Impact of initial mass functions on the dynamical channel of gravitational wave sources

Dynamically formed black hole (BH) binaries (BBHs) are important sources of gravitational waves (GWs). Globular clusters (GCs) provide a major environment to produce such BBHs, but the total mass of the known GCs is small compared to that in the Galaxy; thus, the fraction of BBHs formed in GCs is also small. However, this assumes that GCs contain a canonical initial mass function (IMF) similar to that of field stars. This might not be true because several studies suggest that extreme dense and metal-poor environment can result in top-heavy IMFs, where GCs may originate. Although GCs with top-heavy IMFs were easily disrupted or have become dark clusters, the contribution to the GW sources can be significant. Using a high-performance and accurate N-body code, PETAR, we investigate the effect of varying IMFs by carrying out four star-by-star simulations of dense GCs with the initial mass of 5 × 10⁵ M_⊙ and the half-mass radius of 2 pc. We find that the BBH merger rate does not monotonically correlate with the slope of IMFs. Due to a rapid expansion, top-heavy IMFs lead to less efficient formation of merging BBHs. The formation rate continuously decreases as the cluster expands because of the dynamical heating caused by BHs. However, in star clusters with a top-heavier IMF, the total number of BHs is larger, and therefore, the final contribution to merging BBHs can still be more than that from clusters with the standard IMF, if the initial cluster mass and density are higher than those used in our model.