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For the de Sitter scale factor in flat slicingchoose the additive constant so thatAn eternal expanding patch has $-\infty <\tau <0$.

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

Differentiate the Friedmann equation and use energy conservation to obtain $\dot{H}=-4\pi G(\rho +P/c^2)$ on the expanding branch, where $H\ne 0$. Since $\ddot{a}/a=\dot{H}+H^2$,For pressureless matter, conservation gives ${\rho }_M={\rho }_{M0}{a}^{-3}$. In terms of the specified cosmological density parameters, flatness implies ${\Omega }_M+{\Omega }_{\Lambda }=1$, andSet $b=a^{3/2}$ and take ${\Omega }_M>0$. Then $\dot{b}=\frac{3}{2}{H}_0\sqrt{{\Omega }_M+{\Omega }_{\Lambda }{b}^2}$. Integrating with the Big Bang at $t=0$ givesFor another time origin replace $t$ by $t-t_B$. The cosmological constant has effective density ${\rho }_{\Lambda }=3H_0^2{\Omega }_{\Lambda }/(8\pi G)$. Equality with the matter density means $a^3={\Omega }_M/{\Omega }_{\Lambda }$, soIf ${\Omega }_M=0$, the expanding solution is pure exponential de Sitter scale factor in flat slicing and there is no finite matter-density equality time or dust Big Bang; the stated hyperbolic-sine formula presumes a nonzero dust component.

With $\Lambda >0$ and $k=-1$, the Friedmann effective potential for pressureless matter isAlthough $V$ has one maximum, it is negative, so $C=1$ lies above it and ${\dot{a}}^2=C-V(a)$ never vanishes. An initially small expanding universe therefore expands forever. At large $a$,so the expansion approaches the exponential de Sitter scale factor in flat slicing. There is no turning point.

Write $H=\dot{a}/a$. The nonzero Christoffel symbols for the flat FLRW metric areThe mixed Einstein tensor isFor a mixed tensor,For $\mu =t$, direct substitution givesFor each spatial $\mu$, homogeneity makes the partial derivatives vanish and the remaining diagonal connection terms cancel pairwise. This verifies the flat FLRW Einstein-tensor divergence directly.

In vacuum,The $tt$ equation isChoosing the positive root gives the expanding de Sitter scale factor in flat slicingand the spatial equation is then automatically satisfied.