Distinguishing Between Formation Channels for Binary Black Holes with LISA
Katelyn Breivik, Carl L. Rodriguez, Shane L. Larson, Vassiliki Kalogera, Frederic A. Rasio
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DISTINGUISHING BETWEEN FORMATION CHANNELS FOR BINARY BLACK HOLES WITH LISA
Katelyn Breivik , Carl L. Rodriguez , Shane L. Larson , Vassiliki Kalogera , Frederic A. Rasio Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Dept. of Physics and Astronomy, NorthwesternUniversity 2145 Sheridan Rd, Evanston, IL 60208, USA The Adler Planetarium, 1300 S Lakeshore Dr, Chicago, IL 60605
ABSTRACTThe recent detections of GW150914 and GW151226 imply an abundance of stellar-mass binary-black-hole mergers in the local universe. While ground-based gravitational-wave detectors are limited toobserving the final moments before a binary merges, space-based detectors, such as the Laser Interfer-ometer Space Antenna (LISA), can observe binaries at lower orbital frequencies where such systemsmay still encode information about their formation histories. In particular, the orbital eccentricity andmass of binary black holes in the LISA frequency band can be used together to discriminate betweenbinaries formed in isolation in galactic fields and those formed in dense stellar environments such asglobular clusters. In this letter, we explore the orbital eccentricity and mass of binary-black-hole pop-ulations as they evolve through the LISA frequency band. Overall we find that there are two distinctpopulations discernible by LISA. We show that up to ∼
90% of binaries formed either dynamically orin isolation have eccentricities measurable by LISA. Finally, we note how measured eccentricities oflow-mass binary black holes evolved in isolation could provide detailed constraints on the physics ofblack-hole natal kicks and common-envelope evolution.
Keywords: gravitational waves – globular clusters: general – stars: black holes INTRODUCTIONThe era of gravitational wave (GW) astrophysics be-gan with the discovery of the binary black hole (BBH)merger, GW150914, by Advanced LIGO (aLIGO) (Ab-bott et al. 2016b). The subsequent detection of BBHmerger, GW151226, with smaller progenitor masses sug-gests diversity in the potential formation channels of theBBHs (Abbott et al. 2016c). Parameter estimation anal-yses of GW150914 were done assuming a circular orbit,but eccentricities of e (cid:46) . e (cid:38) .
001 (Seto 2016; Nishizawa et al.2016). Thus, LISA may be able to discriminate betweenBBH formation channels using eccentricity.In this letter we show that BBHs formed in galacticfields and BBHs formed in GCs have different combi-nations of mass and eccentricity in the LISA frequencyband, thus providing a method to distinguish betweenthe two formation channels. We also show two distinctpopulations within the BBHs evolved in isolation that a r X i v : . [ a s t r o - ph . GA ] S e p Breivik, Rodriguez, Larson, Kalogera, Rasio are separated by higher and lower eccentricities that willbe potentially distinguishable by LISA. In § § § GALACTIC FIELD AND GLOBULAR CLUSTERMODELSWe simulated two populations of BBHs, one evolved inisolation in galactic fields and one formed dynamically inGCs, to explore the evolution of each population fromformation frequencies of f ∼ − − − Hz all theway to the upper end of the aLIGO band at f ∼ Hz. The same binary evolution physics was used to cre-ate both population simulations with stellar dynamicseffects turned on and off.We used a modified version of the binary stellar evo-lution code
BSE (Hurley et al. 2002) with updated stel-lar evolution prescriptions including BH formation andnatal kicks (Belczynski et al. 2002; Fryer & Kalogera2001), updated metallicity-dependent stellar-wind pre-scriptions (Belczynski et al. 2010; Vink et al. 2001; Vink& de Koter 2005), and additional mechanisms to ac-count for fallback in neutrino-driven supernovae(Fryeret al. 2012). For this study, we adopt the fiducial mod-els of Rodriguez et al. (2016) which are based on themost recent stellar evolution prescriptions for galacticfields (Dominik et al. 2013).Our population of dynamically-formed binaries istaken from a collection of 48 GC models developed inRodriguez et al. (2016) with the Cluster Monte Carlo(CMC) code (Pattabiraman et al. 2013, and referencestherein). CMC employs a statistical approach to stel-lar dynamics, first developed by H´enon (1975), whichenables the modeling of star clusters with significantlygreater number of particles than a direct N -body simula-tion, while still employing the necessary physics—singleand binary stellar evolution with BSE, 3- and 4-bodystrong gravitational encounters (Fregeau & Rasio 2007),and dynamical three-body binary formation (Morscheret al. 2013)—to fully characterize the dynamical BBHmerger problem. We neglect long-term secular effects(e.g., Antonini et al. 2014, 2016) and relativistic dynam-ical scattering (e.g., Samsing et al. 2014), which are ex-pected to contribute to the overall BBH population atthe ∼
1% or lower level, according to these studies. For an in depth discussion of a LISA-like detector’s capa-bilities of distinguishing between BBH formation channels usingeccentricity, see Nishizawa et al. (2016a).
These 48 GC models span a range of initial particlenumbers ( N = 2 × , × , × , and 2 × ),initial virial radii ( R v = 1 , Z = 0 . Z (cid:12) , . Z (cid:12) , and 0 . (cid:12) ), with twostatistically-independent models generated for each setof initial conditions. As was done in Rodriguez et al.(2016), we assume that the GC population of the localuniverse is comprised of ∼
44% high-metallicity GCs(0 . Z (cid:12) ), and ∼
56% low-metallicity GCs (0 . Z (cid:12) and0 . Z (cid:12) ). We further assign to each GC a tidal radiusbased on its galactocentric distance, which we assumeto be correlated to its stellar metallicity based on ob-servations of the Milky Way and other galaxies (Harris2010). We finally assume a log-normal GC mass func-tion, based on recent observations of the GC luminosityfunction in brightest-cluster galaxies (Harris et al. 2014)and a mass-to-light ratio of 2 for old stellar systems (e.g.,Bell et al. 2003).We draw a sample of BBHs ejected from our 48 GCmodels by randomly selecting binaries from each model.The number of binaries selected from a given model isdetermined by its weight, which we assign by divid-ing the GC mass function into bins, the midpoints ofwhich are set by the average mass of GC models withthe same initial particle number. The integral of theGC mass function over that bin then determines theweight assigned to that model. In other words, GCmodels with larger masses (3 × M (cid:12) to 6 × M (cid:12) ,corresponding to models with initial particle numbers of1 × and 2 × ) contribute more binaries than clus-ters with smaller masses (see Rodriguez et al. 2016, fordetails). Once we select this population of binaries fromall clusters, we generate a five-dimensional Gaussian ker-nel density estimate (KDE) from the formation masses,separation, eccentricity and ejection times of the BBHpopulation. We sample 1 ,
000 BBHs from the KDE thatevolve to the LISA band at f = 10 − Hz in the last Gyrbefore the present time. We assume that all GCs wereformed 12 ± initial binaries sampled from standard probabil-ity distribution functions to assign each binary with aninitial metallicity ( Z ), primary mass ( m ), mass ratio( q ), orbital separation ( a ), and eccentricity ( e ). Thepopulation models compare four different metallicities( Z = 1 Z (cid:12) , . Z (cid:12) , . Z (cid:12) , and 0 . Z (cid:12) ), with thesub-solar metallicities being consistent with the metal-licities used in the GC models. For the initial primary istinguishing Between BBH Formation Channels with LISA ξ ( m ) ∝ m − . , m ≥ M (cid:12) (Kroupa 2001) with a primary mass limit of 150 M (cid:12) . Weassume a uniform initial mass ratio distribution consis-tent with current observational constraints (Mazeh et al.1992; Goldberg & Mazeh 1994; Kobulnicky et al. 2014).We assume initial orbital separations are distributeduniformly in log( a ) at wide separations (10 R (cid:12) ≤ a ≤ . × R (cid:12) ) and fall off linearly at small separationsas ζ ( a ) ∝ ( a/a ) . , a < R (cid:12) (Han 1998). The initialeccentricities are distributed thermally (Heggie 1975) as η ( e ) = 2 e .We evolve the galactic field population for 13.87 Gyrusing the same binary evolution models as the GC popu-lation, creating an equivalent population to the GC pop-ulation but without dynamics. We log the birth param-eters of each BBH, including important formation pro-cesses like the number of common-envelope episodes andthe natal kicks imparted to the binary from the birth ofeach black hole. As with the dynamically-formed BBHs,we require the low metallicity ( Z < Z (cid:12) ) BBHs to enterthe LISA band in the last Gyr before the present time.We retain any solar metallicity BBHs that evolve to theLISA band over the last 10 Gyr.Due to the eccentric nature of BBHs formed in GCs,it is useful to consider the frequency of GWs emitted athigher harmonics of a binary’s orbital frequency. Thefrequency of maximum GW power emission from an ec-centric binary is estimated as (Wen 2003) f GW = (cid:112) G ( m + m ) π (1 + e ) . [ a (1 − e )] . . (1)If we consider the peak GW frequency of Eq. 1 forBBHs formed in GCs, we find the GW frequencies ofthe population are substantially higher than the circularGW frequencies. Since the peak sensitivity of LISA fallsnear f ∼ − − − Hz, shifts to higher frequency aidin the detectability of these sources. ECCENTRICITIES ACROSS THE LISA BAND3.1.
BBH orbital evolution
Since the dynamically-formed BBHs are ejected fromtheir host GC and the galactic field BBHs evolve in iso-lation, only GW emission will affect the evolution ofeach binary. Using the simulated populations from § Hz and each dynamically-formed BBHfrom the time of ejection from its host cluster to 10 Hzusing the quadrupole approximated GW orbital evolu-tion equations (Peters 1964).BBHs with frequencies larger than 10 − Hz are ex-pected to have measurable frequency evolution due toGW emission known as the GW chirp. The chirp mass, M c = ( m m ) / / ( m + m ) / , and eccentricity (or lack therof) can be measured for every BBH with a mea-sured chirp. Fig. 1 illustrates the evolution of eccentric-ity and GW frequency for each binary in the modelsfrom each formation channel. The minimal eccentric-ities measurable by a LISA-like detector are shown inred (Nishizawa et al. 2016). The LISA frequency rangeis highlighted in grey and the frequency range whereBBHs with chirp mass M c (cid:38) M (cid:12) have measurablechirps is highlighted in light blue.In addition to dynamically-formed BBHs, we considertwo populations of BBHs formed in isolation: those in-cluding either a single or no common-envelope episodes(hereafter 1CE and 0CE). BBHs formed both dynami-cally (black) and in isolation (green, blue) have measur-able eccentricities and frequencies above 10 − Hz, withdynamically-formed binaries having larger eccentricitiesthan those in galactic fields. Above 10 − Hz, only BBHsformed in GCs fall above the e = 0 .
01 line. This suggeststhat any BBH detected above frequency − Hz with ec-centricity e ≥ . formed dynamically in a dense stellarenvironment .3.2. Eccentricity Distributions dynamically-formed BBHs are ejected from the clusterwith a thermal eccentricity distribution. Once ejected,BBHs evolve only through the emission of GWs whichcircularize and shrink the binary orbit. This leads toBBHs with eccentricities of e (cid:38) . f GW ∼ − Hz) and eccentrici-ties of e (cid:38) .
001 in the high end of the LISA frequencyband ( f GW ∼ − Hz). BBHs formed in isolation gen-erally form at lower eccentricities, but with also lowerchirp masses than BBHs formed in GCs.Fig. 2 shows the cumulative distribution of the ec-centricities of the simulated BBHs formed dynamically(black) and in isolation (blue, green) at different pointsin their orbital evolution. Highly eccentric dynamically-formed binaries are recently ejected from the GC, whilethe less eccentric binaries have had more time to cir-cularize through GW emission. At 10 − Hz, 92% ofdynamically-formed BBHs have e ≥ . e ≥ .
01. At 10 − Hz, the eccentricity ofthe dynamically-formed BBHs has decreased such that30% have e ≥ .
001 and 7% have e ≥ .
01. For the1CE BBHs, 91% have e ≥ .
001 and 23% have e ≥ . − Hz. Again, at 10 − Hz the eccentricity of the1CE BBHs has decreased, with 19% having e ≥ . e ≥ .
01. There are no 0CE BBHs withmeasurable eccentricities at GW frequencies above 10 − Hz. 3.3.
Chirp Mass and Eccentricity Correlations
A particularly useful way to separate BBHs formeddynamically or in isolation is through the correlations
Breivik, Rodriguez, Larson, Kalogera, Rasio -7 -6 -5 -4 -3 -2 -1 GW Frequency (Hz) -6 -5 -4 -3 -2 -1 E cc e n t r i c i t y globular clustergalactic field: 1CEgalactic field: 0CE Figure 1 . Eccentricity evolution tracks as a function of GW frequency for BBHs formed both dynamically in dense stellarenvironments and in isolation in galactic fields. Black lines denote BBHs ejected from GCs and green and blue lines denote1CE and 0CE BBHs evolved in galactic fields. The lower horizontal red line denotes the measurable eccentricity ( e ≤ . T obs = 5 yrs (2 yrs). The upper horizontal red line shows the eccentricity ( e ≤ .
01) thatwill always be measurable for any observed BBH. The grey band highlights the LISA frequency range and and the blue bandhighlights the frequency range where BBHs with chirp mass M c (cid:38) M (cid:12) are expected to have measurable frequency evolution. -5 -4 -3 -2 -1 Eccentricity: F r a c t i o n o f Figure 2 . Cumulative fraction of dynamically-formed(black), 1CE (blue), and 0CE (green) BBHs with an ec-centricity greater than a minimum eccentricity for variouspoints in the orbital evolution of the binaries. The left redline denotes the measurable eccentricity ( e ≤ . T obs = 5 yrs (2 yrs). The rightred line shows the eccentricity ( e ≤ .
01) that will always bemeasurable for any observed chirping BBH. between the chirp mass and eccentricity of each popu-lation. BBHs with chirp masses M c < M (cid:12) exclu-sively form in isolation in galactic fields, though we notethat young, high-metallicity clusters (not included in ourmodels) are capable of producing BBHs with lower chirpmasses (Chatterjee et al. 2016, in prep).If only the chirp mass of a BBH with M c > M (cid:12) isobserved, it is impossible to discern which population itoriginated in since the chirp masses of the GC, 1CE, and0CE populations overlap in this region. However, if theeccentricity is also measured, the three populations canbe resolved. Fig. 3 shows the eccentricity vs chirp massplots of each population at f GW = 10 − Hz. The shapeof the distributions stays constant but the eccentric-ity decreases as the BBHs evolve to higher frequenciesthrough GW emission. For each population, the estima-tion error on the chirp mass is ∆ M c / M c (cid:39) f / ˙ f ) / f = 0 . SN R/ − T − obs (Takahashi & Seto2002). In all cases, the chirp mass estimation error issmaller than the width of the data points.The chirp masses of the GW150914 and GW151226progenitors are plotted in Fig. 3. The GW150914 and istinguishing Between BBH Formation Channels with LISA Log ( ) -6-5-4-3-2-10 L o g E cc e n t r i c i t y GC1CE0CE02468 P r o b a b i l i t y (a) (b) (c) Probability
Figure 3 . Scatter plot and histograms of the chirp mass andeccentricity of the BBHs formed in isolation (green, blue) anddynamically in dense stellar environments (black) at GW fre-quency f GW = 10 − Hz. The red lines denote the chirp massof the (a) GW151226, (b) LVT151012, and (c) GW150914progenitors with 90% confidence limits. The histograms arenormalized separately for each population and do not reflectthe relative number of sources.
LVT151012 progenitors are consistent with both dynam-ical and isolated formation channels, and the GW151226progenitor is consistent with the 1CE isolated formationchannels.
If the GW150914, LVT151012, or GW151226progenitors had been observed by LISA, an eccentricitymeasurement (or lack thereof ) could have aided in iden-tifying their formation histories .3.4.
Multiple Field Populations
We find that the population isolated binaries thatevolve into the LISA frequency band is split into higherand lower eccentricity populations. The higher eccen-tricity population is comprised entirely of BBHs whichhave undergone a single common-envelope episode whilethe lower eccentricity population is comprised of a com-bination of 0CE and 1CE BBHs. The higher-eccentricity1CE population, while still distinct from the GC BBHs,has potentially measurable eccentricities.The natal kick imparted to a BH at birth is connectedto kicks thought to be imparted to neutron stars (NSs).NS natal kicks are assumed to follow a Maxwellian dis-tribution with a dispersion of 265 km/s (Hobbs et al.2005). The BH natal kick is modified to be depen-dent on the mass of the pre-collapse stellar core throughfallback processes, with higher kicks imparted to lower-mass remnants(Fryer et al. 2012). A binary can be dis- rupted if the overall energy imparted through the natalkicks is higher than the binding energy of the binaryorbit.In the case of the higher-eccentricity population, the1CE binaries are driven to small separations through thecommon-envelope mechanism. Since the masses of the1CE BBH components are generally low due to low fall-back, the natal kick speeds are high enough to producehigher eccentricities, but not so high as to disrupt theBBHs. This suggests that the eccentricity of the 1CEpopulation is dependent on the natal kick physics of theformation of the second BH.We simulated three additional populations with vary-ing natal kick prescriptions to explore the effect of natalkicks on the eccentricity of BBHs formed in isolation.The first prescription sets the BH natal kick equal tozero, thus any orbital changes imparted to the binaryare due to momentum conservation from instantaneousmass loss in the BH formation. We also include twovariants of the BH natal kick prescription: one usingstandard NS kicks for BHs and one that modifies theNS kick by the mass fraction ( M NS /M BH ), where M BH is the mass of the newly formed BH and we assume M NS = 1 . M (cid:12) .We compare our results for four BH natal kick pre-scriptions: no natal kicks, fallback modulated kicks,fractional NS kicks weighted by mass, and full NS kicks.We did not generate new GC simulations for each kickprescription since BBHs formed dynamically in densestellar environments form with thermally distributed ec-centricities, losing the memory of kick effects on the ec-centricity distribution.Fig. 4 plots the eccentricity vs chirp mass for the 0CEand 1CE formed in isolation for the four natal kick pre-scriptions mentioned above at their birth orbital fre-quency and at 10 − Hz. Again, in all cases, the chirpmass estimation error bars are smaller than the widthof the data points. The lower mass BBHs in all caseshave preferentially higher kicks and thus higher birth ec-centricities. The 0CE BBHs generally have larger birthseparations and masses which results in low eccentrici-ties by the time the BBH evolves through GW emissionto the LISA frequency band.As expected, BBHs formed with full NS natal kicks re-tain the largest eccentricities through their GW drivenorbital evolution, followed by fractional NS kicks, thenfallback modulated kicks and no natal kicks. More mas-sive BBHs evolve to lower eccentricities at a given or-bital frequency than the lower mass systems in the samepopulation because of the mass dependence of orbitalfrequency. This is responsible for diminishing the higheccentricities found at birth in the 0CE population forboth the full and fractional NS kicks once they reach theLISA band.
Breivik, Rodriguez, Larson, Kalogera, Rasio -2 -1 (a) (b) (c) (d) -5 -4 -3 -2 -1 (a) E cc e n t r i c i t y (b) (c) (d)Chirp Mass (M ) Figure 4 . Eccentricity vs chirp mass at birth (top row) and 10 − Hz (bottom row) of 0CE (green) and 1CE (blue) BBHs formedin isolation in galactic fields. The panels are (a) no natal kicks, (b) fallback modulated kicks, (c) fractional NS kicks weightedby mass, and (d) full NS kicks. The magnitude of the natal kick increases from right to left.4.
DISCUSSIONWe have shown that BBHs formed both dynamicallyand in isolation may have measurable eccentricity inthe LISA band. If BBHs are detected by LISA witheccentricities of e (cid:38) .
01 at frequencies above 10 − Hz, they will have almost certainly originated from dy-namical processes in old, dense stellar environments. IfBBHs with eccentricities of e (cid:38) .
01 and chirp masses of M c (cid:46) M (cid:12) are detected by LISA at low frequencies,they likely originated from a common-envelope forma-tion scenario.In the future, we plan to extend this study by imple-menting a full treatment of the formation-redshift dis-tribution of the BBHs observable by LISA originating from both dynamical processes in dense stellar environ-ments and isolated binary evolution in galactic fields.We also plan to properly account for the detectability ofeach BBH using eccentricity dependent signal-to-noiseratios (Breivik et al. 2016a, in prep).KB and SLL acknowledge support from NASA GrantNNX13AM10G. CR and FAR acknowledge supportfrom NSF Grant AST-1312945 and from NASA GrantNNX14AP92G. VK acknowledges support from NSFGrant PHY-1307020/002 and from Northwestern Uni-versity. VK and FAR also acknowledge support fromNSF Grant PHY-1066293 at the Aspen Center forPhysics.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, ApJL,818, L22Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, PhysicalReview Letters, 116, 061102Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, PhysicalReview Letters, 116, 241103Antonini, F., Murray, N., & Mikkola, S. 2014, ApJ, 781, 45Antonini, F., Chatterjee, S., Rodriguez, C. L., et al. 2016, ApJ,816, 65Belczynski, K., Kalogera, V., & Bulik, T. 2002, ApJ, 572, 407Belczynski, K., Benacquista, M., & Bulik, T. 2010, ApJ, 725, 816Belczynski, K., Holz, D. E., Bulik, T., & O’Shaughnessy, R.2016, arXiv:1602.04531Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003,ApJS, 149, 289 Bird, S., Cholis, I., Mu˜noz, J. B., et al. 2016, Physical ReviewLetters, 116, 201301Breivik et al. 2016, In prep.Chatterjee et al. 2016, In prep.Correnti, M., Gennaro, M., Kalirai, J. S., Brown, T. M., &Calamida, A. 2016, ApJ, 823, 18Dominik, M., Belczynski, K., Fryer, C., et al. 2013, ApJ, 779, 72Fregeau, J. M., & Rasio, F. A. 2007, ApJ, 658, 1047Fryer, C. L., & Kalogera, V. 2001, ApJ, 554, 548Fryer, C. L., Belczynski, K., Wiktorowicz, G., et al. 2012, ApJ,749, 91Goldberg, D., & Mazeh, T. 1994, A&A, 282, 801Han, Z. 1998, MNRAS, 296, 1019Harris, W. E. 2010, Philosophical Transactions of the RoyalSociety of London Series A, 368, 889 istinguishing Between BBH Formation Channels with LISA7