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For an expanding pressureless matter universe with positive curvature, zero cosmological constant and present scale factor one, set $u_0=\arccos (2/{\Omega }_0-1)\in (0,\pi )$. The closed matter-dominated Friedmann solution then gives the displayed age of an FLRW universe. Its flat limit is $2/(3H_0)$, despite the individually vanishing numerator and denominator. To see this, write $\delta ={\Omega }_0-1$; then $u_0=2\arctan \sqrt{\delta }$ and $u_0-\sin u_0=(4/3){\delta }^{3/2}+O({\delta }^{5/2})$.

For an expanding pressureless matter universe with zero cosmological constant, normalize $x=a/a_0$. Dust relation between deceleration and density and mass conservation give ${\dot{x}}^2=H_0^2(1-2q_0+2q_0/x)$. Integrating from the Big Bang yields the age integral above.

For $q_0>1/2$, set $x=q_0(1-\cos \theta )/(2q_0-1)$ to obtain $H_0{t}_0=q_0({\theta }_{\ast }-\sin {\theta }_{\ast })/(2q_0-1{)}^{3/2}$, with $\cos {\theta }_{\ast }=(1-q_0)/q_0$ on the expanding branch. For $0<q_0<1/2$, the corresponding hyperbolic expression is $q_0(\sinh {\eta }_{\ast }-{\eta }_{\ast })/(1-2q_0{)}^{3/2}$ with $\cosh {\eta }_{\ast }=(1-q_0)/q_0$. The flat-dust and empty Milne universe limits are respectively $2/3$ and $1$.

For positive pressureless matter density, positive cosmological constant, negligible radiation and zero curvature, the Friedmann equation gives $dt=\sqrt{a}da/[H_0\sqrt{{\Omega }_m+{\Omega }_{\Lambda }{a}^3}]$. Substituting $u=\sqrt{{\Omega }_{\Lambda }/{\Omega }_m}{a}^{3/2}$ gives the displayed age of an FLRW universe measured from $a=0$. Its zero-vacuum limit is $2a^{3/2}/(3H_0\sqrt{{\Omega }_m})$; the pure positive-vacuum flat patch has no finite big-bang age.

For an expanding pressureless matter universe with zero cosmological constant, fixed current Hubble parameter $H_0$ and current matter cosmological density parameter $\Omega >0$, the age of a dust universe without a cosmological constant is $H_0{t}_0={\int }_0^1\sqrt{x}/\sqrt{\Omega +(1-\Omega )x}dx$. For every $0<x<1$ the integrand decreases strictly as $\Omega$ increases. Thus open models with $0<\Omega <1$ are older than the flat model, and closed expanding models with $\Omega >1$ are younger. The flat value is $2/3$ and the empty Milne universe limit is $1$. Specifying what is held fixed matters: at fixed present scale factor and energy density, the integral $t_0={\int }_0^{a_0}da/\sqrt{C/a-k}$ instead increases with $k$, provided all three models reach $a_0$ on their expanding branches.

For an axial Killing vector $\xi$ and zero cosmological constant, set $F^{ab}={\nabla }^b{\xi }^a$. The Killing equation and Ricci identity give ${\nabla }_b{F}^{ab}=-R^a{}_b{\xi }^b$. In a vacuum region this vanishes, so Stokes theorem makes the Komar angular momentum equal on homologous enclosing surfaces. The region may contain gravitational waves with nonzero Weyl curvature: Ricci flatness, rather than complete flatness, is what removes the flux. This proves absence of axial angular-momentum transport by exactly axisymmetric gravitational radiation, without assuming that a radiating spacetime is stationary.

A real cosmological deformation adds $3m^{2}e/{\kappa }^2+2me{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu }{\psi }_{\nu }$ and changes the gravitino variation to ${\mathcal{D}}_{\mu }ϵ/\kappa$. The modified commutator identity is ${\gamma }^{\mu \nu \rho }{\mathcal{D}}_{\nu }{\mathcal{D}}_{\rho }ϵ=(G^{\mu \nu }+\Lambda g^{\mu \nu }){\gamma }_{\nu }ϵ/2$. It fixes the negative cosmological constant in terms of the gravitino mass parameter. A positive cosmological constant is not obtained by this real unbroken minimal deformation.

In the exterior orthonormal coframe $E^0=\sqrt{V}dt$, $E^1=dr/\sqrt{V}$, $E^2=rd\theta$, $E^3=r\sin \theta d\varphi$, use ${\Omega }^a{}_b=\frac{1}{2}{R}^a{}_{bcd}{E}^c\wedge E^d$. The independent mixed-index curvature 2-forms are ${\Omega }^0{}_1=-V^{\prime \prime }{E}^0\wedge E^1/2$, ${\Omega }^0{}_A=-V^{\prime }{E}^0\wedge E^A/(2r)$, ${\Omega }^1{}_A=-V^{\prime }{E}^1\wedge E^A/(2r)$ for $A=2,3$, and ${\Omega }^2{}_3=(1-V)E^2\wedge E^3/r^2$. The orthonormal Ricci tensor is diagonal with entries $V^{\prime \prime }/2+V^{\prime }/r$, $-V^{\prime \prime }/2-V^{\prime }/r$, and twice $(1-V-r{V}^{\prime })/r^2$. The Einstein field equations with negative cosmological constant then give $V=1-2M/r+r^2/l^2$.

// Target: general-relativity.bigb

For positive curvature and zero cosmological constant, the single-fluid Friedmann solution has a bang-to-crunch conformal time interval $2\pi /(1+3w)$ for $w\ge 0$. A radial null geodesic in FLRW spacetime travels that same great-circle comoving distance. Relative to a three-sphere circumference $2\pi$, it is $1/(1+3w)$ circuits: one for dust and one half for radiation, with the final point reached only in the singular-endpoint limit.

A positively curved pressureless matter universe with zero cosmological constant has the cycloid parametrization displayed above. In natural units, if the Friedmann equation is ${\dot{a}}^2=C/a-k$ with $C,k>0$, then $A=C/(2k)$ and $u=\sqrt{k}\tau$ for conformal time measured from the Big Bang. Substituting in $a^{\prime 2}=Ca-k{a}^2$ verifies the scale factor; integrating $dt=ad\tau$ gives cosmic time. The universe reaches maximum size at $u=\pi$ and recollapses into a Big Crunch at $u=2\pi$.

With positive curvature, zero cosmological constant and radiation density ${\rho }_0{a}^{-4}$ in units $8\pi G=1$, $A=\sqrt{{\rho }_0/3}$. The scale factor is positive for $0<\tau <\pi$, reaches its maximum at $\pi /2$, and has a Big Crunch at conformal time $\pi$ or proper time $2A$. This complements the closed matter-dominated Friedmann solution.

The vacuum Einstein field equations with positive cosmological constant admit a closed spherical slicing with scale factor $a(t)=\ell \cosh [(t-t_0)/\ell ]$. The time-symmetric slice is at $t=t_0$. The substitution $T=2\arctan (e^{\tau })-\pi /2$ gives $dT/d\tau =\mathrm{sech}\tau$ and $\cos T=\mathrm{sech}\tau$, so the metric is ${\ell }^2{\sec }^{2}T[-d{T}^2+d{\Omega }_3^2]$. The conformal metric is a finite-time slab of the Einstein static universe, with spacelike conformal infinities $T=\pm \pi /2$.

In an FLRW metric with zero separate cosmological constant, $\varrho +3p\ge 0$ makes the Friedmann acceleration equation imply $\ddot{a}\le 0$. If $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$, then the scale factor is a concave function andthroughout its regular past interval. Thus a regular positive scale factor cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0{)}^{-1}$, where $H$ is the Hubble parameter.

The continuation condition matters: without restrictions on the equation of state, an earlier curvature singularity can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt{t}$, $t>0$, with $k=0$, defines energy density $\varrho =3(\dot{a}/a{)}^2/(8\pi G)$ and pressure $p=-\varrho /3-\ddot{a}/(4\pi Ga)$. These satisfy the Friedmann equations and $\varrho +3p>0$, yet $a\to 1$ at the singular past boundary. Conversely the Milne model reaches zero without divergent curvature. A finite duration, a zero of the scale factor, and a curvature singularity are distinct conclusions.

Nonrelativistic matter dilutes as $a^{-3}$, while a cosmological constant has constant energy density. The ratio of vacuum energy density to matter density consequently scales as $a^3$, so comparable contributions occur during a limited epoch. The cosmic coincidence problem asks why observers find these very different components comparable near their own epoch. This timing puzzle is distinct from the cosmological constant problem, which concerns the tiny absolute vacuum-energy scale.

A smooth cosmological bounce is a positive minimum of the scale factor: contraction changes to expansion, with $a(t_b)>0$, $\dot{a}(t_b)=0$ and $\ddot{a}(t_b)>0$ in the nondegenerate case. The closed slicing of de Sitter spacetime gives $a=H_{\Lambda }^{-1}\cosh [H_{\Lambda }(t-t_b)]$, whose minimum is $H_{\Lambda }^{-1}$. The Friedmann acceleration equation with zero separate cosmological constant requires $\varrho +3p<0$ there.

An expanding-universe regime in which a positive cosmological constant supplies most of the Friedmann equation. The approximately constant positive expansion rate gives exponential growth of the scale factor. Since matter, radiation, and curvature contributions decay with expansion, a positive cosmological constant eventually dominates a perpetually expanding model containing them; this does not guarantee that curvature dominates at an intermediate time.

Contract the Einstein field equations with a null vector. Both the trace term and the cosmological constant term multiply $g_{ab}{\ell }^a{\ell }^b=0$ and disappear. The null energy condition therefore still implies the null convergence condition, independently of the sign of the cosmological constant. A black-hole area theorem still needs the appropriate global horizon assumptions as well as this local focusing condition.

The cosmological constant problem is the unexplained smallness of the gravitationally observable vacuum energy compared with the contributions expected from quantum field theory. In units with $\hslash =c=1$, an ultraviolet cutoff ${\Lambda }_{\mathrm{UV}}$ generically gives contributions of order ${\Lambda }_{\mathrm{UV}}^4$, whereas the observed dark-energy density is of order $10^{-47}{\mathrm{GeV}}^4$. The Einstein equation depends on the sum of a bare cosmological constant and these contributions; subtracting a constant from a nongravitational Hamiltonian does not remove its gravitational effect. Exact global supersymmetry gives zero energy to a supersymmetric vacuum, but supersymmetry breaking generally reintroduces a much larger positive scale.

For a spatially flat universe consisting only of dust and constant cosmological constant, $A_0={\ddot{a}}_0/a_0$ and $H_0$ determine the two model parameters through the Friedmann equations. Predicting the future also assumes those matter laws persist. Local derivatives alone do not determine arbitrary future stress evolution: $a_{+}(t)=1+t+t^2/2$ and $a_{-}(t)=1+t+t^2/2-t^3$ agree through second derivative at zero, but the first expands forever and the second recollapses.

For positive spatial curvature and positive cosmological constant, the effective potential has its maximum atThe critical case $V(a_E)=-1$ hasand contains the Einstein static universe.

Normalize the present scale factor to one. In a dust-only open cosmology with no cosmological constant, the Friedmann equation gives $-k{c}^2=H_0^2(1-{\Omega }_m)$. The present comoving and physical curvature radii are therefore $r_c=c/[H_0\sqrt{1-{\Omega }_m}]$. At another epoch the physical curvature radius is $a{r}_c$.

// Target: cosmic-microwave-background-anisotropy.bigb