On the initial spin periods of magnetars born in weak supernova explosions and their gravitational wave radiation
The initial spin periods of newborn magnetars are \textbf{strongly associated with the origin of their strong magnetic fields, both of which can affect the electromagnetic radiation and gravitational waves (GWs) emitted at their birth.} Combining the upper limit $E_{\rm SNR}\lesssim10^{51}$ erg on the explosion energies of \textbf{the supernova (SN) remnants around slowly-spinning magnetars} with a detailed investigation on the evolution of newborn magnetars, we set constraints on the initial spin periods of magnetars \textbf{born in weak SN explosions}. Depending on the conversion efficiency $η$ of the electromagnetic energy of \textbf{these} newborn magnetars into the kinetic energy of SN ejecta, the minimum initial spin periods of \textbf{these} newborn magnetars are $P_{\rm i, min}\simeq 5-6$ ms for an ideal efficiency $η=1$, $P_{\rm i, min}\simeq 3-4$ ms for a possible efficiency $η=0.4$, and $P_{\rm i, min}\simeq 1-2$ ms for a relatively low efficiency $η=0.1$. \textbf{Based on these constraints and adopting reasonable values for the physical parameters of the newborn magnetars, we find that their GW radiation at $ν_{\rm e,1}=ν$ may be undetectable by the Einstein Telescope (ET) since the maximum signal-to-noise ratio (${\rm S/N}$) is only 2.41 even the sources are located at a very close distance of 5 Mpc, where $ν$ are the spin frequencies of the magnetars. At such a distance, the GWs emitted at $ν_{\rm e,2}=2ν$ from the newborn magnetars with dipole fields $B_{\rm d}=5\times10^{14}$ and $10^{15}$ G may be detectable by the ET because ${\rm S/N}$ are 10.01 and 19.85, respectively. However, if these newborn magnetars are located at $20$ Mpc away in the Virgo supercluster, no GWs could be detected by the ET due to low ${\rm S/N}$.