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For pressureless matter, the Friedmann acceleration equation with vanishing cosmological constant isEvaluating at the present cosmic time and using the definition of the deceleration parameter givesThe denominator is the critical density, so the cosmological density parameter satisfies the dust relationThis uses pressureless matter and $\Lambda =0$; spatial flatness has not been assumed.

At the present epoch, define $s=1+z$ and $F(z)=\dot{z}/H_0$. For flat matter plus a cosmological constant, neglecting radiation,In an Einstein-de Sitter universe, $E=s^{3/2}$, henceThe redshift drift is zero at $z=0$, has initial slope $F^{\prime }(0)=-1/2$, and decreases monotonically because $F^{\prime }=1-3\sqrt{s}/2<0$. Its second derivative is $-3/(4\sqrt{s})<0$, so the sketch bends downward, with $F\sim -z^{3/2}$ at large positive redshift.

A pure positive cosmological constant in a flat universe gives constant $H=H_0$, the expanding de Sitter model. ThusThe redshift drift is a straight line through the origin with positive slope. Here the source and observer expansion rates are equal; the factor $1+z$ in the observer contribution makes every positive-redshift comoving source drift toward still larger redshift.

Separate the observables before combining their parameter constraints. The baryon fraction is included within total matter: for the benchmark, ${\Omega }_{\mathrm{CDM}}={\Omega }_m-{\Omega }_b=0.255$, rather than adding ${\Omega }_b$ to ${\Omega }_m$ a second time. With $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})=0.7$, the physical density combinations areFor baryons, Big Bang nucleosynthesis predicts primordial light-element abundances as a function of the baryon-to-photon ratio; deuterium is especially sensitive. Independently, baryon inertia changes the relative heights of compressional and rarefaction peaks in the cosmic microwave background through the baryon loading parameter. Agreement of these two physical-density estimates supports a baryon abundance near the quoted value. For example, the three-year WMAP cosmological analysis found ${\Omega }_b{h}^2=0.02229\pm 0.00073$ in its fitted model.

For total matter, matter-radiation equality affects the microwave-background peak pattern and fixes a characteristic turnover in the matter power spectrum. galaxy clustering therefore constrains matter density, with corrections for galaxy bias and peculiar-velocity distortions. Cluster baryon fractions combine observed gas and stars with total masses inferred dynamically or by gravitational lensing; comparison with the cosmic baryon fraction also constrains ${\Omega }_m$. Galactic rotation, cluster dynamics and gravitational lensing give independent evidence that visible baryons alone cannot supply the gravitating mass. The precise ${\Omega }_m$ estimate comes from quantitative joint fits, rather than from a rotation curve by itself. Different 2006 data combinations gave somewhat different best fits, so $0.3$ is a rounded benchmark rather than a universal exact fit.

For a cosmological constant, standardized high-redshift Type Ia supernovae measure the shape of the luminosity-distance relation. In a flat matter-vacuum model,The observed supernova dimming relative to simple decelerating matter models favours late accelerated expansion. The original evidence is given by Riess and collaborators' 1998 supernova analysis. Combining the distance curve with matter constraints selects a positive vacuum fraction close to the quoted benchmark; it gives $q_0={\Omega }_m/2-{\Omega }_{\Lambda }=-0.55$.

For curvature, the angular scale of the Cosmic microwave background acoustic peaks compares the sound horizon with the angular diameter distance to last scattering. This strongly constrains spatial geometry, but curvature and late-time expansion parameters can partly compensate one another. Independent distances, notably baryon acoustic oscillations and supernova distances, help break that degeneracy. The 2005 SDSS acoustic-peak detection supplies a low-redshift standard ruler complementary to the microwave-background ruler. In combination, these observations support near-flat geometry and the closure relation ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k\simeq 1$.

The benchmark is supported by mutually constraining abundance, clustering, angular-scale and distance measurements. Its uncertainties are correlated and depend on model assumptions. An exactly zero curvature parameter has no meaningful percentage error; observational evidence establishes consistency with zero and an absolute uncertainty, not a literal fractional accuracy better than ten per cent on zero.

Assume the present universe is expanding, $\dot{a}(t_0)>0$. The stated energy inequality and the Raychaudhuri equation imply $\ddot{a}\le 0$ wherever $a>0$. Therefore for $t<t_0$ in that positive branch, $\dot{a}(t)\ge \dot{a}(t_0)$, andNonnegativity of $a(t)$ forces $t_0-t\le a(t_0)/\dot{a}(t_0)=H_0^{-1}$. If the solution remains valid towards the past, its positive branch must thus end in $a=0$ within that time; a positive scale factor extending further would contradict the inequality. HenceThis argument presumes an expanding branch and the stated energy condition throughout it, rather than proving either assumption.

There are two relevant failures of the naive numerical bound. The supplied ten-billion-year Hubble time is an approximate observational input: a smaller inferred present expansion rate gives a larger Hubble time, and comparison with stellar ages must include these uncertainties. More decisively, a positive cosmological constant, or other dark energy with sufficiently negative pressure, has $\rho +3P/c^2<0$ and permits accelerated expansion, so the concavity assumption fails. Observations of acceleration therefore remove this particular upper-bound argument. This is a discussion of the supplied historical estimate, not an assertion that its numerical value is a current measurement.

For a scalar, ${\nabla }_a\varphi ={\partial }_a\varphi$, while its second covariant derivative is ${\nabla }_a{\nabla }_b\varphi ={\partial }_a{\partial }_b\varphi -{\Gamma }_{ab}^c{\partial }_c\varphi$. Partial derivatives commute and the Levi-Civita connection is torsion-free, ${\Gamma }_{ab}^c={\Gamma }_{ba}^c$, proving the asserted symmetry.

Write the Einstein equation without cosmological constant as $G_{ab}=\kappa T_{ab}$, $\kappa =8\pi G/c^4$. Tracing the given Ricci equation gives $R=g^{ab}{\partial }_a\varphi {\partial }_b\varphi$. HenceIn units $\kappa =1$ the prefactor is absent. Metric compatibility, the product rule and the just-proved symmetric Hessian of a scalar field giveThe contracted Bianchi identity makes the left side zero. Wherever the gradient covector is nonzero, at least one component is nonzero, forcing ${\nabla }_a{\nabla }^a\varphi =0$, even when that gradient is null.

Finally ${\Gamma }_{ab}^a={\partial }_b\log \sqrt{-g}$, obtained by differentiating the metric determinant. Thus for any vector $V$, ${\nabla }_a{V}^a=(\sqrt{-g}{)}^{-1}{\partial }_a(\sqrt{-g}{V}^a)$. Applying it to $V^a=g^{ab}{\partial }_b\varphi$ proves

For a flat matter-dominated universe, the Friedmann solution has $a\propto t^{2/3}$, $H=2/(3t)$ and $\bar{\rho }=1/(6\pi G{t}^2)$. Thus the density equation isTrying $t^s$ gives $s(s-1)+4s/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe growing mode is proportional to the scale factor. When smooth dark energy or a cosmological constant dominates, the matter source density falls while expansion provides continued damping, suppressing growth. In the late constant-$\Lambda$ limit, $H\to H_{\Lambda }$ and $\bar{\rho }\to 0$, so $\ddot{\delta }+2H_{\Lambda }\dot{\delta }\simeq 0$ and $\delta$ tends to a constant plus a decaying $e^{-2H_{\Lambda }t}$ term. Matter perturbations therefore cease growing appreciably rather than continuing the matter-era law indefinitely.

The hierarchy problem concerns the enormous separation between the electroweak scale and a possible fundamental ultraviolet scale such as the reduced Planck mass: $v\simeq 246\mathrm{GeV}$ whereas $M_{\mathrm{Pl}}\simeq 2.4\times 10^{18}\mathrm{GeV}$. For an elementary Higgs field, heavy states or a physical high-energy ultraviolet cutoff generically generateHere $\kappa$ represents the appropriate dimensionless couplings. Maintaining a small physical Higgs mass then requires a very accurate cancellation against its bare mass parameter. The problem is sensitivity to actual high-energy physics, rather than merely the presence of a quadratic divergence in one choice of regulator.

The cosmological constant problem concerns the much smaller gravitationally observable vacuum energy. In the Einstein equation, a constant term in the effective action contributes to the cosmological constant, with ${\Lambda }_{\mathrm{eff}}={\Lambda }_{\mathrm{bare}}+{\rho }_{\mathrm{vac}}/M_{\mathrm{Pl}}^2$ in the usual normalization. Generic zero-point contributions scale as ${\Lambda }_{\mathrm{UV}}^4$, and vacuum transitions also change ${\rho }_{\mathrm{vac}}$. The dark-energy density is only of order $10^{-47}{\mathrm{GeV}}^4\sim (\text{a few meV}{)}^4$. The puzzle is why the total remains so small despite much larger individual contributions. A constant shift of the nongravitational Hamiltonian cannot remove its gravitational effect.

Supersymmetry relates bosonic and fermionic degrees of freedom, including their interactions. Their opposite loop signs cancel the quadratic contributions to a scalar field mass in the unbroken theory. With soft supersymmetry breaking, the surviving correction is controlled by the mass splitting, schematicallyThus a sufficiently low soft-breaking scale protects a small electroweak scale against a much higher ultraviolet scale. This is the useful supersymmetric response to the hierarchy problem.

Exact global supersymmetry also cancels vacuum energy: the supersymmetry algebra expresses the Hamiltonian as a positive sum of anticommutators, so a vacuum annihilated by every supercharge has zero energy. Equivalently, with canonical kinetic terms,vanishes in a supersymmetric vacuum. However, ordinary particles and their superpartners cannot have the identical mass spectrum of unbroken supersymmetry, so a useful phenomenological theory needs supersymmetry breaking. Its vacuum energy is generically of order the fourth power of the breaking scale; even a scale of order $10^2\mathrm{GeV}$ gives about $10^8{\mathrm{GeV}}^4$, enormously above the required value. In supergravity, a negative term can cancel positive breaking contributions:but arranging a tiny residual is a new tuning. Consequently supersymmetry can protect the hierarchy after soft breaking, while its exact vacuum-energy cancellation does not by itself survive realistic breaking.

One higher-dimensional proposal uses large extra dimensions. Take $n$ flat compact dimensions of common radius $R$, with gravity propagating in the bulk and Standard Model fields confined to a four-dimensional brane. Reducing the gravitational Einstein-Hilbert action givessoA fundamental gravitational scale $M_{\ast }$ near a TeV then produces apparently weak four-dimensional gravity through dilution into the large compact volume. For $n=2$ and $M_{\ast }=1\mathrm{TeV}$, $R=M_{\mathrm{Pl}}/(2\pi M_{\ast }^2)$ is about $0.075\mathrm{mm}$, using $1{\mathrm{GeV}}^{-1}\simeq 1.97\times 10^{-16}\mathrm{m}$. At distances much smaller than $R$, the gravitational potential scales as $r^{-(n+1)}$; at distances much larger than $R$ it scales as $r^{-1}$. Confining charged Standard Model fields to the brane avoids a light tower of charged Kaluza-Klein modes associated with the large radius. The primary explanatory challenge is radius stabilization: one must generate and stabilize $R{M}_{\ast }\gg 1$, rather than simply postulate a huge volume and transfer the original hierarchy into a modulus.

A second proposal is the Randall–Sundrum model. Compactify on an interval, equivalently an orbifold with branes at $y=0$ and $y=L$, and use the warped metricA negative bulk cosmological constant and suitably related brane tensions support this geometry. Integrating over the full orbifold interval $[-L,L]$ givesFor a scalar field $h$ on the brane at $L$, its induced metric gives the kinetic coefficient $e^{-2kL}$ and mass coefficient $e^{-4kL}{m}_0^2$. The canonical field is $h_c=e^{-kL}h$, henceWith $k$ and $M_5$ of the same fundamental order, an exponent $kL\simeq 35$--$37$ converts a fundamental mass into a weak-scale mass, without requiring a large proper compactification volume.

Again the separation must be selected dynamically. The original warped solution leaves a massless radion, so merely choosing $L$ does not complete the explanation. For example, the Goldberger–Wise stabilization mechanism introduces a bulk scalar field with boundary values $v_0,v_L$ and a small mass $m_{\chi }$. Its static equation and solution areFor small backreaction and large $kL$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0{e}^{-ϵkL}{)}^2$. Its leading minimum obeys $kL\simeq {ϵ}^{-1}\log (v_0/v_L)$, so a modest boundary-value ratio and a small $ϵ$ can give the required exponent. Corrections and the residual four-dimensional cosmological constant must still be controlled. Both proposals need a stable dynamical origin for the compactification geometry; their volume or warp relations alone do not supply it.

The strong equivalence principle asserts universality of free fall, including gravitational self-energy, and local independence of nongravitational and gravitational experiments from a freely falling laboratory's location and velocity. In a sufficiently small local inertial frame, uniform gravitational acceleration is removed and local physics takes its special-relativistic form. The Riemann curvature tensor remains measurable through tidal effects; a finite laboratory is not made exactly flat. Freely falling test bodies follow geodesics, clocks measure proper time, and minimally coupled light follows null geodesics in geometrical optics. Local nongravitational laws retain their special-relativistic form even when different laboratories observe gravitational redshift.

The comma-to-semicolon prescription replaces flat partial derivatives by the Levi-Civita connection, and the Minkowski metric by the curved metric tensor. It implements this local correspondence for minimally coupled matter. However, an identity obtained by commuting partial derivatives need not survive unchanged when they become covariant derivatives.

Raising the indices of the field-strength definition in flat spacetime gives $F^{ab}={\partial }^a{A}^b-{\partial }^b{A}^a$. Substitution into the sourced Maxwell equations yieldsso the first potential equation isPartial derivatives commute, giving the second formNo Lorenz gauge assumption was used.

Apply the comma-to-semicolon prescription before commuting derivatives. The two candidate left-hand sides becomeUse the cover curvature $C$ from Question 1, with $C_{ab}=C^c{}_{acb}$. The vector commutator is $[{\nabla }_c,{\nabla }_d]A^a=-C^a{}_{bcd}{A}^b$. HenceThis is the derivative-order ambiguity in covariant Maxwell potential equations. At a point in Riemann normal coordinates, the connection itself vanishes but its derivatives need not; the strong equivalence principle does not dispose of this curvature term by a coordinate choice.

Covariantizing the antisymmetric field strength and its Maxwell equations directly gives ${\mathcal{P}}_1^a={\mu }_0{J}^a$, orThis formulation has gauge invariance under $A_a\mapsto A_a+{\nabla }_a\chi$, since $F_{ab}$ is unchanged. In contrast, the uncorrected ${\mathcal{P}}_2$ acquires $-C^a{}_b{\nabla }^b\chi$ under that transformation. Thus preserving the original electromagnetic gauge symmetry singles out the directly covariantized Maxwell equations, rather than treating the two potential equations as interchangeable in matter. The equivalence principle alone is not a unique recipe for every possible higher-derivative or curvature coupling.

In a Ricci-flat vacuum spacetime, $C_{ab}=0$, so ${\mathcal{P}}_1={\mathcal{P}}_2$ and this particular ambiguity disappears even though the full Riemann curvature tensor may be nonzero. This uses vacuum with zero cosmological constant; an Einstein vacuum with nonzero cosmological constant is not Ricci flat.

For the terrestrial estimate, the relevant effect is Ricci curvature inside matter, not merely the tidal field outside the Earth. If a typical material density is $\rho \sim 5\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, the Einstein field equations give a characteristic scaleFor a field varying over scale $L$, the derivative terms are $O(A/L^2)$ whereas the disputed term is $O(CA)$. Their relative size is $O(C{L}^2)$: about $10^{-22}$ for a metre-scale experiment. Even substituting the Earth's radius gives onlyA curvature length comparable to $C^{-1/2}\sim 10^{13}\mathrm{cm}$ would be needed for an order-one effect. In principle non-vacuum measurements could distinguish different curvature couplings, but an ordinary terrestrial laboratory has an exceedingly small signal, overwhelmed by its ordinary electromagnetic response to matter. This is a practical scale estimate, not a theorem that an arbitrarily precise experiment is impossible.

Let $H=\dot{a}/a$ be the Hubble parameter, and normalize the scale factor by $a(t_0)=1$. The cosmological continuity equation gives ${\rho }_M={\rho }_{M0}{a}^{-3}$ for pressureless matter, whereas the cosmological constant has constant energy density ${\rho }_{\Lambda }$. The spatially flat Friedmann equation therefore becomesThe present Hubble constant is $H_0=H(t_0)$. In terms of the present critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$, the matter cosmological density parameter is ${\Omega }_M={\rho }_{M0}/{\rho }_{\mathrm{crit},0}$. Spatial flatness gives ${\rho }_{\Lambda }/{\rho }_{\mathrm{crit},0}=1-{\Omega }_M$, so

An orthonormal coframe is a collection of independent differential forms $E^{\hat{a}}=e^{\hat{a}}{}_{\mu }d{x}^{\mu }$ satisfying $g_{\mu \nu }={\eta }_{\hat{a}\hat{b}}{e}^{\hat{a}}{}_{\mu }{e}^{\hat{b}}{}_{\nu }$, with $\eta =\mathrm{diag}(-1,1,1,1)$. Locally one can diagonalize the metric tensor and rescale the resulting covectors, or use Gram-Schmidt orthogonalization for a symmetric bilinear form, choosing a timelike direction first. The coframe is not unique: a local Lorentz transformation preserves the metric. For the Levi-Civita connection, solve Cartan's first structure equation and metric compatibility together:These determine the connection 1-forms. The antisymmetry applies to the lowered pair of indices; a mixed time-space pair instead has ${\omega }^0{}_i={\omega }^i{}_0$. Next Cartan's second structure equation gives the curvature 2-forms, and reading off their coefficients determines the Riemann curvature tensor:This also states the curvature-sign convention. Contracting gives the Ricci tensor, and its coordinate components are $R_{\mu \nu }=e^{\hat{a}}{}_{\mu }{e}^{\hat{b}}{}_{\nu }{R}_{\hat{a}\hat{b}}$.

On a static patch with $V>0$, chooseWrite $h=\sqrt{V}$, $a=h^{\prime }$, $b=h/r$, and $c=\cot \theta /r$. The exterior derivative givesThe nonzero independent connection 1-forms are consequentlywith ${\omega }^1{}_0={\omega }^0{}_1$, spatial mixed pairs antisymmetric, and the remaining forms zero. For example,Similarly ${\Omega }^0{}_2={\omega }^0{}_1\wedge {\omega }^1{}_2=-ab{E}^0\wedge E^2$, and the angular derivatives in $d{\omega }^1{}_3$ cancel against ${\omega }^1{}_2\wedge {\omega }^2{}_3$. The resulting Cartan curvature forms for a static spherical metric areFor the last entry, $d(-c{E}^3)=E^2\wedge E^3/r^2$ and ${\omega }^2{}_1\wedge {\omega }^1{}_3=-V{E}^2\wedge E^3/r^2$. The remaining curvature 2-forms follow from ${\Omega }_{ab}=-{\Omega }_{ba}$.

Put $A=V^{\prime \prime }/2$, $B=V^{\prime }/(2r)$, and $C=(1-V)/r^2$. Reading the Riemann curvature tensor from the forms and contracting gives the diagonal orthonormal Ricci tensorFor example $R_{00}=R^1{}_{010}+R^2{}_{020}+R^3{}_{030}=A+2B$, while $R_{22}=-B-B+C$. All off-diagonal entries vanish because the curvature 2-forms only involve their own index planes. In the coordinate basis the nonzero components areThe angular Einstein field equations with the negative cosmological constant give $(rV{)}^{\prime }=1+3r^2/l^2$. Integrating introduces one real constant, conventionally written $-2m$:Substitution gives $V^{\prime \prime }/2+V^{\prime }/r=3/l^2$, so the time and radial Einstein field equations hold as well. The solution is the Schwarzschild--anti-de Sitter black hole family, including its zero and negative mass limits.

For $m>0$, $V$ increases strictly from $-\infty$ at $r=0$ to $+\infty$ at infinity. Its unique positive simple zero $r_h$ satisfies $r_h^3+l^2{r}_h=2m{l}^2$. This zero is a coordinate singularity, not a curvature singularity: with the ingoing coordinate $v=t+\int dr/V$, the radial metric is $-Vd{v}^2+2dvdr$ and is regular there. Inside the future black-hole region, $\nabla r={\partial }_v+V{\partial }_r$ is future timelike, so every future-directed causal trajectory decreases $r$ and cannot return to the asymptotic anti-de Sitter boundary. Conversely, every exterior radius admits an outward radial null geodesic reaching that boundary, since ${\int }_r^{\infty }dr/V$ converges. Thus $m>0$ gives an event horizon at the unique positive zero of $V$. Here the event horizon is defined relative to the asymptotic timelike boundary of Anti-de Sitter spacetime. For $m=0$, $V=1+r^2/l^2>0$ and the spacetime is pure Anti-de Sitter spacetime, with no black-hole event horizon. For $m<0$, $V=1+2|m|/r+r^2/l^2>0$, so there is also no event horizon; the mass term produces a naked singularity at $r=0$. The field equation itself does not impose a positive mass integration constant.

On an expanding branch with $\dot{a}>0$, minimizing $\dot{a}$ is equivalent to minimizing its square. Differentiating the preceding Friedmann equation with respect to $a$ givesFor positive matter density and a positive cosmological constant, it vanishes uniquely atThe second derivative is $H_0^2[2{\Omega }_{m,0}{a}_0^3/a^3+2{\Omega }_{\Lambda ,0}]>0$. Thus this is the minimum expansion speed in a matter-Lambda universe. The matter and vacuum densities satisfy ${\rho }_m=2{\rho }_{\Lambda }$ there, precisely the transition from deceleration to acceleration in the Friedmann acceleration equation.

The conclusion concerns an expanding branch that actually reaches this value of $a$. A sufficiently closed model can recollapse before reaching it; positive $\Lambda$ alone does not rule that out. In flat or open models with the stated positive densities the minimum is reached. The degenerate zero-speed limiting case requires separate treatment and does not describe a finite-time passage through a positive-speed minimum.

Use the stipulated common normalization $A=H_0{\Omega }_{m,0}$ and plot $R=A{r}_h/c$. The Einstein-de Sitter universe has $H_{0,E}=A$, whereas the $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$ model has $H_{0,\Lambda }=A/0.3$. Their comoving particle horizons are thereforeBoth decrease with increasing $z$, or increase with cosmic time. The Einstein-de Sitter universe curve diverges at $z=-1$ and has $R_E(0)=2$, $R_E(2)=2/\sqrt{3}$. The matter-plus-vacuum curve remains finite at $z=-1$: the integrand approaches a constant there, and its high-redshift tail is integrable. Moreover it lies below the Einstein-de Sitter curve for this particular normalization, since, with $u=1+z^{\prime }>0$,The high-redshift ratio of the two curves tends to $\sqrt{0.3}$, not one. Holding $H_0\sqrt{{\Omega }_{m,0}}$ fixed would give a different comparison; it is $H_0{\Omega }_{m,0}$ that the original question specifies.

A positive cosmological constant eventually makes the scale factor grow exponentially. The future integral ${\int }_t^{\infty }cd{t}^{\prime }/a(t^{\prime })$ then converges, so the remaining conformal time is finite. The comoving particle horizon approaches a finite limiting visibility radius; it does not shrink. The remaining distance to that limit is the comoving radius of the cosmological event horizon. Thus a signal emitted sufficiently far away today can never reach us, even though the proper particle-horizon radius continues to grow. This is the total conformal lifetime of a flat matter-Lambda universe.

In a spatially flat FLRW metric, a radial null geodesic satisfies $d\chi =cdt/a(t)$. Normalize $a_0=1$. The comoving particle horizon at time $t$ is therefore $r_h=c{\int }_0^{t}d{t}^{\prime }/a(t^{\prime })$, using the initial singularity as the time origin. The cosmological redshift relation $1+z=1/a$ givesFor matter and a cosmological constant in a spatially flat model, the Friedmann equation gives $H(z)=H_0[{\Omega }_{m,0}(1+z{)}^3+{\Omega }_{\Lambda ,0}{]}^{1/2}$. The initial singularity corresponds to $z\to \infty$, so reversing the integration limits yieldsThis is a comoving particle horizon, not a distance to the source currently observed at redshift $z$, and not a future-directed cosmological event horizon. The corresponding proper horizon radius at that epoch is $r_h(z)/(1+z)$.

Identify the ambient real four-dimensional vector space with $2\times 2$ real matrices byFor $A,B\in SL(2,\mathbb{R})$, the linear action $X\mapsto AX{B}^{-1}$ preserves this quadratic form. It gives a group homomorphism to the special orthogonal group of signature $(2,2)$; its determinant on matrix space is $(\det A{)}^2(\det B{)}^{-2}=1$. The group $SL(2,\mathbb{R})$ is connected: an oriented Gram–Schmidt factorization writes every element as a rotation times an upper triangular matrix with positive diagonal and determinant one, and each factor varies in a connected space. Thus the image lies in the identity component $S{O}_0(2,2)$.

If $AX{B}^{-1}=X$ for every $X$, taking $X=I$ gives $A=B$. A matrix commuting with every real $2\times 2$ matrix is scalar, and its determinant-one condition then gives $A=B=\pm I$. The kernel is therefore exactly $\{(I,I),(-I,-I)\}$. Infinitesimally the action is $X\mapsto aX-Xb$, with $a,b$ traceless. If this vanishes for every $X$, then $a=b$ and it is scalar and traceless, hence zero. The differential is injective; source and target Lie algebras each have dimension six, so it is an isomorphism. The image is therefore an open subgroup of the connected target, which must be the entire target. ConsequentlyThe finite quotient is by the diagonal central subgroup, not by a sign in just one factor. The connected-component subscript is a genuine qualification of the source's displayed formula: the full $SO(2,2)$ is disconnected and cannot be isomorphic to a quotient of the connected product.

Let the anti-de Sitter radius be $\ell >0$. Its quadric is $-u^2-v^2+x^2+y^2=-{\ell }^2$, or $\det X={\ell }^2$. Thus the map $X\mapsto g=X/\ell$ identifies this quadric smoothly and globally with $SL(2,\mathbb{R})$. This is the group model of three-dimensional anti-de Sitter spacetime. The induced Lorentzian metric is bi-invariant, since the two matrix multiplication actions preserve the ambient quadratic form. At the identity, tangent matrices $a,b$ are traceless andIndeed, for a traceless matrix $a$, $\det a=-\frac{1}{2}\mathrm{Tr}(a^2)$, and polarizing $-\det (\ell a)$ gives this metric. For ${\mathfrak{sl}}_2(\mathbb{R})$ the Killing form is $K(a,b)=4\mathrm{Tr}(ab)$: in the basis $H=\mathrm{diag}(1,-1)$, $E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ and $F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$, the brackets are $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$, giving $K(H,H)=8$, $K(E,F)=4$ and the remaining independent pairings zero. Therefore $g_{\ell }={\ell }^{2}K/8$.

The canonical torsion-free connection on a Lie group is the Levi-Civita connection of this bi-invariant pseudo-Riemannian metric. Its Ricci tensor, already calculated from the double bracket, isSubstitution into the vacuum Einstein field equations givesThus the group metric solves the three-dimensional vacuum equations with negative cosmological constant.

The antipodal map $X\mapsto -X$ is $g\mapsto -g$, multiplication by the central element $-I$. Hence the antipodal quotient is $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I\}$, with the same local curvature and cosmological constant. To see the global time issue, useThe induced metric isOn the quadric $t$ has period $2\pi$; curves with fixed $\rho ,\phi$ are closed timelike curves. The antipodal identification is $(t,\phi )\sim (t+\pi ,\phi +\pi )$, and does not eliminate this problem. At $\rho =0$, even the half-period time shift closes after the antipodal quotient.

If global time is not identified, take the universal cover, with $t\in \mathbb{R}$. The topology of $SL(2,\mathbb{R})$ is $S^1\times {\mathbb{R}}^2$, as follows from the same rotation–triangular decomposition, so the universal cover is diffeomorphic to ${\mathbb{R}}^3$ and carries the lifted group structure $\widetilde{SL(2,\mathbb{R})}$. The metric and Einstein equation are unchanged locally. Here $t$ is a global time function with timelike gradient, excluding closed causal curves. Unwrapped anti-de Sitter spacetime is the universal covering group, rather than the original embedded quadric or its antipodal quotient; retaining the antipodal time-shift identification would wrap time again.

The age of an FLRW universe at an epoch is the elapsed cosmic time since the Big Bang, not the lookback time from today. For positive matter and positive cosmological constant, integrating $dt=da/\dot{a}$ givesPut $u=\sqrt{{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}}{a}^{\prime 3/2}$; then $du=(3/2)\sqrt{{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}}\sqrt{a^{\prime }}d{a}^{\prime }$. The integral becomes the age of a flat matter-Lambda universe:Choose $0<\theta <\pi /2$ such that $\tan \theta =\sqrt{{\Omega }_{m,0}/{\Omega }_{\Lambda ,0}}{a}^{-3/2}$. The upper-limit value of $u$ is $\cot \theta$, andConsequentlyThe integration constant is fixed by $t\to 0$ as $z\to \infty$, where $\theta \to \pi /2$. Equivalently, differentiating $u=\cot \theta$ changes the last integral to $-\int d\theta /\sin \theta$, recovering the supplied trigonometric identity. The formula assumes positive ${\Omega }_{\Lambda ,0}$ and ${\Omega }_{m,0}$; its zero-vacuum limit is regular even though this parametrization is singular.