  We consider the relative canonical line bundle $K_{\mathcal{X}/\mathcal{T}}$
and a relatively ample line bundle $(L, e^{-\phi})$ over the total space $
\mathcal{X}\to \mathcal{T}$ of fibration over the Teichm\"uller space by
Riemann surfaces. We consider the case when the induced metric
$\sqrt{-1}\partial\bar{\partial}\phi|_{\mathcal{X}_y}$ on $\mathcal{X}_y$ has
constant scalar curvature and we obtain the curvature asymptotics of
$L^2$-metric and Quillen metric of the direct image bundle
$E^k=\pi_*(L^k+K_{\mathcal{X}/\mathcal{T}})$. As a consequence we prove that
the second variation of analytic torsion $\tau_k(\bar{\partial})$ satisfies
\begin{align*}
  \partial\bar{\partial}\log\tau_k(\bar{\partial})=o(k^{-l})
  \end{align*} at the point $y\in\mathcal{T}$ for any $l\geq 0$ as
$k\to\infty$.
