Analysis

We analyze the rates and properties of BBH mergers by modelling environments where these stellar runaway events take place frequently. We also discuss how probable the detection of these events would be with current and planned GW instruments.

We use a semi-analytic framework to model the repeated, hierarchical mergers of BBHs in dense star clusters. We model the evolution of stellar clusters using parameters such as initial cluster mass, cluster radius, metallicity etc. and returns the masses of the binary components, their generations, spins, times of mergering etc. (Fragione & Rasio 2023). Our method captures all the essential features of N-body and MCMC results for BBH mergers Fragione & Rasio 2023. It also allows us to sample and readily access wide regions of the parameter space. For a detailed summary of our methods, see Section 2 of (Fragione & Rasio 2023).

We add a new parameter to the code describing the runaway effect of stars. This parameter is the fraction of the initial cluster mass that undergoes the collisional runaway effect. Thus, the total mass that undergoes the runaway is expressed as:

m_run = f x M_CL
We use a half-mass radius r_h = 1 pc and run simulations for three different metallicities z = 0.0002, 0.002, 0.02 and four values of the fraction f = 0, 0.001, 0.005, 0.01. For each fraction, we combine 5,000 simulations per metallicity (15,000 total). We then calculate the rates of BBH mergers as a function of redshift. The case for f = 0.01 is considered the extreme and the cases for f = 0.001, 0.005 are considered more likely.

We calculate the binary black hole merger rates as a function of redshift. This is shown in Figure 1 below. For any given primary mass (m₁ > 100, 500, 1000 ... M_⊙) the highest fraction of f = 0.01 has the largest merger rate.

To better understand the nature of these runaway effects on BH growth and evolution, we study the number of clusters that form a massive BH with mass over a given threshold. This is shown in the figure below. From this plot, we see that for higher fractions of runaway rates, most clusters have a BH over 100 M_⊙. As we consider higher thresholds, the number of clusters increases for a larger runaway rate.

With our analysis, we hope to answer questions about BH growth and possible evolution into the IMBH regime. We study the most massove BH formed in each cluster to find that the highest runaway fraction of 0.01 has the most massive BH formed. This is shown in the plot below.

We calculated the detection fraction of these runaway events through current and planned GW instruments. We consider four GW observatories: LISA, LIGO's Voyager, the Einstein Telescope, and the Cosmic Explorer. We use the imbhistory code created by Fragione 20203 to compute cosmological merger rates by considering parameters such as the number of samples, redshift ranges, sample IMBH masses, and BBH mass ratios. For the fraction f = 0.01, the detection fraction as a function of primary mass and mass ratio is shown in the figure below.

I am continuing to work with Dr Fragione after the summer to complete this project in order to have a larger body of analysis which would convey a more complete story. Here is an outline of a few of our next steps:

1. Analyze the most massive BH formed in each cluster for different half-mass radii, like r_h = 2, 4 pc.
2. Account for the delay time in merger and re-create the detection fraction plot.
3. Address other fractions of runaway such as 0.0005 and so on.
4. Study the change in detection fraction as a function of primary mass.

I am excited for what comes next. Please check back here for more updates!