LISA and the Existence of a Fast-Merging Double Neutron Star Formation Channel
Jeff J. Andrews, Katelyn Breivik, Chris Pankow, Daniel J. D'Orazio, Mohammadtaher Safarzadeh
DDraft version October 30, 2019
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LISA and the Existence of a Fast-Merging Double Neutron Star Formation Channel
Jeff J. Andrews, Katelyn Breivik, Chris Pankow, Daniel J. D’Orazio, and Mohammadtaher Safarzadeh Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy,Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064
Submitted to ApJLABSTRACTUsing a Milky Way double neutron star (DNS) merger rate of 210 Myr − , as derived by the LaserInterferometer Gravitational-Wave Observatory (LIGO), we demonstrate that the Laser InterferometerSpace Antenna (LISA) will detect on average 240 (330) DNSs within the Milky Way for a 4-year (8-year) mission with a signal-to-noise ratio greater than 7. Even adopting a more pessimistic rate of 42Myr − , as derived by the population of Galactic DNSs, we find a significant detection of 46 (65) MilkyWay DNSs. These DNSs can be leveraged to constrain formation scenarios. In particular, traditionalNS-discovery methods using radio telescopes are unable to detect DNSs with P orb (cid:46) (cid:46)
10 Myr). If a fast-merging channel exists that forms DNSs at these short orbital periods, LISAaffords, perhaps, the only opportunity to observationally characterize these systems; we show that toymodels for possible formation scenarios leave unique imprints on DNS orbital eccentricities, which maybe measured by LISA for values as small as ∼ − . Keywords: binaries: close – stars: neutron – supernovae: general INTRODUCTIONCurrent population synthesis models predict a doubleneutron star (DNS) merger rate within the Milky Wayof ≈ − (Vigna-G´omez et al. 2018; Kruckowet al. 2018; Chruslinska et al. 2018; Mapelli & Giacobbo2018). Alternatively, the DNS merger rate can be calcu-lated from the merger times of the known DNSs in theMilky Way field, accounting for survey selection effects(Phinney 1991; Kim et al. 2003). The latest applica-tion of this method, using 17 DNSs in the Milky Wayfield, finds a rate of 42 +30 − Myr − although this estimateis sensitive to pulsar luminosities, lifetimes, assump-tions about the contribution from elliptical galaxies, andbeaming correction factors (Pol et al. 2019). In compar-ison, a volumetric DNS merger rate of 920 +2220 − Gpc − yr − (which translates into a Milky Way rate of ≈ − ; Kopparapu et al. 2008), can be derived from the jeff[email protected] second Laser Interferometer Gravitational-Wave Obser-vatory (LIGO)/Virgo observing run (O2; Abbott et al.2017; The LIGO Scientific Collaboration et al. 2018).While the relatively large errors make the two observa-tional rate estimates consistent, with values dependenton various assumptions about NS population properties,the rate derived from LIGO (The LIGO Scientific Col-laboration et al. 2018) is nevertheless somewhat largerthan the analogous rate derived from the DNS MilkyWay field population.One possible origin of this difference could arise fromthe methods by which DNSs are detected within pul-sar surveys. The known DNSs in the Milky Way haveorbital periods ( P orb ) ranging from as large as 45 days(J1930 − t merge ∼ P orb8 / , it is much more likely to observe DNSswhich form with longer orbital periods. Second, Doppler a r X i v : . [ a s t r o - ph . H E ] O c t Andrews, J. J. et al. smearing reduces the sensitivity of pulsar surveys to bi-naries with orbital periods (cid:46) hours (Bagchi et al. 2013).The detection of such short-period binaries typically re-quires acceleration searches, which are both technicallychallenging and computationally expensive (see e.g., Nget al. 2015). The difficulty of identifying DNSs at shorterperiods than a few hours will cause DNS merger rate es-timates based on Milky Way populations, such as thoseby Pol et al. (2019), to be systematically underestimatedby an amount that depends on the number of systemsformed at these short orbital periods; while these esti-mates account for detection biases for the observed sys-tems, they cannot account for systems that are formedwith P orb so short that selection effects make them ef-fectively un-observable.While radio observations are sensitive to DNSswith P orb (cid:38) hrs and LIGO/Virgo detects DNSs atmerger, the Laser Interferometer Space Antenna (LISA;Amaro-Seoane et al. 2017) is sensitive to binaries with P orb ∼ minutes, bridging the gap between these tworegimes. We use the equations from Peters (1964) toshow in the top panel of Figure 1 how the orbits of the20 known DNSs in the Milky Way (17 in the field, 3in globular clusters; see Ridolfi et al. 2019; Andrews& Mandel 2019, and references therein for a list) willevolve over the next 10 Gyr as GR causes them to cir-cularize and decay. Depending on eccentricity, DNSsborn with orbital periods longer than ≈
18 hours takelonger than the age of the Universe to merge due toGR. The bottom panel of Figure 1 shows the evolutionof the orbital eccentricity with the gravitational wavefrequency ( f GW = 2 /P orb ), of these same 20 systems.If sufficiently close to the Solar System, binaries with10 − Hz < f GW < − (Nishizawa et al.2016). Pre-empting our quantitative results in Section3, we find that LISA will have have only a slightly de-graded precision for DNS binaries, measuring the ec-centricities of typical systems as small as a few 10 − at f GW ≈ − . Hz. This is in agreement with re-cent results by Lau et al. (2019) who also study thedetectability of DNSs with LISA. Comparison with thetracks in the bottom panel of Figure 1 shows this preci-sion is sufficient to measure DNS eccentricities as theyevolve through the LISA band. Thus, eccentricity mea-surements by LISA (Lau et al. 2019) may be used toinform DNS formation scenarios in a similar fashion to − − Orbital Period (days)0 . . . . . . E cce n tr i c i t y Globular ClusterMW Field10 − − − − − − GW Frequency10 − − − E cce n tr i c i t y Figure 1. Top panel:
The sample of 20 DNSs in theMilky Way (red points). Uncertainties on the measured P orb and e are smaller than the data points. Black lines indi-cate the evolution of these orbits as these systems circularizeand decay due to gravitational wave radiation in a Hubbletime. Bottom panel:
These DNSs retain residual eccen-tricities ( e (cid:38) − ) as they evolve through the LISA band(the relative sensitivity of LISA is represented by the orangebackground). Pre-empting our results in Section 2.2, LISAcan measure binary eccentricities as small as ∼ − (blue,dashed line) for typical DNSs. Therefore, LISA will mea-sure the eccentricities of many DNSs (which typically have f GW ≈ − . Hz; see Figure 2). binary black holes (Breivik et al. 2016; Nishizawa et al.2017; Samsing & D’Orazio 2018; D’Orazio & Samsing2018).If no DNSs are formed with orbital periods (cid:46) P orb (cid:46) f GW − e plane. In thiswork, we demonstrate that LISA may be able to mea- NSs in LISA DETECTING DNS WITH LISAObservatories such as LISA detect gravitational wavesby measuring the slight perturbations they cause tospace-time as they propagate through the detector. Fora circular binary at a distance, d , and with a chirpmass, M c (for two stars with masses M and M , M c = M / M / ( M + M ) − / ), the amplitude of thisstrain can be determined as a function of f GW : h ( f GW ) = 8 √ πf GW ) / ( GM c ) / c d , (1)where G is the Gravitational constant and c is the speedof light. The coefficient is set to account for averagingof the wave polarization, sky position, and DNS orienta-tion. Keeping f GW and h ( f GW ) constant, we find thatthe observable volume for a gravitational wave signalscales with M c . Despite the expectation that DNSsought to be more than an order of magnitude morecommon in the Universe than black hole binaries (TheLIGO Scientific Collaboration et al. 2018), the differenthorizon distances between the two types of binaries willmake DNSs significantly rarer within LISA than theirmore massive black hole binary counterparts.To quantitatively determine the detectability of DNSsby LISA, we use the LISA sensitivity curve described byCornish & Robson (2017) and Robson et al. (2019). Theblue line in Figure 2, which includes both the intrinsicdetector sensitivity as well as the contribution from thedouble white dwarf foreground (Korol et al. 2017), de-notes the sensitivity curve in terms of the characteristicstrain ( f GW S LISA ) / (Robson et al. 2019) as a functionof f GW . Below, in turn, we first describe peculiarities ofhow an individual binary is detected by LISA using theHulse-Taylor binary as an example. We then calculatethe expectation from a Milky Way population as wellas populations from the nearby M31 and M81 galaxygroups.2.1. Calculating the SNR for a LISA Detection
The signal-to-noise ratio (SNR) of a LISA detec-tion can be calculated from the binary’s strain am-plitude h ( f GW ), LISA’s noise power spectral density S LISA ( f GW ) and LISA’s lifetime T LISA (Robson et al.2019): SNR = h ( f GW ) T LISA S LISA ( f GW ) . (2)Most definitions of LISA’s SNR include an integral over f GW , since gravitational waves cause a binary’s orbit todecay over LISA’s lifetime. However, for binaries thatevolve slowly, such as the DNSs that LISA will detect,the integral can be accurately approximated by Equa-tion 2 (Robson et al. 2019), where we have absorbedvarious coefficients and prefactors into the definition of h ( f GW ) in Equation 1.In stark contrast with circular binaries, eccentric bi-naries emit gravitational waves at multiple harmonicsof the orbital frequency. Therefore, for a binary withan eccentricity e , the SNR can be calculated from thequadrature sum of the SNRs for each f n harmonic of theorbital frequency (see e.g., D’Orazio & Samsing 2018;Kremer et al. 2018)SNR ≈ ∞ (cid:88) n =1 h n ( f n ) T LISA S LISA ( f n ) . (3)The strain amplitude h n ( f n ) can be calculated as a func-tion of e and orbital frequency harmonic n : h n ( f n ) = 8 √ (cid:18) n (cid:19) / ( πf n ) / ( GM ) / c d (cid:112) g ( n, e ) , (4)where g ( n, e ) provides the relative amplitude at eachharmonic (Peters & Mathews 1963).2.2. Measuring the Eccentricity
For LISA to measure the eccentricity of a binary, thepower of at least two harmonics must be measured; theharmonics at 2/ P orb and 3/ P orb are strongest for bina-ries with e (cid:46) . e ) ≈ e (1 / SNR ) + (1 / SNR ) , (5)where SNR and SNR are the signal-to-noise ratios forthe 2/ P orb and 3/ P orb harmonics, respectively. For e (cid:46) .
3, the ratio of the amplitudes of the two harmonics S LISA is taken from Robson et al. (2019) who denote thisfunction as S n . We opt to adopt a different subscript to avoidconfusion between their n for “noise” and the index specifying theorbital harmonic. Andrews, J. J. et al. − − − − − f GW (Hz) − − − − − − l og C h a r a c t e r i s t i c S tr a i n B1913+16 M31MilkyWayM81 l og T i m e t o M e r g e r( y r) M y r M y r y r y r y r d a y Time to Merger LISA sensitivityChirp frequency limitDNSs in a 4-year missionGalactic DNS distribution
Figure 2.
We compare the distribution of DNSs expected within the Milky Way (grey contours; see Section 2.4 for details)against the LISA’s characteristic strain (blue line). The blue contour extends the sensitivity curve to approximately account foran SNR= 7 detection. For one random Milky Way realization, green points indicate the DNSs with SNR >
7, calculated usinga DNS merger rate of 210 Myr − . We indicate the Hulse-Taylor binary, B1913+16, as a red point, and follow its evolutionby straight lines as the system merges due to GR. The line’s color indicates the time to merger. Placing B1913+16 at largerdistances corresponding to M31 or M81 shows LISA may be able to detect a few DNSs outside of the Milky Way (Seto 2019).Note that all binaries are plotted as though they are circular. We properly account for eccentricity when calculating the SNRfor a LISA detection (see Section 2.1). scale with (9 / e (Seto 2016). The SNR of a particularharmonic depends both on the amplitude of the GWsignal as well as LISA’s sensitivity at that frequency.Using a spectral index, α , to describe LISA’s sensitivityas a function of frequency, we can determine the relativeSNR for the two harmonicsSNR SNR = (cid:18) (cid:19) − α − e. (6)In the limit that eccentricities are small (so thatSNR >> SNR and therefore the 1 / SNR term canbe ignored in Equation 5), we find:∆ e ≈ (cid:18) (cid:19) α +2 (cid:18) (cid:19) . (7)For a DNS with 10 − . Hz < f GW < − Hz, theDWD foreground causes α = 3. Therefore, a DNS with SNR = 10 has ∆ e ≈ .
01. For more eccentric binaries,higher order harmonics become relevant and Equation7 is no longer applicable. However, with multiple de-tected harmonics the measurement precision on e willonly improve.2.3. The Hulse-Taylor Binary as an Example
In Figure 2 we show the characteristic strain ( h c = h ( f GW ) f / T / ) of the Hulse-Taylor binary (redpoint) with its current orbital parameters at its dis-tance to the Sun. Note that the characteristic strainsshown in Figure 2 are calculated using Equation 1 (forplotting purposes only, we assume binaries are circular).Over the course of the next ≈
200 Myr, the system willinspiral due to GR orbital decay. We show the path theHulse-Taylor binary will take in f GW - h c space (assum-ing a constant distance) in Figure 2 by a line, whose NSs in LISA
The Milky Way Population of DNSs
We first focus on the Milky Way. Using the DNSmerger rate estimate of 210 Myr − , as derived by LIGO(The LIGO Scientific Collaboration et al. 2018), we canestimate where the DNSs would lie in h c - f GW space for arandom realization of the Milky Way, assuming a steady-state merger rate. Whereas Kyutoku et al. (2019) haverecently calculated the number of detectable DNSs inthe Galaxy by LISA using an analytic approximation, weuse a Monte Carlo method which allows us to account forthe spatial distribution of DNSs throughout the MilkyWay. Lau et al. (2019) also use a Monte Carlo methodto generate LISA detection predictions for a populationof Milky Way DNSs; however these authors base theirresults on the population synthesis results from Vigna-G´omez et al. (2018), who find a significantly lower MilkyWay DNS merger rate of 33 Myr − − ×
10 Myr = 2100). Extendingthis procedure to longer merger times is unnecessary asLISA is only sensitive to DNSs that will merge in thenext ∼
10 Myr (see Figure 2). We initialize each of thesesystems with very small P orb and e , with values equalto those that B1913+16 will take immediately prior tomerger ( P orb = 0 . e = 5 × − ), then integratetheir orbital evolution backwards in time for each of therandomly chosen times. The resulting set of backwards-evolved ( P orb , e ) values represents a Milky Way popula-tion of DNSs that produces an average merger rate of210 Myr − .We then place each simulated DNS in a random po-sition in the Milky Way, following the Galactic modelfrom Nelemans et al. (2001): P ( r, z ) ∼ e − R/L sech ( z/β ) , (8) using a scale length, L , of 2.5 kpc and a characteristicscale height, β , of 200 pc. We assume that the Solar Sys-tem is located 8.5 kpc from the Galactic Center and fallsalong the x=0, z=0 plane. Finally, using the integrated f GW and e combined with randomly chosen positions inthe Galaxy, we calculate the h c and corresponding SNRfor detection by LISA for each of these 2100 systemsusing Equation 1. Note that during this procedure andthroughout this study, we assume all NSs have a massof 1.4 M (cid:12) .Green points in Figure 2 show the subset of those 2100systems that will have an SNR > T LISA , will have measurable chirpmasses, M c , allowing the heavier DNSs ( M c = 1 . M (cid:12) for two 1.4 M (cid:12) NSs) to be differentiated from theirlower-mass white dwarf analogs ( M c = 0 . M (cid:12) for two0.6 M (cid:12) white dwarfs) (Kyutoku et al. 2019). The limit-ing frequency allowing this measurement can be deter-mined (Nelemans et al. 2001): f chirp ≥ . × − (cid:18) M c . M (cid:12) (cid:19) − / (cid:18) T LISA (cid:19) − / Hz . (9)The vertical dotted line in Figure 2, which shows thislimit on f GW for a four-year LISA mission, demonstratesthat a subset of these DNSs ought to have measurablechirp masses. Given LISA’s sensitivity, those DNSs with f GW > f chirp will be easily identifiable as being com-prised of two NSs (Seto 2019). On the other hand, DNSswith f GW < f chirp can be confused with Galactic doublewhite dwarfs; from Equation 1, h c ∼ M / c f / d − , anda binary comprised of two 0.6 M (cid:12) WDs will need to be ≈ h and f GW . Since most DNSs will be found withinthe Galactic Plane at ∼
10 kpc, the intrinsic faintness ofWDs at distances larger than a few hundred pc, the poorposition determination on the sky by LISA and confu-sion in the densely packed Galactic Plane all combineto make it unlikely that optical follow-up will be able torule out a double WD scenario for these systems.In addition to the green points in Figure 2 represent-ing a single Milky Way realization, we generate 10 sep-arate DNSs using the same procedure except with ran-dom merger times within the last 100 Myr. The greycontours in this Figure represent the overall distribu-tion of DNSs in this plane, expected from a constantmerger rate. Andrews, J. J. et al.
We run 100 separate Milky Way realizations to de-termine the statistical distribution of the number of ex-pected Milky Way DNSs observable by LISA. We fit thenumber of DNS detections to a Gaussian distribution,finding the best fit mean of the distribution from our100 Milky Way realizations: N DNS = (cid:16) R MW
210 Myr − (cid:17) , MW : 4 − year mission330 (cid:16) R MW
210 Myr − (cid:17) , MW : 8 − year mission . (10)For a more pessimistic rate estimate of 42 Myr − , as de-rived by the Milky Way population of DNSs (Pol et al.2019), we find LISA detection rates of 46 (64) for a 4-year (8-year) LISA mission. Since these rates are deter-mined from random sampling, uncertainties on N DNS within the Milky Way scale with √ N DNS . Roughly 25%of the DNSs detected in the MW by LISA will havemeasurable chirp masses.Based on the current estimate of the Milky Way DNSmerger rate, we therefore conclude that LISA will al-most certainly observe a handful of DNSs. Since theseare Poisson processes, the rates in Equation 10 can bescaled up and down arbitrarily as rate estimates im-prove with future observations and analysis. Althoughwe opt to not include it here, one can trivially propagatethis Poisson distribution with uncertainties on the DNSmerger rate.2.5.
DNSs in Other Nearby Galaxies
What about the nearby M31 and M81 galaxies? Re-turning to Figure 2, we see that systems in these galaxiesare detectable by LISA for a much smaller time, sincethese systems have h c above the LISA sensitivity curveonly once f GW (cid:38) − . , corresponding to a merger timeof ∼ yr. Even with the optimal orbital f GW , systemswithin M81 typically do not have an SNR above 2. Onthe other hand, Figure 2 shows that systems within An-dromeda may produce detectable h c (see also Seto 2019).Using the same rate of DNS mergers in Andromeda asin the Milky Way (this is likely an underestimate, asAndromeda is somewhat more massive than the MilkyWay), we repeat the procedure used for the Milky Way,generating 100 random realizations of M31. Setting alimit of SNR >
7, we find: N DNS = . (cid:16) R M31
210 Myr − (cid:17) , M31 : 4 − year mission4 . (cid:16) R M31
210 Myr − (cid:17) , M31 : 8 − year mission . (11)Uncertainty on these rates follow a Poisson distribution,and therefore can be scaled up and down arbitrarily asestimates on the DNS merger rate in M31 are refined. Note that the DNS nature of these sources will be im-mediately apparent from M c , since these systems willall be “chirping” and furthermore the distance to An-dromeda is well-determined.The detectability of DNSs in M31 has been recentlydiscussed by Seto (2019), who find ≈ − , ≈ >
7. Therefore,the expected number of DNSs we find here are consistentwith those found by Seto (2019). A FAST-MERGING CHANNEL?Several studies have argued that if merging DNSs areresponsible for the nucleosynthesis of r -process materialin the Universe, a fast-merging channel that creates,evolves, and merges DNS within ∼
10 Myr is required(e.g., Komiya et al. 2014; Matteucci et al. 2014; Sa-farzadeh & Scannapieco 2017; Safarzadeh et al. 2019b).More recent studies were able to reproduce the enrich-ment of Milky Way stars with a delay time as long as ∼
100 Myr (see discussion in Vangioni et al. 2016). How-ever, the discovery of r -process enrichment in two ultra-faint dwarf galaxies, Reticulum II (Ji et al. 2016) andTucana III (Hansen et al. 2017), suggests that a fast-merging channel is again required to form DNS merg-ers early enough so that a second generation of starscan incorporate the merger products (Beniamini & Pi-ran 2016; Safarzadeh et al. 2019a). Zevin et al. (2019)have recently invoked similar arguments to explain the r -process enrichment observed in many globular clus-ters.What might cause these different evolutionary chan-nels? One possibility deals with the formation of DNSsthrough isolated binary evolution including a phase ofCase BB mass transfer, in which a NS accretor entersa second mass transfer phase when its stripped heliumstar companion evolves into a giant star (Delgado &Thomas 1981). The most up-to-date simulations pre-dict that this Case BB mass transfer phase ought to bestable, forming DNSs with orbital periods as short as ≈ NSs in LISA − . − . − . . Log P orb (days) . . . . . E cce n t r i c i t y No Fast-Merging Channel − . − . − . . Log P orb (days) with Fast-Merging Isolated Binaries − . − . − . . Log P orb (days) with Fast-Merging Dynamically Formed Binaries10 − − − − − f GW − − − − E cce n t r i c i t y − − − − − f GW − − − − − f GW Figure 3. Top Row: P orb − e distributions for three separate models for the formation of DNSs. Blue and green distributionsshow representative models for the formation of the low- and high-eccentricity DNSs observed in the Milky Way (red points).Black contours in the second and third panels show two different ad hoc distributions for a putative fast-merging DNS evolu-tionary channel. Bottom Row:
The evolution of DNSs in f GW − e over as they circularize and inspiral due to GR. Dependingon the characteristics of the fast-merging channel, DNSs will evolve along separate tracks in f GW − e space. Another option was proposed by Andrews & Mandel(2019), who suggest that the high eccentricity subpop-ulation of DNSs in Figure 1 is consistent with beingformed dynamically in globular clusters then kicked outinto the field. Indeed, B2127+11C is a member of theglobular cluster M15 and has parameters consistent withthe high-eccentricity, short-orbital period DNSs in thefield (see Figure 1). Initial population studies of glob-ular clusters suggested that LISA may be sensitive todynamically formed compact object binaries, including,based on crude scaling estimates, tens of double neu-tron stars (Benacquista 1999; Benacquista et al. 2001).Later, more detailed globular cluster models found onlya few dynamically formed DNSs would merge within aHubble time (Grindlay et al. 2006; Ivanova et al. 2008;Lee et al. 2010; Belczynski et al. 2018; Ye et al. 2019b,a)and may produce of order one system detectable byLISA (Kremer et al. 2018). Nevertheless, the similar-ity of B2127+11C with other DNSs in the field suggeststhat the dynamical formation scenario may still be rele-vant (Andrews & Mandel 2019). Since dynamical forma-tion tends to produce systems with eccentricities drawnfrom a thermal distribution (Heggie 1975), this putativefast-merging channel ought to have a much higher ec- centricity distribution than those formed through caseBB mass transfer in isolated binaries.Despite the circularizing effects of GR, most sys-tems will maintain a residual eccentricity as they evolvethrough the LISA band, with a value depending on theexact scenario forming a putative fast-merging chan-nel. Those DNSs with only upper limits on e neces-sarily formed with low eccentricities. LISA’s ability todetect eccentricities in binary orbits as small as 10 − affords a unique opportunity to discern between vari-ous evolutionary channels forming DNSs, analogous towhat has already been shown for double white dwarfs(e.g., Willems et al. 2007) and double black holes (e.g.,Breivik et al. 2016; Nishizawa et al. 2017; Samsing &D’Orazio 2018; D’Orazio & Samsing 2018).To quantitatively test the eccentricity distributions fordifferent DNS formation scenarios, we use toy models forDNS formation. We first include functional models forthe formation through isolated binary evolution of theobserved low- and high-eccentricity DNSs in the MilkyWay. Andrews & Mandel (2019) show that these sepa-rate populations can be reasonably modeled through iso-lated binary evolution, by randomly generating systemsimmediately prior to the second SN, then dynamicallyevolving them through core collapse (see also, Andrews Andrews, J. J. et al. & Zezas 2019). The low-eccentricity systems are mod-eled with a circular pre-SN orbit, with a log-normal or-bital separation distribution ( µ = 0 . σ = 0 .
4, in unitsof R (cid:12) ), and an isotropic SN kick of 50 km s − . Wereproduce the high-eccentricity DNSs in a similar way,but using a log-normal pre-SN orbital separation distri-bution ( µ = 0, σ = 0 .
2, in units of R (cid:12) ) and a SN kickvelocity of 25 km s − . These models for low- and high-eccentricity DNSs (which are adapted from Andrews &Mandel 2019) immediately after the SN are shown in thetop row of panels in Figure 3 as blue and green contours,respectively.In the first column of panels in Figure 3, we showonly the two functional models to reproduce the ob-served low- and high-eccentricity DNSs in the MilkyWay. The next two columns of panels additionally con-tain toy models for the two putative fast-merging for-mation scenarios (black contours). Detailed simulationsof both isolated binary evolution through Case BB masstransfer and dynamical formation within globular clus-ters are outside of the scope of this Letter. For thesetwo toy models, we randomly generate DNSs with alog-normal distribution in orbital separation ( µ = − . σ = 0 . R (cid:12) ) and a normal distribution ineccentricity ( µ =0.2 and 0.7 for the two models, bothwith σ = 0 . h c distribution of DNSs shown inFigure 2: we generate 10 separate systems for eachpopulation (so they are well sampled), evolve them for-ward for a randomly drawn time corresponding to thetime until merger (selected from a uniform distributionwithin the past 100 Myr) due to GR, place them in arandom position in the Milky Way, and calculate theSNR of a LISA detection. We record the system’s ec-centricity if it produces an SNR > − is sufficient to discernbetween the different DNS formation scenarios. DISCUSSION & CONCLUSIONS − . − . − . − . − . . e N o r m a li ze d D i s tr i bu t i o n Isolated Binary EvolutionDynamical FormationLow EccentricityHigh Eccentricity
Figure 4.
Eccentricities of DNSs identified within the LISAband with SNR >
7. Blue and green distributions are designedto match the low and high eccentricity DNSs in the MilkyWay, respectively. The two black distributions (different linestyles) demonstrate the eccentricities expected from two adhoc models for putative fast-merging DNSs.
Using the Milky Way DNS merger rate of 210 Myr − derived from LIGO (The LIGO Scientific Collaborationet al. 2018), we find that a 4-year (8-year) LISA missionwill detect on average 240 (330) DNSs. Approximately25% of those will be “chirping,” allowing for their char-acterization as DNSs through their chirp mass. Theremaining ≈