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We give an explicit formula for the second variation of the logarithm of theSelberg zeta function, $Z(s)$, on Teichm\"uller space. We then use this formulato determine the asymptotic behavior as $\text{Re} (s) \to \infty$ of thesecond variation. As a consequence, for $m \in \mathbb{N}$, we obtain thecomplete expansion in $m$ of the curvature of the vector bundle $H^0(X_t,\mathcal K_t)\to t\in \mathcal T$ of holomorphic m-differentials over theTeichm\"uller space $\mathcal T$, for $m$ large. Moreover, we show that thiscurvature agrees with the Quillen curvature up to a term of exponential decay,$O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closedhyperbolic geodesic.

Second variation of Selberg zeta functions and curvature asymptotics