GW190814: Impact of a 2.6 solar mass neutron star on nucleonic equations of state
GGW190814: Impact of a 2.6 solar mass neutron staron nucleonic equations of state
F. J. Fattoyev, ∗ C. J. Horowitz, † J. Piekarewicz, ‡ and Brendan Reed § Department of Physics, Manhattan College, Riverdale, NY 10471, USA Center for Exploration of Energy and Matter and Department of Physics,Indiana University, Bloomington, IN 47405, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA Department of Astronomy, Indiana University, Bloomington, Indiana 47405, USA (Dated: July 9, 2020)Is the secondary component of GW190814 the lightest black hole or the heaviest neutron starever discovered in a double compact-object system [R. Abbott et al.
ApJ Lett., 896, L44 (2020)]?This is the central question animating this letter. Covariant density functional theory providesa unique framework to investigate both the properties of finite nuclei and neutron stars, whileenforcing causality at all densities. By tuning existing energy density functionals we were able to:(a) account for a 2 . M (cid:12) neutron star, (b) satisfy the original constraint on the tidal deformabilityof a 1 . M (cid:12) neutron star, and (c) reproduce ground-state properties of finite nuclei. Yet, for theclass of models explored in this work, we find that the stiffening of the equation of state required tosupport super-massive neutron stars is inconsistent with either constraints obtained from energeticheavy-ion collisions or from the low deformability of medium-mass stars. Thus, we speculate thatthe maximum neutron star mass can not be significantly higher than the existing observational limitand that the 2 . M (cid:12) compact object is likely to be the lightest black hole ever discovered. PACS numbers: 21.60.Jz, 24.10.Jv, 26.60.Kp, 97.60.Jd
The first direct detection of gravitational waves fromthe binary collision of two black holes launched the newera of gravitational-wave astronomy [1].Two years later,the detection of gravitational waves from GW170817—a binary neutron star merger [2]—in association with itselectromagnetic counterpart [3–6], greatly advanced mul-timessenger astronomy. And two years after GW170817,the LIGO-Virgo collaboration continues to mesmerize thephysics community after reporting the detection of grav-itational waves from the coalescence of a binary systemwith the most extreme mass ratio ever observed: a 23solar mass black hole and a 2.6 solar mass “compact” ob-ject [7]. Although data from all three instruments (LIGO-Livingston, LIGO-Hanford, and Virgo) allowed good skylocalization of the source, no electromagnetic counter-part has been reported. Moreover, unlike GW170817, nomeasurable tidal signature was imprinted on the gravi-tational waveform, which seems consistent with the rela-tively large mass of the 2.6 M (cid:12) compact object. Hence,one is left speculating whether the compact object is ei-ther the most massive neutron star or the lightest blackhole ever discovered [8, 9].The discovery paper suggests that, based on severalcurrent estimates of the maximum neutron star mass,“GW190814 is unlikely to originate in a neutron star-black hole (NSBH) coalescence” [7]. Yet the paper leavesopen the possibility that improved knowledge of the equa- ∗ Electronic address: ff[email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] tion of state or further observations could alter this as-sessment. The absence of super-massive neutron starsis consistent with the analysis by Margalit and Metzgerwho argue against their formation based on the lack of ev-idence of a large amount of rotational energy in the ejectaduring the spin-down phase of GW170817 [10]. Interest-ingly, the suggested upper limit of M max (cid:46) . M (cid:12) [10] isin full agreement with the recent observation by Cromar-tie and collaborators of the heaviest 2 . +0 . − . M (cid:12) neu-tron star to date [11]. However, given that one can notdefinitively exclude the existence of super-massive neu-tron stars, we explore here the impact of a 2.6 M (cid:12) neu-tron star on nucleonic equations of state, particularly inthe framework of covariant density functional theory.It has been known for more than two decades thatthe class of covariant energy density functionals (EDFs)used in this work can reproduce nuclear observables atnormal nuclear densities and also generate neutron starswith maximum masses that differ by more than one so-lar mass [12]. Hence, stable neutron stars with 2.6 M (cid:12) —and even higher—can be readily generated. The chal-lenge, however, is not to reconcile super-massive neutronstars with the properties of finite nuclei, but rather, withneutron-star properties that are sensitive to the equationof state (EOS) at two-to-three times nuclear densities,such as stellar radii and tidal deformabilities that favora rather soft EOS [2, 13].In particular, GW170817 has provided stringent con-straints on the EOS of neutron rich matter at a few timesnuclear densities from the determination of the tidal de-formability of a M = 1 . M (cid:12) neutron star [13–27]. Con-straints from GW170817 seem to favor compact starswith relatively small stellar radii, suggesting a relatively a r X i v : . [ nu c l - t h ] J u l soft EOS. These constraints are consistent with the re-cent determination of both the mass and radius of PSRJ0030+0451 by the Neutron star Interior CompositionExplorer (NICER) [28, 29]. Pulse-profile modeling of thethermal emission from the pulsar’s hot spots suggest amass of about 1.4 M (cid:12) and a radius of nearly 13 km, witha ±
10% uncertainty on both quantities [28, 29]. Notethat although consistent with GW170817, NICER resultscan accommodate slightly stiffer equations of state. How-ever, a real tension develops as one aims to reconcile arelatively soft EOS as demanded by GW170817, withthe much stiffer EOS required to account for heavy neu-tron stars with masses in the vicinity of 2 M (cid:12) [11, 30, 31].Based on this combined evidence, suggestions have beenmade for the existence quark matter cores in massive neu-tron stars [32]. Although the claim may be premature,after all there are purely nucleonic EOS that satisfy allexperimental and observational constraints to date [15],the prospect of identifying an assumed phase transitionin the stellar cores is exciting. As shown below, the ten-sion is exacerbated if the EOS must be stiffened evenfurther to account for the possible existence of super-massive neutron stars.To test the possible existence of a 2.6 M (cid:12) neutron starwe rely on density functional theory [33]. The energydensity functional employed here is defined in terms of anunderlying Lagrangian density that has been extensivelydiscussed in earlier publications [34–36], so we limit our-selves to highlight those terms of relevance to the high-density component of the equation of state. That is, L = . . . + ζ g ( V µ V µ ) + Λ v (cid:16) g ρ b µ · b µ (cid:17)(cid:16) g V ν V ν (cid:17) . (1)The basic degrees of freedom of the model are neutronsand protons interacting via the exchange of photons and“mesons”. Besides the conventional Yukawa couplings(not shown) the model includes non-linear meson interac-tions that serve to simulate the complicated many-bodydynamics and that are required to improve the predictivepower of the model. The two terms shown in Eq. (1) fallinto this category and are of critical importance to thebehavior of dense, neutron-rich matter. The first term inthe expression describes a quartic self-interaction of theisoscalar-vector field V µ which affects the EOS of sym-metric nuclear matter at high densities [12]. In turn, the last term includes a mixed quartic coupling between V µ and the isovector-vector field b µ . This term was intro-duced to modify the density dependence of the symmetryenergy, which plays a critical role in the structure of bothneutron-rich nuclei and neutron stars [34]. Note thatthe symmetry energy is to a very good approximationequal to the difference in the energy per nucleon betweenpure neutron matter and symmetric nuclear matter. Co-variant density functional theory provides a relativisticconsistent framework as one extrapolates to dense mat-ter as it ensures—unlike non-relativistic formulations—that the EOS remains causal at all densities. Finally,the structure of neutron stars will be explored by en-forcing both charge neutrality and chemical equilibrium.As such, the basic constituents of the model are nucleonsand leptons (both electrons and muons). No “exotic” de-grees of freedom—such as hyperons, meson condensates,or quarks—will be considered.As already alluded, tuning the ζ parameter in Eq. (1)allows one to stiffen the symmetric-matter EOS to pro-duce super-massive neutron stars. For example, two ofthe EDFs used in Ref. [15]—IUFSU [37] and FSUGar-net [38])—that are consistent with both the 2 M (cid:12) con-straint [11, 30, 31] and the tidal deformability of a 1.4 M (cid:12) neutron star [2], can be adjusted to produce maximumstellar masses of at least 2.8 M (cid:12) . However, we find noneed to strain the model to such an extreme, so we tune ζ to produce a maximum neutron star mass of 2.6 M (cid:12) . Re-ducing the ζ parameter requires tuning the other modelparameters. To do so, we start with the bulk propertiespredicted by IUFSU—together with the analytical trans-formation described in [36]—to connect the bulk proper-ties to the model parameters. Next, we slightly decreasethe value of the slope of the symmetry energy to obtaina neutron skin of 0 .
15 fm for
Pb—and adjusted thevalue of the incompressibility coefficient to K = 227 MeVto maintain agreement with the centroid energy of thegiant monopole resonance in Pb. Finally, we re-adjustthe values of m s , the saturation density ρ , and the bind-ing energy at saturation (cid:15) to ensure that the bindingenergies and charge radii of both Ca and
Pb remainintact. We refer to this nuclear EDF as “BigApple” anddisplay its parameters, as defined in Ref. [36], in Table Ialongside the other two covariant EDFs used in this work.
Model m s g g g ρ κ λ ζ Λ v IUFSU 491.500 99.4266 169.8349 184.6877 3.38081 +0.000296 0.03000 0.046000FSUGarnet 496.939 110.3492 187.6947 192.9274 3.26018 − − m s and κ are in MeV. Thevector-meson masses are fixed at m v = 782 . m ρ = 763 . M = 939 MeV. Nucleus
B/A (MeV) R ch (fm) R skin (fm) Ca 8.552 (8.551) 3.452 (3.478) − . Ca 8.536 (8.666) 3.476 (3.477) 0 . Ni 8.643 (8.682) 3.875 (3.887) 0 . Zr 8.666 (8.710) 4.255 (4.269) 0 . Sn 8.294 (8.355) 4.708 (4.709) 0 . Pb 7.868 (7.867) 5.503 (5.501) 0 . (cid:0) . +0 . − . (cid:1) TABLE II: Theoretical predictions alongside experimentaldata (in parentheses) for the binding energy per nucleon [40]and charge radii [41] for a representative set of doubly-magicand semi-magic nuclei; the experimental charge radius of Niis from Ref. [42]. The last column displays predictions forthe neutron skin thickness. With the exception of
Pb [43],no electroweak measurements of neutron skins are presentlyavailable.
One of the central tenets of nuclear density functionaltheory is to provide a “universal” EDF that can repro-duce nuclear observables over an enormous range of den-sities and isospin asymmetries. The goal is to build anuclear EDF that can be used to explore both the prop-erties of finite nuclei as well as the structure of neutronstars—dynamical objects that differ in length scales bymore that 18 orders of magnitude. In this context, wedisplay in Table II binding energies and charge radii aspredicted by the BigApple. Although the agreement be-tween theory and experiment is fairly good, it is certainlynot as good as some modern nuclear EDFs that have beenfitted to a host of nuclear observables.Before moving on to discuss neutron star properties,we display in Table III some bulk properties of infiniteneutron-rich matter that encode its behavior in the vicin-ity of the saturation density; the definitions are as inRef. [44]. The saturation point of symmetric nuclear mat-ter is defined in terms of the saturation density ρ andthe value of the energy per particle (cid:15) . Because the pres-sure at saturation density vanishes, the rate at whichthe energy per particle increases is controlled by the in-compressibility coefficient K , a quantity that is stronglycorrelated to the centroid energy of the giant monopoleresonance [45]. In turn, J and L are fundamental param-eters of the symmetry energy that encode the increase inthe energy per particle and its density dependence as thesystem becomes neutron rich. In particular, given thatthe pressure of symmetric nuclear matter vanishes at sat-uration, L is closely related to the pressure of pure neu-tron matter at saturation density. As such, it is stronglycorrelated to both the neutron skin thickness of heavynuclei and the radius of neutron stars [46].We observe from Table III that, with the exception of L ,all three models are in very close agreement. It is impor-tant to note that the calibration of FSUGarnet relied ex-clusively on physical observables that can be either mea-sured in the laboratory or extracted from observation, so Model ρ (cid:15) K J L
IUFSU 0.155 − − − J and its slope L at saturation density. All quantities are given in MeV exceptfor ρ which is given in fm − . the bulk properties listed in Table III are genuine modelpredictions.
10 12 14 16 18 R (cid:43) (km) M (cid:43) / M s un RMF022FSUGold2RMF032 C a u s a lit y NL3 B a u s w e i n e t a l F S U G a r n e t IU-FSU T A M U c T A M U bT A M U a J0348 (2013)
J1614 (2010)
J0740 (2019)
NICER (2019)
Fattoyev et al
BigApple
FIG. 1: (Color online). Mass-vs-Radius relation predictedby a variety of covariant EDFs. The figure has been adaptedfrom Ref. [15] and is supplemented by the maximum mass con-straint from Ref. [11], NICER constraint on the stellar radiusof a 1.4 M (cid:12) neutron star [28, 29], and our predictions fromBigApple. Displayed in Fig. 1 is the mass- vs -radius relation aspredicted by a variety of covariant EDFs—including Bi-gApple. The figure has been adapted from Ref. [15] andincludes, besides earlier constraints on the maximumneutron star mass [30, 31], the newest constraint fromCromartie et al. [11] as well as the recent NICER re-sult. All models support a ∼ M (cid:12) neutron star and,with couple of exceptions, agree with the NICER re-sult. Yet, several of these models generate stellar radii formedium mass neutron stars that are inconsistent with thetidal deformability extracted from GW170817 [2]. Note,however, that BigApple—with a maximum neutron-starmass of 2.6 M (cid:12) satisfies all observational constraints; seealso Table IV. Model R . Λ . M max R max Λ max IUFSU 12.528 499.2 1.939 11.265 20.9FSUGarnet 12.869 624.8 2.066 11.706 18.2BigApple 12.960 717.3 2.600 12.417 5.0TABLE IV: Stellar properties as predicted by the three co-variant EDFs used in this work. The maximum mass is givenin solar masses, both radii in km, and the tidal deformabilityis dimensionless. (cid:108) / (cid:108) P ( M e V / f m ) IUFSU FSUGarnetBigApple
NS EOS crust-core (cid:108) / (cid:108) c s FIG. 2: (Color online) Equation of state—pressure as a func-tion of baryon density—for neutron-star matter in chemicalequilibrium. Here ρ (cid:39) .
154 fm − is the density of nuclearmatter at saturation (see Table III), and the inset displaysthe associated speed of sound in units of the speed of light. We now proceed to examine the microphysics respon-sible for the macroscopic properties displayed in Fig. 1.The underlying neutron-star matter EOS alongside thesquare of the speed of sound—defined as the derivative ofthe pressure with respect to the energy density—is dis-played in Fig. 2. The kink in the pressure at around 2 / ρ ) thepressure is generated exclusively by the symmetry en-ergy. This pressure determines the thickness of the neu-tron skin in neutron-rich nuclei [47–50]. In turn, the pres-sure at two-to-three times saturation density appears tocontrols the radius of medium-mass neutron stars [23–25, 51]. Indeed, all three models share the same pres-sure just below 2 ρ and, as consequence, predict similarradii for a canonical 1.4 M (cid:12) neutron star. However, themaximum mass of stable neutron stars stars is highlysensitive to the pressure at the highest densities [27]. In-deed, a dramatic rate of increase in the pressure—bestreflected in the speed of sound—is required to support asuper-massive 2.6 M (cid:12) neutron star. Based on the evidence presented so far, there seems tobe no compelling argument against the possible existenceof a super-massive 2.6 M (cid:12) neutron star. Note, however,that whereas the relatively large tidal deformability pre-dicted by BigApple is consistent with the limits presentedin the discovery paper [2], the revised limit of Λ . =190 +390 − [13] presents a more serious challenge. However,the most serious evidence against such a stiff EOS comesfrom laboratory experiments involving the energetic col-lision between two gold nuclei, a violent encounter thatcompressed matter to pressures in excess of 10 Pa [52];note that 1Pa =10 dyn/cm = 6.242 × − MeV / fm .We display in Fig. 3 constraints on the EOS of sym-metric nuclear matter as extracted from the analysisof particle flow in heavy-ion collisions [52]. We ob-serve that IUFSU and FSUGarnet—with predictions of1.94 M (cid:12) and 2.07 M (cid:12) for the maximum neutron starmass—already sit near the upper edge of the allowed re-gion. In contrast, the very stiff EOS predicted by theNL3 parametrization [53] was explicitly ruled out by theheavy-ion data [52]. Following a similar trend as NL3, itis clear that the stiff EOS predicted by BigApple and re-quired to account for super-massive neutron stars is alsoruled out. Given that FSUGarnet is approaching theupper boundary allowed by the heavy-ion data, it is un-likely that the maximum neutron star mass can go muchbeyond the present observational limit of 2.14 M (cid:12) [11]. ρ / ρ P ( M e V / f m ) Danielewicz et al.
IUFSU FSUGarnetBigAppleNL3
Symmetric Matter
FIG. 3: (Color online) Equation of state—pressure as a func-tion of baryon density—of symmetric nuclear matter. Theblue shaded area represents the EOS extracted from the anal-ysis of flow data using a value of ρ (cid:39) .
16 fm − for the satu-ration density [52]. In summary, motivated by the recent identification ofa compact object in the 2 . − . M (cid:12) mass gap [7], we haveexplored the possibility that such an object could be asuper-massive neutron star. Given the lack of an elec-tromagnetic counterpart to GW190814 and the absenceof tidal distortions to the gravitational waveform, it isunlikely that the nature of the 2.6 M (cid:12) compact objectwill ever be resolved by a further analysis of the data. Inthis letter we have adapted a class of modern covariantEDFs to account for a stable 2.6 M (cid:12) neutron star, whileensuring that earlier constraints on the structure of neu-tron stars as well as ground-state properties of finite nu-clei are accurately reproduced. Indeed, we demonstratedthat such an EDF—dubbed “BigApple”—successfullyaccounts for the binding energy and charge radii of arepresentative set of spherical nuclei, is consistent withthe bulk properties of infinite nuclear matter, reproducesthe recent NICER data, and is compatible with the limitson the tidal deformability of a 1.4 M (cid:12) neutron star, as re-ported in the GW170817 discovery paper. In particular,our predictions are well aligned (see Table IV) with theconstraints extracted from GW190814 under the NSBHscenario: R . = 12 . +0 . − . and Λ . = 616 +273 − [7].However, despite the considerable success of the model,two sets of data strongly disfavor such a stiff EOS. First,the revised upper limit from GW170817 of Λ . = 190 +390 − (at the 90% level) [13] is significantly lower than theΛ . (cid:39)
720 value predicted by BigApple. This would re-quire a softening of the EOS, particularly of the symme-try energy. Second, constraints on the EOS of symmet-ric nuclear matter extracted from particle-flow in high-energy nuclear collisions rule out an overly stiff nuclearmatter EOS. For example, BigApple predicts a pressureat four times saturation density that is nearly twice aslarge as the upper limit extracted from the flow data. In principle, by adding additional interactions at high densi-ties one could soften the EOS of symmetric nuclear mat-ter to bring it into agreement with the heavy-ion dataat the expense of needing to stiffen the symmetry en-ergy. Whereas such a procedure may still result in anoverall EOS that can account for a 2.6 M (cid:12) neutron star,it may require considerable fine tuning to keep the pres-sure of nuclear matter and the tidal deformability lowenough. Hence, we conclude that the low deformabil-ity demanded by GW170817 combined with the heavy-ion data for symmetric nuclear matter make it highlyunlikely that the maximum mass could be as large as2.6 M (cid:12) , at least for the class of models used in this work.So as one is left speculating whether the 2.6 M (cid:12) compactobject in GW190814 is either the most massive neutronstar or the lightest black hole ever detected, our analysispoints strongly in favor of the latter. Acknowledgments
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