Solution
= Solution
With an approximately constant <Hubble parameter> $H$, the <scale factor> is $a(t)=e^{Ht}$ after a choice of normalization. Integrate $d\tau=dt/a(t)$ and choose the integration constant so that the future endpoint is $\tau=0$:
$$
\tau=-\frac{e^{-Ht}}H,\qquad\boxed{a(\tau)=-\frac1{H\tau},\quad-\infty<\tau<0.}
$$
Substituting $dt=a\,d\tau$ in the <FLRW metric> gives $ds^2=a^2(\tau)(-d\tau^2+d\mathbf x^2)$. This is <Conformal time during de Sitter expansion>.