Dynamics and Merger Rate of Primordial Black Holes in a Cluster

The PBH clusters can be sources of gravitational waves, and the merger rate depends on the spatial distribution of PBHs in the cluster which changes over time. It is well known that gravitational collisional systems experience the core collapse that leads to significant increase of the central density and shrinking of the core. After core collapse, the cluster expands almost self-similarly (i.e., density profile extends in size without changing its shape). These dynamic processes affect the merger rate of PBHs. In this paper, the dynamics of the PBH cluster is considered using the Fokker–Planck equation. We calculate the merger rate of PBHs on cosmic time scales and show that its time dependence has a unique signature. Namely, it grows by about an order of magnitude at the moment of core collapse which depends on the characteristics of the cluster, and then decreases according to the dependence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>∝</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>1.48</mn></mrow></msup></mrow></semantics></math></inline-formula>. It was obtained for monochromatic and power-law PBH mass distributions with some fixed parameters. Obtained results can be used to test the model of the PBH clusters via observation of gravitational waves at high redshift.