Letter

The first gravitational-wave source from the isolated evolution of two stars in the 40–100 solar mass range

Received:
Accepted:
Published online:

Abstract

The merger of two massive (about 30 solar masses) black holes has been detected in gravitational waves1. This discovery validates recent predictions2,3,4 that massive binary black holes would constitute the first detection. Previous calculations, however, have not sampled the relevant binary-black-hole progenitors—massive, low-metallicity binary stars—with sufficient accuracy nor included sufficiently realistic physics to enable robust predictions to better than several orders of magnitude5,6,7,8,9,10. Here we report high-precision numerical simulations of the formation of binary black holes via the evolution of isolated binary stars, providing a framework within which to interpret the first gravitational-wave source, GW150914, and to predict the properties of subsequent binary-black-hole gravitational-wave events. Our models imply that these events form in an environment in which the metallicity is less than ten per cent of solar metallicity, and involve stars with initial masses of 40–100 solar masses that interact through mass transfer and a common-envelope phase. These progenitor stars probably formed either about 2 billion years or, with a smaller probability, 11 billion years after the Big Bang. Most binary black holes form without supernova explosions, and their spins are nearly unchanged since birth, but do not have to be parallel. The classical field formation of binary black holes we propose, with low natal kicks (the velocity of the black hole at birth) and restricted common-envelope evolution, produces approximately 40 times more binary-black-holes mergers than do dynamical formation channels involving globular clusters11; our predicted detection rate of these mergers is comparable to that from homogeneous evolution channels12,13,14,15. Our calculations predict detections of about 1,000 black-hole mergers per year with total masses of 20–80 solar masses once second-generation ground-based gravitational-wave observatories reach full sensitivity.

Main

We study the formation of coalescing black-hole binaries using the StarTrack population synthesis code16,17. This method has been updated to account for the formation of massive black-hole systems in isolated stellar environments. The new key factors include an observationally supported star-formation rate, chemical enrichment across cosmic time and a revised initial condition for evolution of binary stars. Hitherto, simulations have been unable to achieve the desired predictive power because of the limitations on the input physics (for example, limited metallicity range) and numerical accuracy. To ensure the dominant contribution from intrinsically rare low-metallicity star-forming environments are adequately sampled, we use a dense grid of metallicities (32 metallicities) with high precision (20 million binaries each).

Although binary population synthesis is dependent on a number of uncertain physical factors, there has been recent progress in reducing this uncertainty and understanding how it affects predictions. In light of this, we consider the following three models to encompass major sources of uncertainty (Methods): M1 represents our ‘standard’ classical formation model for double compact objects composed of two black holes (BH–BH), two neutron stars (NS–NS), or one of each (BH–NS); M2 is our ‘optimistic’ model, in which Hertzsprung-gap stars may initiate and survive common-envelope evolution, leading to many more binaries being formed; and M3 is our ‘pessimistic’ model, in which black holes receive large natal kicks, which disrupts and thereby reduces the number of BH–BH progenitor binaries.

For each generated double compact object merger, with its intrinsic component masses and the redshift of the merger, we estimate the probability that such a merger would have been detectable in the first observing run (O1) of the Laser Interferometer Gravitational-Wave Observatory (LIGO) advanced detectors. We adopt a self-consistent model of evolution of stellar populations in the Universe3,4, and we take the representative noise curve for O1 (https://dcc.ligo.org/LIGO-G1501223/public) and assume 16 days of coincident science-quality observational time1.

In Fig. 1 we show the formation and evolution of a typical binary system that result in a merger with similar masses and at a similar time to GW150914. Stars that form such mergers are very massive (40M–100M; M is the mass of the Sun), and at the end of their lives they collapse directly to form black holes18. Because there is no associated supernova explosion, there is also no mass ejection. We allow 10% of the collapsing stellar mass to be emitted in neutrinos. If natal kicks are associated with asymmetric mass ejection (as in our standard model), then our prediction is that these massive black holes do not receive natal kicks and that their spin directions are the same as that of their progenitor collapsing stars. The binary evolution removes the hydrogen-rich envelope from both binary components, making both stars compact and luminous Wolf–Rayet stars before they collapse to black holes. The first binary interaction is a dynamically stable Roche-lobe overflow phase, whereas the second interaction consists of a common-envelope phase that produces a compact binary. After the common-envelope phase, the progenitor binary resembles two known high-mass X-ray binaries hosting massive black holes: IC10 X-1 and NGC 300 X-1 (ref. 19). A massive BH–BH binary (each with a mass of approximately 30M) is formed in approximately 5 Myr of evolution, with a relatively wide orbit (semi-major axis a ≈ 50R; R is the radius of the Sun), leading to a long time to coalescence of tmerger ≈ 10 Gyr. The accretion onto the first black hole in the common-envelope phase is only modest (approximately 1.5M), whereas accretion from stellar wind of its companion is rather small (less than 0.1M).

Figure 1: Example binary evolution leading to a BH–BH merger similar to GW150914.
Figure 1

A massive binary star (96M (blue) + 60M (purple)) is formed in the distant past (2 billion years after Big Bang; z ≈ 3.2; top row), and after 5 million years of evolution forms a BH–BH system (37M + 31M; second-last row). For the ensuing 10.3 billion years, this BH–BH system is subject to loss of angular momentum, with the orbital separation steadily decreasing, until the black holes coalesce at redshift z = 0.09. This example binary formed in a low-metallicity environment (Z = 0.03Z). MS, main-sequence star; HG, Hertzsprung-gap star; CHeB, core-helium-burning star; BH, black hole; a, orbital semi-major axis; e, eccentricity.

To investigate general aspects of the formation history of GW150914, we select a population of GW150914-like BH–BH mergers with a total redshifted mass of Mtot,z = 54M–73M, and then further restrict our sample to binaries that would be detectable in O1. The formation channels typical for these massive BH–BH mergers are summarized in Extended Data Table 1.

We find that the most likely progenitor of GW150914 consists of a primary star in the mass range 40M–100M and a secondary in the mass range 40M–80M. In our standard model, the binary formed in a low-metallicity environment (Z < 0.1Z; Z is the metallicity of the Sun; see Extended Data Fig. 1) and either in the early Universe (2 Gyr after the Big Bang) or very recently (11 Gyr after the Big Bang).

The distribution of birth times of these massive BH–BH mergers is bimodal (Fig. 2 and Extended Data Fig. 2), with a majority of systems originating from the distant past (55% of binaries; about 2 Gyr after the Big Bang, corresponding to z ≈ 3) and a smaller contribution from relatively young binaries (25%; formed about 11 Gyr after the Big Bang, corresponding to z ≈ 0.2). This bimodality arises from two naturally competing effects: on the one hand, most low-metallicity star formation occurs in the early Universe; on the other hand, in contrast to previous work3,4, significantly more low-metallicity star formation is currently expected to occur in the low-redshift Universe20. Therefore, as is the case with binary neutron stars, we anticipate a significant contribution to the present-day binary-black-hole merger rate from binary black holes formed in low-redshift, low-metallicity star-forming regions. The delay-time distribution of BH–BH binaries in our simulations follows a 1/t distribution. The birth times therefore naturally pile up at low redshifts (z ≈ 0.1–0.3) and this gives rise to a low-z peak (Extended Data Fig. 2a). However, the low-metallicity (Z < 0.1Z) star formation responsible for the production of massive BH–BH mergers peaks at a redshift of z ≈ 3 (Extended Data Fig. 2b). The convolution of these two effects produces the bimodal birth-time distribution (Extended Data Fig. 2c).

Figure 2: Birth times of GW150914-like progenitors across cosmic time.
Figure 2

dRdet/dt represents the contribution to the detection rate from binaries in a given 0.1-Gyr bin of birth time. Half of the binaries that form BH–BH mergers detectable in O1 with total redshifted mass in the range Mtot,z = 54M–73M were born within 4.7 Gyr of the Big Bang (corresponding to z > 1.2). The birth and merger times of the binary depicted in Fig. 1 are marked in blue; this binary follows the most typical evolutionary channel for massive BH–BH mergers (BHBH1 in Extended Data Table 1). The merger redshift of GW150914 is z = 0.088. The bimodal shape of the distribution originates from a combination of the BH–BH delay-time distribution and the low-metallicity star-formation history (see Extended Data Fig. 2 for details).

These massive GW150914-like mergers consist of black holes with comparable masses. The vast majority (99.8%) of mergers are found with mass ratios in the range q = 0.7–1.0 (Extended Data Fig. 3), with the mass ratio of GW150914 () falling near the centre of the expected region. The formation of low-mass-ratio objects is suppressed because low-mass-ratio progenitors tend to merge during the first mass-transfer event when the more massive component overfills its Roche lobe21. However, with decreasing total merger mass, the mass ratio extends to lower values. In particular, for the lower mass bin of Mtot,z = 25M–37M, mass ratios as low as q = 0.3 are also found.

We now use our full sample of double compact object mergers to make predictions for the merger-rate density, detection rates and merger mass distribution. The results are shown in Fig. 3 and Extended Data Table 1, in which we compare them to the measured values inferred from O1 LIGO observations. We find an overall detection rate that is consistent with the detection of one significant candidate (GW150914) during the principal 16-day double coincident period (when both LIGO gravitational-wave interferometers are operating simultaneously) for our standard model (M1), but that is inconsistent for our other two models (optimistic M2 and pessimistic M3; more detail below).

Figure 3: Comparison of merger rates and masses with O1 LIGO results.
Figure 3

Results are shown for standard (M1; red solid lines), optimistic common-envelope phase (M2; pink dash-dotted lines) and pessimistic large black-hole kicks (M3; green/black solid/dash-dotted line) models. a, Distribution of total redshifted binary mass. The merger-rate density of GW150914 (70.5M) is indicated by the blue square (with 90% confidence interval in mass, and its vertical position arbitrary). The blue solid line shows the fiducial estimate of the sensitivity (or upper limits) of the 16-day O1 run. A comparison of the shapes of the blue and red lines suggests that the most likely detections for M1 are BH–BH mergers with masses in the range 25M–73M. NS–NS mergers (first bin) and BH–NS mergers (next five bins) are well below the estimated sensitivity and thus detections in O1 are not expected. The rate densities are in the detector rest frame. b, Comparison of the LIGO estimate of the BH–BH merger rate with our models. The LIGO value of 2–400 Gpc−3 yr−1 (90% credible range) compares well with our standard (M1) and large black-hole natal kicks (M3) models. The rate densities are in the source rest frame. An updated version of Fig. 3, including additional gravitational-wave detections as they occur, can be found at http://www.syntheticuniverse.org/stvsgwo.html.

The BH–BH merger rates inferred from the 16 days of O1 LIGO observations are in the range 2–400 Gpc−3 yr−1 (ref. 22). For comparison, we estimate the rate density of binary black holes from our population synthesis dataset. We consider the full population of binary black holes within a redshift of z = 0.1 (that is, not weighted by their detection probability) and calculate their average source-frame merger-rate density. We find a value of 218 Gpc−3 yr−1 for our standard model (M1), which is in good agreement with the inferred LIGO rate22. By contrast, our optimistic model (M2) predicts too many mergers, with a rate density of 1,303 Gpc−3 yr−1, and our pessimistic model (M3) is at the very bottom end of the allowable range with a predicted rate of 6.6 Gpc−3 yr−1. In our models, the BH–BH merger-rate density increases with redshift (Extended Data Fig. 4). This increase is modest; our predicted source-frame BH–BH merger-rate density would double if the cut-off redshift was increased from z = 0.1 to z = 0.6.

The merger-rate density for the model with an optimistic common-envelope phase (M2) is an order of magnitude larger than the rate estimate from LIGO. This implies that unevolved massive stars (during main sequence and Hertzsprung gap) do not initiate/survive the common-envelope phase9,23. In our classical BH–BH formation scheme, only evolved stars (during core helium burning) with well-developed convective envelopes are allowed to initiate and survive the common-envelope phase.

Our predictions for the pessimistic model (M3) imply that large natal kicks (with average magnitudes of more than about 400 km s−1) are unlikely for massive black holes. This model predicts that an event such as GW150914 would happen only 1% of the time, with the detection of any BH–BH system happening less than 10% of the time (Table 1). In principle, this conclusion applies to the formation of only the first black hole in the binary, because large natal kicks lead to disruption of BH–BH progenitors while the binaries are wide. During the formation of the second black hole, the progenitor binaries are on very close orbits (Fig. 1) and are not disrupted by natal kicks. In Extended Data Fig. 4 we show a sequence of models with intermediate black-hole natal kicks; future observations may allow us to discriminate between these models and to constrain the natal-kick distribution. Future observations converging on M1 would indicate no natal kicks nor supernova explosions in massive black-hole formation18. A striking ramification of this is the prediction that hot and luminous Wolf–Rayet progenitors of massive black holes24 should disappear from the sky as a result of direct collapse to a black hole (that is, with no supernova explosion). Targeted observational campaigns to search for such phenomena are already underway25.

Table 1: Expected detection rate and number of detections

Figure 3 shows the relative contribution to the overall merger-rate density associated with each bin of total redshifted merger mass Mtot,z. For comparison, Fig. 3 also shows the fiducial sensitivity (see Methods) as a function of mass, assuming equal-mass zero-spin binary black holes. Figure 3 demonstrates that the intersection of the strongly mass-dependent sensitivity and the intrinsic detectable mass distribution strongly favours sources with total redshifted masses of 25M–73M, consistent with recent work4 and total redshifted mass of GW150914 (Mtot,z = 70.5M). In our simulations, the maximum intrinsic mass of a merging BH–BH binary is Mtot = 140M. When accounting for cosmological redshift (Mtot,z = (1 + z)Mtot) and taking into account the advanced O1 horizon redshift for this most massive binary (z = 0.7), the highest possible observed mass within O1 would be approximately 240M.

Spin magnitudes and directions of merging black holes are potentially measurable by LIGO1. The second-born black hole in a BH–BH binary does not accrete mass, and its spin at merger is unchanged from its spin at birth. The first-born black hole, on the other hand, has a chance to accrete material from the stellar wind of the unevolved companion or during common-envelope evolution. However, because this is limited either by the very low efficiency of accretion from stellar winds or by inefficient accretion during common-envelope evolution26,27, the total accreted mass onto the first-born black hole is expected to be rather small (about 1M–2M). This is insufficient to significantly increase the spin, and thus the spin magnitude of the first-born black hole at merger is within about 10% of its birth spin.

In our modelling, we assume that stars that are born in a binary have their spins aligned with the angular-momentum vector of the binary. If massive black holes do not receive natal kicks (for example, in our standard model M1), then our prediction is that black-hole spins are aligned during the final massive BH–BH merger. We note that our standard model includes natal kicks and mass loss for low-mass black holes (less than about 10M), and therefore BH–BH binaries with one or two low-mass black holes may show misalignment. Alternatively, binaries could be born with misalignment and retain it, misalignment could be caused by the third body or by interaction between the radiative envelope and the convective core28, or misalignment could result from a large natal kick on the second-born black hole. Several binaries are reported with misaligned spins29. Therefore, spin alignment of massive merging black holes suggests isolated field evolution, while misaligned spins do not elucidate formation processes.

As shown in Fig. 1, we find that the formation of massive BH–BH mergers is a natural consequence of isolated binary evolution. Our standard model (M1) of BH–BH mergers fully accounts for the observed merger-rate density and merger mass (Fig. 3), and for the mass ratio of two merging black holes (Extended Data Fig. 3) inferred from GW150914.

Our standard formation mechanism (M1) produces significantly more binary black holes than do alternative, dynamical channels associated with globular clusters. A recent study11 suggests globular clusters could produce a typical merger rate of 5 Gpc−3 yr−1; our standard model (M1) BH–BH merger-rate density is about 40 times larger: 218 Gpc−3 yr−1.

However, one non-classical isolated binary evolution channel involving rapidly rotating stars (homogeneous evolution) in very close binaries may also fully account for the formation of GW150914 (refs 12, 13, 14, 15). In particular, typical rates of 1.8 detections in 16 days of O1 observations are found13, which is comparable to our prediction of 2.8 (Table 1). Only very massive BH–BH mergers with total intrinsic masses of more than about 50M are formed in this model12,13, whereas our model predicts mergers with masses in a broader range, down to greater than about 10M. Future LIGO observations of BH–BH mergers may allow us to discriminate between these two very different mass distributions/models.

Methods

Online Methods

Our Monte Carlo evolutionary modelling is performed with the StarTrack binary population synthesis code16. In particular, we incorporate a calibrated treatment of tidal interactions in close binaries17, a physical measure of the common envelope (CE) binding energy30,31, and a rapid-explosion supernova model that reproduces the observed mass gap between neutron stars and black holes (BHs)18,32. Our updated mass spectrum of BHs shows a strong dependence on the metallicity of the progenitor stars (Extended Data Fig. 5). In galaxies with metallicities similar to the Milky Way (Z = Z = 0.02), BHs that formed out of single massive stars (initial mass MZAMS = 150M) reach a maximum mass of MBH = 15M, whereas, for very low metallicity (Z = 0.0001 = 0.005Z), the maximum mass becomes MBH = 94M. The above input physics represents our standard model (M1), which is representative of our classical formation scheme for double compact objects (BH–BH, BH–NS and NS–NS).

We have adopted specific values for a number of evolutionary parameters. Single stars are evolved with calibrated formulae based on detailed evolutionary calculations33. Massive star winds are adopted from detailed studies of radiation-driven mass loss34. For the Luminous Blue Variable phase, a high rate of mass loss is adopted (1.5 × 10−4M yr−1). Binary interactions and, in particular, the stability of Roche-lobe overflow (RLOF) is judged on the basis of binary parameters: mass ratio, evolutionary stage of donor, response to mass loss, and behaviour of the orbital separation in response to mass transfer. The orbital separation is additionally affected by gravitational radiation, magnetic braking, and loss of angular momentum associated with systemic mass loss. During stable RLOF, we assume that half of the mass is accreted onto the companion, while the other half (1 − fa = 0.5) is lost with specific angular momentum (dJ/dt = jloss[Jorb/(Mdon + Macc)](1 − fa)dMRLOF/dt with scaling factor jloss = 1.0 where fa is the fraction of the mass accreted, Jorb is the orbital angular momentum, Mdon is the donor mass, Macc is the accretor mass, and dMRLOF/dt is the mass transfer rate; ref. 35). The CE is treated by considering the energy balance with fully effective conversion of orbital energy into envelope ejection (conversion efficiency α = 1.0), whereas the envelope binding energy for massive stars is calibrated by a parameter λ, which depends on star radius, mass and metallicity. For massive stars, λ ≈ 0.1 is adopted31. During CE evolution, compact objects accrete at 10% of the Bondi–Hoyle rate as estimated by recent hydrodynamical simulations26,27. Our CE evolution is instantaneous, so the time at the beginning and end of the CE phase is exactly the same (see Fig. 1); the time duration of the CE phase has no impact on our results.

We consider two extra variations of the input physics of binary evolution. In one model (M2), we test highly uncertain CE physics36 and we allow for Hertzsprung-gap stars to initiate and survive CE evolution. This is an optimistic assumption, because these stars may not allow for CE evolution23, nor survive as a binary if a CE forms9. For comparison, in our standard model, we allow only evolved stars with a deep convective envelope (core-helium-burning stars) to survive a CE phase.

In the opposite extreme, we use a model (M3) in which BHs receive large natal kicks. In particular, each BH gets a natal kick with its components drawn from a Maxwellian distribution with a one-dimensional root-mean-square σ = 265 km s−1, independent of BH mass. Such large natal kicks are measured for Galactic pulsars37. This is a pessimistic assumption, because large natal kicks tend to disrupt BH–BH progenitor binaries. This assumption is not yet excluded on the basis of electromagnetic observations4. By contrast, in our standard model, BH natal kicks decrease with BH mass. In particular, for massive BHs that form through direct collapse of an entire star to a BH with no supernova explosion (MBH 10M for Z = Z, MBH 15M for Z = 0.1Z and MBH 15M–30M for Z = 0.01Z), we assume no natal kicks18. We also calculated a series of models with intermediate BH kicks (see Extended Data Fig. 4): σ = 200 km s−1 (model M4), σ = 130 km s−1 (model M5) and σ = 70 km s−1 (model M6).

For each evolutionary model we compute 2 × 107 massive binaries for each point on a grid of 32 sub-models covering a wide range of metallicities: Z = 0.0001, 0.0002, 0.0003, 0.0004, 0.0005, 0.0006, 0.0007, 0.0008, 0.0009, 0.001, 0.0015, 0.002, 0.0025, 0.003, 0.0035, 0.004, 0.0045, 0.005, 0.0055, 0.006, 0.0065, 0.007, 0.0075, 0.008, 0.0085, 0.009, 0.0095, 0.01, 0.015, 0.02, 0.025 and 0.03. We assume that stellar evolution at even lower metallicities proceeds in the same way as the evolution at Z = 0.005Z. However, stars with very low metal content (for example, Population III) may evolve differently to metal-rich stars38.

Each sub-model is computed with initial distributions of orbital periods P (proportional to [log(P)]−0.5), eccentricities e (proportional to e−0.42) and mass ratios q (proportional to q0) appropriate for massive stars39. We adopt an initial mass function that is close to flat for low-mass stars (proportional to M−1.3 for 0.08M ≤ M < 0.5M and to M−2.2 for 0.5M ≤ M < 1.0M) and that is top-heavy for massive stars (proportional to M−2.3 for 1.0M ≤ M ≤ 150M), as guided by recent observations40. The adopted initial mass function generates higher BH–BH merger-rate densities as compared with the steeper initial mass function (proportional to M−2.7 for 1.0M ≤ M ≤ 150M) adopted in previous studies4,41, because there are more BH–BH merger progenitors in our simulations42.

A moderate binary fraction (fbi = 0.5) is adopted for stars with masses MZAMS < 10M, whereas we assume that all more massive stars are formed in binaries (fbi = 1.0), as indicated by recent empirical estimates39,43.

We adopt an extinction-corrected cosmic star-formation rate (SFR) based on numerous multi-wavelength observations44:This SFR declines rapidly at high redshifts (z > 2). This may be contrasted with some SFR models used previously45, which generated a greater number of stars at high redshifts. This revision will thus reduce the BH–BH merger-rate densities at all redshifts. Even though the formation of BH–BH binaries takes a very short time (about 5 Myr), the time to coalescence of two BHs may be very large (Fig. 1 and Extended Data Fig. 2).

In our treatment of chemical enrichment of the Universe, we follow the mean metallicity increase with cosmic time (since Big Bang until present). The mean metallicity as a function of redshift is:with a return fraction R = 0.27 (mass fraction of each generation of stars that is put back into the interstellar medium), a net metal yield y = 0.019 (mass of new metal created and ejected into the interstellar medium by each generation of stars per unit mass locked in stars), a baryon density with Ωb = 0.045 and h0 = 0.7, a SFR given by Equation (1), and E(z) = with ΩΛ = 0.7, ΩM = 0.3, Ωk = 0 and H0 = 70.0 km s−1 Mpc−1. The shape of the mean-metallicity dependence on redshift follows recent estimates44, although the level was increased by 0.5 dex to better fit observational data46. At each redshift, we assume a log-normal distribution of metallicity around the mean, with a standard deviation of σ = 0.5 dex (ref. 47). Our prescription (Extended Data Fig. 6) produces more low-metallicity stars than previously41. Because BH–BH formation is enhanced at low-metallicity2, our new approach increases the predicted rate densities of BH–BH mergers.

Here we discuss caveats of evolutionary calculations. First, we consider only isolated binary evolution, and thus our approach is applicable to field stars in low-density environments. It is possible that dynamical interactions enhance BH–BH merger formation in dense globular clusters11, offering a completely independent channel.

Second, our predictions are based on a ‘classical’ theory of stellar and binary evolution for the modelling of massive stars that we have compiled, developed and calibrated over the last 15 years. We do not consider exotic channels for the formation of BH–BH mergers, such as the one from rapidly rotating stars in contact binaries48.

Third, our modelling includes only three evolutionary models: a standard model consisting of our best estimates for reasonable parameters (M1), as well as optimistic (M2) and pessimistic (M3) alternative models. The optimistic model consists of only one change from the standard model: we allow all stars beyond the main sequence to survive the CE phase. Alternatively, the pessimistic model also consists of only one change: larger BH natal kicks. We have not investigated other possible deviations from the standard model (for example, different assumptions of mass and angular-momentum loss during stable mass-transfer evolution) nor have we checked inter-parameter degeneracies (for example, models with large BH kicks and an optimistic CE phase). Precursor versions of these computationally demanding studies have already been performed49, albeit with low statistics and limited scope; these calculations indicate that our three models probably cover the range of interesting effects.

Fourth, our observations are severely statistically limited. We are attempting to draw inferences about our models on the basis of a single detection (GW150914).

In was argued50 that the formation of GW150914 in isolated binary evolution requires a metallicity lower than 0.5Z. This argument was based on single stellar models51; stars in close binaries are subject to significant mass loss during RLOF/CE, and they form BHs with lower mass than BHs formed by single stars. Thus, in binaries, the metallicity threshold for massive BH formation is lower than in single stellar evolution. For example, formation of a single 30M BH requires Z < 0.25Z (Extended Data Fig. 5, whereas formation of two such BHs in a binary requires Z < 0.10Z (Extended Data Fig. 1). The value of this threshold depends on assumptions for the model of stellar evolution, winds and BH formation processes. The physical models we have adopted yield a threshold of Z < 0.10Z, the same as that obtained with MESA (http://mesa.sourceforge.net/) for homogeneous stellar evolution12. Our model was calibrated using known masses of BHs and, in particular, we do not exceed 15M for Z (the highest-mass stellar BH known in our Galaxy). By contrast, single stellar models used to derive the high metallicity threshold produce 25M for Z (ref. 51). The highest threshold obtained with binary evolution was reported at the level of 0.5Z (ref. 14). Such a high value of the metallicity threshold for the progenitor of GW150914 implies that stars at approximately solar metallicity (Z = 0.014) produce BHs as massive as 40M (ref. 14). This is neither supported nor excluded by available electromagnetic BH mass measurements (https://stellarcollapse.org/bhmasses).

In the following, we present calculation of the gravitational radiation signal. The output of StarTrack is a binary merger at a given time. We then calculate the gravitational waveform associated with this merger, and determine whether this binary would have been observable by LIGO in the O1 configuration3,4.

We model the full inspiral–merger–ringdown waveform of the binaries using the IMRPhenomD gravitational waveform template family52,53. This is a simple and fast waveform family that neglects the effects of spin (which are not relevant for GW150914). We consider a detection to be given by a threshold of SNR> 8 in a single detector, and we use the fiducial O1 noise curve (https://dcc.ligo.org/LIGO-G1501223/public). We calculate the face-on, overhead SNR for each binary directly from equation (2) of ref. 3. We then calculate the luminosity distance at which this binary would be detected with SNR = 8. As the distance to the binary changes, the observer-frame (redshifted) mass also changes, and therefore calculating the horizon redshift requires an iterative process. Once this has been calculated, we then determine the predicted detection rates using equation (9) of ref. 3; the effects of the antenna power pattern are incorporated in the pdet term in this equation.

An estimate of fiducial advanced LIGO sensitivity during the 16-day GW150914 analysis is shown in Fig. 3. We estimate the sensitivity to coalescing compact binaries using a reference O1 noise curve. We assume that both detectors operate with the fiducial O1 noise curve, which is the same sensitivity we adopted to calculate compact binary detection rates. For comparison, this model agrees reasonably well with the ‘early-high’ sensitivity model54. Our expression is a 50th percentile upper limit, assuming no detections. The critical application of this expression is not related to its overall normalization; we are instead interested in its shape, which characterizes the strongly mass-dependent selection biases of LIGO searches.

Using these inputs, our fiducial estimate of the advanced LIGO sensitivity during the first 16 days of O1 for a specific mass bin ΔMi is:where T = 16 days corresponds to the analysis of GW1509141, the volume:is the sensitive volume averaged over mass bin ΔMi, and pdet(w, M) is the orientation-averaged detection probability3,4. The function pdet(w, M) depends on the coalescing binary redshifted mass M through the maximum luminosity distance (‘horizon distance’) at which a source could produce a response of SNR >8 in a single detector through a projection parameter w, which is maximum (w = 1) for a face-on, overhead source, and minimum (w = 0) for sky locations and orientations where the LIGO detector has no response to the source. To calculate this distance, we adopt the IMRPhenomD gravitational waveforms52,53 that we also used to estimate compact binary detection rates. Extended Data Fig. 7 shows our estimated horizon redshift as a function of the total redshifted binary merger mass for equal-mass mergers.

Code availability

We have opted not to release the population synthesis code StarTrack used to generate binary populations for this study.

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Acknowledgements

We are indebted to G. Wiktorowicz, W. Gladysz and K. Piszczek for their help with population synthesis calculations, and to H.-Y. Chen and Z. Doctor for their help with our LIGO/Virgo rate calculations. We thank the thousands of Universe@home users that have provided their personal computers for our simulations. We also thank the Hannover GW group for letting us use their ATLAS supercomputer. K.B. acknowledges support from the NCN grant Sonata Bis 2 (DEC-2012/07/E/ST9/01360). D.E.H. was supported by NSF CAREER grant PHY-1151836. D.E.H. also acknowledges support from the Kavli Institute for Cosmological Physics at the University of Chicago through NSF grant PHY-1125897 as well as an endowment from the Kavli Foundation. T.B. acknowledges support from the NCN grant Harmonia 6 (UMO-2014/14/M/ST9/00707). R.O’S. was supported by NSF grant PHY-1505629.

Author information

Affiliations

  1. Astronomical Observatory, Warsaw University, Ujazdowskie 4, 00-478 Warsaw, Poland

    • Krzysztof Belczynski
    •  & Tomasz Bulik
  2. Enrico Fermi Institute, Department of Physics, Department of Astronomy and Astrophysics, and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA

    • Daniel E. Holz
  3. Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, New York 14623, USA

    • Richard O’Shaughnessy

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Contributions

All authors contributed to the analysis and writing of the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Krzysztof Belczynski.

Reviewer Information Nature thanks M. Cantiello and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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