HOLISMOKES - VII. Time-delay measurement of strongly lensed SNe Ia using machine learning

The Hubble constant ($H_0$) is one of the fundamental parameters in cosmology, but there is a heated debate on the $>$4$\sigma$ tension between the local Cepheid distance ladder and the early Universe measurements. Strongly lensed Type Ia supernovae (LSNe Ia) are an independent and direct way to measure $H_0$, where a time-delay measurement between the multiple supernova (SN) images is required. In this work, we present two machine learning approaches to measure time delays in LSNe Ia, namely, a fully connected neural network (FCNN) and a Random Forest (RF). For the training of the FCNN and the RF, we simulate mock LSNe Ia from theoretical SN Ia models including observational noise and microlensing. We test the transfer learning capability of both machine learning models, by using a final test set based on empirical LSN Ia light curves not used in the training process, and we find that only the RF provides low enough bias to achieve precision cosmology, which is therefore preferred to our FCNN approach for applications to real systems. For the RF with single-band photometry in the $i$-band, we obtain an accuracy better than 1 % in all investigated cases for time delays longer than 15 days, assuming follow-up observations with a 5$\sigma$ point-source depth of 24.7, a two day cadence with a few random gaps and a detection of the LSNe Ia 8 to 10 days before peak in the observer frame. In terms of precision, we can achieve under the same assumptions on the $i$ band $\sim$1.5 days uncertainty for a typical source redshift of $\sim$0.8. To improve the measurement we find that three bands, where we train a RF for each band separately and combine them afterwards, help to reduce the uncertainty to $\sim$1.0 day. The dominant source of uncertainty is the observational noise and therefore the depth is an especially important factor when follow-up observations are triggered.