Effective spin distribution of black hole mergers in triples
Giacomo Fragione Bence Kocsis


Dynamical variational approach to Bose polarons at finite temperatures
    Michael Fleischhauer
    December 30, 2023
======================================================================




Many astrophysical scenarios have been proposed to explain the several black hole (BH) and neutron star binary mergers observed via gravitational waves (GWs) by the LIGO-Virgo collaboration. Contributions from various channels can be statistically disentangled by mass, spin, eccentricity and redshift distributions of merging binaries. In this paper, we investigate the signatures of BH-BH binary mergers induced by a third companion through the Lidov-Kozai mechanism in triple systems. We adopt different prescriptions for the supernovae natal kicks and consider different progenitor metallicities and initial orbital parameters. We show that the typical eccentricity in the LIGO band is 0.01-0.1 and that the merger rate is in the range 0.008-9  Gpc^-3 yr^-1, depending on the natal kick prescriptions and progenitor metallicity. Furthermore, we find that the typical distribution of effective projected spin is peaked at χ_ eff∼ 0 with significant tails. We show that the triple scenario could reproduce the distribution of χ_ eff. We find that the triple channel may be strongly constrained by the misalignment angle between the binary component spins in future detections with spin precession.



galaxies: kinematics and dynamics – stars: black holes – stars: neutron – stars: kinematics and dynamics – Galaxy: kinematics and dynamics




§ INTRODUCTION



Many astrophysical scenarios have been proposed to explain the several black hole (BH) and neutron star (NS) binary mergers observed via gravitational wave (GW) emission by the LIGO-Virgo collaboration. Possibilities include isolated binary evolution <cit.>, mergers catalyzed by stellar envelope expansion <cit.>, mergers in star clusters <cit.>, mergers in galactic nuclei <cit.>, mergers in gaseous disks <cit.>, and Lidov-Kozai (LK) mergers in isolated triple and quadruple systems <cit.>.

With the improving sensitivity of LIGO-Virgo and the expected commissioning of KAGRA and LIGO India, hundreds of detections of merging systems is expected within the decade. Thus, it is fundamental to provide tools to distinguish among the mergers that originate in different astrophysical channels. It has been shown that useful parameters that can help doing so are the masses, spins, eccentricity and redshift of the merging binaries. Their distributions can be used as an indicator to statistically disentangle among the contributions of the several scenarios. This includes identifying correlations between mass and eccentricity
<cit.>, mass and spins <cit.>, 
the mass distribution <cit.>, the spin distribution <cit.>, eccentricity distribution in the LIGO/Virgo band <cit.> and in the LISA band <cit.>, spin orientations <cit.>, the projected effective spin parameter <cit.>, 
and other waveform features <cit.>.





GW emission is highly efficient at circularizing the orbit of an inspiraling binary. As a consequence, BHs that merge in isolation are expected to enter the LIGO frequency band (10 Hz) almost circular, with typical eccentricities in the range e_10 Hz∼ 10^-7–10^-6. In the case the BH binary merges in a hierarchical system as a result of the LK mechanism, a number of authors showed that the typical eccentricity has much higher values, in the range ∼ 10^-2–10^-1 <cit.>. If the BH binary is dynamically assembled in a cluster environment, the spectrum of possible eccentricities is rich, with three possible outcomes <cit.>: (i) binaries that are ejected and merge outside the cluster have eccentricities ∼ 10^-7–10^-6, as in the isolated binary case; (ii) binaries that merge as a result of a GW capture process have eccentricities ∼ 10^-2–10^-1, as in the LK-induced mergers; (iii) binaries that merge within the cluster have intermediate eccentricities ∼ 10^-5–10^-3. The recent non-detection of eccentric sources by LIGO-Virgo was used to place an upper limit on the rate of eccentric mergers with e_10 Hz>0.1 of ℛ≤ 300 f_ ecc  Gpc^-3 yr^-1, where f_ ecc is the fraction of mergers with e_10 Hz>0.1 <cit.>. In comparison, the rate density of circular mergers is between 28-104  Gpc^-3 yr^-1 for a power-law mass distribution prior <cit.>. The eccentricity of dynamically formed BH binaries is much higher in the GW frequency band of LISA. 

Another powerful quantity to discriminate among the contributions of different astrophysical merger channels is χ_ eff, the effective spin of the BH binary at merger defined as the mass-weighted average of the binary components' spins projected onto the orbital angular momentum vector of the binary. Binaries that evolve and merge in isolation are expected to have spin vectors that are aligned with the orbital angular momentum vector of the binary mainly due to tidal dissipation effects <cit.>. BHs that merge in star clusters are instead expected to have spin vectors distributed isotropically, as a consequence of the fact that the dynamical binary assembly does not prefer any specific direction <cit.>. For what concerns the focus of this work, LK-induced mergers in triple systems, <cit.> and <cit.> claimed that the BH binary component spins typically align in the perpendicular direction with respect to the orbital angular momentum of the binary due to the tertiary companion in a triple system, implying near-zero effective spins at merger. However, even more recently, <cit.> have shown that these findings are sensitive to the applied approximations and orbit-averaging in the equations of motion. They showed that the BH spin exhibits a wide range of evolutionary paths, and different distributions of final spin-orbit misalignments can be produced depending on the system parameters.

In this paper, we study the dynamical evolution of BH triples by means of high-precision direct N-body simulations, including post-Newtonian (PN) terms up to 2.5PN order. We start from the main sequence progenitors of the BHs and model the supernova (SN) events that lead to the formation of the BH triple. We adopt different prescriptions for the SN natal kicks, and consider different progenitor metallicities and orbital parameters. We determine the expected distributions of various properties of merging systems, including masses, eccentricities, spin-orbit misalignment, and merger times.

The paper is organized as follows. In Section <ref>, we discuss the initial conditions adopted in this paper. In Section <ref>, we discuss the distribution of masses, orbital parameters, eccentricities and spin-orbit misalignements of merging systems. Finally, in Section <ref>, we discuss the implications of our findings and draw our conclusions.



§ INITIAL CONDITIONS



The stellar triples in our simulations are initialized as described in what follows. In total, we consider nine different models (see Table <ref>).

We consider a triple system comprised of an inner binary of mass m_ in=m_1+m_2 and a third body of mass m_3 that orbits the inner binary. The semi-major axis and eccentricity of the inner orbit are  and , respectively, and the semi-major axis and eccentricity of the outer orbit are  and , respectively. The inner and outer orbital plane have initial relative inclination i_0. 

We sample the mass m_1 of the most massive star in the inner binary from an initial mass function

    dN/dm∝ m^-β ,

in the mass range 20-150, reflecting the progenitor of the BH. We fix β=2.3 <cit.>. We adopt a flat mass ratio distribution for the inner binary, m_2/m_1, and the outer binary, m_3/(m_1+m_2) <cit.>. The mass of the secondary in the inner binary and of the third companion are sampled from the range 20-150. For comparison, we also estimate how the final rate changes if the mass ratio distribution is assumed to be log-uniform <cit.>[From observations, <cit.> found that f(q)∝ q^1.16±0.16 and q^-0.01±0.03 for solar type stars with period less than or larger than 10^5.5day, respectively, while <cit.> found f(q)∝ q^-1.0± 0.4 for massive O-type stars.].

We take the distribution of the inner and outer semi-major axis,  and , respectively, log-uniform <cit.>. In order to make sure that the periapsis distance of the inner binary is large enough that no common-envelope phase or mass transfer occurs[This would also shrink the inner orbit and make the LK mechanism strongly suppressed by the relativistic precession <cit.>.], we set a minimum orbital separation (1-^2)≳ 10 AU <cit.>. We adopt three values for the maximum separation of the triple respectively in different simulation sets, =2000 AU, 3000 AU, and 5000 AU, see Table <ref> <cit.>. For the orbital eccentricities of the inner binary, , and outer binary, , we assume a flat distribution. The initial mutual inclination i_0 between the inner and outer orbit is drawn from an isotropic distribution, while the other relevant angles (i.e. argument of periapsis, argument of node, mean anomaly) are drawn randomly.

After sampling the relevant parameters, we check that the initial configuration satisfies the stability criterion of hierarchical triples of <cit.>.

Given this set of initial conditions for the stellar triples, we assume that each of the three stars in the triple undergoes a SN event sequentially. We assume that SNe take place instantaneously (on a time-scale shorter than the orbital period), during which a given star looses mass instantaneously and collapses to a BH <cit.>[We ignore the SN-shell impact on the companion stars.]. We determine the final masses of the BHs by using the fitting formulae to the results of the parsec stellar evolution code <cit.>. We adopt five different values of the progenitor metallicity as shown in Table <ref>, Z=0.0001, 0.001, 0.005, 0.01, and 0.015, which ultimately sets the final mass of the BH remnant.

As a result of the mass loss, the exploding star is imparted a kick to its center of mass <cit.>, and the system receives a natal kick due to recoil from an asymmetric supernova explosion. We assume that the BH natal velocity kick is drawn from a Maxwellian distribution

    p()∝^2 e^-^2/σ^2 ,

with a velocity dispersion σ. In our fiducial model, we consider σ=260 for NSs, consistent with the distribution deduced by <cit.>. We run an additional model where we set σ=100, consistent with the distribution of natal kicks found by <cit.>. We also adopt a model where no natal kick is imparted during BH formation (see Table <ref>). For BHs, we implement momentum-conserving kicks, in which we assume that the momentum imparted to a BH is the same as the momentum given to a NS <cit.>. As a consequence, the kick velocities for the BHs is lowered by a factor of 1.4/ m_ BH with respect to neutron stars. The value of σ is highly uncertain. 



After each SN event, the orbital elements of the triple are updated as appropriate <cit.>, to account both for mass loss and natal kicks. The final masses of the BHs in the inner binary are  and , and the outer companion . We also check that the stability criterion of hierarchical triples of <cit.> is satisfied and the triple is stable following mass loss and natal kicks. After all the SNe took place, we integrate the triple systems by means of the ARCHAIN code <cit.>, including PN corrections up to order 2.5PN. We perform ∼ 750–1000 simulations for each model in Table <ref>. We fix the maximum integration time as <cit.>,

    T=min(10^3 × T_ LK, 10 ) ,

where T_ LK is the triple LK timescale <cit.>

    T_ LK=8/15πm_ tot/P_ out^2/P_ in(1-e_ out^2)^3/2 ,

where m_ tot=++ and P_ in and P_ out are the inner and outer orbital period, respectively. 



§ RESULTS



In our simulations, we find that ∼ 5%–8% of the triple systems lead to a merger of the inner binary. In the following subsections, we investigate the distribution of these mergers with respect to masses, orbital parameters, and spins.



 §.§ Mass distribution


Figure <ref> shows the cumulative distribution function (CDF) of the total BH mass of BH-BH binaries in triples that lead to a merger, for different values of Z (top panel; σ=260) and σ (bottom panel; Z=0.01). Both Z and σ affect the distribution of total BH masses.

For Z≲ 0.005, the distributions do not depend significantly on the progenitor metallicity and the total BH mass is limited to ∼ 60. For higher metallicities, the progenitors can collapse to more massive BHs, up to ∼ 120–140 <cit.>. We find that ∼ 50% of the mergers have total mass ≳ 90 and ≳ 120 for Z=0.001 and Z=0.0001, respectively. This is consistent with the findings of <cit.>. For comparison, we report the distribution of total masses inferred from LIGO-Virgo mergers and the IAS group <cit.> in Table <ref>. We note that population of very massive binaries we find in our simulations may be due to the fact that we are using fitting formulae to single stellar evolutionary tracks to determine the BH mass from the mass of its progenitor <cit.>. We are not modelling the mass loss during neither possible episodes of Roche-lobe overflows nor possible common evolution phases. Nevertheless, we set in our initial conditions a minimum orbital separation (1-^2)≳ 10 AU to ensure that the periapsis distance of the inner binary is large enough that no common-envelope phase or mass transfer occurs <cit.>. These processes are modeled in binary systems, but not entirely comprehended in triple systems  <cit.>.





Concerning the role of σ, we find that the larger the kick velocity the larger the total BH mass on average. This is explained in relation to our assumption of momentum-conserving kicks, where ∝ m_ BH^-1. In our simulations, for Z=0.01, we find that ∼ 50% of the mergers have total mass ≳ 25 and ≳ 40 for σ=0 and σ 100, respectively. Increasing the mean kick velocity to σ=260 does not change considerably the distribution of total masses. Other prescriptions for the natal kicks may lead to different total mass distributions.

We report in Figure <ref> the CDF of the chirp mass, the combination of masses measured most accurately during the inspiral,

    m_chirp=()^3/5/(+)^1/5 ,

of BH-BH binaries in triples that lead to a merger, for different values of Z (top panel; σ=260) and σ (bottom panel; Z=0.01). As in the case of the total mass, the distribution of chirp masses is not significantly affected by the progenitor metallicity for Z≳ 0.005, ∼ 90% of the BH-BH mergers have m_ chirp≲ 20. We also find that ∼ 50% fo the mergers have m_ chirp≳ 35 and m_ chirp≳ 45 for Z=0.001 and Z=0.0001, respectively. Moreover, lower natal kicks predict lower chirp masses. We find that ∼ 50% of the BH-BH mergers have m_ chirp≳ 10 and m_ chirp≳ 18 for σ=0 and σ= 100, respectively. Furthermore, increasing the mean kick velocity to σ=260 does not change the distribution of chirp masses significantly. This is consistent with the results of <cit.>, who studied the BH-NS mergers in triples.



 §.§ Inner and outer semi-major axis


Figure <ref> illustrates the CDF of inner (top panel) and outer (bottom panel) semi-major axis of BH-BH binaries in triples that lead to a merger for different values of the progenitor metallicity (σ=260) and mean natal kick velocities (Z=0.01). The progenitor metallicity does not affect the distribution of the inner and outer semi-major axes. On the other hand, the value of σ highly affects their distribution, since higher kicks unbind wider triples. We find that ∼ 50% of the systems have a_ in≲ 25 AU, ≲ 60 AU, ≲ 100 AU for σ=0, 100, 260, respectively, and ∼ 50% of the systems have a_ out≲ 500 AU, ≲ 1500 AU, ≲ 2500 AU for σ=0, 100, 260, respectively. Similar conclusions were found in <cit.> and <cit.>.



 §.§ Eccentricity


For the BH-BH binaries that merge in our simulations, we compute a proxy for the GW frequency, i.e. the frequency corresponding to the harmonic that gives the maximum GW emission <cit.>

    f_ GW=√(G(+))/π(1+)^1.1954/[(1-e_ in^2)]^1.5 .

In Figure <ref>, we illustrate the probability distribution function (PDF) of eccentricities at the moment the BH binaries enter the LIGO frequency band, for Models A1, B1 and C1. We also show the minimum eccentricity e_ 10Hz=0.081 where the LIGO/VIRGO/KAGRA network may distinguish eccentric sources from circular sources <cit.>.[Note that the detection threshold of e_10 Hz varies for different binary mass. For binaries with masses of those in the LIGO observing run O1 and O2, the minimum detectable eccentricity ranges between e_ 10Hz=0.023 and 0.081 <cit.>. ] This value is consistent with the recent LIGO/VIRGO analysis <cit.>.  A large fraction of merging BH-BH in triples retain a significant eccentricity at 10 Hz, ∼ 42% (∼ 9%) of the merging systems have e_ 10Hz higher than 0.023 (0.081). We note that a similar signature could be found for BH binaries that merge near supermassive and intermediate-mass black holes <cit.>, in the GW capture scenario in star clusters <cit.>, and in triple <cit.> and quadruple <cit.> systems.





 §.§ Spin


In our simulations, we incorporate the spin-orbit coupling effect by introducing
the spin vector S=SŜ, where S=(Gm_ BH^2/c)χ, where 0≤χ≤ 1 is the dimensionless Kerr parameter. The leading order (1.5PN) de Sitter precession of the spins S_1 and S_2 of the BHs in the inner binary around the inner binary angular momentum J is given by <cit.>

    dS_1/dt   =   Ω_1×S_1=[2Gμ/c^2 r^3(1+3m_2/4m_1)r×v] ×S_1
    dS_2/dt   =   Ω_2×S_2=[2Gμ/c^2 r^3(1+3m_1/4m_2)r×v] ×S_2 ,

where μ=m_1m_2/(m_1+m_2) is the inner binary reduced mass, r=r_1-r_2 and v=v_1-v_2[We neglect the backreaction of S_1 and S_2 on J and the spin-spin precessional terms <cit.>.]. GW measurements are especially sensitive to the following combination of the two spins <cit.>

    =χ_1cosθ_1+χ_2cosθ_2/+ ,

where cosθ_1=(Ŝ_1·J)/J and cosθ_2=(Ŝ_2·J)/J, where J is the total angular momentum of the inner BH binary, which is well approximated by the orbital angular momentum.









The magnitudes of the BH spins are expected to be set by the physics governing the stellar collapse, which depends on the progenitor star metallicity and mass loss. However, since the specific predictions on how the progenitor properties set the remnant spin is still highly uncertain, therefore we run simulations for seven different models for the spins in order to understand how the results depend on the initial spin of the BHs. Each model differs in the initial magnitude of the Kerr spin parameter and/or the initial orientation of the spins (see Table <ref>)


  * Model S1: χ_1 and χ_2 are drawn independently from an uniform distribution and angles between 0^∘≤cosθ_1,2^ ini≤ 20^∘;[Here cosθ_1,2=Ŝ_1,2·𝐋̂ describes the spin misalignment angle relative to the orbital angular momentum vector.]

  * Model S2: χ_1 and χ_2 set by the BH mass <cit.>

    χ_1,2=p_1-p_2/2tanh(p_3-M_ BH1,2/)+p_1+p_2/2 ,

where p_1=0.86± 0.06, p_2=0.13± 0.13, and p_3=29.5± 8.5. Following <cit.>, spins are generated by drawing random samples uniformly in the region in between the two curves given by the upper and lower limits of the parameters. Eq. <ref> captures some of the key features found in <cit.>, that is heavier BHs tend to have smaller spins, and reflects the uncertainties of this model <cit.>. The misalignment angles are drawn from 0^∘≤cosθ_1,2^ ini≤ 20^∘ uniformly;

  * Model T1: χ_1 and χ_2 are drawn independently from an uniform distribution and the spins are aligned with the inner angular momentum;

  * Model T2: χ_1 and χ_2 are drawn independently from an uniform distribution and the initial spin-orbit misalignents of the BHs are drawn from an isotropic distribution;

  * Models U1, U2, U3: χ_1 and χ_2 are fixed at 0.2, 0.5, 0.8, respectively, and 0^∘≤cosθ_1,2^ ini≤ 20^∘ is drawn uniformly.


In Figure <ref>, we show the final  distributions of merging BH-BH binaries in triples for different values of the progenitor metallicity. In the right panel of Figure <ref>, we show the PDFs of the effective spins for Models U1, U2, and U3, where we fix the initial Kerr parameters of the BHs to 0.2, 0.5, 0.8, respectively. We find that mergers with high BH spins have a broadly distributed , and mergers with low absolute spins have  peaked around zero. The distributions do not depend significantly on the value of Z. 

In the left panel, we illustrate the PDFs for Models S1, S2, T1, T2, in which the magnitude of initial spins are sampled from a broad distribution (Table <ref>). For these distributions, the  distributions have a peak at ∼ 0 and broad tails,
which is not significantly affected by the initial distribution of the spins. A possible exception is Model S2 for Z=0.0001 and Z=0.001. In these models, we sample  from Eq. (<ref>). Since low metallicities produce heavy BHs and Eq. (<ref>) implies small initial spins on average, the final PDF results peak at ∼ 0 with less important tails.



In Figure <ref>, we show the PDFs of  of as function of σ. The final distributions do not depend significantly on the initial choice of the spins of the BHs in Models S1, S2, T1, T2, and presents a peak at ∼ 0 with broad tails. The only outlier is the  distribution in the case σ=0 for Model S2, where the distribution is flatter. In the models where we fix the Kerr parameters, the effective spin PDFs do not depend on σ and present a similar behaviour described above for different Z's. In Table <ref>, we report the  inferred from observed BH-BH mergers <cit.>. In general, some of the models are consistent with the observed distributions.

Finally, we show the distributions of the absolute misalignements θ_1 and θ_2 between S_1 and J and S_2 and J, respectively, and |θ_1-θ_2| in Figure <ref>. The distributions are similar to each other for the spin Models S1, S2, T1, showing that |θ_1-θ_2| is peaked narrowly at zero, while Model T2 shows a broader tail. Note that this is due to the fact that the initial orientation of the BH spins were drawn in a correlated way for models S1 and S2 while they are drawn independently from an isotropic distribution in Model T2. The initial correlations may be due to the interaction among progenitor stars, particularly tidal dissipation. Figure <ref> shows that the relative orientation of the BH spins do not randomize during the evolution, but they retain the information on the initial conditions. However merging systems in triples do not show counteralignment with |θ_1-θ_2|>90^∘ irrespective of the initial condition.

We do not find any significant correlation between the spin and the eccentricity in our different models. The only significant correlation is in model S2, where we use Eq. <ref>. In this case, heavier BHs tend to have smaller spins <cit.>.



 §.§ Merger rates




Figure <ref> shows the merger time CDFs of BH-BH binaries in triples that lead to merger for different progenitor metallicities (left; σ=260) and different kick velocities and triple orbital parameters (right; Z=0.01). In our simulations, the distribution of merger times does not depend significantly on the progenitor metallicity nor on the possible maximum separation of the triple . The only parameter on which the merger time depends significantly is the mean kick velocity, since larger values of σ unbind more wider triples. Therefore, triples surviving the SNe are wider for lower σ on average and merge on a longer timescale since their LK timescale is longer.

In order to compute the merger rate of BH-BH binaries in triples, we follow the method outlined in <cit.> and <cit.>. We assume that the local star formation rate is 0.025 Mpc^-3 yr^-1 <cit.>, thus the number of stars formed per unit mass, volume, and time is given by,

    ṅ(m)    = η_ SFR  f(m)/⟨ m⟩ =
    
       =
    5.2× 10^6 (m/)^-2.3 M_⊙^-1 Gpc^-3yr^-1 ,

where ⟨ m⟩ = 0.38 is the average stellar mass for a Kroupa mass function. Assuming a constant star-formation rate[The star-formation rate depends on the cosmic time. <cit.> adopted a redshift-dependent star-formation rate from <cit.> and found a similar range for the overall merger rate of triple BHs.], the BH-BH merger rate in triple systems is then,

    ℛ_BH-BH   =
    (1-ζ) f_ 3 f_ stable f_ merge∫_20^150ṅ(m_1) dm_1=
    
       =
    7.4× 10^4 η (1-ζ) f_ 3 f_ stable f_ BH-BH Gpc^-3 yr^-1 ,

where f_ 3=0.25 is the fraction of stars in triples, f_ stable is the fraction of systems that remain stable after the SN events take place, f_ merge is the fraction of systems that merge (see Table <ref>), and η is the conditional probability that the secondary is also a progenitor of a BH given that the primary is a BH progenitor:

    η = ∫_20^150 d m_1 f_ IMF(m_1) ∫_20/m_1^1 dq_12  f_q(q_12) 
        /∫_20 ^150 d m_1 f_ IMF(m_1)  ,

where f_q(q_12) is the mass ratio distribution of the inner binary, which we assume to be constant. For a <cit.> initial mass function f_ IMF we get η= 0.38. In Eq. (<ref>), f_ merge∼ 0.06 for all the models, we find that the fraction of stable systems depends both on the mean natal kick and on Z, because lower progenitor metallicities lead to more massive BHs, which are on average imparted lower kicks at birth as a result of our assumption of momentum-conserving kicks (see Tab.<ref>). A factor of uncertainty is the possible KL dynamics during the evolution of the stellar triples before they form a BH-BH system in the inner binary, which we have not modeled here. Some fraction of the parameter space can be removed by the earlier evolution of the system <cit.>. To estimate this uncertainty, we assume very conservatively that any stellar triple whose initial KL timescale is less than the lifetime of the primary star <cit.> in the inner binary will merge, and, as a consequence, will not form a triple system with an inner BH-BH binary <cit.>. We find that the fraction of these triples is ζ∼ 0.83 on average, except for Model A2 where we find ζ∼ 0.55. Using the minimum and maximum values of f_ stable in Table <ref> and plugging numbers into Eq. (<ref>),

    Γ_BH-BH=0.008-9  Gpc^-3 yr^-1 .

This is consistent with the merger rates in triples derived by <cit.> and <cit.>.
The higher merger rates correspond to low natal kicks and low progenitor metallicities. For a log-uniform distribution of mass ratios, we estimate a rate ∼ 2 times larger.



§ DISCUSSION AND CONCLUSIONS



Many astrophysical scenarios have been proposed to explain the BH and NS mergers observed via GW emission by the LIGO-Virgo collaboration. A promising way to disentangle the contributions from different channels is to statistically compare the distributions of masses, spins, eccentricity and redshift of the merging binaries as discussed in Section <ref>.
In this paper, we have studied the dynamical evolution of BH triples in isolation by means of high-precision N-body simulations, including post-Newtonian terms up to order 2.5PN. We started from the main sequence progenitors of the BHs and modeled the SNe that lead to the formation of the BH triple. We adopted different prescriptions for the SN natal kicks, and considered different progenitor metallicities and orbital parameters. We have shown that the typical eccentricity of BH-BH binaries merging in triple systems when reaching the 10 Hz is 0.01-0.1 and that the merger rate is in the range 0.02-24  Gpc^-3 yr^-1, depending on the natal kick prescriptions and progenitor metallicity, consistent with the rates inferred by <cit.> and <cit.>. Higher rates correspond to low natal kicks and low metallicities. We find that the fraction of mergers with a detectable nonzero eccentricity for the LIGO–VIRGO-KAGRA network at design sensitivity is in the range ∼ 9–42%.

We confirm the findings of <cit.> that the BH spin exhibits a wide range of evolutionary paths, and different distributions of final spin-orbit misalignments can be produced depending on the system parameters. The effective spin parameter  is broadly distributed if the merging BHs are highly spinning. Nevertheless,  is peaked at zero for all triple population models that we have investigated, due to the fact that many merging systems have low intrinsic BH spins in these models, consistent with <cit.> and <cit.>. We have also discussed that the triple scenario we studied in this paper could reproduce the distribution of effective spins inferred from LIGO-Virgo mergers and the IAS group <cit.>.

We note that when we check that the triple systems remain stable after each SN event, systems that are deemed unstable by the <cit.> criterion could still merge, thus possibly enhancing our inferred merger rates. In our calculations, we assumed that the SN events take place instantaneously and do not simulate the systems during the main sequence lifetime of the progenitors. Nevertheless, we have used fits to single stellar evolutionary tracks to determine the final BH mass <cit.>. However, the details of the specific evolutionary paths, which depend on stellar winds, metallicity and rotation, of the stellar progenitors could reduce the parameter space <cit.>. The situation becomes more complicated if mass loss during possible episodes of Roche-lobe overflows and common evolution phases in the triple are considered <cit.>. We do not model these possible effects and leave the detailed study to future investigations.

Ongoing and future observations promise to observe hundreds of merging BHs, thus providing a sufficiently high number to constrain the different formation scenarios. The GW observation of a merging BH-BH binary which enters the LIGO band with a non-negligible eccentricity and with a nearly zero effective spin would be consistent with triple scenario. 

While our results show that the mass-weighted sum of the spin vectors projected on the orbital angular momentum vector, , is approximately symmetrically distributed around ∼ 0 in the triple channel, we find that the angle between the individual spins is strictly less than ∼ 90^∘ (Figure <ref>). While this parameter is poorly constrained in the O1 and O2 observing runs, it is potentially measurable through the effects of spin-precession in future longer duration detections at lower GW frequencies <cit.>. The misalignment angle between the two spins represents a powerful constraint to test the triple channel of BH mergers.



§ ACKNOWLEDGEMENTS


GF thanks Seppo Mikkola for helpful discussions on the use of the code archain, and Barak Zackay for providing data on the effective spins of binary black hole mergers. We thank Fabio Antonini for useful discussions. GF acknowledges support from a CIERA postdoctoral fellowship at Northwestern University. This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Programme for Research and Innovation ERC-2014-STG under grant agreement No. 638435 (GalNUC) and from the Hungarian National Research, Development, and Innovation Office under grant NKFIH KH-125675 (to BK). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. 

mn2e


