Error bound for the asymptotic expansion of the Hartman-Watson integral

This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $\theta(r,t)$ in the regime $rt=\rho$ constant. The leading order term has the form $\theta(\rho/t,t)=\frac{1}{2\pi t}e^{-\frac{1}{t} (F(\rho)-\pi^2/2)} G(\rho) (1 + \vartheta(t,\rho))$, where the error term is bounded uniformly over $\rho$ as $|\vartheta(t,\rho)|\leq \frac{1}{70}t$.