Globular cluster formation histories, masses and radii inferred from gravitational waves

Globular clusters (GCs) are found in all types of galaxies and harbor some of the most extreme stellar systems, including black holes that may dynamically assemble into merging binaries (BBHs). Uncertain GC properties, including when they formed, their initial masses and sizes, affect their production rate of BBH mergers. Using the gravitational-wave catalog GWTC-3, we measure that dynamically-assembled BBHs -- those that are consistent with isotropic spin directions -- make up ${61^{+29}_{-44}\%}$ of the total merger rate, with a local merger rate of ${10.9^{+16.8}_{-9.3}}$ Gpc$^{-3}$ yr$^{-1}$ rising to ${58.9^{+149.4}_{-46.0}}$ Gpc$^{-3}$ yr$^{-1}$ at $z = 1$. We assume this inferred rate describes the contribution from GCs and compare it against the Cluster Monte Carlo (CMC) simulation catalog to directly fit for the GC initial mass function, virial radius distribution, and formation history. We find that GC initial masses are consistent with a Schechter function with slope ${\beta_m = -1.9^{+0.8}_{-0.8}}$. Assuming a mass function slope of $\beta_m = -2$ and a mass range between $10^4$--$10^8\,M_\odot$, we infer a GC formation rate at $z = 2$ of ${5.0^{+9.4}_{-4.0}}$ Gpc$^{-3}$ yr$^{-1}$, or ${2.1^{+3.9}_{-1.7}}\times 10^6\,M_\odot$ Gpc$^{-3}$ yr$^{-1}$ in terms of mass density. We find that the GC formation rate probably rises more steeply than the global star formation rate between $z = 0$ and $z = 3$ ({82\%} credibility) and implies a local number density that is ${f_\mathrm{ev} = 22.6^{+29.9}_{-16.2}}$ times higher than the observed density of survived GCs. This is consistent with expectations for cluster evaporation, but may suggest that other environments contribute to the rate of BBH mergers with significantly tilted spins.