Implications of PREX-II on the equation of state of neutron-rich matter
Brendan T. Reed, F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz
IImplications of PREX-II on the equation of state of neutron-rich matter
Brendan T. Reed,
1, 2, ∗ F. J. Fattoyev, † C. J. Horowitz, ‡ and J. Piekarewicz § Department of Astronomy, Indiana University, Bloomington, Indiana 47405, USA Center for Exploration of Energy and Matter and Department of Physics,Indiana University, Bloomington, IN 47405, USA Department of Physics, Manhattan College, Riverdale, NY 10471, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA (Dated: February 25, 2021)Laboratory experiments sensitive to the equation of state of neutron rich matter in the vicinityof nuclear saturation density provide the first rung in a “density ladder” that connects terrestrialexperiments to astronomical observations. In this context, the neutron skin thickness of
Pb( R ) provides the most stringent laboratory constraint on the density dependence of the symmetryenergy. In turn, the cleanest and most precise value of R has been reported recently by the PREXcollaboration. Exploiting the strong correlation between R and the slope of the symmetry energy L , we report a value of L = (106 ±
37) MeV—that systematically overestimates current limits based onboth theoretical approaches and experimental measurements. The impact of such a stiff symmetryenergy on some critical neutron-star observables is also examined.
PACS numbers: 21.60.Jz, 24.10.Jv, 26.60.Kp, 97.60.Jd
The updated Lead Radius EXperiment (PREX-II) hasdelivered on the promise to determine the neutron radiusof
Pb with a precision of nearly 1%. By combiningthe original PREX result [1, 2] with the newly announcedPREX-II measurement, the following value for the neu-tron skin thickness of
Pb was reported [3]: R skin = R n − R p = (0 . ± . , (1)where R n and R p are the root-mean-square radii of theneutron and proton density distributions, respectively.Such a clean—purely electroweak—measurement is ofcritical importance in constraining both models of nu-clear structure as well as the equation of state (EOS) ofneutron-rich matter in the vicinity of nuclear saturationdensity ( ρ ≈ .
15 fm − ). In turn, the EOS around satura-tion density provides the first rung in a “density ladder”that connects laboratory experiment to astronomical ob-servations that probe the EOS at higher densities. It isthe aim of this letter to explore the impact of PREX-II oncertain fundamental parameters of the EOS that, in turn,dictates the behavior of several neutron-star observables.For two decades the neutron skin thickness of Pb hasbeen identified as an ideal laboratory observable to con-strain the EOS of neutron rich matter, particularly thepoorly determined density dependence of the symmetryenergy [4–7]. The EOS of infinite nuclear matter at zerotemperature is enshrined in the energy per particle whichdepends on both the conserved neutron ( ρ n ) and proton( ρ p ) densities; here we assume that the electroweak sec-tor has been “turned off ”. Moreover, it is customary to ∗ Electronic address: [email protected] † Electronic address: ff[email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] separate the EOS into two contributions, one that repre-sents the energy of symmetric ( ρ n = ρ p ) nuclear matterand another one that accounts for the breaking of thesymmetry. That is, EA ( ρ, α ) − M ≡ E ( ρ, α ) = E SNM ( ρ )+ α S ( ρ )+ O ( α ) , (2)where ρ = ( ρ n + ρ p ) is the total baryon density given by thesum of neutron and proton densities, and α = ( ρ n − ρ p ) /ρ is the neutron-proton asymmetry. The first-order correc-tion to the energy of symmetric nuclear matter E SNM ( ρ )is encoded in the symmetry energy S ( ρ ). The symmetryenergy quantifies the increase in the energy per particle ofinfinite nuclear matter for systems with an isospin imbal-ance (e.g., more neutrons than protons). Further, giventhe preeminent role of nuclear saturation, the energy ofsymmetric nuclear matter and the symmetry energy maybe described in terms of a few bulk parameters that char-acterize their behavior around saturation density. In thisletter we focus on the density dependence of the symme-try energy [8]: S ( ρ ) = J + L ( ρ − ρ )3 ρ + . . . (3)The first term ( J ) represents the correction to the bind-ing energy of symmetric nuclear matter, whereas the sec-ond term ( L ) dictates how rapidly the symmetry energyincreases with density. It is the slope of the symmetry en-ergy L that displays a strong correlation to the neutronskin thickness of Pb. Given that symmetric nuclearmatter saturates, namely, its pressure vanishes at sat-uration, the slope of the symmetry energy L is closelyrelated to the pressure of pure neutron matter at satura-tion density. That is, P PNM ( ρ ) ≈ Lρ . (4) a r X i v : . [ nu c l - t h ] F e b R skin (fm) L ( ρ )[ M e V ] ρ = ρ ρ =(2/3) ρ . . (a) (b)
00 5050 100100 150150 200200 250250
L(MeV) χ EFT(2020) χ EFT(2013)Ab-initio(CC)Skins(Sn)QMC α D (RPA) (106 ±37) MeV
FIG. 1: (Color online)
Left : Slope of the symmetry energy atnuclear saturation density ρ (blue upper line) and at (2 / ρ (green lower line) as a function of R . Right : Gaussianprobability distribution for the slope of the symmetry energy L = L ( ρ ) inferred by combining the linear correlation in theleft figure with the recently reported PREX-II limit. The sixerror bars are constraints on L obtained by using differenttheoretical approaches. To assess the impact of the combined PREX–PREX-II measurements (henceforth referred simply as “PREX-II”)—we provide predictions for several observables usinga set of 16 covariant energy density functionals. Theseare FSUGold2 [9] together with a set of eight systemat-ically varied interactions—FSUGold2–L047, L050, L054,L058, L069, L076, L090, L100—with identical isoscalarproperties as FSUGold2, but isovector properties definedby the associated value of the slope of the symmetry en-ergy L . For example, FSUGold2=FSUGold2–L113 pre-dicts a slope of the symmetry energy of L = 113 MeV.Another set of accurately calibrated density functionals isgiven by RMF012, RMF016, RMF022, and RMF032 [10],where now the labels are associated to the predictedvalue of R . For example, RMF032 predicts a neu-tron skin thickness of R = 0 .
32 fm. Finally, TFa, TFb,and TFc, with R = 0 . , . , and 0 .
33 fm, respectively,were created to test whether the large central value of R = 0 .
33 fm originally reported by the PREX collabo-ration [1] was incompatible with other laboratory exper-iments and/or astrophysical observations [11]. We foundthen, that there was no compelling reason to rule outmodels with large neutron skins. This finding has nowbeen validated by the new PREX-II measurement.The strong correlation between L and R in the con-text of the new PREX-II measurement is illustrated inFig. 1. The left-hand panel displays the well-known cor-relation between the slope of the symmetry energy at sat-uration density and the neutron-skin thickness of Pb.Also shown in Fig. 1(a) is the even stronger correlationbetween R skin and the slope of the symmetry energy atthe slightly lower density of ρ = (2 / ρ ≈ . − [5, 12–16]. At such a lower density, which represents an averagevalue between the central and surface densities, the sym-metry energy is well constrained by the binding energy of heavy nuclei with a significant neutron excess. Rely-ing on the strong R skin - L correlation together with theimproved PREX-II limit, one obtains the gaussian proba-bility distribution for L displayed in Fig. 1(b). Using thesame analysis on both J and (cid:101) L —the latter representingthe slope of the symmetry energy at ρ = (2 / ρ —we de-rive the following limits: J = (38 . ± . , (5a) L = (105 . ± . , (5b) (cid:101) L = (71 . ± . . (5c)As indicated in Fig. 1(b), these limits are systematicallylarger than those obtained using either purely theoreti-cal approaches or extracted from a theoretical interpre-tation of experimental data [16–23]. However, we notethat theoretical interpretations of elastic nucleon-nucleusscattering cross sections together with quasielastic re-actions to isobaric analog states obtained limits on L that are consistent with our findings [24]. We also un-derscore that the models used in this letter represent aparticular class of relativistic EDFs. If one expands theset of models as in Ref. [7], then the inferred value of L is even larger. The PREX-II result is also consider-ably larger—and in many cases incompatible—with ex-perimental determinations of R by methods that arehighly model dependent [25–28]. A notable exception isthe dispersive optical model analysis of the WashingtonUniversity group that reported a neutron skin thicknessof R = (0 . ± .
05) fm [29]; a revised lower value of R = (0 . ± .
07) fm—still consistent with [29]—was re-ported shortly thereafter in Ref. [30].To further underscore the tension between PREX-IIand our current understanding of the EOS, we displayin Fig. 2 a summary of simultaneous constraints on both J and L as reported in Refs. [18, 31]. We have adaptedFigure 2 from Ref. [18] by including the PREX-II limitson both J and L derived in Eq.(5). Note that with theexception of the analysis of Ref. [20], all other approachessuggest a positive correlation between L and J . In thecontext of density functional theory, such a positive cor-relation is easy to understand. Using Eq.(3) at a densityof (cid:101) ρ = 2 ρ / S ( (cid:101) ρ ) = J − L → J ≈ (cid:18)
26 MeV + L (cid:19) . (6)The value of S ( (cid:101) ρ ) ≈
26 MeV [12] follows because the sym-metry energy at (cid:101) ρ is tightly constrained by the bindingenergy of heavy nuclei. The PREX-II inferred value for L yields a corresponding value of J = (37 . ± .
10) MeV,that is entirely consistent with the limit obtained inEq.(5). Note that the “Intersection” region in Fig. 2 ob-tained from a variety of experimental and theoretical ap-proaches lies outside the 1 σ PREX-II limits.Next, we explore the impact of PREX-II on a fewneutron-star observables. We start by displaying in Fig. 3the minimum central density and associated neutron star
26 30 34 38 42
J (MeV) L ( M e V ) PREX-II quantum chromodynamics, c s ð n ≫ n Þ ¼ [63]. Theuncertainties, however, are sizeable at the maximumdensity: c s ð n Þ ≃ . $ . (N LO) and c s ð n Þ ≃ . $ . (N LO). Precise measurements of neutronstars with mass ≳ M ⊙ [64 –
67] indicate that the limithastobeexceededinsomedensityregimebeyond n [68].Our σ uncertainty bands are consistent with this happen-ing slightly above n , especially since the downward turnof c s ( n ≳ . fm − ) is likely an edge effect that willdisappear if we train on data at even higher densities. Comparison to experiment. — Figure 2 depicts con-straints in the S v – L plane. The allowed region we derivefrom χ EFT calculations of infinite matter is shown asthe yellow ellipses (dark: σ , light: σ ) and denoted “ GP-B ” (Gaussian process – BUQEYE collaboration).Also shown are several experimental and theoretical con-straints compiled by Lattimer et al. [69 – ð S v ;L j D Þ , where the training data D are the order-by-order predictions of ð E=N Þð n Þ and ð E=A Þð n Þ up to n .The distribution is accurately approximated by a two-dimensional Gaussian with mean and covariance ! μ S v μ L " ¼ ! . . " and Σ ¼ ! . . .
27 4 . " : ð Þ We consider all likely values of n via pr ð S v ;L j D Þ ¼ R pr ð S ;L j n ; D Þ pr ð n j D Þ dn . Here, pr ð S ;L j n ; D Þ describes the correlated to-all-orders predictions at a par-ticular density n , and pr ð n j D Þ ≈ . $ . fm − is theGaussian posterior for the saturation density, includingtruncation errors, determined in Ref. [28]. If the canonicalempirical saturation density, n ¼ . fm − , is usedinstead the posterior mean shifts slightly downwards: S v → S v − . MeV and L → L − . MeV. This shift is wellwithin the uncertainties computed using our internallyconsistent n . In contrast to experiments, which extract S v – L from measurements over a range of densities, ourtheoretical approach predicts directly at saturation density,thereby removing artifacts induced by extrapolation.Our σ ellipse falls completely within constraintsderived from the conjecture that the unitary gas is a lowerlimit on the EOS [69] (solid black line). The same workalso made additional simplifying assumptions to derive ananalytic bound — only our σ ellipse is fully within thatregion(dashedblackline).Figure2alsoshowstheallowedregions obtained from microscopic neutron-mattercalculations by Hebeler et al. [79] (based on χ EFT NN and3Ninteractionsfittofew-bodydataonly)andGandolfi et al. [80] (where 3N interactions were adjusted to a rangeof S v ). The predicted ranges in S v agree with ours, but wefind that L is ≈ MeV larger, corresponding to a strongerdensity-dependence of S ð n Þ . References [79,80] quoterelatively narrow ranges for S v – L , but those come fromsurveying available parameters inthe Hamiltonians andso,unlike our quoted intervals, do not have a statisticalinterpretation. Summary and outlook. — We presented a novel frame-work for EFT truncation errors that includes correlations
FIG. 2. Constraints on the S v – L correlation. Our results( “ GP – B ” ) are given at the 68% (dark-yellow ellipse) and 95%level (light-yellow ellipse). Experimental constraints are derivedfromheavy-ioncollisions(HIC)[72],neutron-skinthicknessesofSnisotopes[73],giantdipoleresonances (GDR)[74],thedipolepolarizability of Pb [75,76], and nuclear masses [77]. Theintersection is depicted by the white area, which only barelyoverlaps with constraints from isobaric analog states and iso-vectorskins(IAS þ Δ R )[78].Inaddition,theoreticalconstraintsderived from microscopic neutron-matter calculations byHebeler et al. (H) [79] and Gandolfi et al. (G) [80] as well asfromtheunitarygas(UG)limitbyTews etal. [69].Thefigurehasbeen adapted from Refs. [70,71]. A Jupyter notebook thatgenerates it is provided in Ref. [42]. PHYSICAL REVIEW LETTERS
FIG. 2: (Color online). Constraints on the J – L correlationobtained from a variety of experimental and theoretical ap-proaches. The figure was adapted from Refs. [18, 31] and no-ticeably displays the tension with the recent PREX-II result. mass required for the onset of the direct Urca process.Neutron stars are born very hot ( T (cid:39) K (cid:39)
10 MeV)and then cool rapidly via neutrino emission through thedirect Urca process that involves neutron beta decay fol-lowed by electron capture: n → p + e − + ¯ ν e , (7a) p + e − → n + ν e . (7b)After this rapid cooling phase is completed, neu-trino emission proceeds in the standard cooling scenariothrough the modified Urca process—a process that maybe millions of times slower as it requires the presenceof a bystander nucleon to conserve momentum at theFermi surface[32]. The transition into the much slowermodified Urca process is solely based on the expectationthat the proton fraction in the stellar core is too lowto conserve momentum at the Fermi surface. However,given that the proton fraction is controlled by the poorlyknown density dependence of the symmetry energy [33],the minimal cooling scenario may need to be revisited.In particular, a stiff symmetry energy—as suggested byPREX-II—favors large proton fractions that may triggerthe onset of the direct Urca process at lower central densi-ties. This analysis is particularly timely given that x-rayobservations suggest that some neutron stars may requiresome form of enhanced cooling. Indeed, the detected x-ray spectrum of the neutron star in the low-mass x-raybinary MXB 1659-29 strongly suggests the need for a fastneutrino-cooling process [34]. For a comprehensive re-port that explores the interplay between the direct Urcaprocess and nucleon superfluidity in transiently accreting R skin (fm) DU r ca T h r e s ho l d ρ ★ (fm -3 ) Μ ★ /M sun - . - . - . PREX-II
FIG. 3: (Color online). Direct Urca thresholds for the onsetof enhanced cooling in neutron stars. The threshold density isdepicted by the lower blue line and the corresponding stellarmass for such a central density with the upper green line.The shaded area represents PREX-II 1 σ confidence region.For each of these two quantities, the best-fit line is displayedtogether with their associated correlation coefficients. neutron stars, see Ref. [35]. The shaded area in Fig. 3 dis-plays the region constrained by PREX-II. In particular,the 1 σ lower limit of R skin = 0 .
212 fm suggests a thresholdmass for the onset of direct Urca cooling of M (cid:63) ≈ . M (cid:12) and a corresponding central density of ρ (cid:63) ≈ .
42 fm − .However, if instead one adopts the larger PREX-II cen-tral value of R skin = 0 .
283 fm, then one obtains the con-siderably lower threshold values of M (cid:63) ≈ . M (cid:12) and ρ (cid:63) ≈ .
24 fm − , or a threshold density just slightly higherthan saturation density. Although some stars are likelyto require enhanced cooling, observations of many iso-lated neutron stars are consistent with the much slowermodified URCA process [36]. This may be because thedirect URCA neutrino emissivity is reduced by nucleonpairing.We close the section by displaying in Fig. 4 the dimen-sionless tidal polarizability of a 1.4 M (cid:12) neutron star asa function of both the stellar radius R . (cid:63) and R skin . Al-though not shown, for the set of density functionals usedin this work a very strong correlation (of about 0.98) isobtained between R . (cid:63) and R skin . However, because thecentral density of a 1.4 M (cid:12) neutron star may reach den-sities as high as 2-to-3 times saturation density, the ro-bustness of such a correlation should be examined in thecontext of alternative theoretical descriptions. Moreover,a precise knowledge of the EOS of the crust is neededto minimize possible systematic uncertainties [37]. Asin Fig. 3, the 1 σ confidence region is indicated by theshaded area in the figure. Also shown are NICER con-strains on the radius of PSR J0030+0451 [38, 39], thatare depicted by the two horizontal error bars and which PREX-II
Allowed . NICER R skin (fm) Λ ★ . R ★ (km) FIG. 4: (Color online). Showcase of neutron star observablesas a function of R as predicted by the set of energy densityfunctionals considered in this work. The tidal polarizabilityΛ . (cid:63) of a 1.4 M (cid:12) neutron star is displayed with blue dots andconnected by a best fit power law that scales as the 4 . ≈ R . The combined PREX-II result together withNICER constraints on the stellar radius is depicted by thesmall (blue) window of models allowed. suggest an upper limit of R . (cid:63) ≤ .
26 km. Invoking thestrong R . (cid:63) – R skin correlation observed in our models,one obtains an upper limit on the neutron skin thicknessof R skin (cid:46) .
31 fm and a lower limit on the stellar radius of R . (cid:63) (cid:38) .
25 km. The region that satisfies both PREX-IIand NICER constraints is indicated by the narrow (blue)rectangle in Fig. 4, which excludes a significant numberof models. In turn, given that the tidal deformabilityapproximately scales with the fifth power of the stellarradius [40], one can also set limits on the tidal deforma-bility of a 1 . M (cid:12) neutron star. Indeed, based on thesecombined constraints one obtains:0 . (cid:46) R skin (fm) (cid:46) .
31 (8a)13 . (cid:46) R . (cid:63) (km) (cid:46) .
26 (8b)642 (cid:46) Λ . (cid:63) (cid:46) . (8c)The allowed region for the tidal deformability falls com-fortably within the Λ . (cid:63) (cid:46)
800 limit reported in the GW170817 discovery paper [41]. Yet, the revised limitof Λ . = 190 +390 − (cid:46)
580 [42] presents a more seriouschallenge. To confirm whether this tension is real, itwill require a multi-prong approach involving a moreprecise determination of R , additional NICER ob-servations, and more multi-messenger detections of neu-tron star mergers. The prospect of a more precise elec-troweak determination of R is challenging as it mayrequire the full operation of the future Mainz Energy-recovery Superconducting Accelerator (MESA) which isforeseen to start until 2023 [43]. Future determinationsof stellar radii by NICER for neutron stars with knownmasses, such as J0437-4715 [44], could be made at a ± ± . R [28]. By assessingthe impact of PREX-II at higher densities, we were ableto provide limits on both the radius and deformabilityof a 1.4 M (cid:12) neutron star. Given that our analysis of thetidal deformability reveals some tension with the revisedlimit of Λ . (cid:46)
580 [42], we eagerly await the next gener-ation of terrestrial experiments and astronomical obser-vations to verify whether the tension remains. If so, thesoftening of the EOS at intermediate densities, togetherwith the subsequent stiffening at high densities requiredto support massive neutron stars, may be indicative of aphase transition in the stellar core [40].
Acknowledgments
This material is based upon work supported bythe U.S. Department of Energy Office of Science,Office of Nuclear Physics under Awards DE-FG02-87ER40365 (Indiana University), Number DE-FG02-92ER40750 (Florida State University), and Number DE-SC0008808 (NUCLEI SciDAC Collaboration). [1] S. Abrahamyan, Z. Ahmed, H. Albataineh, K. Aniol,D. S. Armstrong, et al., Phys. Rev. Lett. , 112502(2012).[2] C. J. Horowitz, Z. Ahmed, C. M. Jen, A. Rakhman, P. A.Souder, et al., Phys. Rev.
C85 , 032501 (2012).[3] D. Adhikari et al. (2021), 2102.10767. [4] B. A. Brown, Phys. Rev. Lett. , 5296 (2000).[5] R. J. Furnstahl, Nucl. Phys. A706 , 85 (2002).[6] M. Centelles, X. Roca-Maza, X. Vi˜nas, and M. Warda,Phys. Rev. Lett. , 122502 (2009).[7] X. Roca-Maza, M. Centelles, X. Vi˜nas, and M. Warda,Phys. Rev. Lett. , 252501 (2011). [8] J. Piekarewicz and M. Centelles, Phys. Rev.
C79 , 054311(2009).[9] W.-C. Chen and J. Piekarewicz, Phys. Rev.
C90 , 044305(2014).[10] W.-C. Chen and J. Piekarewicz, Phys. Lett.
B748 , 284(2015).[11] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. Lett. ,162501 (2013).[12] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. ,5647 (2001).[13] C. Ducoin, J. Margueron, and C. Providencia, Europhys.Lett. , 32001 (2010).[14] C. Ducoin, J. Margueron, C. Providencia, and I. Vidana,Phys.Rev. C83 , 045810 (2011).[15] C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch,R. Michaels, et al., J. Phys.
G41 , 093001 (2014).[16] Z. Zhang and L.-W. Chen, Phys. Lett.
B726 , 234 (2013).[17] K. Hebeler, J. Lattimer, C. Pethick, and A. Schwenk,Astrophys. J. , 11 (2013).[18] C. Drischler, R. Furnstahl, J. Melendez, and D. Phillips,Phys. Rev. Lett. , 202702 (2020).[19] G. Hagen et al., Nature Phys. , 186 (2015).[20] L.-W. Chen, C. M. Ko, B.-A. Li, and J. Xu, Phys.Rev. C82 , 024321 (2010).[21] A. Steiner and S. Gandolfi, Phys.Rev.Lett. , 081102(2012), 1110.4142.[22] S. Gandolfi, J. Carlson, S. Reddy, A. Steiner, andR. Wiringa, Eur. Phys. J.
A50 , 10 (2014), 1307.5815.[23] X. Roca-Maza, X. Vi˜nas, M. Centelles, B. K. Agrawal,G. Col`o, N. Paar, J. Piekarewicz, and D. Vretenar, Phys.Rev.
C92 , 064304 (2015).[24] P. Danielewicz, P. Singh, and J. Lee, Nucl. Phys. A ,147 (2017).[25] A. Trzci´nska, J. Jastrzebski, P. Lubi´nski, F. Hartmann,R. Schmidt, et al., Phys.Rev.Lett. , 082501 (2001).[26] J. Zenihiro, H. Sakaguchi, T. Murakami, M. Yosoi, Y. Ya-suda, et al., Phys.Rev. C82 , 044611 (2010). [27] C. Tarbert et al., Phys. Rev. Lett. , 242502 (2014).[28] M. Thiel, C. Sfienti, J. Piekarewicz, C. J. Horowitz, andM. Vanderhaeghen, J. Phys.
G46 , 093003 (2019).[29] M. C. Atkinson, M. H. Mahzoon, M. A. Keim, B. A. Bor-delon, C. D. Pruitt, R. J. Charity, and W. H. Dickhoff,Phys. Rev. C , 044303 (2020).[30] C. D. Pruitt, R. J. Charity, L. G. Sobotka, M. C. Atkin-son, and W. H. Dickhoff, Phys. Rev. Lett. , 102501(2020).[31] J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. , 485 (2012).[32] D. Page, J. M. Lattimer, M. Prakash, and A. W. Steiner,Astrophys. J. Suppl. , 623 (2004).[33] C. J. Horowitz and J. Piekarewicz, Phys. Rev. C66 ,055803 (2002).[34] E. F. Brown, A. Cumming, F. J. Fattoyev, C. Horowitz,D. Page, and S. Reddy, Phys. Rev. Lett. , 182701(2018).[35] A. Potekhin, A. Chugunov, and G. Chabrier, Astron.Astrophys. , A88 (2019).[36] D. Page, J. M. Lattimer, M. Prakash, and A. W. Steiner,Astrophys. J. , 1131 (2009).[37] J. Piekarewicz, F. J. Fattoyev, and C. J. Horowitz, Phys.Rev.
C90 , 015803 (2014).[38] T. E. Riley et al., Astrophys. J. Lett. , L21 (2019).[39] M. C. Miller et al., Astrophys. J. Lett. , L24 (2019).[40] F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys.Rev. Lett. , 172702 (2018).[41] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 161101 (2017).[42] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 161101 (2018).[43] D. Becker et al. (2018), 1802.04759.[44] D. Reardon et al., Mon. Not. Roy. Astron. Soc. , 1751(2016).[45] H. T. Cromartie et al., Nat. Astron.4