Constrain magnetar parameters by taking into account the evolutionary effects of radius and moment of inertia with \emph{Swift}/XRT data
A newly born millisecond magnetar has been proposed as one possible central engine of some GRBs with X-ray plateau emission. In this work, we systematically analyzed the Swift/XRT data of long GRBs with plateau emission that were detected before 2023 December, and estimated the physical parameters by considering the $R/I$ evolutionary effects. We found that neglecting the $R/I$ evolutionary effects can lead to systematic overestimation or underestimation of magnetar parameters such as $B_p$, $P_0$, and $ε$ from 20\% to 50\%. We also found that some tight correlations, which can be approximately expressed as $ε\propto P_0^{1.57\pm0.22}$, $ε\propto B_p^{0.97\pm0.13}$, $B_p\propto P_0^{1.30\pm0.16}$, $E_{\rm wind}\propto E_{\rm jet,iso}^{0.83\pm0.07}(E_{\rm jet}^{0.76\pm0.06})$, $P_0\propto E_{\rm jet,iso}^{-0.29\pm0.03}(E_{\rm jet}^{-0.26\pm0.02})$, $B_p\propto E_{\rm jet,iso}^{-0.58\pm0.06}(E_{\rm jet}^{-0.55\pm0.05})$, and $ε\propto E_{\rm jet,iso}^{-0.55\pm0.07}(E_{\rm jet}^{-0.52\pm0.06})$ for our selected EoSs. The universal correlations suggest that a nascent magnetar with the faster $P_0$, lower $B_p$, and lower $ε$ are more inclined to power a more energetic GRB jet, and the $ε$ and $P_0$ of newborn magnetar are likely to originate from the magnetically induced distortion and correspond to the equilibrium spin period as a result of interaction between the magnetar and its accretion disk, respectively. Finally, we found that the GW signals from the remnants of those GW-dominated GRBs with redshift measurements cannot reach aLIGO sensitivity threshold, and only two cases (GRBs 150323A and 170607A) can reach ET sensitivity threshold. Future GW observations could not only offer the first smoking gun that a protomagnetar can serve as the central engine of GRBs but also play a crucial role in precisely constraining the neutron star EoS.