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A radiation-dominated universe has expansion controlled mainly by radiation in cosmology. When spatial curvature and the cosmological constant are negligible, the Friedmann equation and density scaling $\rho \propto a^{-4}$ give the displayed scale factor, with Hubble parameter $H=1/(2t)$.

In a flat universe containing separately conserved radiation in cosmology and pressureless matter, the Friedmann equation becomes $(a^{\prime }{)}^2=H_0^2({\Omega }_{r,0}+{\Omega }_{m,0}a)$ in conformal time. Setting $a=0$ at $\tau =0$ gives the stated quadratic-plus-linear scale factor. The linear term controls radiation domination and the quadratic term controls matter domination. The formula presumes $a_0=1$, ${\Omega }_{r,0}+{\Omega }_{m,0}=1$ and no cosmological constant.

For radiation in cosmology with zero cosmological constant, $p=\rho /3$. The Friedmann acceleration equation gives $\ddot{a}/a=-8\pi G\rho /3$, so the deceleration parameter equals the cosmological density parameter, regardless of spatial curvature. This is a relation for a pure radiation fluid, not a general identity for a mixture.

The Schwarzschild-de Sitter spacetime is the spherically symmetric vacuum solution with positive cosmological constant. In static coordinates its metric function is $f(r)=1-2m/r-H^2{r}^2$, with a black-hole horizon and a cosmological horizon when $mH<1/\sqrt{27}$.

For $m\ne 0$ and positive $\beta$, the Polonyi model has a unique zero-energy global minimum at $\beta =2-\sqrt{3}$, $⟨z⟩=\sqrt{3}-1$. Writing $u=\mathrm{Re}z-(\sqrt{3}-1)$ and $y=\mathrm{Im}z$, its potential bracket is $(u^2+y^2+\sqrt{3}u{)}^2+(2\sqrt{3}-3)u^2+(4-2\sqrt{3})y^2$, a sum of nonnegative terms. The real and imaginary curvatures are $4\sqrt{3}$ and $8-4\sqrt{3}$. The other zero-energy stationary point with $\beta =2+\sqrt{3}$ and $z=-\sqrt{3}-1$ is a saddle point. The stable vacuum has nonzero supergravity auxiliary field, so it has supersymmetry breaking even with zero cosmological constant. The $m=0$ branch does not fix these values.

For a point mass in a positive cosmological constant, the inward Newtonian acceleration $GM/r^2$ balances the outward cosmological acceleration $\Lambda r/3$ at the static radius $r_{\mathrm{s}}=(3GM/\Lambda {)}^{1/3}$.

The 1998 High-Z Supernova Search Team paper by Adam Riess, Brian Schmidt and collaborators presented supernova evidence for acceleration.^[ref]

The 1999 Supernova Cosmology Project paper by Saul Perlmutter and collaborators independently analysed 42 distant events. Alexei V. Filippenko coauthored both papers.^[ref]

The High-Z Supernova Search Team search used the CTIO Blanco 4-m telescope and a $2048\times 2048$ charge-coupled device, comparing fields across dark runs. Difference images located rising candidates; Keck, MMT and ESO spectroscopy classified them and measured host redshifts. Follow-up photometry used filters approximating emitted B and V bands. Calibrated field stars transferred the magnitude zero point; late host images removed galaxy light, point spread function fits measured brightness, and inserted artificial stars tested recovery.^[ref]

The Supernova Cosmology Project likewise discovered rising events with CTIO 4-m charge-coupled device imaging, then followed their light curves with several telescopes. Keck LRIS and ESO spectroscopy supplied classification and redshift, preferably from narrow host lines. Cross-filter K corrections connected received R/I observations to emitted B/V light.^[ref]

Why does fainter mean accelerating? First it means a larger inferred luminosity distance only at fixed standardized luminosity and corrected attenuation. Next it favours acceleration only after a comparison family is specified: an FLRW metric, representative propagation of light, common nearby and distant standardization, and controlled measurement calibration and selection. For the usual general relativity models containing nonnegative pressureless matter, spatial curvature of an FLRW universe and a cosmological constant, the shape of the Hubble diagram constrains ${\Omega }_{m,0},{\Omega }_{\Lambda ,0}$ and thus $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$. A positive residual relative to one decelerating model rejects that prediction; it does not alone prove $q_0<0$ across arbitrary expansion histories or arbitrary opacity laws.

The essential advance is that a nearby sample fixes an intercept while a range of distant redshifts measures the shape. The two teams supplied independent observations and analysis, although common nearby calibrators and Type Ia supernova physics can produce shared uncertainties. The dark-energy acceleration criterion explains the interpretation within general relativity; the worked supernova distance comparison shows the size and sign of the effect without fitting any survey data.

Use $c=1$, $a_0=1$, positive pressureless matter and positive cosmological constant, with ${\Omega }_m+{\Omega }_{\Lambda }=1$ and no radiation. The Friedmann equation gives ${\chi }_{\max }=H_0^{-1}{\int }_0^{\infty }dy/({\Omega }_{\Lambda }+{\Omega }_m{y}^3{)}^{1/2}$. Set $y=({\Omega }_{\Lambda }/{\Omega }_m{)}^{1/3}u$, then $v=u^3$. The remaining integral is $\frac{1}{3}{\int }_0^{\infty }{v}^{-2/3}(1+v{)}^{-1/2}dv=\frac{1}{3}B(1/3,1/6)$, proving the formula by the beta function integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.

In a region with vanishing stress-energy tensor, the Einstein field equations reduce to $R_{\mu \nu }=0$ when the cosmological constant vanishes.

A vacuum spacetime has vanishing matter stress-energy tensor. In $D>2$ dimensions, tracing the Vacuum Einstein equations gives $R=2D\Lambda /(D-2)$ and hence $R_{ab}=2\Lambda g_{ab}/(D-2)$; in four dimensions this reduces to $R_{ab}=\Lambda g_{ab}$. With zero cosmological constant it is Ricci flat, although its full Riemann curvature tensor can be nonzero: a vacuum Kerr black hole and vacuum gravitational waves illustrate that distinction.