= Solution
\b[Definitions and <pressure> convention.] $R(t)$ is the <scale factor>, $t$ <cosmic time> and dots <time derivatives>. $G$ is the <gravitational constant>, $c$ the <speed of light>, $\rho$ the mass-equivalent <density> of the material fluid and $k=+1,0,-1$ the spatial-curvature sign in this normalization. $\Lambda$ here has units of inverse time squared, rather than inverse length squared. Comparison of the acceleration equation with $\ddot R/R=\Lambda/3-4\pi G(\rho+3P/c^2)/3$ fixes
$$
\boxed{\epsilon=\frac{3P}{\rho c^2}=3w}.
$$
This $\epsilon$ is a <pressure> parameter, unrelated to Question 1's radiative efficiency. <pressureless matter> has $P=0$ and $\epsilon=0$. When solving the generalized model below, $\epsilon$ is assumed constant.
\b[Evidence for a positive vacuum component.] Standardized <Type Ia supernovae> measure the luminosity-distance–<redshift> relation. Their corrected apparent magnitudes at cosmological distances imply greater distances than a decelerating matter-dominated model of comparable parameters, favoring late accelerated expansion. Light-curve standardization, <interstellar extinction>, selection bias and possible population evolution must be controlled rather than treating every supernova as an identical candle.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[Abstract and distance–redshift analysis]{ref}<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[Abstract and cosmological likelihood analysis]{ref}
The <cosmic microwave background> and <baryon acoustic oscillation> measurements constrain geometry and matter <density> in combination. Near spatial flatness together with a subcritical matter <density> leaves room for an additional smooth component; cosmic ages and structure growth supply consistency tests. A positive <cosmological constant> fits these observations within the standard framework. They do not establish that every allowed dark-energy theory must be a strictly constant vacuum <energy>; the inference depends on the cosmological and gravitational model.
\b[<pressureless matter> evolution.] Set $k=0$ and $\epsilon=0$. Since $\dot H=\ddot R/R-H^2$, eliminating $\rho$ gives
$$
\dot H=-\frac32\left(H^2-\frac{\Lambda}{3}\right),\qquad
\boxed{\frac23\dot H+H^2=\frac{\Lambda}{3}}.
$$
For $H=b\coth(at)$, $\dot H=-ab\,\mathrm{csch}^2(at)$ and $\coth^2x=1+\mathrm{csch}^2x$. Therefore
$$
b^2+\left(b^2-\frac23ab\right)\mathrm{csch}^2(at)=\frac{\Lambda}{3}.
$$
On the expanding Big-Bang branch with positive $\Lambda$, choose
$$
\boxed{b=\sqrt{\frac{\Lambda}{3}},\qquad
a=\frac32b=\frac{\sqrt{3\Lambda}}2,\qquad
H(t)=\sqrt{\frac{\Lambda}{3}}
\coth\left(\frac{\sqrt{3\Lambda}}2t\right)}.
$$
The origin $t=0$ is the <Big Bang>; an arbitrary time translation would shift this origin. Integrating gives $R\propto\sinh^{2/3}(at)$. At early times $H\sim2/(3t)$, while at late times $H\to\sqrt{\Lambda/3}$.
\b[Age and the <density> fractions.] Dividing the flat constraint by $H^2$, with $\rho_{\rm crit}=3H^2/(8\pi G)$, gives
$$
\boxed{\Omega_m+\Omega_\Lambda=1,\qquad
\Omega_\Lambda=\frac{\Lambda}{3H^2}}.
$$
The solution implies $\sqrt{\Omega_\Lambda}=\tanh(at)$, so
$$
\boxed{t_0=\frac{2}{3H_0\sqrt{1-\Omega_{m0}}}
\operatorname{artanh}\sqrt{1-\Omega_{m0}}}.
$$
Here $H_0=2.3331\times10^{-18}\ \mathrm{s^{-1}}$, using the printed megaparsec conversion. Taking one year as $365.25$ days gives
$$
\boxed{t_0\simeq1.309\times10^{10}\ \mathrm{yr}=13.09\ \mathrm{Gyr}.}
$$
For the same $H_0$, an <Einstein-de Sitter universe> has
$$
\boxed{t_{\rm EdS}=\frac{2}{3H_0}\simeq9.05\ \mathrm{Gyr}.}
$$
A cosmic model must be older than its oldest objects, with additional time allowed for their formation. Old stellar populations are dated through stellar-evolution fitting, particularly the <globular cluster> <main-sequence turnoff>, with uncertainties from distances, composition and input stellar physics. A pre-examination <globular-cluster age constraint on cosmology> obtained a $95\%$ lower cosmic-age limit of $11\ \mathrm{Gyr}$ after allowing for formation time. This conflicts with the $9.05\ \mathrm{Gyr}$ model but is compatible with the $13.09\ \mathrm{Gyr}$ calculation.<New Globular Cluster Age Estimates and Constraints on the Cosmic Equation of State and The Matter Density of the Universe by Lawrence M. Krauss and Brian Chaboyer>[Abstract and stellar-age/formation-time analysis]{ref} Individual <stellar age> estimates are not exact clocks, and the numerical $13.09\ \mathrm{Gyr}$ is the result for the specified $H_0$ and matter fraction, not a claim to be a contemporary best-fit age.
\b[Constant nonzero <pressure> parameter.] Eliminating $\rho$ before setting the <pressure> to zero instead gives
$$
\begin{aligned}
\dot H
&=\frac{\Lambda}{3}-H^2-\frac{1+\epsilon}{2}
\left(H^2-\frac{\Lambda}{3}\right)\\
&=-\frac{3+\epsilon}{2}
\left(H^2-\frac{\Lambda}{3}\right).
\end{aligned}
$$
Hence
$$
\boxed{\frac{2}{3+\epsilon}\dot H+H^2-\frac{\Lambda}{3}=0}.
$$
For constant $\epsilon>-3$, the positive-density expanding Big-Bang branch is
$$
\boxed{H(t)=H_\Lambda\coth\left(\frac{3+\epsilon}{2}H_\Lambda t\right),
\qquad H_\Lambda=\sqrt{\frac{\Lambda}{3}}}.
$$
Equivalently use $t'=(3+\epsilon)t/3$ in the <pressureless matter> solution. The <scale factor> obeys $R\propto\sinh^{2/(3+\epsilon)}[(3+\epsilon)H_\Lambda t/2]$, and the early limit is $H\simeq2/[(3+\epsilon)t]$. The <age of a flat barotropic matter-vacuum universe> with the same present <density> fractions and $H_0$ is
$$
t_0(\epsilon)=\frac{2\,\operatorname{artanh}\sqrt{\Omega_{\Lambda0}}}
{(3+\epsilon)H_0\sqrt{\Omega_{\Lambda0}}}
=\frac{3}{3+\epsilon}t_0(0).
$$
Thus
$$
\boxed{t_0(0.2)\simeq12.28\ \mathrm{Gyr}.}
$$
The small positive <pressure> makes the early universe expand with a smaller age at the same late expansion parameters.
\b[The acceleration transition.] Normalize the present <scale factor> to one. Constant $\epsilon$ and the continuity equation give $\rho=\rho_0(1+z)^{3+\epsilon}$. Dividing the acceleration equation by $H_0^2$ gives
$$
\frac{\ddot R}{R H_0^2}
=\Omega_{\Lambda0}-\frac{1+\epsilon}{2}
\Omega_{m0}(1+z)^{3+\epsilon}.
$$
At the <barotropic matter-vacuum acceleration transition> its right-hand side vanishes. For the parameters here, $1+\epsilon>0$, so the past transition is
$$
\boxed{z_{\rm tr}=
\left[\frac{2(1-\Omega_{m0})}{(1+\epsilon)\Omega_{m0}}\right]^{1/(3+\epsilon)}-1
\simeq0.529}.
$$
At greater <redshift> the material term dominates and expansion decelerates; at smaller <redshift> vacuum <energy> dominates and expansion accelerates. The <pressure> factor $1+\epsilon$ belongs in the numerator of the gravitating material term; omitting it would confuse this model with <pressureless matter>.
\Image[/past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2015/paper-1-cosmic-expansion.png]
{title=Original constant-pressure-parameter expansion histories and acceleration transitions}
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