F(s;\rho):=\ln \det\left(I-K^{\mathrm{Pe}}_{s,\rho}\right)$$ the logarithm of Fredholm determinant associated with the Pearcey kernel, our main result is the following theorem.

\[main-thm\] With $F(s;\rho)$ defined in , we have, as $s\to +\infty$, $$\label{main-F-asy}
F(s;\rho)= -\frac{9 s^{\frac83}}{2^{\frac{17}3}} + \frac{\rho s^2}{4} - \frac{\rho^2 s^{{\frac43}}}{2^{{\frac{10}3}}} - \frac{2}{9} \ln s +\frac{\rho^4}{216} + C + {\mathcal{O}}(s^{-\frac{2}{3}}),$$ uniformly for $\rho$ in any compact subset of $\mathbb{R}$, where $C$ is an undetermined constant independent of $\rho$ and $s$.

In the literature, the large $s$ asymptotics of $F(s;\rho)$ was formally derived in [@BH] for $\rho = 0$, based on the coupled nonlinear differential equations satisfied by $F(s;0)$. Moreover, the asymptotics therein contains the leading term alone, without providing any information about the error estimate or the sub-leading terms. Our asymptotic expansion includes more terms and improves the result in [@BH]. After a change of variable $s\mapsto s/2$, the leading term of the asymptotic formula , i.e., $-9 s^{\frac83}/2^{\frac{17}3}$, agrees with that obtained in [@BH (3.36)]. Furthermore, we wish to emphasize that our derivation is rigorous, which makes use of integrable structure of the Pearcey kernel in the sense of Its-Izergin-Korepin-Slavnov [@IIKS90] and involves a steepest descent analysis of the revelent Riemann-Hilbert (RH) problem.

As also observed in [@BH], the leading 