= Solution
Set $c=1$ and let $H=\dot a/a$ denote the proper-time <Hubble parameter>. The <Friedmann equation> is the expansion constraint: density drives expansion, while spatial curvature and the <cosmological constant> contribute separately. The <Friedmann acceleration equation> shows that the active gravitational density is $\rho+3P$; positive pressure gravitates, while positive vacuum energy accelerates expansion.
Subtracting the constraint from the acceleration equation gives $\dot H=-4\pi G(\rho+P)+k/a^2$. Differentiating the constraint gives $2H\dot H=(8\pi G/3)\dot\rho+2kH/a^2$. Eliminate $\dot H$ to obtain
$$
\boxed{\dot\rho+3H(\rho+P)=0}.
$$
This <cosmological continuity equation> is local energy conservation. For a comoving volume $V\propto a^3$, it is $d(\rho V)=-P\,dV$, the adiabatic first law: expansion does work against pressure.
Define the <deceleration parameter> $q=-a\ddot a/\dot a^2$, the <critical density> $\rho_c=3H^2/(8\pi G)$, and the <cosmological density parameters> $\Omega_m=\rho_m/\rho_c$ and $\Omega_\Lambda=\Lambda/(3H^2)$. The dust-plus-vacuum acceleration equation then gives \b[$q=\Omega_m/2-\Omega_\Lambda$.] Dust conservation gives $\rho_m\propto a^{-3}$, while $\rho_\Lambda=\Lambda/(8\pi G)$ is constant. At the acceleration transition, $\rho_m=2\rho_\Lambda$, so
$$
\boxed{1+z_{\rm tr}=\left(\frac{2(0.7)}{0.3}\right)^{1/3},\qquad z_{\rm tr}\simeq0.671}.
$$
For spatial flatness and $a_0=1$, the constraint today sets $\Omega_{\Lambda0}=1-\Omega_{m0}$. Thus $H^2=H_0^2[\Omega_{m0}a^{-3}+1-\Omega_{m0}]$. Put $M=\Omega_{m0}$, $L=1-M>0$ and $y=a^{3/2}$. Then $\dot y=(3H_0/2)\sqrt{M+Ly^2}$. Integrating from $y(0)=0$ gives $\operatorname{arsinh}(y\sqrt{L/M})=(3H_0/2)\sqrt L\,t$, hence the <flat dust-and-vacuum expansion>
$$
\boxed{a(t)=\left(\frac ML\right)^{1/3}\sinh^{2/3}\!\left(\frac32H_0\sqrt L\,t\right)}.
$$
For small $t$, $a\simeq[(3/2)H_0\sqrt M\,t]^{2/3}$, the matter-dominated law. At late times, $a\simeq[M/(4L)]^{1/3}e^{H_0\sqrt L\,t}$, approaching <de Sitter spacetime>. Negative spatial curvature adds a positive $a^{-2}$ contribution to $H^2$. This is subdominant to dust's $a^{-3}$ at small $a$ and to the constant vacuum term at large $a$, so it does not change either leading limit.
The model describes the transition from matter domination to the accelerated late-time expansion indicated by cosmological distance observations, with nearly flat geometry and vacuum-like dark energy. Its formal $t\to0$ dust limit is not the actual earliest universe: radiation, neglected here, dominates then and gives $a\propto t^{1/2}$.
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