tokens: within these haloes, we can quantify the expected rate of DCBH mergers that will be detectable by LISA due to SMS multiplicity. 2 NUMBER DENSITY DISTRIBUTION OF DCBHS In order to obtain the abundance of DCBH seeds, we use the results of A12, D14 and H16. We focus on two scenarios with J c = 30 J 21 1 and 300 J 21 respectively, where J c is the critical Lyman–Werner (LW) flux required to induce DCBH formation. The exact value of J c that can facilitate DCBH formation in an ACH is still unknown. What complicates the matter further is that a single value of J c is not representative of the chemothermodynamical processes that lead to the formation of a DCBH, namely the photodestruction of H − and H 2 (Wolcott-Green Haiman 2012 ; Sugimura, Omukai Inoue 2014 ; Agarwal Khochfar 2015 ; Agarwal et al. 2016 ). Therefore, we use abundance estimates from the literature and employ two values of J c as our extreme limits. In doing so, we are implicitly assuming that other process may either augment or replace the effect of an external LW field to produce similar results (e.g. streaming velocities, see Tanaka Li 2014 ; Hirano et al. 2017 ; Schauer et al. 2017 , or rapid accretion, see Yoshida et al. 2003 ). Thus, J c = 30 J 21 can be viewed as a scenario in which DCBHs are relatively common and would result in a sufficient number of massive seeds for the entire MBH population. On the other hand, J c = 300 J 21 would result in a number density of heavy seeds that could only seed a subpopulation of massive black holes (perhaps the highest redshift quasars) and can thus be viewed as a scenario where large initial mass black holes are rare. We now describe our steps going from the theoretical estimates of abundance of DCBH seeds to the observed number of events. Using the models and predictions from the redshift distributions of the newly formed DCBH seeds per unit comoving Mpc 3 found in A12, D14, H16 we calculate a ‘rate’ of formation at time-step ‘i’ in units of cMpc −3 where N new is the number of newly formed DCBH sites identified in the simulations for a given value of J c , and d V denotes the comoving volume normalization. To get the DCBH formation rate per unit time per unit redshift from any given redshift in the observer’s frame of reference (i.e. as seen at z = 0), we follow Magg et al. 2016 (equation 8) where R z is the comoving distance at a given redshift. In Fig. 1 we plot the number of DCBH seeds per unit time per unit volume i.e. equation ( 1 ), generating a fit using a second-order polynomial. The fit is obtained from the estimate for the number density of DCBHs based on J c = 30 J 21 (in blue filled circles, A12 , D14 , and H16 ), and can be written as where t is the age of the Universe at which the rate is computed. The number density of pristine ACHs scales approximately as J (D14, Inayoshi Tanaka 2015 ), hence rescaling from J c = 30 to J c = 300 requires a normalization factor of 10 −5 . This is shown using the red filled circles. The
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