Detectability of "Merger-nova" emission from long-lived magnetar in short gamma-ray bursts

One possible progenitor of short gamma-ray bursts (GRBs) is thought to be from double neutron star (NS) merger, and the remnant of such merger may be a supra-massive NS which supported by rigid rotation with surviving hundreds of seconds before collapsing into a black hole (BH). If this is the case, an optical/infrared transient (namely merger-nova) generated from the ejected materials and it powered by radioactive decay from r-process, spin-down energy from supra-massive NS, as well as the magnetic wind from new-born BH. In this paper, we systematically search for signature of supra-massive NS central engine by analysing the X-ray emission of short GRBs with internal plateau observed by {\em Swift}, and find that five candidates of short GRBs have such feature with redshift measurement. Then, we calculate the possible merger-nova emission from those candidates for given the typical model parameters by considering above three energy sources, and compare its brightness with the sensitivity of some optical telescopes. We find that the merger-nova emission of GRB 060801 in K-, r-, and U-band with variation of $M_{\rm ej}$ ($10^{-4}-10^{-2} M_{\odot}$), $\kappa$ ($0.1-10 ~\rm cm^{2}~g^{-1}$), and $\beta$ ($0.1-0.3$) is very difficult to be detected by Vera C. Rubin, Pan-STARRS, ZTF, as well as Roman Space Telescope, except for the case of large ejecta mass $M_{\rm ej}=10^{-2} M_{\odot}$. However, it is very hopeful to detect the merger-nova emission of GRBs 090515, 100625A and 101219A by more sensitive instruments of Vera C. Rubin, Pan-STARRS, and Roman. Moreover, the merger-nova emission of GRB 160821B is bright enough to be detected in our calculations, and it is also consistent with current real observations of merger-nova emission.