Astrophysical Constraints on the Symmetry Energy and the Neutron Skin of ^{208}Pb with Minimal Modeling Assumptions
LLA-UR-21-20527
Astrophysical Constraints on the Symmetry Energy and the Neutron Skin of
Pbwith Minimal Modeling Assumptions
Reed Essick,
1, 2, ∗ Ingo Tews, † Philippe Landry, ‡ and Achim Schwenk
5, 6, 7, § Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5 Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637, USA Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Gravitational-Wave Physics & Astronomy Center, California State University,Fullerton, 800 N State College Blvd, Fullerton, CA 92831 Technische Universität Darmstadt, Department of Physics, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
The symmetry energy and its density dependence are crucial inputs for many nuclear physicsand astrophysics applications, as they determine properties ranging from the neutron-skin thicknessof nuclei to the crust thickness and the radius of neutron stars. Recently, PREX-II reported avalue of 0.29 ± . fm for the neutron-skin thickness of Pb, implying a slope parameter L =110 ± MeV, larger than most ranges obtained from microscopic calculations and other nuclearexperiments. We use a nonparametric equation of state representation based on Gaussian processesto constrain the symmetry energy S , L , and R Pb skin directly from observations of neutron stars withminimal modeling assumptions. The resulting astrophysical constraints from heavy pulsar masses,LIGO/Virgo, and NICER clearly favor smaller values of the neutron skin and L , as well as negativesymmetry incompressibilities. Combining astrophysical data with PREX-II and chiral effective fieldtheory constraints yields S = 34 +3 − MeV, L = 58 +19 − MeV, and R Pb skin = 0 . +0 . − . fm. Introduction–
The symmetry energy S ( n ) is a cen-tral quantity in nuclear physics and astrophysics. Itcharacterizes the change in the nuclear-matter energyas the ratio of protons to neutrons is varied and thusimpacts, e.g., the neutron-skin thickness of nuclei [1–3],their dipole polarizability [4, 5], and the radius of neu-tron stars (NSs) [6, 7]. This information is encoded inthe nuclear equation of state (EOS), described by thenucleonic energy per particle, E nuc /A , a function of to-tal baryon density n and proton fraction x = n p /n forproton density n p . The energy per particle is connectedto the bulk properties of atomic nuclei for proton frac-tions close to x = 1 / , i.e., symmetric nuclear matter(SNM) with E SNM /A = ( E nuc /A ) | x =1 / . As the neutron-proton asymmetry increases (or the proton fraction x decreases) the energy per particle increases, reaching amaximum for x = 0 , i.e., pure neutron matter (PNM)with E PNM /A = ( E nuc /A ) | x =0 . PNM is closely relatedto NS matter. The symmetry energy characterizes thedifference between these two systems: S ( n ) = E PNM A ( n ) − E SNM A ( n ) . (1)Crucial information is encoded in the density depen-dence of S ( n ) , which is captured by the slope parameter L and the curvature K sym defined at nuclear saturationdensity, n ∼ . fm − , L = 3 n ∂S ( n ) ∂n (cid:12)(cid:12)(cid:12)(cid:12) n , K sym ( n ) = 9 n ∂ S ( n ) ∂n (cid:12)(cid:12)(cid:12)(cid:12) n . (2)As d ( E SNM /A ) /dn = 0 at n , L describes the pressure ofPNM around n . S = S ( n ) and L are of great interest to nuclear physics [5, 8, 9] and astrophysics [10–12]. Ex-perimental [4, 5, 13, 14] and theoretical [15–18] determi-nations consistently place S in the range of – MeVand L in the range of – MeV. Recently, however,the PREX-II experiment reported a new result for theneutron-skin thickness of
Pb [19], R Pbskin , a quantitystrongly correlated with L (see, e.g., [1–3]). The measure-ment of R Pbskin = 0 . ± . fm (mean ± standard devia-tion), including PREX-I and PREX-II data, led Ref. [20]to conclude that L = 110 ± MeV. This value is largerthan previous determinations, and thus presents a chal-lenge to our understanding of nuclear matter, should ahigh L value be confirmed precisely.In this Letter, we address this question by constrain-ing S , its density dependence L , and R Pbskin directlyfrom astrophysical observations. We adopt a nonpara-metric representation for the EOS [21, 22] to minimizethe model dependence of the analysis, in contrast to otherastrophysical inferences, e.g., Refs. [23–26]. Nonparamet-ric inference allows us to explore a multitude of EOSsthat are informed only by a NS crust model at densities n < . n , where the EOS uncertainty is small, com-bined with the requirements of causality and thermody-namic stability at higher densities. Following Ref. [27],the possible EOSs are weighted based on their compati-bility with gravitational-wave (GW) and electromagneticobservations of NSs (massive pulsars and X-ray timingwith NICER). By calculating S , L , K sym and R Pbskin foreach of these EOSs, we obtain astrophysically informedposterior distributions for these key nuclear properties.Furthermore, we study how L and R Pbskin change as con-straints from nuclear theory are included up to progres- a r X i v : . [ nu c l - t h ] F e b L [ M e V ] − K s y m [ M e V ]
25 30 35 40 45 50 55 60 S [MeV]0 . . . . . . . . R P b s k i n [ f m ] L [MeV] −
500 0 500 K sym [MeV] Nonparametric PriorNonparametric Astro Posterior χ EFT Astro Posterior
PREX-IIFigure 1. Correlations between the symmetry energy S , the slope parameter L , the symmetry incompressibility K sym , andthe neutron skin thickness of Pb: R Pbskin . We show the nonparametric prior ( grey ), the nonparametric posterior conditionedon astrophysical observations ( green ), and the nonparametric posterior conditioned on four χ EFT calculations (up to ∼ n )and astrophysical observations ( blue ). Joint distributions show the 68% ( shaded ) and 90% ( solid lines ) credible regions. Shadedbands ( pink ) show the approximate 68% credible region for parameters constrained by PREX-II: R Pbskin [19] and the resultingconstraints on L using the correlation from Ref. [2]. Note how the inclusion of the astrophysical observations shifts the peak inthe marginal distributions for S , L and R Pbskin , a trend that is reinforced by the addition of χ EFT information. sively higher densities.
Nonparametric inference for the EOS–
We connect NSobservables to S and L using a nonparametric rep-resentation of the EOS based on Gaussian processes(GPs) [21, 22]. The GPs model the uncertainty in thecorrelations between the sound speed in β -equilibriumat different pressures, but do not specify the exactfunctional form of the EOS, unlike other parameteriza-tions [28–36]. The nonparametric EOSs consequently ex-hibit a wider range of behavior than parametric EOSs,mitigating the impact of modeling assumptions.The nonparametric EOS inference proceeds through Monte-Carlo sampling from a prior constructed as a mix-ture of GPs to obtain a large set of EOS realizations.Each EOS is then compared to astrophysical observa-tions via optimized kernel density estimates (KDEs) ofthe likelihoods, resulting in a discrete representation ofthe posterior EOS process as a list of weighted samples(see [22, 27] for more details). The posterior probabilityof a given EOS realization ε β is calculated as P ( ε β |{ d } ) ∝ P ( ε β ) (cid:89) i P ( d i | ε β ) , (3)where { d } = { d , d , . . . } is the set of observations, E PNM /A [MeV] S [MeV] L [MeV] K sym [MeV] R Pbskin [fm]
Nonparametric Prior +15 − +15 − +110 − − +847 − . +0 . − . Nonparametric Astro Posterior +12 − +12 − +66 − − +586 − . +0 . − . Nonparametric Astro+PREX-II Posterior +12 − +12 − +66 − − +686 − . +0 . − . χ EFT Astro Posterior +3 − +3 − +20 − − +132 − . +0 . − . χ EFT Astro+PREX-II Posterior +3 − +3 − +19 − − +128 − . +0 . − . Table I. Medians and 90% highest-probability-density credible regions. We compute R Pbskin from L using the linear fit reportedin Ref. [2], approximating the uncertainty in the fit as described in the text. P ( d i | ε β ) are the corresponding likelihood models, and P ( ε β ) is the EOS realization’s prior probability. Thespecific likelihoods used in this work are as follows:(a) Pulsar timing measurements of masses for thetwo heaviest known NSs (PSR J0740+6620 [37], PSRJ0348+0432 [38]) modeled as Gaussian distributionswith means and standard deviations . ± . M (cid:12) and . ± . M (cid:12) , respectively; (b) GW measurements ofmasses and tidal deformabilities in the binary NS mergerGW170817 [39] from advanced LIGO [40] and Virgo [41],modeled with an optimized Gaussian KDE [22]; and (c)X-ray pulse-profile measurements of PSR J0030+0451’smass and radius assuming a three-hotspot configura-tion [42] (see also Ref. [43], which yields comparable re-sults [27]), similarly modeled with an optimized GaussianKDE [22].Our basic nonparametric prior can also be conditionedself-consistently on theoretical calculations of the EOSat nuclear densities, while retaining complete model free-dom at higher densities [44]. Here we marginalize overthe uncertainty bands from four different chiral effectivefield theory ( χ EFT) calculations: quantum Monte Carlocalculations using local χ EFT interactions up to next-to-next-to-leading order (N LO) [45], many-body perturba-tion theory (MBPT) calculations using nonlocal χ EFTinteractions up to next-to-next-to-next-to-leading order(N LO) of Refs. [16, 46], and MBPT calculations withtwo-nucleon interactions at N LO and three-nucleon in-teractions at N LO (based on a broader range of three-nucleon couplings) [31, 47]. This allows us to account fordifferent nuclear interactions and many-body approaches,increasing the robustness of our results.To translate the EOS posterior process into distribu-tions for the nuclear physics properties, we establish aprobabilistic map from ε β to E PNM /A , S , L , and K sym (described below). Marginalization over the EOS thenyields a posterior P ( E PNM /A, S , L, K sym |{ d } ) = (cid:90) D ε β P ( ε β |{ d } ) P ( E PNM /A, S , L, K sym | ε β ) (4)informed by the astrophysical observations. Constraintson R Pbskin are obtained from empirical correlations with L [2] calculated from a broad range of nonrelativisticSkyrme and relativistic mean-field density functionals;see also Refs. [1, 3]. To account for the theoretical uncer-tainty in the fit of Ref. [2] and mitigate its model depen-dence, we adopt a probabilistic mapping: P ( R Pbskin | L ) = N ( µ R , σ R ) with µ R [fm] = 0 .
101 + 0 . × ( L [MeV]) and σ R = 0 . . Reconstructing the symmetry energy–
Because our non-parametric EOS realizations are not formulated in termsof S , L , or K sym , we discuss how to extract the nuclearparameters near n directly from the EOS. The nonpara-metric inference provides the individual EOSs in terms ofthe baryon density n , and pressure p β and energy density ε β in β -equilibrium. Each realization is matched to theBPS crust [48] around . n . The EOS quantities arerelated to E nuc /A through ε = n · ( E nuc /A + m N ) withthe average nucleon mass m N . To reconstruct E nuc /A ,we correct ε β by the electron contribution ε e , E nuc A ( n, x ) = ε β ( n ) − ε e ( n, x ) n − m N . (5)The proton fraction x ( n ) is unknown and needs to bedetermined self-consistently for each EOS by enforcing β -equilibrium, µ n ( n, x ) = µ p ( n, x ) + µ e ( n, x ) , where µ i ( n, x ) is the chemical potential for particle species i .This leads to the condition for β -equilibrium (see [31] fordetails), m n − m p − ∂ ( E nuc /A ) ∂x − µ e ( n, x ) . (6)To extract the symmetry energy from each EOS realiza-tion, we need to know the dependence of E nuc /A withproton fraction. Here, we approximate the x dependenceusing the standard quadratic expansion, E nuc A ( n, x ) = E SNM A ( n ) + S ( n )(1 − x ) . (7)Non-quadratic terms are small at n and can be neglectedgiven current EOS uncertainties [49, 50]. Because wework around n , we can characterize the SNM energyusing the standard expansion, E SNM A ( n ) = E + 12 K (cid:18) n − n n (cid:19) + · · · , (8) .
28 0 .
48 0 .
64 0 .
81 0 . median { n ( p max ) /n } a priori L [ M e V ] .
88 1 . .
17 0 .
45 2 . p max [ MeV / fm ] .
28 0 .
48 0 .
64 0 .
81 0 . median { n ( p max ) /n } a priori . . . . . . R P b s k i n [ f m ] .
88 1 . .
17 0 .
45 2 . p max [ MeV / fm ] PREX-II α D PREX-II α D Figure 2. Prior ( gray, unshaded ), Astro posterior( green , left/unshaded ), and Astro+PREX-II posterior ( red,right/shaded ) distributions for L ( top ) and R Pbskin ( bottom )as a function of the maximum pressure ( top axis ) or density( bottom axis ) up to which we trust theoretical nuclear-physicspredictions from χ EFT (see text for details). Shaded bandsshow the approximate 68% credible region from PREX-II [19]( pink ) and of Ref. [13] based on the electric dipole polarizabil-ity α D ( light blue ). where uncertainty in the saturation energy E , n , andthe incompressibility K is based on the empirical rangesfrom Ref. [9]. Combining Eqs. (1) and (5)–(8), we findthat β -equilibrium must satisfy − x β m p − m n + µ e ( n, x β ))= (cid:18) ε β − ε e ( n, x β ) n − m N − E SNM A ( n ) (cid:19) . (9)We use the relations for a relativistic Fermi gas for theelectron energy density and chemical potential [51]. Tosummarize, given a nonparametric EOS realization and afair draw from the empirical distributions for the param-eters E , K , and n , we reconstruct the proton fraction x β self-consistently at each density around nuclear satu-ration. We then calculate E PNM /A , S , L , and K sym asa function of n and report their values at the referencedensity n (ref)0 = 0 .
16 fm − . The neutron-skin thickness is L [MeV]0 . . . . . . . . R P b s k i n [ f m ] L [MeV]910111213141516 R . [ k m ] Astro+PREX-IINonparametricPosterior χ EFT Astro+PREX-IIPosteriorPREX-IIFigure 3. Correlations between R Pbskin , L , and the radiusof a . M (cid:12) NS, R . . In addition to the priors and poste-riors shown in Fig. 1, we show the nonparametric ( red ) and χ EFT (trusted up to n ; light blue ) posteriors conditioned onboth astrophysical observations and PREX-II. Astro+PREX-II posteriors are shaded in the one-dimensional distributionsto distinguish them from the Astro-only posteriors. Joint dis-tributions show the 68% ( shaded ) and 90% ( solid lines ) cred-ible region. Shaded bands ( pink ) show the approximate 68%credible region from PREX-II. estimated via the empirical fit between R Pbskin and L , asdiscussed above. Results and discussion–
The constraints on S , L , K sym , and R Pbskin are shown in Fig. 1. We plot the non-parametric prior, the posterior constrained by astrophys-ical data, and the posterior additionally constrained bythe χ EFT calculations up to n ∼ n . As our GPs areconditioned on χ EFT up to a maximum pressure ( p max ) ,we report the median density at that pressure (the exactdensity at p max varies due to uncertainty in the EOS from χ EFT). Prior and posterior credible regions are providedin Tb. I. We find that the PREX-II result for R Pbskin andthe extracted range for L of Ref. [20], – MeV at σ ,are in mild tension with the GP conditioned on χ EFTcalculations up to n , while the GP conditioned only onastrophysical observations is consistent with both resultsand cannot resolve any tension due to its large uncertain-ties. However, the Astro-only and χ EFT posteriors peakat similar values for L ( – MeV), below the PREX-IIresult. The astrophysical data does not strongly con-strain K sym , but suggests it is negative.In Fig. 2, we show the evolution of our constraints on L and R Pbskin as a function of the maximum density upto which we condition on χ EFT, from no conditioningon χ EFT to conditioning on χ EFT up to n . The morewe trust χ EFT constraints, the larger the tension withPREX-II results becomes. We estimate a probabil-ity ( p -value) that the true R Pbskin differs from the PREX-II mean at least as much as the Astro+ χ EFT posteriorsuggest, given the uncertainty in PREX-II’s measure-ment. However, if a hypothetical experiment confirmedthe PREX-II mean with half the uncertainty, this p -valuewould be reduced to . . We also show the estimate for R Pbskin obtained from an analysis of dipole polarizabilitydata [13], which finds R Pbskin = 0 . – .
19 fm . The lat-ter agrees very well with both the χ EFT results and thenonparametric GP.In Fig. 3, we present the modeled correlation between L and R Pbskin as well as the radius of a . M (cid:12) NS, R . .In addition to Fig. 1, we show posteriors that are alsoconditioned on the PREX-II result. Even though the re-sults for L and R Pbskin are very different for the variousconstraints, R . does not significantly change. Indeed,the mapping from L to R . is broader than often as-sumed [6], and we find that R . is nearly independentof our range for L . Hence, the findings of Ref. [20], in-dicating that PREX-II requires large radii, include somemodel dependence.Given the mild tension between the PREX-II value of R Pbskin and that inferred from the astrophysical inferencewith χ EFT information, we investigate what kind of EOSbehavior is required to satisfy both the PREX-II and as-trophysical constraints. In Fig. 4 we show the speed ofsound c s as a function of density for the nonparametricGP conditioned only on astrophysical data for all valuesof L , for MeV < L ≤ MeV, and for
L >
MeV.We find that the speed of sound generally increases withdensity. However, if we assume
L >
MeV, we finda local maximum in the median c s ( n ) just below n , al-though the uncertainties in c s are large. The reason forthis feature is that EOSs that are stiff at low densities(large L ) need to soften beyond n to remain consistentwith astrophysical data. Should the PREX-II constraintsbe confirmed with smaller uncertainty in the future, thismight favor the existence of a phase transition between – n .In summary, we have used nonparametric GP EOSinference to constrain the symmetry energy, its den-sity dependence, and R Pbskin directly from astrophysicaldata, leading to S = 35 +12 − MeV, L = 57 +66 − MeV, and all L
30 MeV < L ≤
70 MeV100 MeV < L
Figure 4. Median and 90% one-dimensional symmetric pos-terior credible regions for c s at each density n with astro-physical observations for all L ( shaded green ),
30 MeV < L ≤
70 MeV ( unshaded blue hatches ), and
100 MeV < L ( shadedpurple ). R Pbskin = 0 . +0 . − . fm . Folding in χ EFT constraints re-duces these ranges to S = 34 +3 − MeV, L = 52 +20 − MeV,and R Pbskin = 0 . +0 . − . fm . While these results pre-fer values below the recent PREX-II values [19, 20], ingood agreement with other nuclear physics information,the PREX-II uncertainties are still broad and any ten-sion is very mild. Our nonparametric analysis suggeststhat a R Pbskin uncertainty of ± .
04 fm could challenge as-trophysical and χ EFT constraints. Finally, our resultsdemonstrate that the correlation between R . and L (or R Pbskin ) is looser than analyses based on a specific classof EOS models would suggest. Extrapolating neutron-skin thickness measurements to NS scales thus requiresa careful treatment of systematic EOS model uncertain-ties. In particular, the PREX-II result does not requirelarge NS radii. However, if the high L values of PREX-II persist, this may suggest a peak in the sound speedaround saturation density. Acknowledgements–
R.E. was supported by thePerimeter Institute for Theoretical Physics and the KavliInstitute for Cosmological Physics. Research at Perime-ter Institute is supported in part by the Government ofCanada through the Department of Innovation, Scienceand Economic Development Canada and by the Provinceof Ontario through the Ministry of Colleges and Uni-versities. The Kavli Institute for Cosmological Physicsat the University of Chicago is supported by an endow-ment from the Kavli Foundation and its founder FredKavli. The work of I.T. was supported by the U.S. De-partment of Energy, Office of Science, Office of NuclearPhysics, under contract No. DE-AC52-06NA25396, bythe Laboratory Directed Research and Development pro-gram of Los Alamos National Laboratory under projectnumber 20190617PRD1, and by the U.S. Department ofEnergy, Office of Science, Office of Advanced ScientificComputing Research, Scientific Discovery through Ad-vanced Computing (SciDAC) program. P.L. is supportedby National Science Foundation award PHY-1836734 andby a gift from the Dan Black Family Foundation to theGravitational-Wave Physics & Astronomy Center. Thework of A.S. was supported in part by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) – Project-ID 279384907 – SFB 1245. The authorsalso gratefully acknowledge the computational resourcesprovided by the LIGO Laboratory and supported by NSFgrants PHY-0757058 and PHY-0823459. 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