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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 byRiemann surfaces. We consider the case when the induced metric$\sqrt{-1}\partial\bar{\partial}\phi|_{\mathcal{X}_y}$ on $\mathcal{X}_y$ hasconstant 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 thatthe 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$k\to\infty$.

The asymptotics of the $L^2$-curvature and the second variation of analytic torsion on Teichm\"uller space