  Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and
$L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic
of the curvature of $L^2$-metric and Qullien metric on the direct image bundle
$\pi_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than
$k^{n-1}$ for large $k$. As an application we prove that the analytic torsion
$\tau_k(\bar{\partial})$ satisfies
$\partial\bar{\partial}\log(\tau_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is
the dimension of fibers.
