GW170817 and GW190814: tension on the maximum mass
DDraft version January 25, 2021Typeset using L A TEX twocolumn style in AASTeX63
GW170817 and GW190814: tension on the maximum mass
Antonios Nathanail, Elias R. Most,
2, 3, 4 and Luciano Rezzolla
1, 5, 6 Institut fΓΌr Theoretische Physik, Max-von-Laue-Strasse 1, 60438 Frankfurt, Germany Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA Princeton Gravity Initiative, Princeton University, Princeton, NJ 08544, USA School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt, Germany School of Mathematics, Trinity College, Dublin 2, Ireland (Received January 25, 2021; Revised; Accepted)
ABSTRACTThe detection of the binary events GW170817 and GW190814 has provided invaluable constraints on themaximum mass of nonrotating conο¬gurations of neutron stars, π TOV . However, the large diο¬erences in theneutron-star masses measured in GW170817 and GW190814 has also lead to a signiο¬cant tension betweenthe predictions for such maximum masses, with GW170817 suggesting that π TOV (cid:46) . π (cid:12) , and GW190814requiring π TOV (cid:38) . π (cid:12) if the secondary was a (non- or slowly rotating) neutron star at merger. Using a geneticalgorithm, we sample the multidimensional space of parameters spanned by gravitational-wave and astronomicalobservations associated with GW170817. Consistent with previous estimates, we ο¬nd that all of the physicalquantities are in agreement with the observations if the maximum mass is in the range π TOV = . + . β . π (cid:12) within a 2- π conο¬dence level. By contrast, maximum masses with π TOV (cid:38) . π (cid:12) , not only require eο¬cienciesin the gravitational-wave emission that are well above the numerical-relativity estimates, but they also lead toa signiο¬cant under-production of the ejected mass. Hence, the tension can be released by assuming that thesecondary in GW190814 was a black hole at merger, although it could have been a rotating neutron star before. Keywords: equation of state β gravitational waves β methods: analytical β stars: neutron INTRODUCTIONThe recent detection of the gravitational-wave (GW) eventGW190814 has seen involved the merger of black hole (BH),with a mass of 22 . β . π (cid:12) , with a compact object havinga much smaller mass of 2 . β . π (cid:12) (The LIGO Scientiο¬cCollaboration et al. 2020). The unclear nature of the sec-ondary component has raised questions about the astrophys-ical evolutionary paths that would yield objects with thesemasses in a binary system. When assigning a NS nature tothe secondary in GW190814, two scenarios are possible. Inthe ο¬rst one, the secondary was a nonrotating or slowly ro-tating NS at merger, so that GW190814 should eο¬ectively beconsidered a BH-NS merger (see, e.g., Zevin et al. 2020; Sa-farzadeh et al. 2020; Kinugawa et al. 2020; Liu & Lai 2020;Lu et al. 2020; Ertl et al. 2020, for some possible formationscenarios). In this case, the maximum mass π TOV of nonrotat-ing NSs needs to reach values as large as (cid:38) . π (cid:12) (Fattoyevet al. 2020; Sedrakian et al. 2020; Tan et al. 2020; Tsokaroset al. 2020; Godzieba et al. 2020; Biswas et al. 2020). In thesecond scenario, the need for a large maximum mass can bereplaced by the presence of rapid rotation. In fact, it has been shown that uniformly rotating NS can support about 20%more mass than nonspinning ones (Breu & Rezzolla 2016;Shao et al. 2020). Note that in the case of NSs with a phasetransition, universal relations are still present, but depend onthe properties of the phase transition (Bozzola et al. 2019;Demircik et al. 2020).Based on these universal relations, Most et al. (2020b) andZhang & Li (2020) have pointed out that a massive (rapidly)rotating NS with a mass > . π (cid:12) is perfectly consistentwith a maximum mass π TOV (cid:39) . π (cid:12) inferred from theGW170817 event (see, e.g., Rezzolla et al. 2018; Shibataet al. 2019). Given the diο¬culty of sustaining rapid rotationover the very long timescales associated with the inspiral ofthe binary, the secondary must have collapsed at one pointbefore merger, so that in this second scenario GW190814should eο¬ectively be considered a BH-BH merger.While a priori both scenarios are plausible, shedding lighton which of them is the most likely is important from severalpoints of view. To this scope, we here exploit the rich vari-ety of GW and electromagnetic observables that have beenobtained with GW170817 to explore the two scenarios com- a r X i v : . [ a s t r o - ph . H E ] J a n Nathanail, Most and Rezzollabining the constraints set from the GW and electromagneticsignal from GW170817. In particular, we employ a geneticalgorithm to sample through the distributions of maximummasses, ejected matter (Drout et al. 2017; Cowperthwaiteet al. 2017; Kasen et al. 2017; Villar et al. 2017; Cough-lin et al. 2018), and GW emission from numerical-relativity(NR) simulations (Zappa et al. 2018). Consistent with pre-vious results (Rezzolla et al. 2018; Shibata et al. 2019) weο¬nd that GW170817βs observations clearly set an upper limitfor the maximum mass of π TOV (cid:46) . π (cid:12) . When forcingthe algorithm to allow for maximum masses π TOV (cid:38) . π (cid:12) ,we ο¬nd that this requires unrealistically large GW eο¬cienciesfrom the merger remnant and a deο¬cit in the ejected matter. FRAMEWORK FOR THE GENETIC ALGORITHMThe observations of a bright blue kilonova has providedconvincing evidence that the merger remnant in GW170817could not have collapsed promptly to a BH. Rather, it musthave survived for a timescale of the order of one second (Gillet al. 2019; Lazzati et al. 2020; Hamidani et al. 2020), and suf-ο¬ciently large so that the hypermassive NS (HMNS) producedby the merger has reached uniform rotation at least in its core(Margalit & Metzger 2017; Rezzolla et al. 2018). FollowingRezzolla et al. (2018), we recall that quasi-universal relationsexist between the masses of uniformly rotating stellar modelsalong the stability line to BH formation and the correspond-ing dimensionless angular momentum π coll normalised to themaximum (Keplerian) one π Kep (Breu & Rezzolla 2016). Wehere express this relation as π ( π coll / π Kep ) : = π crit π TOV = + πΌ (cid:18) π coll π Kep (cid:19) + πΌ (cid:18) π coll π Kep (cid:19) , (1)where πΌ = . Γ β and πΌ = . Γ β , and the valueof the Keplerian speciο¬c angular momentum is approximatelygiven by π Kep βΌ .
68 (see Eq. (4) in Most et al. 2020b, formore accurate estimates). The function π is deο¬ned between0 and 1 and describes all models with a mass that is criticalfor collapse to a BH. To ο¬x ideas, in the case of nonorotatingmodels, π coll = π ( ) =
1, while for maximally rotatingmodels π coll = π Kep and (Breu & Rezzolla 2016) π ( ) : = π max / π TOV β . + . β . , where π max is the maximum massthat can be sustained through uniform rotation (see Weih et al.2018, for diο¬erentially rotating stars). Note that range π ( ) isbased on a speciο¬c set of hadronic equations of state (EOSs)and that a diο¬erent estimate suggests π ( ) = . + . β . (Shaoet al. 2020).Because Eq. (1) expresses a relation between gravitationalmasses, while the electromagnetic emission from GW170817informs us about the ejected baryonic mass, we need a relationbetween gravitational and baryonic mass π π for uniformlyrotating NSs at the mass-shedding limit (see, e.g., Timmeset al. 1996; Gao et al. 2020, for a detailed discussion). Also in this case, this relation obeys a quasi-universal relation near thevalues of the maximum mass that, with a 2- π uncertainty, isgiven by π : = π π, max / π max β . Β± .
014 at the maximum-mass limit (Rezzolla et al. 2018). Note that π is in principlea function of π and that the value reported above is for π = π max . However, π is almost constant in the neighbourhoodof π max β where all of our considerations are made β so thathereafter we simply write the conversion between the twomasses as π π = ππ .The total gravitational mass of GW170817 as inferred fromthe GW signal is π π = . + . β . (The LIGO Scientiο¬c Col-laboration et al. 2019), whose corresponding baryonic mass π π soon after the merger can be thought of as being given bythe combination of the baryonic mass in the HMNS π π, HMNS β itself composed of the mass in the core and in a Kepleriandisk β and of the mass ejected dynamically, i.e., π π = π π, core + π π, disk + π dynej = ππ β π , (2)where π β π : = π π β π inspGW and π inspGW is the energy lost to GWsin the inspiral. Here, the last equality relates the baryonic andgravitational mass of the merger remnant. Deο¬ning now π asthe fraction of the HMNS baryonic mass in the core, the twocomponents of the HMNS shortly after merger can be writtenas π π, core : = π (cid:16) π π β π dynej (cid:17) = π (cid:16) ππ β π β π dynej (cid:17) (3)The fraction π is in principle unknown, but numerical simula-tions have shown that this ratio is actually weakly dependenton the EOS and given by π β . + . β . (Hanauske et al.2017). As time goes by, the merger remnant will loose partof its baryonic mass via the emission of magnetically drivenor viscous-driven winds, so that at collapse it will have abaryonic mass π π = π coll π, core + π coll π, disk + π dynej + π blueej + π redej , (4)where the last equality follows from rest mass conservationand π coll π, core and π coll π, disk are the respective values of the coreand the disk at the time when BH formation of the core istriggered, while π blueej ( π redej ) is the part of the ejected matterleading to the blue (red) emission in the kilonova and diο¬ersfrom the dynamical ejecta from the timescale over which thematerial is lost. The two components also diο¬er in the typicalvelocities of the matter, which is larger in the blue component( π£ / π (cid:46) . π£ / π (cid:28) . π π , which is again largerin the blue component (0 . (cid:46) π π (cid:46) . π π (cid:46) . π disk : = π redej / π π, disk (cid:39) . β . , (5)representing the unbound fraction of the disk mass, whichcan be estimated based on numerical simulations (Siegel& Metzger 2017; FernΓ‘ndez et al. 2019; Fujibayashi et al.2018; Nathanail et al. 2020). In a merger scenario as that ofGW170817, where the remnant may have lived for about onesecond (Gill et al. 2019), BH formation is triggered when thegravitational mass is reduced by the emission of GWs and theremnant hits the stability line for uniformly rotating modelswith a massive core π coll π, core π coll π, core = ππ β π β π π, disk β π dynej β π blueej = π ππ TOV , (6)where the last equality relates the baryon mass of the remnantcore to the maximum mass π TOV of nonrotating NS via Eq.(1). Indeed, when expressed as as a constraint equation onthe maximum mass, Eq. (6) can also be written π ππ
TOV = π (cid:16) ππ β π β π dynej (cid:17) β π blueej . (7)Two more equations can be used for consistency π redej = π disk ( β π ) (cid:16) ππ β π β π dynej (cid:17) , (8) ππ TOV = π β π β π β (cid:16) π coll π, disk + π dynej + π blueej + π redej (cid:17) β π post GW , (9)where the ο¬rst one expresses the conservation of rest-massleading to the kilonova emission and the second one the con-servation of gravitational mass since π post GW is the mass lostto GWs after the merger. Expression (9) does not constrain π post GW , which thus remains undetermined. As a way around(see also Fan et al. (2020)), we use an approximate quasi-universal relation between the total mass lost to GWs π totGW and the speciο¬c angular momentum of the remnant after themerger (Zappa et al. 2018) π totGW βΌ π + π π rem , + π (cid:0) π rem , (cid:1) , (10)where π totGW : = π totGW /( π π π ) , π rem , : = π½ rem , /( π π π ) is thespeciο¬c angular momentum of the remnant within βΌ
20 msfrom the merger, and π : = π π /( π + π ) is the symmetricmass ratio. Note that π = . , π = β . π = . π and π are chosen considering the low-spin prior for GW170817,i.e., π β [ . , . ] . By splitting total mass lost to GWsinto the two components relative to the inspiral and post-merger, i.e., π totGW = π insp GW + π post GW , Eq. (10) allows us tointroduce an additional constraint between π coll β which wederive from Eq. (1) β and π post GW . Two more remarks before concluding the presentation ofour methodology. First, not all of the merger remnantβs an-gular momentum will end up in the collapsed object. Anumber of physical processes will move part of the angu-lar momentum outwards, placing it on stable orbits relativeto the newly formed BH. Because the eο¬ciency of this pro-cess depends on microphysics that is poorly understood, weaccount for this by introducing a fudge factor π π΅ deο¬ned as π coll = : ( β π π΅ ) π rem , , so that the speciο¬c angular momentumof the disk is π disk : = π π΅ π rem , . Second, since in Eqs. (7), (8)and (9) the function π always appears together with π TOV , itis diο¬cult to set reasonable ranges for π . However, numericalsimulations have revealed that the dimensionless spin of theBH produced by the merger π BH (and hence π coll (cid:46) π BH ) is actu-ally constrained in a rather limited range, i.e., 0 . (cid:46) π BH (cid:46) . π coll = . π BH , we can eο¬ectively constrain π to be inthe range 1 . β€ π (cid:46) .
22. Very similar results are obtainedwhen making the more drastic assumption that only half ofthe BH spin comes from the remnant, i.e., π coll = . π BH ,which further reduces the lower limit to be π = .
05 (seeSupplement Material for details).In summary, we need to solve a multidimensional para-metric problem as expressed by Eqs. (7), (8) and (9) aftervarying in the appropriate ranges the (ten) free parameters inthe system: π, π, π, π insp GW , π blueej , π dynej , π TOV , π, π disk and π π΅ .While we treat all these parameters equally, some of them( π, π dynej , π ), vary in very narrow ranges and their variationsdo not signiο¬cantly aο¬ect the overall results. In practice, atany iteration of the genetic algorithm we ensure that: (i) thetotal gravitational mass of the system is π π = . + . β . (TheLIGO Scientiο¬c Collaboration & The Virgo Collaboration2017; The LIGO Scientiο¬c Collaboration et al. 2019); (ii) the total ejected mass is π totej = . Β± . π (cid:12) (Arcaviet al. 2017; Nicholl et al. 2017; Chornock et al. 2017; Cow-perthwaite et al. 2017; Villar et al. 2017; Drout et al. 2017;Kasen et al. 2017; Tanaka et al. 2017; Waxman et al. 2018;Coughlin et al. 2018); (iii) the dynamically ejected mass is π dynej β β π (cid:12) (Sekiguchi et al. 2015; Bovard et al. 2017;Radice et al. 2018; Poudel et al. 2020); (iv) the blue/red ejectedcomponents are respectively 0 . < π blueej / π (cid:12) < .
02 and0 . < π redej / π (cid:12) < .
55 (Drout et al. 2017; Cowperthwaiteet al. 2017; Smartt & Chen 2017; Kasliwal et al. 2017; Kasenet al. 2017; Villar et al. 2017; Tanaka et al. 2017; Waxmanet al. 2018; Coughlin et al. 2018) we have also explored alarger upper bound on the blue ejecta, i.e., π blueej / π (cid:12) < . (v) the maximum mass is taken to be in the range1 . π (cid:12) < π TOV < π (cid:12) , note that the posterior lower boundis consistent with pulsar observations (Antoniadis et al. 2013; Nathanail, Most and Rezzolla . . . . . . . . . M TOV [ M (cid:12) ]0246 P r o b a b ili t y d i s t r i bu t i o n f un c t i o n Ο . M (cid:12) Ο . M (cid:12) Rezzolla + (2018)genetic algorithm
Figure 1.
Uniform posterior from the analysis of Rezzolla et al.(2018) (magenta) and the posterior obtained with the multidimen-sional genetic algorithm (blue) discussed here. Indicated with ver-tical lines are the 1- π (dotted) and 2- π (dashed) values. Cromartie et al. 2020); (vi) the energy radiated in GWs beforethe merger is constrained to be 0 . (cid:46) π insp GW / π (cid:12) (cid:46) . (i) β (vi) are uniform. RESULTSFigure 1 provides a ο¬rst important impression of the re-sults of the genetic-algorithm. In particular, shown with amagenta-shaded area is the maximum-mass estimate madeby Rezzolla et al. (2018) and which is a simple uniformposterior for π TOV = . + . β . π (cid:12) . Shown instead with ablue-shaded area is the posterior distribution obtained withthe genetic algorithm. The median of the distribution is π TOV = . + . β . π (cid:12) , where the errors reported here arefor 2- π uncertainty. Overall, this yields a lower bound of π TOV > . π (cid:12) and an upper bound of π TOV < . π (cid:12) at 2- π level (vertical dashed lines), and thus in good agree-ment with massive-pulsar measurements (Antoniadis et al.2013; Cromartie et al. 2020) and previous estimates (Rez-zolla et al. 2018; Shibata et al. 2019). Interestingly, ourresults are in good agreement with the conclusions reachedby Shao et al. (2020) and Fan et al. (2020), who have alsoconsidered the post-merger GW emission to deduce boundson the maximum mass.As a consistency check, we can use set of parameters thatyield the maximum-mass distribution in Fig. 1, to estimate theGW energy lost both in the inspiral and in the post-merger.This is shown in the left panel of Fig. 2, where we re-port the posterior distributions for π insp GW (black dotted line)and π postGW (black dashed line), as well as their sum, π tot GW (black solid line). Also marked with a vertical lavender- shaded is the upper limit estimated by Zappa et al. (2018), πΈ totGW / π π β€ . π (cid:12) π , on the basis of a large numberof NR simulations, with an associated uncertainty of 20%.A similar consistency is found in the right panel of Fig. 2,where we report the posterior of the total ejected mass consis-tent with the maximum-mass distribution in Fig. 1. Indicatedwith a green-shaded area are the constraints obtained fromthe kilonova observations of GW170817. More speciο¬cally,the width of the shaded area represents the standard deviationusing various estimates for the total ejected mass π totej esti-mated for GW170817 (Arcavi et al. 2017; Nicholl et al. 2017;Chornock et al. 2017; Cowperthwaite et al. 2017; Villar et al.2017; Drout et al. 2017; Kasen et al. 2017; Tanaka et al. 2017;Waxman et al. 2018; Coughlin et al. 2018). Clearly, also theejected-mass distribution is in perfect agreement with obser-vational bounds when the maximum mass is below 2 . π (cid:12) .Given these results, it is natural to ask: does anything breakdown when larger maximum masses are considered? Thepositive answer to this question is contained in Fig. 3, whosepanels are similar to those in Fig. 2, when however the geneticalgorithm is forced to consider two speciο¬c values of themaximum mass, namely, π TOV = . π (cid:12) and π TOV = . π (cid:12) .Concentrating ο¬rst on the left panel of Fig. 3, it is clearthat when allowing for large maximum masses, the massradiated in GWs after the merger, π post GW (red dashed line), issigniο¬cantly larger than what NR simulations predict; this istrue both for π TOV = . π (cid:12) and for π TOV = . π (cid:12) . Thisbehaviour is due to the fact that remnants with a given π willradiate more GWs if they are more massive [cf., Eq. (9)].Next, when considering the right panel of Fig. 3 it is alsoeasy to realize that large maximum masses lead to a deο¬citin the ejected matter. This is simply due to the fact thatconsidering large-mass stars inevitably reduces the portion ofthe budget available for the ejecta. We have conο¬rmed that,even if (unrealistically) large additional angular momentumtransport was assumed, these results remain unchanged for π TOV (cid:38) . π (cid:12) , (see Supplemental Material).In summary, while a value π TOV (cid:46) . π (cid:12) is fully con-sistent with the GW emission from NR simulations and theobserved ejected mass, a value π TOV (cid:38) . π (cid:12) requires eο¬-ciencies in the GW emission that are well above the estimatesfrom a large number of accurate NR simulations and, overall,leads to an under-production of ejected mass. CONCLUSIONSWe have carried out a systematic investigation to ascer-tain whether the tension on the maximum mass followingthe detections of GW170817 and GW190814 can in someway be resolved or at least attenuated. In particular, we haveemployed a genetic algorithm to sample through the multidi-mensional space of parameters that can be built on the basis ofthe astronomical observations (i.e., ejected mass in the vari-W170817 and GW190814: tension on the maximum mass 5 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . M GW [ M (cid:12) ]050100150200250 p r o b a b ili t y d i s t r i bu t i o n f un c t i o n NR upper limiton M tot GW M TOV β€ . M (cid:12) M insp GW M post GW M tot GW .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 .
06 0 .
07 0 . M totej [ M (cid:12) ]050100150200 P r o b a b ili t y d i s t r i bu t i o n f un c t i o n M totej from GW170817kilonova M TOV β€ . M (cid:12) Figure 2.
Left panel:
Posteriors for the mass radiated in GWs consistent with the distribution in Fig. 1; the lavender-shaded area reportsthe upper limit coming from NR simulations.
Right panel:
Posterior for the total ejected mass consistent with the distribution in Fig. 1; thegreen-shaded area reports the range estimated in the literature. .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . M GW [ M (cid:12) ]050100150200250 p r o b a b ili t y d i s t r i bu t i o n f un c t i o n NR upper limiton M tot GW M TOV = 2 . M (cid:12) M TOV = 2 . M (cid:12) .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 .
06 0 .
07 0 . M totej [ M (cid:12) ]050100150200 P r o b a b ili t y d i s t r i bu t i o n f un c t i o n M totej from GW170817kilonova M TOV = 2 . M (cid:12) M TOV = 2 . M (cid:12) Figure 3.
Left panel:
The same as in the left panel of Fig. 2, but when considering two ο¬xed values for the maximum mass, i.e., π TOV = . π (cid:12) (orange) and π TOV = . π (cid:12) (red). Right panel:
The same as in the right panel of Fig. 2, but for two ο¬xed maximum-mass values. ous components), GW observations (i.e., gravitational massesof the binary components), and of NR simulations (i.e., prop-erties of the remnant and eο¬ciency of GW emission).The results of this investigation have allowed us to re-ο¬ne in a probabilistic manner the previous estimates ofthe maximum mass (Rezzolla et al. 2018), obtaining that π TOV = . + . β . π (cid:12) within a 2- π conο¬dence level. Inthis range, all of the physical quantities are in very goodagreement with the estimates coming from the observations.By contrast, we ο¬nd that considering maximum masses with π TOV (cid:38) . β . π (cid:12) requires eο¬ciencies in the GW emis-sion well above the NR estimates and leads to a signiο¬cantunder-production of the ejected mass, well below the valuesexpected from the observations. Although robust, our results can be strengthened in a number of ways. Improved post-merger modeling and long-term NR simulations (Fujibayashiet al. 2018) can help to narrow the uncertainties in the pa-rameter ranges for π and π rem . Reο¬ned universal relationsof uniformly rotating NSs including temperature dependence(Koliogiannis & Moustakidis 2020), will also help narrowingdown the errors in π and the upper limit of π . Such im-provements will be crucial to understand the viability of themaximum-mass constraint for π TOV (cid:46) . π (cid:12) .In light of these considerations, we conclude that the sec-ondary in GW190814 was most likely a BH at merger, al-though it may well have been a rotating NS at some stageduring the evolution of the binary system. Nathanail, Most and Rezzolla Acknowledgements.
It is a pleasure to thank C. Ecker, J. Pa-penfort, and L. Weih for useful comments. Support comes inpart also from βPHAROSβ, COST Action CA16214 and theLOEWE-Program in HIC for FAIR. ERM gratefully acknowl- edges support from a joint fellowship at the Princeton Centerfor Theoretical Science, the Princeton Gravity Initiative andthe Institute for Advanced Study.
Software:
Scipy (Virtanen et al. 2020),
Corner (Foreman-Mackey 2016),REFERENCES
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Nathanail, Most and RezzollaSUPPLEMENTAL MATERIALIn what follows we provide additional information that complements the one provided in the main text. While the detailsillustrated below do not vary the conclusions drawn in the main text, they provide additional technical details on the geneticalgorithm employed in our analysis. In addition, they help investigate how the results change when the parameters are variedbeyond the (reasonable) ranges assumed so far.For the solution of our multidimensional parametric problem the procedure we adopt is as follows (see also Fromm et al.2019; Nathanail et al. 2020, for additional information). We start by recalling that genetic algorithms are designed to generatehigh-quality solutions to problems of this type where a searching optimization is sought. The name follows from the operators ofmutation, crossover and selection that are normally found in biological systems. Our choice of a genetic algorithm in place of amore traditional Bayesian analysis based on a Markov-Chain Monte Carlo approach is motivated mostly by the overall simplicityof our problem and the reduced computational costs that are associated with a genetic algorithm.In practice, our algorithm samples through the parameter space of the ten free parameters. From those it computes the π redej through Eqs. (3) and (5), π post GW through Eq. (10) and the speciο¬c angular momentum at collapse π coll via Eq. (1), using thesampled value of π . Subsequently, Eqs. (7), (8), and (9) are solved to match the observed values of π π and π totej within the errors,ο¬nding the best-ο¬t values. The genetic algorithm employed here makes use of Python packages from the SciPY software library(Virtanen et al. 2020).As a corollary to the discussion made in the main text and relative to Figs. 1β3, we provide with the corner plots in Fig. 4information on the probability distribution functions of the various quantities involved in our analysis. More speciο¬cally, Fig.4 shows the corner plot relative to maximum-mass posterior shown in Fig. 1 and should therefore accompany the informationpresented in Fig. 2. On the other hand, Fig. 5 refers to the case when the genetic algorithm is forced to consider π TOV = . π (cid:12) .In this case, the maximum-mass is set to vary uniformly in the very small interval around π TOV = . + . β . π (cid:12) , leaving all theother parameters free to be adjusted till a best-ο¬t is found. In this sense, the information in Fig. 5 complements what is reported inFig. 3 and shows that all the posterior distributions are pushed to be very narrow at the edges of the allowed ranges. For instance,the dimensionless spin π is narrowly peaked around its minimum value 1 .
1, the mass in the disk is much smaller and of the orderof (cid:39) . π (cid:12) , while the blue and red ejecta are comparable and equal to (cid:39) . π (cid:12) .Note that to avoid having a large number of small panels, we have limited ourselves either to the most salient ones, omittingthose quantities for which the distributions are either almost constant or restricted to a very small region. More speciο¬cally, inFig. 4 the values found are: π = . + . β . , π = . + . β . , π dynej = . + . β . π (cid:12) , π disk = . + . β . , and π = . + . β . ,which corresponds to π = . + . β . . We have also explored a modiο¬ed scenario in which the blue ejecta are larger thaninferred from observations. In particular, we have adjusted the upper bound on the blue ejecta from π blueej / π (cid:12) < .
02 to π blueej / π (cid:12) < .
05. In this case, we ο¬nd that the blue ejecta converge to a distribution with a median around βΌ . π (cid:12) , while thered ejecta component decreases to βΌ . π (cid:12) . At the same time, the changes in the posterior for the maximum mass are minute,i.e., π TOV = . + . β . π (cid:12) .Finally, in Fig. 6 we provide information that is similar in content to that in Fig. 3, but when we allow for the dimensionlessspin to attain even smaller values, i.e., 1 . β€ π (cid:46) .
22. Note that in this case, the ejected mass for π TOV = . π (cid:12) is within theobservational bounds, but the excess in radiated mass is more severe. The disagreement becomes even stronger for π TOV = . π (cid:12) .As a concluding remark we note that the interpretation of the nature of GW190425 is likely unaο¬ected by our ο¬ndings on themaximum masses of neutron stars. While a BH-NS nature cannot be fully ruled out, the most plausible case of a NS-NS natureof the system is perfectly compatible with our ο¬ndings on the maximum mass, as the initial masses in GW190425 are both wellbelow the maximum-mass limit we have presented here (see also Most et al. 2020a, for a discussion on GW190425). On the otherhand, an indirect impact that our results have on GW190425 is on whether the merger led to a prompt collapse (i.e., where thehypermassive neutron star collapses to a black hole either at or shortly after merger), or to a stable long lived remnant. Using theresults of Koeppel et al. (2019) (but see also Bauswein et al. 2017), and given the values for the maximum mass found here, aprompt or delayed collapse scenario seems likely for GW190425.W170817 and GW190814: tension on the maximum mass 9 Ο = . +0 . β . . . . M b , d i s k M b, disk = . +0 . β . . . . M b l u ee j M blueej = . +0 . β . . . M r e d e j M redej = . +0 . β . . . . M T O V M TOV = . +0 . β . .
12 1 .
16 1 . Ο . . j d i s k / j r e m . . . M b, disk .
010 0 .
015 0 . M blueej .
06 0 . M redej . . . M TOV . . j disk /j rem j disk /j rem = . +0 . β . Figure 4.
Corner plot reporting the posterior distributions of the most important parameters in our analysis. Indicated with the two outermostvertical dashed lines are the corresponding 2- π values, while the labels on the diagonal cells report the average values (central vertical dashedline). Ο = . +0 . β . . . . M b , d i s k M b, disk = . +0 . β . . . . M b l u ee j M blueej = . +0 . β . . . . M r e d e j M redej = . +0 . β . . . . M T O V M TOV = . +0 . β . .
12 1 .
16 1 . Ο . . j d i s k / j r e m .
06 0 .
12 0 . M b, disk .
010 0 .
015 0 . M blueej .
03 0 .
06 0 . M redej . . . M TOV . . j disk /j rem j disk /j rem = . +0 . β . Figure 5.
Same as Fig. 4 but when the maximum mass is held ο¬xed at the value π TOV = . π (cid:12) . W170817 and GW190814: tension on the maximum mass 11 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . M GW [ M (cid:12) ]050100150200250 p r o b a b ili t y d i s t r i bu t i o n f un c t i o n NR upper limiton M tot GW M TOV = 2 . M (cid:12) M TOV = 2 . M (cid:12) .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 .
06 0 .
07 0 . M totej [ M (cid:12) ]050100150200 P r o b a b ili t y d i s t r i bu t i o n f un c t i o n M totej from GW170817kilonova M TOV = 2 . M (cid:12) M TOV = 2 . M (cid:12) Figure 6.
The same as Fig. 3 but when we allow for the dimensionless spin to attain even smaller values, i.e., 1 . β€ π (cid:46) .
22. Note that inthis case disagreement in the radiated and ejected mass becomes even stronger for π TOV = . π (cid:12)(cid:12)