Probing the redshift evolution and sub-populations of binary neutron stars with the Einstein Telescope

The formation channels of binary neutron stars (BNSs) currently remain uncertain, but important information can be gathered by observing their mergers with gravitational-wave detectors. The processes that lead to BNS coalescence are encoded in the time-delay distribution between stellar binary formation and BNS coalescence, and therefore in the BNS merger rate. Moreover, the detection of GW190425 by LIGO/Virgo/KAGRA (LVK) suggests a sub-population of massive BNSs, possibly formed through unstable 'case BB' mass transfer with short merger delays. We investigate whether next-generation detectors such as the Einstein Telescope (ET) can constrain the time-delay distribution of BNSs and identify such sub-populations. Using the latest LVK constraints, we generated mock ET catalogues that contain a mixture of light and heavy sub-populations. We modelled the redshift distribution of each sub-population as the convolution of the cosmic star formation rate with a time-delay distribution. We first considered a scenario where the time-delay distribution is common to all BNSs and follows a power law with indices α=-0.5,-1,-1.5. In the second scenario, heavy BNSs have fixed short delays, while light BNSs follow power-law delays with the same set of indices. Hierarchical Bayesian analyses were then performed on catalogues of 100–5 , 000 events. With thousands of events, ET will be able to accurately characterise the time-delay distribution for the α=-0.5 and α=-1 cases. We find that with hundreds of detections from ET, we will be able to establish that the total mass distribution is bimodal. A few thousand events are sufficient to disentangle the redshift distributions of the two sub-populations for moderate time-delay indices (α_ ̊m L =-0.5 or -1). For steeper indices (α_ ̊m L =-1.5), the differences are more subtle and require larger catalogues, which was beyond what we could explore given our computational resources. Next-generation detectors should enable the detection of multiple BNS sub-populations and their redshift evolution, and provide valuable insights into their formation pathways.