The mass gap, the spin gap, and the origin of merging binary black holes

Two of the dominant channels to produce the black-hole binary mergers observed by LIGO and Virgo are believed to be the isolated evolution of stellar binaries in the field and dynamical formation in star clusters. Their relative efficiency can be characterized by a “mixing fraction.” Pair instabilities prevent stellar collapse from generating black holes more massive than about $45{\mathrm{M}}_{\odot }$. This “mass gap” only applies to the field formation scenario, and it can be filled by repeated mergers in clusters. A similar reasoning applies to the binary’s effective spin. If black holes are born slowly rotating, the high-spin portion of the parameter space (the “spin gap”) can only be populated by black-hole binaries that were assembled dynamically. Using a semianalytical cluster model, we show that future gravitational-wave events in either the mass gap, the spin gap, or both can be leveraged to infer the mixing fraction between the field and cluster formation channels.

Figure 1

Illustration of the “mass gap” in the primary mass $m_1$ (top panel) and of the “spin gap” in the effective spin ${\chi }_{\mathrm{eff}}$ (bottom panel). Solid (dashed) lines are computed under the assumption that the maximum individual BH spin at birth is ${\chi }_{\max }=0.1(0.2)$. Only 2g events can populate the regions of the parameter space with high values of $m_1$ and/or ${\chi }_{\mathrm{eff}}$.

Figure 2

Timescales involved in the merger of $(10+10){\mathrm{M}}_{\odot }$ (top) and $(50+50){\mathrm{M}}_{\odot }$ (bottom) BBHs. The timescales related to three-body interactions, binary-single interactions, mass segregation, GW inspiral, and critical hardening are indicated in green, yellow, red, purple, and blue, respectively. The thick black line marks the sum $t_{\mathrm{tot}}$ of Eq. (26). The gray shaded region marks time delays larger than the age of the Universe.

Figure 3

Dominant timescales in the $(M_{\mathrm{cl}},m_1)$ plane for an equal-mass binary $(m_1=m_2)$. Regions where three-body interactions, mass segregation, hardening, and GW inspiral dominate are indicated in green, red, blue, and purple, respectively.

Figure 4

Contour plot of the total delay time for the merger of an equal-mass BBH system $(m_1=m_2)$ as a function of the cluster mass. The white dashed line is the boundary between the regions where most mergers happen ex situ (left) and in situ (right).

Figure 5

Distribution of host cluster masses (left) and primary BBH masses (right) for the $1\mathrm{g}+1\mathrm{g}$ populations. Black curves show the full sample of 1g mergers (higher generations are excluded from this plot). Red curves show the fraction of binaries that survive dynamical kicks and are able to merge inside the cluster. Blue curves show systems that further survive GW kicks and remain available to assemble the second generation of BH mergers. We assume ${\chi }_{\max }=0.1$ (solid) and ${\chi }_{\max }=0.5$ (dashed).

Figure 6

Fraction of BBHs retained after kicks due to either (i) dynamical interactions before merger (solid line) or (ii) GW recoil at merger (dashed lines). The largest spin of 1g BHs ${\chi }_{\max }$ increases from top (red, ${\chi }_{\max }=0.1$) to bottom (purple, ${\chi }_{\max }=0.7$).

Figure 7

Distribution of merger redshifts $z$. Red, green, and blue curves indicate the $1\mathrm{g}+1\mathrm{g}$, $2\mathrm{g}+1\mathrm{g}$, and $2\mathrm{g}+2\mathrm{g}$ populations, respectively. Solid (dashed) histograms are obtained with ${\chi }_{\max }=0.01$ (${\chi }_{\max }=0.5$).

Figure 8

Fraction of events that lie in one or both gaps. The total contributions to the mass (${\lambda }_{\mathrm{M}}$) and spin (${\lambda }_{\mathrm{S}}$) gap are given by the thick solid red and thick dashed blue curves, respectively. The other curves indicate contributions from binaries that are in one gap but not in the other one (${\lambda }_{\mathrm{M}\overline{\mathrm{S}}}$ and ${\lambda }_{\overline{\mathrm{M}}\mathrm{S}}$), in both gaps (${\lambda }_{\mathrm{M}\wedge \mathrm{S}}$), or in either of the two gaps (${\lambda }_{\mathrm{M}\vee \mathrm{S}}$).

Figure 9

Mass-gap efficiency ${\lambda }_{\mathrm{M}}$ as a function of the mass spectral index $\alpha$ (top panel) and the gap edge $m_{\max }$ (bottom panel). The largest 1g spin ${\chi }_{\max }$ is varied from 0 to 0.5 (top to bottom in each panel). In the top panel, black dashed lines show the approximate dependence from Eq. (42) with $\alpha =-1.6$. In the bottom panel, black dashed lines represent our default value $m_{\max }=45{\mathrm{M}}_{\odot }$.

Figure 10

Probability of an event being in only one of the two gaps. The blue (black) curve shows the probability that a binary lies in the spin (mass) gap but not in the mass (spin) gap. The dashed black line corresponds to the approximation ${\chi }_{\max }/0.34$ (see text).

Figure 11

PDF ${\widehat{p}}_{\text{cluster}}(\widehat{\chi })$ for cluster binaries from Eq. (a14) (solid black line) and field binaries ${\widehat{p}}_{\text{field}}(\widehat{\chi })$ from Eq. (a9) (dashed lines). For the field binaries, we assume either ${\theta }_{\max }=0°$ (blue) or ${\theta }_{\max }=60°$ (red).

Figure 12

Relative error $\delta f/f$ on the fraction of dynamical mergers as a function of ${\chi }_{\max }$ considering either only 1g mergers (black), the mass gap (red), or the spin gap (blue). We assume a catalog of $N={10}^4$ observations, a mixing fraction $f=0.5$, and the largest misalignment angles for field binaries ${\theta }_{\max }=30°$ (left) and 60° (right). The contributions due to Poisson counting errors and efficiency uncertainties are marked with dashed and dotted lines, respectively.

Figure 13

Fractional error on the mixing fraction $\sqrt{N}\delta f/f$ obtained using the mass gap (left) and spin gap (right) as a function of ${\chi }_{\max }$ and $f$ assuming ${\theta }_{\max }=60°$. The white dashed line marks the location where the error $\delta f$ from 1g events equals the one obtained with the gaps. In particular, gap (1g) events dominate to the left (right) of the white dashed lines.

Figure 14

The shaded areas mark values of $f$ and ${\chi }_{\max }$ where gaps provide a more accurate measurement of the mixing fraction $f$ compared to 1g events. Results for the spin (mass) gap are indicated with solid (dashed) curves. Darker (lighter) regions show results for ${\theta }_{\max }=10°$ (60°).