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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.

A Type Ia supernova can be standardized as a standard candle by correcting its peak brightness using the light curve shape and colour index. Its host spectrum gives the redshift; the corrected apparent magnitude and calibrated absolute magnitude give the distance modulusThe resulting Hubble diagram measures the shape of the luminosity distance as a function of redshift. A spatially flat matter-and-vacuum model, for example, predictsNonzero cosmological spatial curvature changes both the expansion rate and the conversion from radial to transverse comoving distance. Comparing the observed Hubble diagram with these predictions constrains the matter density parameter, cosmological constant and cosmological spatial curvature; a single low-redshift observation cannot separate them. The nearby sample calibrates the common intercept, while distant Type Ia supernovae supply the redshift-dependent shape. Relative faintness at a specified redshift means a greater inferred luminosity distance.

Type Ia supernova cosmology requires correction for interstellar extinction, possible evolution of the standardized luminosity, and selection bias. The absolute magnitude–Hubble constant degeneracy means that an uncalibrated sample constrains the shape rather than independently determining $H_0$.

Normalize the scale factor to $a(t_0)=1$, use $c=1$, and define the instantaneous critical density by ${\rho }_{\mathrm{cr}}(t)=3H(t{)}^2/(8\pi G)$. Separately conserved pressureless matter and radiation in cosmology obey the cosmological continuity equation,The present cosmological density parameters areThe cosmological constant has equivalent energy density ${\rho }_{\Lambda }=\Lambda /(8\pi G)$. Substituting into the Friedmann equation yieldsAt the present epoch this gives $k=H_0^2({\Omega }_0-1)$, or curvature contribution ${\Omega }_{K0}=-k/H_0^2=1-{\Omega }_0$. Dividing each component by $H_0^2$ provesSpatial curvature is not included in the total physical-density parameter ${\Omega }_0$. In particular, positive vacuum density does not have the radiation or matter scaling.

Interpret $A$ first as an infinitesimal screen area transported along generators of a future event horizon, not as the total area of an arbitrary trapped surface. Horizon generators form an affinely parametrized, hypersurface-orthogonal null geodesic congruence. The two-dimensional screen metric is positive definite. Its optical tensor has zero twist and decomposes asThe scalar here agrees with the supplied derivative definition: nullness and the affine geodesic equation remove the longitudinal terms in the contracted norm, leaving $B_{AB}{B}^{AB}-{\theta }^2/2$.

To derive the focusing equation, vary neighbouring geodesics and let $J$ be their screen Jacobi map. The geodesic deviation equation is $J^{\prime \prime }=-\mathcal{R}J$, where ${\mathcal{R}}_{AB}=R_{AcBd}{p}^c{p}^d$. Hence $B=J^{\prime }{J}^{-1}$ satisfies $B^{\prime }=-B^2-\mathcal{R}$. Taking its screen trace yields the Null Raychaudhuri equationSet $a=A^{1/2}$. Since $a^{\prime }/a=\theta /2$, the area-square-root optical focusing equation isThe PDF's plus sign before ${\sigma }^2$ is inconsistent with its own positive shear definition. This is an actual source error, not an antisymmetrization or curvature-sign convention.

For an explicit check, take a flat-space twist-free beam with screen Jacobi factors $1+\lambda$ and $1-\lambda$, for $|\lambda |<1$. ThenIts Ricci term is zero, so the result equals $-{\sigma }^2{A}^{1/2}$ and has the opposite sign to the printed equation. This is flat-space anisotropic beam shear focusing.

Under the null convergence condition $R_{ab}{p}^a{p}^b\ge 0$, the correct equation gives concavity of each local area square root. Equivalently ${\theta }^{\prime }\le -{\theta }^2/2$. If a horizon generator had $\theta ({\lambda }_0)={\theta }_0<0$, integration gives a focal point within affine distance at most $2/|{\theta }_0|$. Such a generator could not remain on an achronal boundary beyond the focal point. Under the usual global hypotheses that horizon generators remain regular and future complete, this contradicts their being generators of the event horizon. Hence $\theta \ge 0$ and $A^{\prime }=\theta A\ge 0$. Integrating over the horizon patches, and including newly joining generators which can add area, proves Hawking's area theorem. Global predictability/completeness assumptions are essential; the local differential inequality alone is not a theorem about arbitrary trapped-surface areas.

For the perfect fluid stress tensor, null contraction eliminates the pressure-metric term:A nonzero null vector cannot be orthogonal to a timelike fluid velocity. Thus the perfect-fluid null energy condition is preciselySeparate requirements $\rho \ge 0$ or $p_{\mathrm{fluid}}\ge 0$ are not needed for this null focusing argument. With the Einstein equation, $R_{ab}{p}^a{p}^b=8\pi T_{ab}{p}^a{p}^b$.

A cosmological constant contributes only a term proportional to $g_{ab}$, whose null contraction is zero:Therefore a nonzero cosmological constant does not change the fluid condition or the local area-focusing argument, though it can change the global asymptotics and which horizons and completeness hypotheses are appropriate.

Normalize $a_0=1$ and use the open dust model, with no cosmological constant and negligible present radiation. The Friedmann equation at the present epoch reads $H_0^2=H_0^2{\Omega }_m-k{c}^2$. Since $k=-r_{\mathrm{curv}}^{-2}$, the curvature radius of an open FLRW universe isIn the metric's $c=1$ convention the numerator is one. With $a_0=1$ this is also the present physical curvature radius; at scale factor $a$ the physical radius is $a{r}_{\mathrm{curv}}$. Inferring curvature from ${\Omega }_m$ alone would require this matter-only assumption; an additional dark-energy density would also enter the present Friedmann equation.

In the observational setting of this paper, the cosmic microwave background favours nearly flat geometry, while cluster dynamics and large-scale structure favour a matter density near ${\Omega }_m\sim 0.3$. The missing contribution to the critical density is then roughly ${\Omega }_X\sim 0.7$. The supernova luminosity distance measurements require recent accelerated expansion. With present radiation negligible, $q_0<0$ requiresFor an X-dominated universe the threshold would instead be $w<-1/3$. Thus X cannot be ordinary pressureless matter or radiation. It must be comparatively smooth on cluster scales, so that it contributes to the background expansion without being counted as clustered dynamical matter. Its negative pressure makes its density dilute more slowly than matter, allowing late dominance. A cosmological constant, with $w=-1$, or sufficiently potential-dominated quintessence can have these properties.

The comparison with a perfect fluid in general relativity needs a distinction between background stress and perturbations. A homogeneous canonical scalar has canonical scalar stress as a perfect fluid:It has isotropic pressure and no background shear stress, just as a perfect fluid in general relativity does. Negative pressure therefore does not, by itself, prevent a perfect-fluid description. However, an adiabatic barotropic fluid with the constant closure $p=w\rho$ would have $c_s^2=dp/d\rho =w$. For negative $w$, a short-wavelength density mode then has ${\omega }^2\simeq w{k}_{\mathrm{phys}}^2<0$, the negative-w barotropic-fluid gradient instability. Such a closure cannot give a stable, smooth propagating X component.

A canonical scalar instead has rest-frame sound speed squared equal to one: its spatial and temporal kinetic terms have the same coefficient. Its pressure perturbation need not obey the background relation $\delta p=w\delta \rho$. This nonadiabatic response suppresses small-scale clustering without the negative-sound-speed instability. A cosmological constant is a separate limiting case with no independent propagating fluid density mode. Thus the issue is the microphysics and perturbation closure, not whether the background stress-energy tensor has perfect-fluid form.

Type Ia supernovae have similar peak luminosities that can be standardized using their light-curve width and color, rather than assumed to be identical without correction. Their redshifts and corrected apparent magnitudes give a distance modulus,Compare this Hubble diagram with the luminosity-distance prediction of a background cosmology. Restoring $c$, put $E(z)=H(z)/H_0$, $I(z)={\int }_0^{z}d{z}^{\prime }/E(z^{\prime })$ and define ${\mathcal{S}}_{{\Omega }_k}(I)$ as $\sinh (\sqrt{{\Omega }_k}I)/\sqrt{{\Omega }_k}$ for ${\Omega }_k>0$, $I$ for ${\Omega }_k=0$, and $\sin (\sqrt{-{\Omega }_k}I)/\sqrt{-{\Omega }_k}$ for ${\Omega }_k<0$. ThenNeglecting radiation at these redshifts, Friedmann equations and a separately conserved dark energy component with $w(z)=p/(\rho c^2)$ giveFor constant $w$, the last factor is $(1+z{)}^{3(1+w)}$; a cosmological constant has $w=-1$. Thus the redshift dependence of the measured distance constrains matter density, curvature and the dark-energy equation of state. At small redshift, $d_L=(cz/H_0)[1+(1-q_0)z/2+\cdots ]$, showing how departures from the linear Hubble law probe the deceleration parameter.

The early distant-supernova analyses found larger distances than matter-only decelerating models predicted, providing evidence for accelerated expansion; see the 1998 supernova analysis and the 1999 independent supernova analysis. This is a statement about the distance-redshift curve under the standardization and cosmological assumptions, not a direct measurement of a fluid pressure.

The absolute magnitude–Hubble constant degeneracy means uncalibrated supernovae constrain the shape of $d_L(z)$ without separately determining $H_0$ and the common absolute luminosity. The integral over $H(z)$ also makes different matter, curvature and $w(z)$ histories partly degenerate. Wide redshift coverage and independent cosmic microwave background and large-scale structure of the universe information help break these degeneracies. Dust extinction, source evolution, photometric calibration, sample selection and K corrections must be modeled, and constraints on a time-dependent equation of state depend on how that function is parameterized. Standardized supernovae test the expansion history; they do not by themselves uniquely determine every cosmological parameter.

For a uniform sphere of radius $R$, write $M(r)=M(r/R{)}^3$ and $dM=3M{r}^{2}dr/R^3$. The self-gravitational potential energy is assembled by adding shells:The cosmological constant produces the external potential per unit mass ${\Phi }_{\Lambda }=-\Lambda r^2/6$, since $-\nabla {\Phi }_{\Lambda }=\Lambda \mathbf{r}/3$ in units $c=1$. Its energy isThere is no additional factor $1/2$ for this fixed external potential, unlike the pair-counting factor for gravitational self-energy. At the maximum radius $R=r_m$, the kinetic energy vanishes, soThis is the turnaround energy in uniform-sphere collapse with a cosmological constant. The original PDF's shell kinetic term is ${\dot{r}}^2/2$, not the $r^2/2$ produced by the TeX transcription.

Take zero cosmological constant and an open geometry, which means $k<0$ in the metric convention. Evaluating the Friedmann equations today gives $H_0^2=H_0^2{\Omega }_0-k/a_0^2$, soOn a constant-time slice the radial line element is $d\ell =a_{0}dr/\sqrt{1-k{r}^2}$. ThereforeThis is a cosmological proper distance on today's slice, not the distance traveled during the photon flight. The printed condition $k<1$ is insufficient to specify an open model; the derivation uses its intended $k<0$.

The Pound-Rebka experiment measures a small gravitational redshift by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The Mössbauer effect permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\mathrm{m}$ have a predicted fractional frequency difference of order $2.5\times 10^{-15}$. A calibrated Doppler effect from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.

The prediction follows either from an accelerating laboratory and the equivalence principle, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq 1+2\Phi /c^2$. The time-translation Killing vector field gives a conserved photon energy $E_{\xi }=p_a{\xi }^a$, while a static observer has four-velocity $u^a={\xi }^a/N$ and measures $h\nu =E_{\xi }/N$. ThusFor an upward ray over height $H$, $\Delta \nu /\nu \simeq -gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.

The weak equivalence principle asserts universal free fall for test bodies. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The strong equivalence principle extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.

The geometrical realization uses the Levi-Civita connection of a spacetime metric. At any event, normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols vanish. Freely falling test particles follow geodesics; local nongravitational equations take their special-relativistic form at that event. The Riemann curvature tensor need not vanish, so finite laboratories can measure tidal geodesic deviation. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.

For a perfect fluid, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe first is the energy equation, including pressure work; the second is the relativistic Euler equation. Its rest-frame spatial pressure force is $-\nabla p$.

Further assumptions are needed to derive the Einstein field equations. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with stress-energy conservation. The available rank-two tensors are $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; a cosmological constant term remains allowed.

The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[{\nabla }_d,{\nabla }_c]$, and a static weak field has $R_{00}\simeq -{\nabla }^2\Phi /c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C{T}_{ab}$ implies $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, with $T=\rho -3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.

Setting $c=1$ for the action normalization, the same metric equation follows from the Einstein-Hilbert action $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the contracted Bianchi identity are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.

The Friedmann acceleration equation gives the gravitational deceleration from a uniform pressureless matter density, plus the acceleration from a cosmological constant. Here $\rho$ is the matter mass density, $G$ is the gravitational constant, $H=\dot{a}/a$ is the Hubble parameter, and $k$ is the constant spatial-curvature coefficient in the chosen normalization of $a$. The Friedmann equation is the first integral relating expansion to density, curvature and vacuum energy. With $c=1$, the matter, curvature and cosmological-constant contributions to $H^2$ are $8\pi G\rho /3$, $-k/a^2$ and $\Lambda /3$. The cosmological constant has effective density $\Lambda /(8\pi G)$ and equation-of-state parameter $w=-1$; its negative pressure explains its positive acceleration contribution when $\Lambda >0$.

For the Newtonian derivation with $\Lambda =0$, follow a test mass on the edge of a uniform sphere of physical radius $R=a\chi$. Conservation of the enclosed matter gives $M=4\pi \rho R^3/3$ constant. The shell theorem removes the force of external spherical shells and givesIntegrating the force equation once gives the conserved specific mechanical energyDivide by $R^2/2=a^2{\chi }^2/2$ to obtainHomologous motion makes $\epsilon /{\chi }^2$ constant between shells. Newtonian mechanics supplies this energy integration constant; its interpretation as intrinsic spatial curvature belongs to the relativistic FRW metric.

An outward acceleration $\Lambda \mathbf{r}/3$ comes from the Newtonian potential per unit masssince $-\nabla {\Phi }_{\Lambda }=\Lambda \mathbf{r}/3$. With conventional dimensional $\Lambda$, restore $c^2$ in this potential and the corresponding acceleration and Friedmann terms.

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 strong equivalence principle extends universality of free fall to bodies with appreciable gravitational binding energy and requires results of sufficiently local experiments, including gravitational experiments, in freely falling laboratories to be independent of their location and velocity. The Einstein equivalence principle makes the corresponding statement for nongravitational experiments; the weak principle concerns test-body free fall without significant self-gravity. In a locally inertial frame the nongravitational laws take their special-relativistic form, while internal gravitational experiments retain the same local gravitational laws and coupling constants. They are not declared to have no gravity.

In a metric description, one may set $g_{ab}={\eta }_{ab}$ and ${\partial }_c{g}_{ab}=0$ at a point by Riemann normal coordinates, so the Levi-Civita connection vanishes there. This removes a locally uniform external gravitational acceleration, not the Riemann curvature tensor. Tidal effects between separated bodies, or accumulated over a finite-duration experiment, remain. The local limit must control both the laboratory's size and duration. Gravitationally self-bound bodies are included in the strong version, so it is stronger than equality of inertial and passive gravitational mass for ordinary test particles.

A heuristic field-equation argument supplements this principle with further assumptions: gravity is described only by a Lorentzian metric, matter couples universally to it, the affine connection is Levi-Civita, and the leading local field equation is a symmetric covariant equation with at most second metric derivatives and is linear in those second derivatives. Matter obeys stress-energy conservation ${\nabla }^a{T}_{ab}=0$. Curvature therefore supplies the natural candidates $R_{ab}$ and $R{g}_{ab}$, with a constant multiple of $g_{ab}$ also allowed. Write the candidate as $R_{ab}+cR{g}_{ab}+\Lambda g_{ab}=\kappa T_{ab}$. The contracted Bianchi identity and metric compatibility giveTo permit varying scalar curvature consistently with matter conservation, take $c=-1/2$. This gives the Einstein tensor $G_{ab}=R_{ab}-g_{ab}R/2$ andThe cosmological constant is constant so that its divergence vanishes. These assumptions motivate the usual Einstein equation; the equivalence principle alone does not exclude theories with extra fields or higher curvature terms.

The sign and normalization are fixed by the supplied Newtonian limit. Use signature $(+---)$, the curvature commutator convention of Q1, and the Ricci contractionThis is the contraction for which $R_{00}\simeq -{\nabla }^2\phi$ in the question. In four dimensions, tracing the candidate equation gives $-R+4\Lambda =\kappa T$, and substituting back givesFor slowly moving pressureless matter, $T_{00}\simeq \rho$, $T\simeq \rho$ and $g_{00}\simeq 1$. On local scales where the cosmological term is negligible, $R_{00}\simeq \kappa \rho /2$. Compare this with $R_{00}\simeq -{\nabla }^2\phi$ and the Poisson equation ${\nabla }^2\phi =4\pi G\rho$. The Newtonian normalization of the Einstein field equations is thereforeUnits with light speed one are used, as in the paper; for conventional physical stress-energy units the coupling is $-8\pi G/c^4$. If the opposite Ricci/curvature sign convention is adopted, the displayed geometric coupling sign changes accordingly. With the cosmological term retained in the same weak-field calculation, the Newtonian equation becomes ${\nabla }^2\phi =4\pi G\rho -\Lambda$, consistent with this sign choice.

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 $q=-1-\dot{H}/H^2$.