersurfaces generated by null geodesics. Generators may join at past caustics or crease sets, so the horizon need not be smooth everywhere. A past event horizon is defined by the time-reversed escape criterion; in the maximally extended Schwarzschild solution it bounds a white-hole region.

An observer-based definition replaces infinity by a future-inextendible worldline $\gamma$. Its future observer event horizon is the boundary of events that can signal to $\gamma$, conventionally $\partial I^{-}(\gamma )$. A past observer horizon similarly bounds events reachable from its past worldline. Such horizons depend on the observer and need not imply a black hole.

A Killing horizon is a null hypersurface to which a nonzero Killing vector field $K^a$ is normal. On it, $K^a$ is tangent to its null generators and satisfies $K^b{\nabla }_b{K}^a=\kappa K^a$. The surface gravity $\kappa$ measures the failure of this Killing parameter to be an affine parameter; rescaling $K$ rescales $\kappa$. A degenerate Killing horizon has $\kappa =0$. Stationary black-hole horizons often are Killing horizons, whereas a general evolving event horizon need not have any Killing symmetry. The condition $K^a{K}_a=0$ alone is insufficient: the null vector must also be normal to a null hypersurface.

The Schwarzschild spacetime illustrates several of these notions. In units $G=c=1$, let $f=1-2M/r$. Its metric is $d{s}^2=fd{t}^2-f^{-1}d{r}^2-r^{2}d{\Omega }^2$, and $r=2M$ is the future event horizon of the black-hole region in the appropriate extension. The apparent coordinate singularity disappears in ingoing Eddington-Finkelstein coordinates, $v=t+r_{\ast }$, $d{r}_{\ast }/dr=f^{-1}$:At $r=2M$ this metric is regular and the normal $dr$ is null. Future outgoing radial rays satisfy $dr/dv=f/2$; inside the horizon their areal radius decreases even on the outgoing family. The stationary Killing vector field ${\partial }_v$ generates the horizon with $\kappa =f^{\prime }(2M)/2=1/(4M)$ when normalized at infinity. The Kretschmann scalar is $48M^2/r^6$, finite at the horizon and divergent at $r=0$. Thus the event horizon is not the curvature singularity. The photon sphere at $r=3M$, which supports circular null orbits, is also not the horizon.

Quasi-local horizon notions use null expansions. For a closed spacelike two-surface, choose future outgoing and ingoing null normals $\ell ,n$, with $\ell \cdot n=1$ in signature $(+---)$. Each expansion is the logarithmic rate of its positive area element along the corresponding null normal. A future trapped surface has ${\theta }_{\ell }<0$ and ${\theta }_n<0$: both future light fronts initially shrink. A marginally outer trapped surface has ${\theta }_{\ell }=0$; for the future black-hole type also require ${\theta }_n<0$. On a chosen spacelike slice, the outer boundary of the future trapped region is an apparent horizon, ordinarily an outermost marginally outer trapped surface when regularity permits this characterization. It depends on the slice, and a slicing need not display an apparent horizon even when the global spacetime contains an event horizon.

For the Schwarzschild metric in ingoing coordinates choose $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. The spherical null expansion in ingoing coordinates givesThe round spheres inside $2M$ are future trapped, while $r=2M$ is marginal. On standard horizon-penetrating slices its sections give the apparent horizon, coinciding with the stationary event and Killing horizon. This coincidence in a symmetric stationary example does not identify the notions in a general spacetime.

A trapping horizon is a hypersurface foliated by marginal surfaces. Its future outer type obeys ${\theta }_{\ell }=0$, ${\theta }_n<0$ and ${\mathcal{L}}_n{\theta }_{\ell }<0$. Unlike an event horizon, this can be diagnosed from local expansion data and their variation. The null energy condition explains its causal character. At a marginal surface, the Null Raychaudhuri equation, with the present curvature sign and vanishing null twist, givesin units $c=1$. A possible nonaffinity term vanishes here because it is proportional to the zero expansion. Write a horizon evolution vector as $V=\ell -Cn$. Preserving ${\theta }_{\ell }=0$ gives $C=({\mathcal{L}}_{\ell }{\theta }_{\ell })/({\mathcal{L}}_n{\theta }_{\ell })\ge 0$. Hence $V^2=-2C\le 0$: the future outer horizon is spacelike or null. Its area expansion is ${\theta }_V=-C{\theta }_n\ge 0$. These conclusions use the outer condition and the energy condition, not merely marginality.

For the global event horizon, the future-complete horizon focusing proof of the area theorem uses the same Null Raychaudhuri equation. Under the null energy condition, a negative generator expansion forces a future caustic in finite affine parameter. Future completeness and the usual regularity or predictability assumptions exclude such a caustic on the future event-horizon boundary. Its generator expansion is therefore nonnegative and its classical area cannot decrease. These assumptions matter: this is not an unconditional result for every null surface or every matter model.

Rotating and charged solutions provide further distinctions. A nonextremal Kerr black hole has $r_{\pm }=M\pm \sqrt{M^2-a^2}$, with $a=J/M$. Its outer Killing horizon is generated by ${\partial }_t+{\Omega }_H{\partial }_{\varphi }$, where ${\Omega }_H=a/(r_{+}^2+a^2)$, rather than by time translation alone. The ergosphere begins at the stationary-limit surface $g_{tt}=0$, whose radius is $M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the axis this lies outside the event horizon: it forbids static observers but does not forbid escape to infinity. In Reissner-Nordstrom spacetime, the nonextremal radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer surface is an event horizon, whereas the inner surface is a Cauchy horizon in the exact analytic extension.

A Cauchy horizon concerns predictability from initial data. The future domain of dependence $D^{+}(\Sigma )$ of a partial Cauchy surface consists of events for which every past-inextendible causal curve intersects $\Sigma$. Its future Cauchy horizon isBeyond it, causal influences can arrive without meeting the specified initial surface; data on that surface no longer determine the solution there. This is distinct from the inability to send signals to infinity. The inner horizon of the exact charged black hole supplies an example, and the definition necessarily specifies which initial surface is being used.

Flat spacetime itself has acceleration horizons. In the right Rindler wedge, set $t=\xi \sinh (a\tau )$ and $x=\xi \cosh (a\tau )$, with $\xi >0$. The Minkowski spacetime metric becomes $d{s}^2=a^2{\xi }^{2}d{\tau }^2-d{\xi }^2-d{y}^2-d{z}^2$. The null boundaries $t=x$ and $t=-x$ are Rindler horizons of the boost Killing flow. A uniformly accelerated observer cannot receive signals from one side of its future horizon, even though the curvature vanishes and inertial observers encounter no such boundary. These are observer and Killing horizons, not a black-hole event horizon defined using the full Minkowski null infinity.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with noncompact spatial sections, radial light rays have $d\chi /dt=\pm 1/a(t)$. The particle horizon measures how far signals could have travelled since the initial cosmic time; the cosmological event horizon measures how far signals emitted now could ever travel toward a comoving observer. Their comoving and proper radii areThe respective horizon is present only if its integral is finite. The event-horizon integral uses the actual future endpoint, often $t_f=\infty$, so it is global and observer-dependent. For $a(t)=t^q$ on $0<t<\infty$, a finite particle horizon exists for $0<q<1$, while a finite event horizon exists for $q>1$. Neither integral is finite for $q=1$. In the expanding flat patch of de Sitter spacetime, $a=e^{Ht}$ and the proper event-horizon radius is $H^{-1}$. The static form $d{s}^2=(1-H^2{r}^2)d{t}^2-(1-H^2{r}^2{)}^{-1}d{r}^2-r^{2}d{\Omega }^2$ identifies it as a cosmological Killing horizon; its surface-gravity magnitude is $H$ with the central static time normalization.

Finally, the Hubble radius must not automatically be called an event horizon. For a flat expanding cosmology, the two future radial null expansions of a sphere of proper radius $R$ are ${\theta }_{\pm }=2(H\pm 1/R)$. At $R=H^{-1}$ the ingoing expansion vanishes, while outside it both expansions are positive: this is the cosmological marginal, past-trapped type rather than a future black-hole trapped surface. A dust universe with $a\propto t^{2/3}$ has such a Hubble sphere but no future event horizon. Global causal boundaries, slice-dependent trapped-surface boundaries, symmetry horizons and initial-data boundaries must therefore be distinguished, even when several coincide in a special example.

Use units $c=1$ and take the initially expanding branch with ordinary positive matter or radiation density. In the Friedmann-Lemaître-Robertson-Walker metric, $k=+1,0,-1$ respectively describe positive, zero and negative spatial curvature. Separately conserved pressureless matter has ${\rho }_M\propto a^{-3}$ and radiation in cosmology has ${\rho }_R\propto a^{-4}$, so either dominates the curvature and vacuum terms sufficiently far back on a hot Big Bang branch.

With negligible cosmological constant, a spatially flat radiation universe has $a\propto t^{1/2}$ and a flat matter universe has $a\propto t^{2/3}$; both expand forever while decelerating. An open universe also expands forever and eventually becomes approximately curvature dominated, with $a\propto t$. A closed universe containing only ordinary matter/radiation reaches a maximum size and recollapses. For example, dust gives ${\dot{a}}^2=C/a-k$ with $C>0$, so only $k=+1$ admits the positive turning point $a=C$.

Curvature alone does not determine the fate when the cosmological constant is retained. Positive $\Lambda$ can cause late accelerated expansion and prevent even a closed universe from recollapsing; whether a closed expanding branch turns around depends on the relative matter and vacuum terms. Negative $\Lambda$ can cause recollapse even for flat or open spatial sections. These statements follow by inspecting ${\dot{a}}^2=C/a-k+\Lambda a^2/3$ for dust; the curvature sign is a geometrical classification, not by itself a universal destiny criterion.

Define the critical density ${\rho }_c=3H^2/(8\pi G)$ and cosmological density parameters ${\Omega }_M={\rho }_M/{\rho }_c$, ${\Omega }_R={\rho }_R/{\rho }_c$, ${\Omega }_{\Lambda }=\Lambda /(3H^2)$ and ${\Omega }_k=-k/(a^2{H}^2)$. The Friedmann equation gives ${\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }+{\Omega }_k=1$. Dividing the Friedmann acceleration equation by $-H^2$ givesConsequentlyThe deceleration parameter measures the sign and size of the acceleration relative to $a{H}^2$: $q>0$ means deceleration, $q<0$ accelerated expansion and $q=0$ instantaneously zero acceleration. In particular a falling Hubble parameter alone does not imply deceleration, since $q=-1-\dot{H}/H^2$.

For conformal time $d\eta =dt/a$, the conformal Hubble parameter is $\mathcal{H}=aH$. Using matter conservation and differentiating ${\Omega }_M\propto {\rho }_M/H^2$,This is the matter density flow with a cosmological constant. Neglecting ${\Omega }_{\Lambda }$ during matter domination reduces it to the requested relationThat last equation is not exact with nonzero $\Lambda$. For example, a flat dust-plus-vacuum state with $({\Omega }_M,{\Omega }_{\Lambda })=(0.9,0.1)$ gives ${\Omega }_M^{\prime }/\mathcal{H}=-0.27$, whereas the matter-only expression gives $-0.09$.

For the flatness problem, let ${\Omega }_{\mathrm{tot}}={\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }$. Curvature satisfies ${\Omega }_{\mathrm{tot}}-1=k/(aH{)}^2$, soIn decelerating near-flat radiation and matter eras, deviations therefore grow as $a^2$ and $a$ respectively. Maintaining small curvature at a late epoch requires very small early deviations unless a prior mechanism suppresses them. Cosmic inflation, with $q\simeq -1$, drives the deviation down approximately as $a^{-2}$. If vacuum energy is present, flatness means ${\Omega }_{\mathrm{tot}}=1$, not ${\Omega }_M=1$.

The spatial coordinates are cyclic because the flat Friedmann-Lemaître-Robertson-Walker metric depends only on time. Differentiating the printed affine Lagrangian gives the canonical momentumThe Euler-Lagrange equations give $d{P}_i^c/d\tau =\partial L/\partial x^i=0$, soThis is the conserved quantity associated with spatial translation invariance. The overall minus sign in the stated action makes this canonical covector opposite to the usual future-directed mechanical momentum; it does not alter the geodesic trajectories or the conservation result. The canonical momentum sign in the affine particle action will be kept explicit below.

Write ${\gamma }_{ij}={}^{(3)}{g}_{ij}$, $\gamma =\det {\gamma }_{ij}$ and $D_i$ for the spatial covariant derivative. With the negative-shift convention, the extrinsic curvature of a spatial hypersurface is $K_{ij}=-({\dot{\gamma }}_{ij}+D_i{N}_j+D_j{N}_i)/(2N)$. Contracting with ${\gamma }^{ij}$ and using metric compatibility givesThe logarithmic determinant identity makes ${\gamma }^{ij}{\dot{\gamma }}_{ij}=\dot{\gamma }/\gamma$, henceThis is also the fractional volume expansion, with a minus sign from the extrinsic-curvature convention.

Define $a={\gamma }^{1/6}$ so that $\sqrt{\gamma }=a^3$. The useful unit-determinant conformal decomposition is ${\gamma }_{ij}=a^2{\tilde{\gamma }}_{ij}$ with $\det \tilde{\gamma }=1$, equivalently ${\tilde{\gamma }}_{ij}=a^{-2}{\gamma }_{ij}$. Multiplying by $a^2$ instead would be a different conformal rescaling, not this unit-determinant one; no conformal decomposition is needed for the trace identity itself.

With zero shift vector, $\dot{\gamma }/\gamma =6\dot{a}/a$ and the local volume Hubble parameter isAlong a normal trajectory proper time satisfies $ds=Ndt$, so $H=a^{-1}da/ds$ is the local physical volume-expansion rate. The printed equality with $\dot{a}/a$ requires the proper-time gauge $N=1$; zero shift alone does not set the lapse to unity. In a homogeneous Friedmann-Lemaître-Robertson-Walker metric it reduces to the ordinary Hubble parameter, while in an inhomogeneous spacetime it can vary from one normal trajectory to another.