\section{Introduction}
\label{sect:intro}

The LIGO/Virgo Collaboration has recently released the second Gravitational Wave Transient Catalog \citep[GWTC-2, ][]{lvc2020cat}. Together with results from the first two observing runs \citep[GWTC-1, ][]{lvc2019cat}, the full gravitational-wave (GW) catalog is comprised of more than $50$ events, and is transforming our understanding of black holes (BHs) and neutron stars (NSs) across cosmic space and time \citep{lvc2020catb,lvc2020catc}.  Despite this wealth of information, the origin of many of these binary mergers, particularly the binary BHs (BBHs) mergers, remains highly uncertain.

Various formation scenarios for BBH mergers have been proposed over the years, including isolated binary star evolution \citep[e.g.,][]{bel16b,demi2016,gm2018,Breivik2020}, dynamical formation in dense star clusters \citep[e.g.,][]{askar17,baner18,fragk2018,Rodriguez2018,sams18,ham2019,krem2019}, mergers in triple and quadruple systems \citep[e.g.,][]{antoper12,ll18,GrishinPerets2018,Rodriguez2018a,fragg2019,flp2019,fragk2019,liu2019,FragioneLoeb2020,MartinezFragione2020}, mergers of compact binaries in galactic nuclei \citep[e.g.,][]{bart17,sto17,rasskoc2019,mck2020}, and mergers of primordial black holes \citep[e.g.,][]{sasaki2016}. While several formation scenarios can account for some or even all of the BBH merger rate \citep{2021RNAAS...5...19R}, the contribution of different channels must be disentangled using a combination of the mass, spin, redshift, and eccentricity distributions as the number of detected events increases \citep[e.g.,][]{olea09,Fishbachetal2017,gondan2018,perna2019,WongBreivik2020,2021ApJ...910..152Z}.

The majority of the BBHs in GWTC-2 are consistent with having equal mass components \citep{lvc2020catb}. This is also consistent with theoretical models of BBH formation through either isolated binary evolution or dynamical formation, which prefer BBH mergers with mass ratios near unity \citep[e.g.,][]{bel16b,Rodriguez2016}. Nevertheless, GW190412, a BBH with mass ratio of nearly four-to-one, provides evidence for the existence of mergers with small mass ratios. It has been proposed that these systems can be formed through isolated binary evolution \citep[e.g.,][]{OlejakFishbach2020} or as a result of repeated mergers in star clusters \citep[e.g.,][]{2020ApJ...896L..10R}.

Another channel that can potentially produce BBH mergers with smaller mass ratios is the Lidov-Kozai (LK) mechanism in triple systems \citep{Kozai1962,Lidov1962}. In this scenario, a tertiary companion on a sufficiently inclined outer orbit could drive the inner binary to extreme eccentricities, leading to efficient gravitational radiation and orbital decay. In this Letter, we study in detail the mass ratio distribution of BBH mergers induced by tertiary companions in triple systems and compare them to the mass ratio distributions from detailed binary population synthesis and star cluster dynamics. We show that the triple channel predicts a flatter mass ratio distribution than either isolation binary or dynamical formation scenarios, and occupies a distinct region of the of total mass, mass ratio, and spin parameter space for merging BBHs, which can be used to disentangle the triple contribution to the overall observed GW sources.

The Letter is organized as follows. In Section~\ref{sect:methods}, we present our method to build populations of merging binary black holes. In Section~\ref{sect:results}, we compare the distributions of total masses and mass ratios for different channels. Finally, in Section~\ref{sect:concl}, we discuss the implications of our results and draw our conclusions.

\section{Models of Merging Binaries}
\label{sect:methods}

We compare three populations of merging BBHs: those created from mergers from ``isolated'' stellar binaries, those from stellar triples where the merger is induced by the third companion due to secular interactions, and those formed through three- and four-body encounters in dense star clusters.  As much as possible, the three sets of results presented here strive to use identical prescriptions for the evolution of massive stars.  To that end, both the binary and triple evolution are generated using the \texttt{COSMIC} software package for population synthesis \citep{Breivik2020}. The BBHs from GCs are taken from the models presented in \cite{Kremer2020}, which used nearly identical physical assumptions for the evolution of massive stars. See \ref{sec:gcs} for specific differences as well as further details about the cluster models. 

As with any population synthesis study, the choice of initial conditions can change the resulting distributions by several orders of magnitude.  As such, any claims regarding the difference between formation channels must be made using \emph{the same} initial conditions and physical choices.  The binary and triple initial conditions we use here are taken from the same population, whose orbital periods, masses, and mass ratios are chosen to be consistent with observations of stellar triples in galactic fields. This is consistent with observational studies of stellar multiplicity, which suggest that the typical massive O star (the progenitors of BHs) have more than one companion on average \citep{Moe2017,Moe2019}.  We sample a range of stellar metallicities, from $Z=0.0001$ to $0.03$ and, and cosmologically sample both the binaries and triples from the same star formation and metallicity enrichment distributions. See Appendix A for more details. As in our previous work, we cleanly delineate between ``binary'' and ``triple'' BBH mergers by only considering as triple-induced mergers those systems whose inner binary would \emph{not} have merged within a Hubble time due in the absence of a tertiary companion.  In other words, our triple population is specifically limited to those whose mergers are due to the LK effect.  


In addition to the astrophysical populations of merging BBHs, we are also interested in the population of BBHs detectable by advanced GW observatories.  Here we follow the technique presented in \cite{Rodriguez2019}.  We assign each BBH a detectability weight defined as   

\begin{equation}
    w_{\rm det} \equiv f_d (m_1,m_2,z) \frac{dVc}{dz} \frac{dt_s}{dt_o} ~,
    \label{eqn:gwweight}
\end{equation}

\noindent where $f_d$ is the fraction of sources with masses $m_1$ and $m_2$ merging at redshift $z$ that are detectable (defined as a signal-to-noise ratio of 8 or higher) by a three-detector network operating at the design sensitivity of Advanced LIGO \citep{Abbott2018}.  $\frac{dVc}{dz}$ is the amount of co-moving volume in a slice of the universe at redshift $z$, and $\frac{dt_s}{dt_o} \equiv \frac{1}{1+z}$ is the difference in co-moving time between the merger redshift and the observer at $z=0$.  We use the IMRPhenomD GW approximant for computing GW detectability \citep{2016PhRvD..93d4006H,2016PhRvD..93d4007K}. Each of the detectable quantities presented here are weighted using \eqref{eqn:gwweight}.

\section{Results}
\label{sect:results}

\begin{figure*}
\includegraphics[width=0.96\textwidth]{massratio_pdf.pdf}
\caption{Distribution of mass ratios for different BBH formation channels. The merging sample is weighted by star (cluster) formation history for the field (cluster) mergers (left) and detectability (right), to produce realistic astrophysical and detectable populations in the local universe. In addition to the field binary (solid blue) and field triple (dashed black) distributions, we also include the mass ratio distributions from GC clusters using data presented in \cite{Kremer2020}, split into 1G (dotted orange) and 2G (dotted green) channels. The median and 90\% regions for the mass ratios from the Power-law model from GWTC-2 (solid purple line and purple shading) is presented in the left panel.}
\label{fig:massratio}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[]{mtot_q_contours.pdf}
\caption{Contour plots for mass ratio--total mass distributions for different channels. The contours show the regions containing 99.5 (solid), 90 (dashed), and 60 (dotted) percent of all data points. The left column shows astrophysically weighted samples while the right column shows the same samples weighted by detectability. In each row, the field triple channel (black) is compared to the field binary (top, red), cluster 1G (middle, orange), and cluster 2G (bottom, green) channels. Also included in the right column is the detection of GW190412 during O3a \citep{theligoscientificcollaboration2020}. Note the differing range in the y-axis of each row.}
\label{fig:mtot_q}
\end{figure*}

\begin{figure}
\includegraphics[]{zams_triples.pdf}
\caption{Top: median and 50 percentile initial inner binary mass ratio as a function of inner semi-major axis. Bottom: normalized distribution of inner semi-major axis prior to stellar evolution for all initial triples in the sample (solid blue), merging binaries (dash-dotted red), and merging triples (dashed green). }
\label{fig:a1i}
\end{figure}

\begin{figure}
\includegraphics[]{compare_cut.pdf}
\caption{Distribution of mass ratios for two different cuts to the triple sample. The dotted black line matches that of the left panel of Figure \ref{fig:massratio}. The solid blue line represents the sample where all stars which may have experienced LK-driven Roche Lobe overflow are discarded.}
\label{fig:rlof}
\end{figure}

For the purposes of comparison between the different channels, we delineate between Field Binary, Field Triple, Cluster 1G, and Cluster 2G mergers. Cluster 1G mergers are dynamically-assembled mergers which involved only first-generation BHs (those created from stellar collapse), whereas Cluster 2G mergers involve at least one second-generation BH. We do not distinguish between binary and triple mergers in clusters, as the triple companion was shown to have minimal effect on both the mass and mass ratio distribution \citep{MartinezFragione2020}. 

In Figure \ref{fig:massratio}, we compare the mass ratio ($q=m_2/m_1$ for $m_1>m_2$) distributions of BH mergers events from the four populations. On the left side, we show the field (cluster) population weighted by star (cluster) formation history. On the right, we further weight the populations by detectability. On the left side, we also show the power law model (with an arbitrary normalization) for $q$ from the \textsc{Power Law + Peak} model ($\beta_q=1.3^{+1.6}_{-1.3}$) fits from GWTC-2 \citep{lvc2020catb} with the purple line and shaded purple band. Using the \textsc{Broken Power Law} model ($\beta_q=1.4^{+1.7}_{-1.4}$) instead of the \textsc{Power Law + Peak} model gives nearly identical results.

We find that the mass ratio distribution for the field binaries here agree well with previous results \citep[e.g.,][]{Zevin2020}. The cluster 1G and 2G distributions are both shaped by binary-mediated dynamical interactions in the cores of dense stellar systems, which will typically exchange lower-mass components of binaries in favor of producing equal mass BBHs \cite[e.g.,][]{1993ApJ...415..631S}. Thus, there is a preference for equal mass mergers in the 1G case and for mergers with $q\sim0.5$ in the 2G case, as the heaviest BH in the cluster, typically the remnant of a 1G merger, will be roughly twice the mass of the next heaviest BH. Remarkably, we find that the mass ratio distribution for triples is largely flat between roughly 0.3 and 1.0.  This is particularly true compared to binaries, which means that--given the same initial conditions--the LK mechanism preferentially enhances the merger rate of lower mass ratio systems. \cite{2021arXiv210301963S} demonstrated that octupole-order effects in hierarchical triple evolution cause the merger fraction to increase as the mass ratio of the inner binary is decreased. As a result, the mass ratio distribution produced through this channel produces low mass ratio mergers compared to the isolated binary channel when using self-consistent initial conditions. Furthermore, we find that this mass ratio distribution is the one that most closely tracks the inferred distribution function from GWTC-2 \citep{lvc2020catb}. Going from the astrophysical population to the detectable population, many of the same general trends are present, though all the distributions are slightly biased towards equal mass ratios.

In Figure \ref{fig:mtot_q}, we present the same mass ratio distributions as above against the total binary masses of each merging system as contour plots, where we show the regions containing 60\% (dotted), 90\% (dashed), and 99.5\% (solid) of the sample. The left column shows the astrophysical population and the right shows the detectable population. In each row, we compare the field triple populations (black contours) to the field binary (red contours), cluster 1G (yellow contours), and cluster 2G (green contours) populations. We also include in the right column in each row GW190412 \citep{theligoscientificcollaboration2020}. 

Many of the general features from Figure \ref{fig:massratio} are also present here, such as the large number of triple mergers with low mass ratios. In the top row, we show that when comparing mergers with $q\lesssim0.5$, triple mergers tend to have a higher total binary mass compared to the field binary mergers, with the center of the total mass ratio distribution at $\sim50-60\msun$ for triples, whereas binaries appear to always have total mass less than $\sim40\msun$. In the middle row, this observation still roughly holds when comparing the field triple and cluster 1G mergers, where it is clear that the mergers produced by the triple channel tend to have a lower mass ratio. Finally, the bottom row demonstrates that the peak of the cluster 2G distribution nearly occupies the same portion of $M_{\rm tot}-q$ space as the field triples. However, note that if BHs are born with negligible spin \citep{Fuller2019a}, as is assumed in this study and in the GC models \citep{Kremer2020}, then these two populations will be easily distinguished by the spins of the components, since the primary 2G BH should have a spin $\chi\sim0.7$. If BHs are born with significant ($\chi \gtrsim 0.2$) spins from stellar collapse, then the relative contribution of hierarchical mergers from GCs will be greatly diminished \citep[e.g.,][]{Rodriguez2019,FragioneLoeb2020b}. Finally, we observe that GW190412 is consistent (with $99\%$ confidence) with either an isolated triple or 2G origin based solely on its position in $m_{\rm tot}-q$ space. The detection announcement of GW190412 also reported a confident detection of the spin of the primary BH $\chi_1=0.44_{-0.22}^{+0.16}$ while the spin of the secondary BH is largely unconstrained \citep{theligoscientificcollaboration2020}. If BHs are born with high spin, then by the logic above, our models would suggest an isolated triple origin is a more likely explanation between the two.


As stated previously, we found that the majority of merging binaries undergo at least one common envelope phase, whereas only $0.1\%$ merging triples do. In the top panel of Figure \ref{fig:a1i}, we show the median and 50 percentile of the initial mass ratio distribution of our sample as a function of semi-major axis. We show in the bottom panel the distribution of ZAMS inner semi-major axis for our total initial sample (solid blue), as well as the ZAMS inner semi-major axis of the systems that merge as a binary (dash-dotted red) and as a triple (dashed green). In the bottom panel, it can be seen that the two merging populations sample different areas of the initial inner semi-major axis distribution, which also have different initial mass ratios. Whereas the merging binaries tend to have smaller initial semi-major axes, with a very apparent enhancement at $\sim0.1\,\rm{au}$, the triples have a much larger range of initial semi-major axes. Accordingly, the merging triples tend to have smaller initial mass ratios than the binaries. These different mass ratio distributions are then sculpted by further binary stellar evolution and these subtle differences are further amplified, most notably through common envelope phases. In our sample, only a single triple system experienced a common envelope phase, whereas the majority of the binaries ($77.6\%$) experience a common envelope phase. 

In this study we considered only the useful triples. In principle, it is possible that the "non-useful" triples, or those whose inner binary would have merged regardless of the tertiary companion, could still enhance the rate of triple mergers within the local Universe. We find that $15.3\%$ of the binary mergers take place within a stable triple system. However, only $0.03\%$ of these mergers take place in a triple system in which the LK mechanism would not be quenched due to relativistic pericenter precession \citep[see also Figure 4 of][]{Rodriguez2018a}. As a result, we expect that these non-useful triples would account for at most an additional few percent of triple mergers where the LK mechanism allows for a substantial decrease in the delay time. Furthermore, in this study we only considered stable triples where all three components are BHs. We found that, for triples (both useful and non-useful) where the inner binary was composed of two BHs, about $2/3$ of the triples had a BH as the outer companion. While a detailed treatment is outside of the scope of this work due to the possibility of mass transfer from the outer companion \citep[e.g.][]{distefano2020}, assuming a comparable merger fraction for the non-BH companions leads to a roughly 50\% increase in the number of triple mergers.

While our approximate technique for modeling hierarchical triples can track the evolution of the triple configuration due to mass loss, supernova, and the evolution of the inner binary, it does not allow for the possibility of LK oscillations during the main sequence.  As a result, it is possible that many of the systems presented here may actually have merged prior to their formation as BH triples.  However, we can estimate the effect this missing physics may have on our results.  In order to explore the sensitivity of our results to LK oscillations during the main sequence and giant phases, we calculated the minimum pericenter distance of each of the inner binaries of the triples attainable due to LK \citep[by numerically solving Eq.~2 of][]{Liu+2019} from their zero-age main sequence initial conditions.  We then compared these separations to the distance required for each star to fill its Roche lobe \citep[Eq.~2 of][]{Eggleton83}, assuming all the stars acquired an average maximum radius of $500\rsun$ \citep[e.g.,][]{Agrawal+2020}, and discarded any systems that would have passed within their Roche lobe separation. We find that $19.3\%$ of the systems survive this cut. The resulting change to the mass distribution is illustrated in Figure \ref{fig:rlof}, where the additional constraint produces an even flatter mass ratio distribution than our original results. This is to be expected, since excluding systems which may have undergone binary interactions or mergers will also preferentially select the ultra-wide triples illustrated in Figure 3, which we show are more likely to have a smaller mass ratio.

\section{Conclusions}
\label{sect:concl}

In this Letter, we started from a sample of triple massive stellar systems and evolved them to their end state as BH triples, in order to obtain a self-consistent comparison between the binary and triple channels. We also compared this population with a population of mergers from previous studies studying BBHs in dense stellar clusters which used the same assumptions for stellar evolution \citep{Kremer2020}. 

We report that the mass ratio distribution from the isolated triple systems in our sample produce a BBH mass ratio distribution that is nearly flat from unity to $\sim0.3$, producing significantly more low-mass-ratio mergers than either isolated binary evolution of dynamical formation of BBHs. Furthermore, we show that the triple channel seems to agree with the power law implied by the most recent catalog of GW observations better than other channels. As a result, the triple channel may be able to more efficiently produce events with low mass ratios such as GW190412.

However, our results do not suggest that triple channel is able to reproduce the whole catalog of observed events. We simply assert that if isolated BH triple systems are produced in nature, then we may be able to gauge the contribution of the LK mechanism to the total population of merging BH systems by low mass ratio and high total mass. This is in addition to other means of distinguishing LK mergers, such as through eccentricity in the LIGO band or effective spin distribution \citep[e.g.,][]{Antonini2012,Antonini2017,Liu2017,Liu2018,Rodriguez2018c,fk2020}.

Apart from the mass ratio distributions, it is important to quantify the relative contribution of the triple channel to the total BBH merger rate.  While a complete estimate of the BBH merger rate is beyond the scope of this Letter, we can approximate the rate by multiplying the ratio of triple BBH mergers to binary BBH mergers using our initial conditions ($0.003$) by the volumetric merger rate at $z=0$ using the same physics and (binary) initial conditions from the literature \citep[$\sim 1.6\times10^3 \gpcyr$,][]{Zevin2020}.  This suggests a local triple merger rate of $\sim5 \gpcyr$, consistent with previous estimates of $0.02-25 \gpcyr$ \citep[e.g.][]{Silsbee2017,2017ApJ...841...77A,Rodriguez2018a,flp2019}.  The initial conditions and common envelope physics used in \cite{Zevin2020} clearly produce extremely unrealistic volumetric rates for BBH mergers (up to two orders of magnitude larger than the observed rate of $23.9^{+14.3}_{-8.6} \gpcyr$ \citep{lvc2020catb}), meaning our ratio of triple to binary mergers may similarly be a significant underestimate.\footnote{For comparison, the BBH merger rate predicted for the clusters presented in this work is roughly $17 \gpcyr$ \citep[][]{2021RNAAS...5...19R}}. As stated previously, the majority of the binary systems experienced a common envelope phase whereas only a single triple system did. Thus, while the binary channel is heavily influenced by uncertainties in the common envelope physics, the triple channel is not.

As the number of GW observations continues to grow, it will become much easier to distinguish between different merger formation channels. We have shown the potential for future observations to constrain the importance of hierarchical triples to the cumulative BH merger rate given the efficiency of producing high total mass, low mass ratio events. We leave it to a future study to probe the sensitivity of these conclusions to different assumptions, such as those concerning binary evolution or accounting for mass transfer from the tertiary companion.

\software{\texttt{numpy} \citep{walt_numpy_2011}, \texttt{scipy} \citep{2020SciPy}, \texttt{matplotlib} \citep{hunter_matplotlib:_2007}, \texttt{COSMIC} \citep{2020zndo...3905335B}}


\acknowledgments

We thank Kyle Kremer, Scott Coughlin, Michael Zevin, Max Moe, and Rosanne Di Stefano for useful discussions and suggestions. We thank Kyle Kremer for providing the cluster models used in this work. Computations were supported in part through the resources and staff contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. This work also used computing resources at CIERA funded by NSF Grant PHY-1726951 and computing resources provided by Northwestern University and the Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA).

\bibliographystyle{aasjournal}
