Laplacian flow for closed G_2 structures: real analyticity

Let $\varphi(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for
closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed
positive time $t\in (0,T]$, $(M,\varphi(t),g(t))$ is real analytic, where
$g(t)$ is the metric induced by $\varphi(t)$. Consequently, any Laplacian
soliton is real analytic and we obtain unique continuation results for the
flow.