Reconstructing the star formation rate for compact binary populations with the Einstein telescope

The Einstein Telescope (ET) is a proposed third-generation, wide-band gravitational wave (GW) detector. Given its improved detection sensitivity in comparison to the second-generation detectors, it will be capable of exploring the Universe with GWs up to very high redshifts. In this paper, we present a population-independent method to infer the functional form of star formation rate density (SFR) for different populations of compact binaries originating in stars from Population (Pop) I+II and Pop III using ET as a single instrument. We use an algorithm to answer three major questions regarding the SFR of different populations of compact binaries. Specifically, these questions refer to the termination redshift of the formation of Pop III stars, the redshift at peak SFR, and the functional form of SFR at high redshift, all of which remain to be elucidated. We show that the reconstruction of SFR as a function of redshift for the different populations of compact binaries is independent of the time-delay distributions up to $z \sim 14,$ and that the accuracy of the reconstruction only strongly depends on this distribution at higher redshifts of $z\gtrsim 14$. We define the termination redshift for Pop III stars as the redshift where the SFR drops to 1\% of its peak value. In this analysis, we constrain the peak of the SFR as a function of redshift and show that ET as a single instrument can distinguish the termination redshifts of different SFRs for Pop III stars, which have a true separation of at least $\Delta z \sim 2$. The accurate estimation of the termination redshift depends on correctly modelling the tail of the time-delay distribution, which constitutes delay times of $\gtrsim 8$ Gyr.