A Triple Scenario for the Formation of Wide Black Hole Binaries Such as Gaia BH1

Recently, several noninteracting black hole–stellar binaries have been identified in Gaia data—for example, Gaia BH1, where a Sun-like star is in a moderately eccentric (e = 0.44) 185 days orbit around a black hole. This orbit is difficult to explain through binary evolution. The present-day separation suggests the progenitor system would have undergone an episode of common-envelope evolution, but a common envelope should shrink the period below the observed one. Since the majority of massive stars form in higher-multiplicity systems, a triple evolution scenario is more likely for the progenitors of BH binaries. Here we show that such systems can indeed be more easily explained via evolution in hierarchical triple systems. von Zeipel–Lidov–Kozai oscillations or instabilities can delay the onset of the common-envelope phase in the inner binary of the triple, so that the black hole progenitor and low-mass star are more widely separated when it begins, leading to the formation of wider binaries. There are also systems with similar periods but larger eccentricities, where the BH progenitor is a merger product of the inner binary in the triple. Such mergers lead to a more top-heavy black hole mass function.


INTRODUCTION
Approximately one out of every thousand stars will end their lives as a black hole (BH).This means a galaxy like the Milky Way should contain of order 10 8 black holes.However, only of order a hundred black hole systems are known in our Galaxy, predominantly from studies of X-ray binaries (Corral-Santana et al. 2016;Fortin et al. 2023), that can only identify close, interacting systems.
Microlensing (Lam et al. 2022;Mróz et al. 2022;Sahu et al. 2022), spectroscopy (Shenar et al. 2022), and astrometry (Andrews et al. 2022;Shahaf et al. 2023) can be used to identify isolated black holes or noninteracting binaries.Such systems probe high mass binary evolution over a wider range of parameter space than X-ray binaries, providing important constraints on the black hole natal kick distribution and common envelope physics.
Gaia DR3 contains a few×10 5 binary orbital solutions (Gaia Collaboration et al. 2023a,b), prompting searches for astrometric BH binaries (Andrews et al. 2022;Shahaf et al. 2023;El-Badry et al. 2023a,b).To date, two unambiguous BH-stellar binaries have been identified: Gaia BH1 and Gaia BH2.Gaia BH1 contains a Gtype main sequence star in a 185.6-day orbit around a 9.62 ± 0.18 M ⊙ black hole.The star's orbit is moderately eccentric with e = 0.45 (El-Badry et al. 2023a).Gaia BH2 contains a red giant in a 1276.7 day orbit around a 8.94 ± 0.34 M ⊙ black hole.This star's orbit is also moderately eccentric with e = 0.52 (El-Badry et al. 2023b).The uncertainties on the orbital periods and eccentricities are much less than 1%.
These systems pose a challenge to isolated binary evolution models.On the one hand, the present-day separations suggest the progenitor system would have undergone an episode of common envelope evolution.On the other hand, a common envelope would shrink the binary period below the observed one.The tension can be resolved by invoking an unusually large common envelope efficiency (α ≈ 5 for Gaia BH1; El-Badry et al. 2023a), or a very massive progenitor for the black hole that avoids a red giant phase (and hence the common envelope) entirely (El-Badry et al. 2023b).
In any case, an isolated binary is not the most likely initial configuration for the progenitor system.Most O & B stars are in triples or higher order multiples (Moe & Di Stefano 2017).Stellar evolution in triples could give rise to a variety of novel evolutionary channels and produce different types of binaries and merger products (e.g.Perets & Fabrycky 2009;Perets & Kratter 2012;Antognini et al. 2014;Michaely & Perets 2014;Naoz 2016;Antonini et al. 2017;Rose et al. 2019;Stegmann et al. 2022).Motivated by this observation we check whether triple evolution can reproduce the Gaia black hole binaries.We find that triples can reproduce the observed period and eccentricity of Gaia BH1, even for normal common envelope efficiencies (α = 1).
The remainder of this paper is organized as follows.In § 2 we discuss our initial conditions and procedure for evolving triples.In § 3 we present the results from our triple population synthesis and compare them to the results of binary population synthesis.In § 4 we discuss formation rates.In § 5 we discuss alternative formation scenarios.We summarize in § 6.

Initial conditions
We use the empirical distributions for multiple properties from Moe & Di Stefano (2017) to initialize triples, with the modifications described below.
The progenitor of the black hole in Gaia BH1 would be ∼ > 20M ⊙ , while its companion is 0.93 M ⊙ .Thus, the initial mass ratio is ∼ < 0.05.The distribution of binary properties for such low mass ratios is unconstrained by Moe & Di Stefano (2017).Here, we extrapolate the broken power-law mass ratio distribution from their work down to the brown dwarf boundary.This extrapolation affects the distribution of companion periods, as shown in Figure 1.
To generate triples, we follow the procedure below: 1. We draw a primary mass between 18 and 150 M ⊙ from an m −2.3 distribution (corresponding to either a Kroupa or Salpeter mass function).
2. Then we generate two companion periods for this primary mass from its companion frequency distribution (modified by the extrapolation to lower mass ratios; see Figure 1), with the following steps (a) We first generate logarithmically spaced bins in period.The probability of a companion in each bin is the companion frequency multiplied by the bin width.
(b) We generate a random number between 0 and 1 for each bin (starting at the shortest period).If this number is less than the companion probability we generate a companion in this bin (with the period drawn from a loguniform distribution).
(c) The companions' eccentricities follow the distribution in Moe & Di Stefano (2017). 1 The mass ratio is drawn from the extrapolated mass ratio distribution.
(d) We continue until we have a triple.For simplicity, we do not allow for quadruples or higher-order multiples in this study.
To form a Gaia BH-like binary, the progenitor system must contain a low-mass star.Thus, we require either the secondary or tertiary star to be between 0.5 and 2 M ⊙ .Finally, we only consider triples that satisfy the (Mardling & Aarseth 2001) stability criterion.
1 We include a turnover at 80% of the maximum eccentricity, , to ensure the eccentricity distribution is continuous.
The distribution of initial orbital properties for the generated triples is shown in Figure 2.

Triple evolution
We evolve the generated triples with MSE (Hamers et al. 2021).MSE models the evolution of hierarchical multiple systems (binaries, triple, quadruples, or higher order multiples), accounting for gravitational and binary interactions between the stars as well as stellar evolution.The code automatically switches between solving the secular equations of motion and direct N-body integration depending on the configuration of the system.Thus MSE can model both stable and unstable phases of evolution, and associated phenomenology (e.g.collisions, escapes, exchanges; see Toonen et al. 2020Toonen et al. , 2022 for overviews of triple evolution).
We evolve each system for a random time between 10 Myr and 10 Gyr (implicitly assuming a constant star formation rate).We have updated the stellar wind, remnant mass, and supernova kick prescription in MSE, which followed the BSE (Hurley et al. 2002), and were somewhat outdated, to match those in El-Badry et al. (2023a).After these updates, single-star evolution is consistent (at the few percent level) between the updated MSE and the binary population synthesis code COSMIC (Breivik et al. 2020) used there. 2s a control, we also evolve binaries generated via the procedure in § 2 in isolation.Where possible, we match the binary and stellar evolution parameters of previous studies of Gaia BH1 (El-Badry et al. 2023a), that used COSMIC.
There are some differences in the treatment of binary evolution between MSE and COSMIC that are not easily removed.For example, COSMIC has different prescriptions for the onset of unstable mass transfer.On the other hand, MSE has prescriptions for eccentric mass transfer that are not in COSMIC (in particular the Hamers & Dosopoulou 2019 prescription that smoothly transitions from continuous mass transfer for circular orbits to impulsive mass transfer at pericenter for highly eccentric orbits).
In this work, we set the common envelope efficiency (α) to either 1 or 5 for simplicity.We also use the fall-back modulated kick prescription from Fryer et al. (2012).In contrast, the common envelope efficiency and kick magnitude are free parameters in El-Badry et al. (2023a).For each set of parameters, we initialize ∼ 5 × 10 5 systems.
The top, left panels of Figure 3 and Figure 4 show the masses and orbital eccentricities for surviving black holemain sequence binaries, with periods ≤ 1000 days, following triple evolution for different common envelope efficiencies (α).The top, right panels show the orbital period as a function of eccentricity for these systems.The binaries in Figures 3 and 4 are no longer in triples, but the stellar companion in these systems was the tertiary star of the original triple ∼ 80% of the time.The initial period and eccentricity of the companion are shown in Figure 5.The bottom panels of Figures 3 and 4 show the distributions for binaries evolved in isolation.
For low common envelope efficiencies (α = 1), binary evolution is unable to produce systems with periods ∼ > 100 days and moderate eccentricities (e ∼ < 0.7) like Gaia BH1.Triple evolution, on the other hand, can produce such systems.
Gaia BH1-like systems form from the initial primary and secondary star, after a common-envelope phase.(We define a Gaia BH1-like system to be one where the eccentricity is within a factor of 1.5 of the observed one, while the BH mass, secondary mass, and period are within a factor of two of those observed).In binaries, the separation before the common envelope only evolves adiabatically due to wind mass loss (with constant eccentricity).Therefore, the common envelope occurs at the beginning of the giant phase.Afterward, the separation increases much faster than the radius due to mass loss from the giant star.In triples, the eccentricity of the secondary can evolve via the von Zeipel-Kozai-Lidov effect (Kozai 1962;Lidov 1962).Thus, in some cases, the eccentricity can decrease at the beginning of the giant phase, and then increase again so that the common envelope occurs later when the separation between the low-mass star and black hole progenitor is wider, as illustrated in Figures 6 and 7.Moreover, stellar evolution and mass loss can trigger and change such secular evolution (Perets & Kratter 2012;Michaely & Perets 2014).Alternatively, the eccentricity of the secondary can be excited by a triple evolution dynamical instability (TEDI) (Perets & Kratter 2012;Michaely & Perets 2014;Toonen et al. 2020;Hamers et al. 2022).In this case, the stellar eccentricity is excited, as the system becomes less hierarchical due to stellar evolution.This type of evolution is shown in Figure 8.Other formation channels are possible for broader definitions of Gaia BH1-like binaries.For example, allowing the eccentricity and other properties to differ by a factor of two from Gaia BH1.This modified definition picks out a distinct binary population with eccentricity ∼ > 0.7.In most cases (68% for α = 1; 81% for α = 5), such binaries are formed after a stellar merger in the inner binary followed by a common envelope with the initial tertiary star.This evolution is illustrated in Figure 9.This behavior depends on uncertain prescriptions for the stellar radius of high-mass stars.In fact, highmass stars ( ∼ > 50M ⊙ ) may not enter a red giant phase (see § 5).Indeed, it was already suggested that stellar mergers in triples can produce blue straggler binaries (Perets & Fabrycky 2009).
Mergers make the BH mass function more top-heavy.Figure 10 shows the BH mass as a function of period for BH-main sequence binaries originating in triples and binaries.The BH never exceeds 20M ⊙ for a binary initial condition but exceeds this threshold for ∼ 10 − 30% of triple initial conditions.Almost all BH-main sequence with BHs above 20M ⊙ have periods ≥ 10 4 days.Mergers that produce BHs above 20M ⊙ are not triggered by the tertiary companion, as they also occur when it is removed.However, in the binaries, the merger product is a single star and is thus not included in the distribution in Figure 10.Additionally, the semi-major axis distribution affects the production of massive BHs.BHs are above 20M ⊙ , while for triples this fraction is 3%.

RATES
We now estimate the formation efficiency of Gaia BH1-like systems for triples and binaries.We ran additional simulations to reduce the uncertainty in our rate estimates.For efficiency, we require the secondary star to be between 0.5 and 2 M ⊙ in the new triple simulations, considering the results in § 3, where all Gaia BH1-like binaries form from the inner binary of the progenitor triple.
The fraction of Gaia BH1-like binaries in a stellar population, f GBH , is (1) Here f O is the fraction of BH progenitors (per stellar mass); f triple,binary is the triple (or binary) fraction for these stars; f solar is the fraction of massive triples (or binaries) with a companion between 0.5 and 2M ⊙ ; f eff is the fraction of such systems that produce Gaia BH1like binaries.We group the last two terms into an overall efficiency, feff .
For Kroupa mass function f O is ≈ 3.6 × 10 −3 M −1 ⊙ (assuming all stars above 18M ⊙ form BHs). Most massive stars are in triples or higher order multiples (Moe & Di Stefano 2017).However, the exact breakdown by multiples is highly uncertain, especially for mass ratios q < 0.1.We leave f triple,binary unspecified, though f triple is plausibly of order unity.f solar ≈ 0.26 (0.12) for triples (binaries) based on the extrapolated Moe & Di Stefano (2017) distributions. 4 Finally, f eff is estimated from the results in the previous section.
The formation efficiencies are summarized in Table 1, for triples and binaries.The efficiency is at least a factor of a few higher for triples than binaries.5Overall, up to 2.9×10 −7 Gaia BH1-like binaries per solar mass of stars formed.
Previously, Di Carlo et al. ( 2023) computed formation rates for BH-main sequence (BH-MS) binaries from clusters, and from isolated binary evolution (using COS-MIC).They find 4.2 × 10 −7 BH-MS binaries (with periods up to 10 yr6 ) per unit star formation from isolated evolution.To compare with this result, we ran additional binary and triple simulations with no restrictions on the companion masses.The formation efficiency BH-MS binaries (with periods less than 10 yr) in these simu-  2023b) reported the discovery of Gaia BH2: a BH-red giant system with a period of 1277 days and an eccentricity of 0.52.The mass of the black hole is 8.94 ± 0.34 M ⊙ , while the companion is a ∼ 1M ⊙ red giant.
For a common envelope efficiency, α = 1 (α = 5), our triple population synthesis produces 0 (3) Gaia BH2-like systems.These systems form from a merger in the inner binary followed by a common envelope.The primary and secondary then evolve into a black hole and red giant.This suggests ∼ < 7 × 10 −9 Gaia BH2-like systems per star formed.Also, all of our Gaia BH2-like systems are more eccentric than the observed system and have eccentricities ∼ >0.7.Our models suggest that a large common envelope efficiency is required to reproduce Gaia BH2-like binaries, even for triples.However, the upper limit on the frac-Table 1. Formation efficiency of Gaia BH1-like binaries.(We define a Gaia BH1-like system to be one where the eccentricity is within a factor of 1.5 of the observed system, while the BH mass, secondary mass, and period are within a factor of two of the observed system).tion of Gaia BH2-like systems per unit star formation is not very stringent due to small number statistics.(We have run 1.2 × 10 6 systems for α = 1, which gives an upper limit of 2.4 × 10 −9 M −1 ⊙ at 95% confidence).Overall, more simulations are required to test the plausibility of producing Gaia BH2 through triples.

ALTERNATIVES
There are a few alternative scenarios for the observed Gaia BH binaries, besides those already discussed.We describe them here for completeness.
First, these systems could have formed from a binary with a very massive primary ( ∼ > 50M ⊙ ) and a Sun-like  secondary (El-Badry et al. 2023b).Such massive primaries may never become red supergiants (Humphreys & Davidson 1994;Smith & Conti 2008;Higgins & Vink 2019).This behavior is not captured by our models.In this case, the observed periods are easily reproduced, considering Gaia BH1/BH2 need not have gone through a common envelope phase.Secondly, the interaction between the inner binary stars in a triple may have prevented either star from be-coming a giant (Justham et al. 2014).This could lead to a low mass tertiary in orbit around a binary black hole system (El-Badry et al. 2023a).This behavior is also not captured by our models.In our triple synthesis, the only systems with binary black holes and a low mass tertiary have periods ∼ > 10 5 yr.Thus, they are much wider than the observed systems.Recently, a couple of candidate BH-stellar binaries have been identified in Gaia data: Gaia BH1 and Gaia BH2.The former is a Sun-like main sequence star in a 185.6-day orbit around a 9.62 M ⊙ black hole, while the latter is a red giant in a 1277-day orbit around an 8.94 M ⊙ black hole.These systems are difficult to explain with isolated binaries, potentially requiring unusual common envelope evolution (El-Badry et al. 2023a) and/or other scenarios, including formation in clusters.
Here, we consider a triple scenario for the formation of such systems.Since, as known from observations, most black hole progenitors are in triples or higher order multiples (Moe & Di Stefano 2017), this is, to begin with, a more likely scenario than a binary progenitor.
Our main results are summarized as follows: 1. Triple evolution allows for the formation of wider BH-main sequence systems at moderate eccentricities because secular eccentricity oscillations or in-stabilities allow systems to enter a common envelope at wider separations than in the binary case.
2. As a result, systems with periods comparable to Gaia BH1 can be produced for smaller common envelope efficiencies (e.g. for α = 1).
3. Gaia BH2 may require a larger common envelope efficiency, even in the triple scenario.However, more simulations are required to test this.
4. The formation efficiency for BH-main sequence binaries with periods < 10 yr is a factor of 3-4 higher for triples than for binaries.Thus, triples provide a more consistent scenario for such systems, without the need for very low-efficiency common-envelope scenarios.Dynamical encounters in a cluster are also not required.
5. BH-main sequence binaries from triples have a more top-heavy BH mass function than those from binaries, due to mergers within the inner binary.For example, the BH never exceeds 20M ⊙ for a binary initial condition but exceeds this threshold for ∼ 10 − 30% of triple initial conditions.Almost all BH -main-sequence binaries with BH masses above 20M ⊙ have periods ≥ 10 4 days.In isolation, such wide binaries would not give rise to any strong interactions, and would not form X-ray binaries, however, flyby encounters in the field can drive such binaries into interaction (Michaely & Perets 2016).In clusters such wide binaries would be disrupted (soft binaries), but the BHs could be exchanged into other binaries, and even be part of BH-BH binaries that may merge through GW inspirals.Therefore, the elevated BH masses in these systems can affect the overall mass function of GW sources.
6.The observed high triple and quadruple multiplicity among massive stars, and our results, among others, suggest that the use of binary population synthesis for the modeling of massive stars and of BHs may systematically provide incorrect evolutionary scenarios.

Figure 1 .
Figure 1.The solid, black line is the companion frequency (per decade orbital period) from Moe & Di Stefano (2017) for a 20M ⊙ primary.The dashed, grey line shows the companion frequency, extrapolated to the brown dwarf limit. 3

Figure 2 .
Figure 2. Distribution of initial semi-major axes and eccentricities of triples' inner (top) and outer (bottom) orbit.The distributions are based on (Moe & Di Stefano 2017) (see text for details).The inner binary semi-major axis distribution is steeper than log-uniform primarily due to the Mardling & Aarseth (2001) stability criterion.

Figure 3 .
Figure 3. Left panels: Eccentricity as a function of primary mass for surviving black hole-main sequence binaries with periods ≤ 1000 days after triple (top) and binary evolution (bottom).Right panels: Period as a function of eccentricity for surviving black hole-main sequence binaries.The star symbol is Gaia BH1.

Figure 5 .
Figure 5.Initial periods and eccentricities for stellar companions in Figures 3 and 4. For triples, the stellar companion was initially the tertiary star 80% of the time.lations, is summarized in Table 2.For consistency with Di Carlo et al. (2023) we focus on the α = 5 case.The formation efficiency of BH-MS binaries for triples is enhanced by a factor of a few compared to binaries. 7This suggests field triples are competitive with young star clusters for forming BH-MS binaries.(Di Carlo et al. 2023 find that young star clusters are ∼4 times more efficient than isolated binaries in producing BH-MS binaries.)

Figure 6 .
Figure 6.Eccentricity as a function of semi-major axis at the start of the common envelope phase for triples (top) and binaries (bottom).For triples, we show the orbital elements of the innermost binary at the last common envelope phase.The colors indicate the primary mass.The black stars correspond to triples that end their evolution as Gaia BH1-like binaries (see text for definition).For comparison, we also reproduce the starred points from the top panel in gray in the bottom panel.

Multiple(a
Possionian uncertainties are ∼ 10% a Gaia BH1-like binaries formed per massive triple or binary.b Gaia BH1-like binaries per M⊙ of stars formed, divided by the triple (or binary fraction) of massive stars.c 95% confidence upper limit.Table 2. Formation efficiency of BH-main sequence (BH-MS) binaries with periods less than 10 yr.Multiple-type α feff a fBH−MS/f triple,binary b BH-MS binaries formed per massive triple or binary.b BH-MS binaries per M⊙ of stars formed, divided by the triple (or binary fraction) of massive stars.

Figure 7 .
Figure 7. Example triple evolution that results in a Gaia BH1-like binary.The top panel shows the evolution of stars and binaries with mobile diagrams.The bottom panel shows the evolution of the eccentricity and inclination as a function of time.These elements evolve due to the von Zeipel-Lidov-Kozai effect.For reference, we show the quadrupole and octupole Kozai timescales in the bottom panel (Naoz 2016).

Figure 10 .
Figure 10.Mass versus log period (in days) for BH-main sequence binaries from triples (top left) and binaries (top right) for α = 1.The bottom panel shows the BH mass distribution for each.Triples produce a more top-heavy BH mass function due to mergers in the inner binary.
Formation of a BH-main sequence binary, following two common-envelope phases.
Giant CE SNeFigure8.Same as Figure7, except in this case, Gaia BH1-like binary is formed after a triple evolution dynamical instability (TEDI).