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tokens: that models in which the DTD favors long delays. 3. The Stochastic Background The stochastic background is defined as the energy density of gravitational waves per logarithmic frequency interval:where is the critical density of the universe given by . Here H 0 is the Hubble constant, and G is the Newton constant. The spectrum of the SGWB is given by:(Phinney 2001 ), where f and are the observed and rest-frame GW frequencies, respectively. The GW spectrum from a coalescing BNS or BBH is given bywhere is the energy emitted in GWs, is the chirp mass, and f i () and σ are frequencies that characterize the inspiral-merger-ringdown waveforms, are normalization constants chosen to make the waveform continuous, and are the post-Newtonian correction factors (Ajith et al. 2008 ). We model the ensemble distribution of primary black hole masses as a power lawand assume a flat distributionof secondary masses. We set as the lower observed BH mass in low-mass X-ray binaries (Farr et al. 2011 ). We set our fiducial values of α = 1, and ; however, we will explore the effect of varying these two parameters later in the paper. The upper limit on the BH mass is due to the assumption that pair-instability supernovae set an upper limit for the mass of the BHs born from stellar progenitors (Woosley 2017 ). The population analysis using BBH observations from LIGO/Virgo’s first and second observing runs measured the spectral index of the primary BH power law to be at 90% confidence assuming a minimum black hole mass of and a maximum total mass of (Abbott et al. 2018 ). For the BNS systems we assume a uniform distribution between 1.3–1.5 for each of the NSs to be consistent with the range of observed masses of galactic double neutron star systems (Kiziltan et al. 2013 ; Özel Freire 2016 ; Farrow et al. 2019 ), although we note that this choice is in tension with the masses of GW190425 (Abbott et al. 2020 ). 4. Results The results for the two extreme cases of DTDs for both the BBH and BNS systems are shown in Figure 2 . The green shaded region shows the predicted range of for a population of BNS composed of two NSs each with a mass uniformly distributed between 1.3 and 1.5 . The rate of the BNS mergers from LIGO is (Abbott et al. 2020 ). In order to get the full possible range we assign the highest rate to the short DTD model, and the lowest rate to the long DTD model. We do the same for the BBHs where we consider a merger rate between 20 to 110 (Abbott et al. 2020 ). For the BBH systems we considered two models in which (i) the primary BH mass follows bounded between 5 and 50 , and the secondary is uniformly distributed between 5 and m 1 (gray shaded region), and (ii) each BH has a mass of 15 (red shaded region). The average chirp masses of the two BBH population models are the same, , although their predicted differs by a factor of 3 in magnitude. We also show the power-law integrated (PI) curve (Thrane Romano 2013 ) for the detection of the SGWB with an S/N of 3 by