Prospects for detecting proto-neutron star rotation and spindown using supernova neutrinos

After a successful supernova, a proto-neutron star (PNS) cools by emitting neutrinos on $\sim 1-100$ s timescales. Provided that there are neutrino emission `hot-spots' or `cold-spots' on the surface of the rotating PNS, we can expect a periodic modulation in the number of neutrinos observable by detectors. We show that Fourier transform techniques can be used to determine the PNS rotation rate from the neutrino arrival times. Provided there is no spindown, a 1-parameter Discrete Fourier Transform (DFT) is sufficient to determine the spin period of the PNS. If the PNS is born as a magnetar with polar magnetic field strength $B_0 \gtrsim 10^{15}$ G and is `slowly' rotating with an initial spin period $\gtrsim 100$ ms, then it can spindown to periods of the order of seconds during the cooling phase. We propose a modified DFT technique with three frequency parameters to detect spindown. Due to lack of neutrino data from a nearby supernova except the $\sim20$ neutrinos detected from SN1987A, we use toy models and one physically motivated modulating function to generate neutrino arrival times. We use the false alarm rate (FAR) to quantify the significance of the Fourier power spectrum peaks. We show that PNS rotation and spindown are detected with $\rm FAR<2\%$ ($2\sigma$) for periodic signal content $\rm M\gtrsim 13-15\%$ if $5\times 10^{3}$ neutrinos are detected in $\sim 3$ s and with $\rm FAR<1\%$ for $\rm M\geq 5\%$ if $5\times 10^{4}$ neutrinos are detected in $\sim 3$ s. Since we can expect $\sim 10^{4}-10^{5}$ neutrino detections from a supernova at 10 kpc, detection of PNS rotation and spindown is possible using the neutrinos from the next Galactic supernova.