Isolated Binary Black Hole Formation and Merger Rates from Galaxy Evolution

Abstract

The LIGO-Virgo-KAGRA (LVK) collaboration has detected over 150 confirmed gravitational wave events through O4a. Binary black hole (BBH) systems represent the overwhelming majority of these observations. We construct a model for the population of the BBHs based on the distribution of metallicities in galaxies and state-of-the-art stellar evolution models implemented through the Stellar EVolution N-body (SEVN) code. We calculate the redshift evolution of the total merger rate of BBHs and the differential rates with respect to primary mass, secondary mass, and the mass ratio. We explore variations in the delay-time distribution's (DTD) power-law index and show that it affects the total merger rate's spectral shape, but primarily acts as an amplitude shift on the differential rates. When comparing to the primary mass distribution, our results indicate that either the average IMF in dwarf galaxies must be top heavy, or most of the 30-40 $\rm M_\odot$ black holes must be formed through a dynamical capture mechanism. For masses greater than about $50 \, \rm M_\odot$, the predicted number of BBH systems plummet to zero, revealing the well-known mass gap due to the pair instability mechanism and mass loss in binary systems.

Figures (12)

• Figure 1: Probability density functions of stellar and remnant masses across metallicity. The top row shows the zero-age main sequence (ZAMS) masses of the primary (left; $M_{1,\mathrm{ZAMS}}$) and secondary (right; $M_{2,\mathrm{ZAMS}}$) stars in merging binary systems, while the bottom row shows the corresponding black hole masses, where $M_{1,\mathrm{BH}}$ denotes the more massive black hole in the binary. Each curve represents a different metallicity, as indicated by the color scheme, and all distributions are weighted by the initial mass function and normalized to highlight relative contributions.
• Figure 2: Density plots of primary black hole mass ($M_{1,\mathrm{BH}}$) versus the primary ZAMS mass ($M_{1,\mathrm{ZAMS}}$) for six metallicities. Each panel lists the metallicity and the number of surviving systems ($N_{\mathrm{surv}}$), illustrating the steep decline in BBH formation efficiency toward higher metallicities. The color scale shows the logarithmic density of systems in hexagonal bins of $2\,M_\odot$, while the red dashed lines indicate single-star evolution (SSE) predictions. The initial-final mass relation deviates substantially from the SSE case, reflecting the effects of binarity.
• Figure 3: The efficiency for black holes to form given a particular progenitor mass, $M_{\rm ZAMS}$, and metallicity, Z. Low-metallicity progenitors experience a higher efficiency and overall survivability probability as expected. In contrast, higher-metallicity systems, although they may avoid pair instability, have a much lower probability of survival as indicated by the reduced efficiency factor.
• Figure 4: Differential binary black hole (BBH) formation rate density, $\dot{n}_{\rm BBH}(M_\star,M_1,z)$, as a function of galaxy stellar mass for three representative primary black hole masses (left to right: $M_1 = 5$, $20$, and $40\rm \,M_\odot$). Each panel shows four redshifts ($z=0$, $0.5$, $2$, and $6$), illustrating how BBH formation shifts across the galaxy population over cosmic time. Binary black hole formation migrates to lower galactic stellar masses as the Universe evolves, this migration occurs quicker for more massive BBHs. Dashed segments indicate the regime below $M_\star = 10^8\rm \,M_\odot$, where the galaxy stellar mass function is extrapolated to lower stellar masses where observational data is not yet constraining.
• Figure 5: Power-law fit to the merger time distribution from the SEVN simulations. The distribution is steeply peaked at short delays, implying that the merger rate density today will be dominated by binaries formed more recently in cosmic history. We find the best fit to our data is a power-law $p(\tau) \propto \tau^{-\gamma}$ with spectral index $\gamma=0.85$. We include a steeper distribution of $\gamma=1.0$Fishbach:2021mhp and a flatter more extreme value of $\gamma=0.5$ for comparison.