Impact of the neutron-star deformability on equation of state parameters
C.Y. Tsang, M.B. Tsang, Pawel Danielewicz, W.G. Lynch, F.J. Fattoyev
IImpact of the neutron-star deformability on equation of state parameters
C.Y. Tsang, M.B. Tsang, Pawel Danielewicz, and W.G. Lynch
National Superconducting Cyclotron Laboratory and the Department of Physics and AstronomyMichigan State University, East Lansing, MI 48824 USA
F.J. Fattoyev
Department of Physics, Manhattan CollegeRiverdale, NY 10471, USA (Dated: September 14, 2020)We use a Bayesian inference analysis to explore the sensitivity of Taylor expansion parametersof the nuclear equation of state (EOS) to the neutron star dimensionless tidal deformability (Λ)on 1 to 2 solar masses neutron stars. A global power law dependence between tidal deformabilityand compactness parameter (M/R) is verified over this mass region. To avoid superfluous correla-tions between the expansion parameters, we use a correlation-free EOS model based on a recentlypublished meta-modeling approach. We find that assumptions in the prior distribution stronglyinfluence the constraints on Λ. The Λ constraints obtained from the neutron star merger eventGW170817 prefer low values of L sym and K sym , for a canonical neutron star with 1.4 solar mass.For neutron star with mass < . L sym and K sym are highly correlated with the tidaldeformability. For more massive neutron stars, the tidal deformability is more strongly correlatedwith higher order Taylor expansion parameters. I. INTRODUCTION
A neutron star (NS) is the remnant of a supernova ex-plosion of a massive star. The interior of a NS containsthe densest nuclear material in the universe. This matteris so dense that it becomes energetically favorable for pro-tons and electrons to combine and form neutrons. Fromdensities ranging from somewhat below saturation den-sity ( ρ = 0 .
155 fm − ) to 3 ρ , it is reasonable to describeNS matter as locally uniform nuclear matter composedmostly of neutrons. Study of NS is of great relevance tonuclear physics because of the information it can provideregarding the equation of state (EOS) of asymmetric nu-clear matter at high density. Even though the currentpaper is self-contained with relevant materials detailedin appendix and extensive references, for those who areinterested, Refs. [1–4] provide more in depth discussionsof the subjects.Astrophysical NS properties, combined with con-straints from nuclear observations, have provided a roughunderstanding of the EOS. Typical temperatures of NSsare low, k B T < ρ [6–8]. Specifically, the symmetric matter constraints onpressure vs. density were determined in Ref. [6] fromthe measurements of transverse and elliptical flow fromAu+Au collisions over a range of incident energies from0.3 to 1 . / u. More recently, these constraints wereconfirmed in an independent analysis of elliptical flowdata [9]. In Refs. [7, 8], a similar constraint from 1 . ρ to2 . ρ was obtained from the Kaon measurements. These heavy ion constraints are consistent with the Bayesiananalyses of the neutron-star mass-radius correlation inRef. [10].Recent gravitational wave observations from LIGO col-laboration [11] opened a new window for understandingneutron-star matter. Specifically, the LIGO observationprovides estimates for the tidal deformability, also knownas tidal polarizability, a quantity that bears direct rele-vance to the nuclear EOS.The tidal deformability is induced when two NSs orbitaround each other and tidal forces from each NS deformsits companion star. The mass quadrupole that developedin response to the external quadrupole gravitational fieldemerges as: Q ij = − λE ij . (1)Here E ij is the external gravitational field strength and λ is the tidal deformability. The orbital period of theinspiral differs from that of two point masses becausethe additional tidal deformation contributes to an over-all orbital energy loss and changes the rotational phase.This difference is used to extract the dimensionless tidaldeformability (Λ) of a NS [12, 13]. Throughout this pa-per, tidal deformability given below always refers to thedimensionless tidal deformability,Λ = λc G M = 23 k (cid:16) c RGM (cid:17) , (2)where k is the second Love number [14, 15]. This wholeexpression, including the Love number, is sensitive to thenuclear EOS [11, 16, 17]. Steps necessary to calculate Λfor a given EOS are detailed in Appendix A. Recentanalysis of the gravitational wave data constrained thisvalue to Λ = 190 +390 − [18].Since most observables from nuclear structure exper-iments constrain the energy density and its derivatives a r X i v : . [ nu c l - t h ] S e p near or somewhat below saturation density (See, for ex-ample, Refs. [19–22]), it is customary to approximate theEOS by a Taylor expansion about saturation density. Wewill explore the parameter space spanned by the deriva-tives of EOS with respect to density at ρ and examineits correlation with Λ.Other studies have been carried out in placing tidaldeformability constraints on these Taylor expansion pa-rameters. They explored the constraints on different 2Dparameter planes [21, 23], on a diverse set of models [24–26], and with Bayesian analysis on EOSs from chiral ef-fective field theory [27]. In this study, we will expand theanalysis by employing a less restrictive form of EOS andexploring a larger parameter space by including higherorder terms.A family of theoretical EOS is needed to correlate theTaylor expansion parameters with the predicted Λ. Onewidely used family in astrophysics is the piece-wise poly-tropes [28], but it is not suitable in this study because aTaylor expansion assumes that the EOS is analytic overthe range of interest. As long as there is only one poly-trope, a Taylor expansion is valid, but its validity doesnot extend past the point of connection between the orig-inal polytrope and the next.Another commonly used family is the Skyrme typeEOS [29]. It derives from simplified approximate nuclearinteraction and relies on 15 free parameters in its ex-panded form. While it is shown to successfully reproducevarious nuclear properties, it is difficult to explore newphysics from the Taylor expansion parameters becausethey are strongly constrained by the form of the Skyrmeinteraction itself. It is difficult to access functional de-pendencies the Taylor expansion parameters that are notcontained in the original choice for the Skyrme functionalform [30, 31].In this study, an EOS from meta-modeling [32] is used.By construction, their derivatives of different orders areindependent of each other. This paper is organized as fol-lows: In section II, a brief description of Bayesian infer-ence is provided. This is the statistical method employedin the extraction of EOS information from NS tidal de-formability constraints. Section III describes our choiceof EOS from meta-modeling approach in Ref. [32] andhow it is adopted to describe neutron star. In section IV,correlation between EOS parameters and tidal deforma-bility of a 1.4 solar mass NS will be discussed. Section Vextends the study to NSs of different masses and sectionVI summarizes our findings. II. BAYESIAN INFERENCE
We use Bayesian inference to study the influence oftidal deformability constraints from LIGO on nuclear-matter EOS parameters. These parameters are sam-pled with a prior probability distribution based on find-ings from literature and are then transformed into a dis-tribution of neutron-star matter EOS. Through solving TOV equation, we are able to calculate the correspond-ing tidal deformabilities. By combining their prior distri-bution and likelihood, which indicates the compatibilitybetween the calculated and the observed tidal deforma-bility, Bayesian inference will assign probability for eachEOS parameters with Bayes theorem: P ( M ) = 1 V tot w ( M ) p (Λ( M )) (cid:89) i g i ( m i ) . (3)In this equation, M is the set of all EOS parameters, m i ∈ M is one of the EOS parameters, V tot is the fea-ture scaling constant, p (Λ( M )) is the likelihood of a EOScalculated from its predicted Λ, g i is the prior distribu-tion of the i th parameter and w ( M ) is the filter conditionthat filters out EOS parameter space that is nonphysical.The likelihood of EOS is the probability of having theobserved LIGO event with the assumption that the giventheoretical EOS is the ultimate true EOS. We will modelthe likelihood function as an asymmetric Gaussian dis-tribution base on the extracted Λ = 190 +390 − [18] fromGW170817. p (Λ) = (cid:40) V exp( − (Λ − × ) , if Λ ≤ V exp( − (Λ − × ) , if Λ > . (4)In the above, V is the feature scaling constant such thatthe likelihood function integrates to 1.The sought function is the probability distribution ofEOS parameters rather than that for Λ, so prior distri-bution g i is required to convert between the two usingBayes theorem. A commonly used prior is the Gaussiandistribution: g i ( m i ) = 1 (cid:112) πσ i exp (cid:16) − ( m i − m i, prior ) σ i (cid:17) , (5)where m i, prior and σ i are the prior mean and standard de-viation of the free parameters, respectively. They shouldbe chosen to reflect our current understanding of thosefree parameters.Some parameter sets may yield nonphysical EOSs dueto various additional considerations. The filter condition w ( M ) takes that into account; it is set to 1 if the follow-ing three conditions of stability, causality and maximummass, are all satisfied and it is set to 0, if not.The stability condition rejects EOSs whose pressuredecreases with energy density. Above the crust-coretransition density, we require the EOSs to be me-chanically stable with thermodynamical compressibilitygreater than zero, which means that the pressure of ho-mogeneous matter does not decrease with density. ForEOSs with negative compressibilities at density abovethe crust-core transition densities predicted by Eq. (15),they will be rejected as being inconsistent with experi-mental information.The requirement of causality rejects EOSs whose speedof sound is greater than the speed of light in the coreregion of their respective heaviest NS. The maximummass condition rejects EOSs that fail to produce a NSof at least 2.04 solar mass in accordance with observa-tion. [33, 34].Using the fact that the binary NS merger GW170817detected by LIGO did not promptly produce a black hole,Ref. [35] inferred that the heaviest possible NS should bearound 2.17 solar mass. Other sources put the maximummass at around 2.15-2.40 solar masses [36–41]. Neitherof these constraints have been adopted in this work butcan be implemented in the future.The calculated probability distribution from Eq. (3) isreferred to as the posterior distribution. By comparingprior to posterior distribution, we will be able to inferthe sensitivity of various EOS parameters to NS tidal de-formability. By construction, priors of different free pa-rameters in meta-modeling EOS are not correlated witheach other, so any correlations in the posterior reflect thecollective sensitivity of the Taylor expansion parametersto NS tidal deformability. III. NUCLEAR EQUATION OF STATEA. Parameters in nuclear EOS
Nuclear matter is a theoretical construct composed ofprotons and neutrons. It resembles the core of ordinarynuclei where the neutron and proton densities are ap-proximately uniform. Since the number of protons andneutrons are usually not far from each other in nuclei,we often expand EOS into the symmetric nuclear matter(SNM) term (isoscalar term) and a correction term forthe deviation from SNM (isovector term), when protondensities and neutron densities are not identical as shownin Eq. (6) below. SNM refers to an infinite system wherethe density of proton equals to the density of neutrons.The EOS is commonly expanded as: E ( ρ, δ ) = E is ( ρ ) + δ E iv ( ρ ) . (6)In the above, E is is the isoscalar term, E iv is the isovectorterm, ρ is the matter density and δ = ( ρ n − ρ p ) / ( ρ n + ρ p )is called the asymmetry parameter. Nuclear struc-ture probes are generally sensitive to the density regionaround saturation [19–22] and as a result, derivatives ofEOS with respect to density at this point are often usedas empirical parameters to characterize the density andisospin dependence of the EOS. The derivatives are com-monly expressed as parameters in the Taylor expansionwhen EOS is expanded in terms of x = ( ρ − ρ ) / (3 ρ ): E is ( ρ ) = E + 12 K sat x + 13! Q sat x + 14! Z sat x + ... (7) E iv ( ρ ) = S + Lx + 12 K sym x + 13! Q sym x + 14! Z sym x + ... (8) One focus of this paper is to explore the sensitivity be-tween Λ and S , L , K , Q , Z . Some families of EOS de-pend on density and asymmetry in a way that cannot beseparated explicitly into the sum of two terms, but theisoscalar term is always well-defined: E is ( ρ ) = E ( ρ, δ = 0) . (9)The isovector term can be defined as the second orderTaylor expansion coefficient in δ around δ = 0 (not to beconfused with Taylor EOS expansion parameters whichexpands in x ), E iv ( ρ ) = 12 ∂ E ( ρ, δ ) ∂δ (cid:12)(cid:12)(cid:12) δ =0 . (10)Likewise Taylor EOS parameters can always be extractedfrom any nuclear EOS. This allows for comparison of vari-ables across families of EOS.Another important quantity that characterizes nuclearmatter properties is the effective mass m ∗ ( ρ, δ ). It is usedto characterize the momentum dependence of nuclear in-teraction and it can be different for protons m ∗ p ( ρ, δ ) andneutrons m ∗ n ( ρ, δ ) depending on the condition which thenuclear matter is subjected to. It is generally assumedthat m ∗ p = m ∗ n in SNM.Comparison of effective mass is commonly carried outthrough the comparison of two quantities: the nuclear ef-fective mass in SNM at saturation m ∗ sat and the splittingin neutron and proton effective masses in pure neutronmatter (PNM) at saturation ∆ m ∗ = m ∗ n − m ∗ p . The choiceof the two quantities mirrors the spirit of splitting EOSinto isoscalar term and isovector term in Eq. (6) in whichcontribution from SNM is separated from the correctionfactor that arises when matter is not symmetric.Sometimes it is more convenient to express m ∗ sat and∆ m ∗ in terms of κ sat , κ sym and κ v : κ sat = mm ∗ sat − κ s ,κ sym = 12 (cid:16) mm ∗ n − mm ∗ p (cid:17) ,κ v = κ sat − κ sym . (11)The parameter κ v plays the role of the enhancementfactor in Thomas-Reiche-Khun sum rule and it dependson the energy region of the resonance energy [42]. In thisanalysis, the effective masses will be expressed in termsof m ∗ sat /m and κ v . B. EOS from a metamodeling approach
Our studies utilizing the metamodeling analysis fol-low the approach of Ref. [32]. Such metamodels for theEOS can be easily constructed with only Taylor expan-sion parameters and effective masses. The metalmodelEOS resembles Skyrme EOS with the same correspond-ing Taylor parameters to a greater extent than a simplepower law expansion.Four different empirical local density functionals (ELF)meta-models are proposed in Ref. [32]: ELFa, ELFb,ELFc and ELFd. ELFa does not produce vanishing en-ergy as density approaches zero. ELFb does not convergeto a typical Skyrme EOS even when identical Taylor pa-rameters are used. ELFc does not have the shortcomingsof EFLa and ELFb and closely resembles Skyrmes withsimilar Taylor parameters. Although ELFd agrees withSkyrmes better, it relies on high density information thatis not well constrained by experiments.From the above considerations, we adopt ELFc inthis study. Similar choice is also made in other recentstudies [43, 44]. The formulation of ELFc is detailedin Appendix B. As assessed in Ref. [32], the followingchoices of parameters have been accurately constrainedby nuclear experiment and are fixed in the analysis: E sat = − . ρ = 0 .
155 fm − . C. Thermodynamic relations
Additional characteristics of nuclear matter can be in-ferred using thermodynamic equations once an EOS isspecified. The pressure at various densities P ( ρ ) is re-lated to the derivative of the energy: P ( ρ ) = ρ ∂E ( ρ, δ ) ∂ρ . (12)The adiabatic speed of sound can then be calculated [45]: (cid:16) v s c (cid:17) = (cid:16) ∂P∂ E (cid:17) S , (13)where E = ρ ( E + mc ) is the energy density of the mate-rial including mass density. This implies any thermody-namic stable EOS must satisfy (cid:16) ∂P∂ E (cid:17) S >
0. Furthermore,since information cannot travel faster than the speed oflight due to causality, the inequality v s < c must hold forall densities relevant to NS. This may not be always truefor ELFc. To stay physical, we will switch from ELFcto an expression for the stiffest possible EOS whenevercausality is violated: P Stiffest ( E , v s , E , P ) = (cid:16) v s c (cid:17) ( E − E ) + P . (14)This equation represents a EOS with constant speed ofsound v s and v s = c yields the stiffest possible EOS [46].Here E and P are reference values of energy density andpressure, respectively. The reference values can be ad-justed to match the conditions at a specific density whereenergy density and pressure are known. The switch inEOS avoids superfluous rejection when causality is con-sidered. D. Structure of a NS and modifications on thenuclear EOS
Neutron stars are more than a “giant nucleus” de-scribed in Ref. [47]. There are structural changes atvarious density regions as a result of a competition be-tween the nuclear attraction and the Coulomb repulsion.The dynamics of the outermost layers of NSs is describedmostly by the Coulomb repulsion and nuclear masses,where nuclei arrange themselves in a crystalline lattice.As the density increases, it becomes energetically favor-able for the electrons to capture protons, and the nuclearsystem evolves into a Coulomb lattice of progressivelymore exotic, neutron-rich nuclei that are embedded in auniform electron gas. This outer crustal region exists asa solid layer of about 1 km in thickness [17].At intermediate densities of sub-saturation, the spheri-cal nuclei that form the crystalline lattice start to deformto reduce the Coulomb repulsion. As a result, the systemexhibits rich and complex structures that emerge from adynamical competition between the short-range nuclearattraction and the long-range Coulomb repulsion [48].At densities of about half of the nuclear saturation,the uniformity in the system is restored and matter be-haves as a uniform Fermi liquid of nucleons and leptons.The transition region from the highly ordered crystal tothe uniform liquid core is very complex and not well un-derstood. At these regions of the inner crust which ex-tend about 100 meters, various topological structures arethought to emerge that are collectively referred to as “nu-clear pasta”. Despite the undeniable progress [49–80] inunderstanding the nuclear-pasta phase since their initialprediction over several decades ago [81–83], there is noknown theoretical framework that simultaneously incor-porates both quantum-mechanical effects and dynamicalcorrelations beyond the mean-field level. As a result, areliable EOS for the inner crust is still missing.The matter in the core region of NS can be described asuniform nuclear matter where neutron, proton, electronsand muons exist in beta equilibrium [48]. Although aphase change and exotic matter such as hyperons [48,84, 85] could appear in the inner core region, there iscurrently no direct evidence of their existence. In thiswork, we calculate the EOS in this region by assumingthat the neutron-star matter is composed of nucleons andleptons only.Due to the rich structure of NS, the nuclear EOS needsto be contextualized before it can be used for NS prop-erties calculation. To begin with, crustal EOS should beused at density below transition density ρ T . Normallythe determination of ρ T requires complicated thermo-dynamic calculations, but some simple relationship hasbeen found between transition densities and Taylor pa-rameters of the EOS in Ref. [86] that greatly simplifiesits calculation. In this study, the following equation isused to determine ρ T : ρ T = ( − . × − L sym + 0 . − . (15)TABLE I: Summary information of various models [32]. The bottom half shows characteristics of the prior andposterior distribution respectively. L sym ( MeV ) K sym ( MeV ) K sat ( MeV ) Q sym ( MeV ) Q sat ( MeV ) Z sym ( MeV ) Z sat ( MeV ) m ∗ sat m κ v Skyrme Average 49.6 -132 237 370 -349 -2175 1448 0.77 0.44Skyrme σ σ σ σ σ Outer and inner crust exhibit different physical proper-ties and should be described by different EOSs. For theouter crust, EOS provided by Ref. [87] is used in thisanalysis. For the inner crust, spline interpolation the re-gion of 0 . ρ T < ρ < ρ T is reserved for a smooth transitionbetween the outer crust and outer core. While this con-nection region cannot precisely describe crustal dynam-ics, tidal deformability does not appear to be sensitive tothe choice of the crustal details for NS [17, 88, 89].The outer core region ρ > ρ T is characterized by theEOS of a beta equilibrated system of protons, neutrons,electrons and muons. Proton and neutrons are collec-tively described by EFLc while electrons and muons aremodeled as relativistic Fermi gases. Equilibrium is at-tained by minimizing the Helmholtz free energy at dif-ferent densities. If the speed of sound for EOS reachesthe speed of light at density ρ = ρ c , it will switch to thestiffest possible EOS of Eq. (14) at higher densities tocomply with causality condition. If ELFc does not violatecausality at all densities relevant to NSs, then ρ c = ∞ and Eq. (14) is never used.To summarize, EOS of the neutron-star matter is for-mulated as follows: P ( E ) = P crust ( E ) , if 0 < ρ < . ρ T P spline ( E ) , if 0 . ρ T < ρ < ρ T P EFLc ( E ) , if ρ T < ρ < ρ c P stiffest ( E , c, E , P ) if ρ c < ρ. (16)In the above equation, E and P are the energy densityand pressure from beta-equilibrated EFLc at ρ c respec-tively, P crust is the pressure from crustal EOS and P ELFc is the pressure from beta-equilibrated ELFc EOS. P spline and E crust govern the cubic spline that smoothly connects P crust to P ELFc and E crust to E EFLc respectively.
IV. RESULTS FOR A 1.4-SOLAR MASS NS
A total of 1,500,000 EOSs have been sampled and682,652 of them satisfy all of our constraints. Only11,711 EOSs apply to all densities without switching tothe stiffest EOS.Prior distributions of the parameters should reflect ourinitial belief of those quantities before information ontidal deformability is taken into account. For this, werely on Ref. [32] which summarizes the distributions ofEOS parameters from three phenomenological families,Skyrme, relativistic mean field (RMF) and relativisticHartee-Fock (RHF). The mean and standard deviationof the parameters for each family are tabulated in thefirst six rows of Table I. In this study, the prior meansand standard deviations are the weighted average valuesof the 3 families, with weights of 0.500, 0.333, 0.167 re-spectively. The weights reflect our confidence in of themodels. We give Skyrme EOS the most weight as it isthe most heavily employed parametrization in a myriadof nuclear predictions [29]. These relative weights are adhoc, but should cover most plausible parameter spaces.Prior means and standard deviations are listed in theseventh and eighth row in table I respectively.The posterior distributions of Taylor expansion param-eters are represented in Fig. 1. The lower triangularplots show the bivariate distributions for two parame-ters. The diagonal plots show the prior (blue curves)and marginalized posterior distributions (red curves) forindividual parameters. The upper triangle displays thePearson correlation coefficients for parameter pairs: ρ X,Y = E [( X − ¯ X )( Y − ¯ Y )] σ X σ Y , (17)where E is the expectation value and σ X and σ Y are thestandard deviations of the parameters distributions. ThePearson coefficient ranges from -1 to 1 and its absolutevalue reflects the strength of the correlation. A positive L s y m ( M e V ) ··· ··· ··· K s y m ( M e V ) ··· ··· ··· K s a t ( M e V ) ··· ··· Z s y m ( M e V ) -0.31 L sym (MeV)20004000 Z s a t ( M e V ) K sym (MeV) K sat (MeV) Z sym (MeV) Z sat (MeV) FIG. 1: Bivariate characteristics of posterior likelihood distributions. Three regions can be distinguished. The lowertriangle panels show likelihood distributions, with intensity proportional to distribution value, for pairs of Taylorparameters. The diagonal panels display prior (blue) and marginalized posterior (red) distributions for eachparameter. The upper triangular region shows Pearson correlation coefficient for parameter pairs. Three dotsindicate weak correlations with magnitude less than 0.1.value close to 1 indicates a strong correlation and a neg-ative value close to -1 indicates strong anti-correlationwhile a value close to 0 indicates lack of correlation [90].Only bivariate distributions between L sym , K sym , K sat , Z sym and Z sat are shown because the higher order param-eters do not seem to be influenced by our tidal deforma-bility constraints. The full correlation plot is included inAppendix C. Characteristics of the probability distribu-tion are summarized in the bottom two rows of Table.I. Fig. 2 shows the mean and 2 σ region spanned by theEOS in the posterior. The 2 σ region converge to a linefor E (cid:46)
20 MeV / fm , which corresponds to the outercrust. Since we connect all EOSs to the crustal EOS givenby Ref. [87], this convergence is expected. From around20 MeV / fm to 70 MeV / fm , the spline connection kicksin and manifests in the broadening of pressure.The cut-offs in the lower left corner of Z sym vs. Z sat distribution and the upper left corner of K sym vs. L sym distribution in Fig. 1 are the consequence of stability con- (MeV/fm ) P ( M e V / f m ) mean2 region FIG. 2: Distribution of EOSs sampled from theposterior. The divergence above energy density ∼ >
20 MeV / fm coincides with the transition fromouter crust to spline connection.dition. At such extreme values, speed of sound may beimaginary when extrapolating to NS of 2 solar masses.This is evident in Fig. 3 in which 50 randomly selectedEOSs from the cut-off region in K sym vs. L sym are shownin the lower panel. The pressure for those EOSs do notincrease monotonically with the energy density and be-come mechanically unstable. These EOSs are discarded.The posterior distributions of K sym and Z sym differfrom the prior distributions significantly. The tidal de-formability constraint favors lower K sym region. The in-ference also narrows the range of possible L sym . Parame-ters such as K sat and Z sat , whose posterior distributionsare not altered significantly reflect that they are not sen-sitive to the tidal deformability constraints.While this Bayesian analysis is well suited to discussthe sensitivity of the deformability to the Taylor expan-sions parameters L sym , K sym , K sat , etc., it has some lim-itations. In particular, we note that the prior and postdistributions of Λ as shown in Fig. 7 (row 2 column 10in Appendix C) are drastically different, probably as aconsequence of the narrow prior distributions of the Tay-lor expansion parameters listed in Table I. This reflectsthe strong sensitivity of Λ to the prior distributions ofthe EOS. Furthermore, the posterior distribution of Λ ismuch sharper and peaked at 624 ±
129 which exceeds thevalue of 190 +390 − obtained in Ref. [18] from the analy-sis of the GW170817. While the GW constraint reflectsthe high density of NS core, the prior distributions ofthe Taylor expansion parameters do not have rigorouslaboratory constraints at high density region where Λ isdetermined.
25 50 75 100 L sym (MeV) K s y m ( M e V ) (MeV/fm ) P ( M e V / f m ) FIG. 3: (Upper panel) The 50 dots in the upper lefthand corner of K sym vs. L sym correspond to 50randomly chosen parameter space within the stabilitycut-off region. (Lower panel) Unstable EOScorresponding to the 50 dots. The red and blue linecorrespond to the red and blue point in the upper panelrespectively. They are highlighted to showcase how atypical EOS in the cut-off region looks like.TABLE II: Predicted tidal deformability for NS ofdifferent masses Λ(1 .
2) Λ(1 .
4) Λ(1 .
6) Λ(1 .
8) Λ(2 . σ
310 129 61 31 17
V. NEUTRON STAR WITH DIFFERENTMASSES
While the chirp mass of GW170817 has been deter-mined quite accurately [11], the exact masses of the twoneutron stars or their mass ratios are not known [11].In anticipation that more merger events involving differ-ent NS masses than the nominal NS mass of 1.4 solarmass are observed in the future [91], we use the poste-rior EOS distributions to predict the deformability of NSwith different masses . The posterior EOS distributionscan be used to predict the deformability of NS with dif-ferent masses. In Table II, we provide our predictionsfor the tidal deformabilities for NS with 1.2, 1.4, 1.6, 1.8and 2 solar mass using this group of EOSs weighted bytheir posterior distributions. To show the sensitivity ofthese predictions to the Taylor parameters, Fig. 4 showsthe bivariate distributions between the Taylor parame-ters of the posterior distributions and the predicted tidaldeformabilities of different stellar masses. We find that Λis more strongly correlated with L sym and K sym than it iswith higher order Taylor expansion parameters. The sen-sitivity to K sym increases, while the sensitivity to L sym decreases, with stellar mass.To quantify this dependence of sensitivity on mass, thePearson correlation coefficients for a few selected Taylorparameter pairs are shown in Fig. 5. A gradual reductionin correlation between L sym and tidal deformability isobserved as the mass of a NS increases. This is expectedas relevant average density for more massive stars shiftupward and away from those most directly impacted by L sym . A high density parameter P (2 ρ ), the pressurefor pure neutron matter at twice the saturation density,is also included in Figs. 4 and 5. The strong correlationbetween tidal deformability and P (2 ρ ) is consistent withprior work [18, 27, 28, 92]. While this strong correlationis maintained for both heavy and light NS, the slope ofthe correlation becomes smaller reflecting the decrease inaverage values and variations of Λ with stellar mass.Such decrease is correlated with an increase in stellarcompactness. Using the posterior probability distribu-tions for the Taylor expansion parameters, we can alsomake predictions on the relation between stellar mass andinverse compactness ( R/M ). Fig. 6 shows tidal deforma-bility plotted against inverse compactness, with calcula-tion results for 1.2, 1.4, 1.6 and 1.8 solar mass NS allcombined together. It is consistent with Eq. (2) whereΛ ∝ k ( R/M ) . The best fitted power law has an in-dex of 5.84 due to additional interdependence of tidalLove number k and R/M . The result is consistent withRefs. [17, 93, 94].After submission of this work for publication, we foundthat independently and in parallel, Ref. [44] conducts avery similar analysis using ELFc. Our work examinescorrelations between more parameters and our study ex-tends to higher mass neutron star. Ref. [44] uses muchwider priors while our prior is more restrictive and pro-vide finer details in a smaller phase-space. In addition,they apply additional constraints on the EOS using datafrom χ EFT approach and ISGMR collective mode. Eventhough their extracted Q sat and K sym values are consis-tent with our extracted values, details in the correlationsare not the same. The subtle differences suggest thatBayesian analysis results depend on the choice of priorsand constraints applied to the EOS. VI. CONCLUSION
In this paper, ELFc form of Metamodeling is used ina sensitivity study of NS tidal deformability to Taylorexpansion parameters of the nuclear equation of state.Constraints on the isoscalar parameters, such as K sat arefound to be less affected by NS properties. For the isovec-tor parameters, L sym is found to be most correlated withtidal deformability, closely followed by K sym , althoughthe importance of the former dwindles and reverses asNS mass increases to above 1.6 solar mass.We have further demonstrated the global relation be-tween tidal deformability and compactness of NS withdifferent masses. When more merger events involvingdifferent NS masses are observed in the future, one canverify the relation of tidal deformability and inverse com-pactness in Fig. 6 and to provide independent constraintson L sym and K sym .A strong correlation with the pressure of matter at 2 ρ is observed. This highlights the need for high-densityobservables from nuclear physics, as constraints on tidaldeformability can be tightened with accurate high den-sity observations. A strong experimental constraint onpressure for PNM at 2 ρ complements pressure [95, 96]constraints from future measurement of NS mergers. VII. ACKNOWLEDGMENTS
This work was partly supported by the US NationalScience Foundation under Grant PHY-1565546 and bythe U.S. Department of Energy (Office of Science) un-der Grants DE-SC0014530, DE-NA0002923 and DE-SC001920. All the NS model calculations with High Per-formance Computers were performed at the Institute forCyber Enabled Research Center at Michigan State Uni-versity.
Appendix A: TOV equation
The Tolman–Oppenheimer–Volkoff (TOV) equationset predicts the structure of a static spherical object un-der general relativity for any given EOS. The equationsare: dP ( r ) dr = − ( E ( r ) + P ( r ))( M ( r ) + 4 πr P ( r )) r (1 − M ( r ) /r ) ,dM ( r ) dr = 4 πr E ( r ) . (A1)Here geometrized units G = c = 1 are used, E ( r ) is theenergy density given by EOS, P ( r ) is the internal pres-sure at given depth and M ( r ) is the integral of gravita-tional mass from the core up to radius r . The surface isdefined as the radial distance R at which P ( R ) = 0. ( . ) ( . ) L sym (MeV)200400 ( . ) K sym (MeV) K sat (MeV) Q sym (MeV) Q sat (MeV) P(2 ) (MeV/fm ) FIG. 4: Bivariate distributions between deformabilities with NS of different masses and Taylor parameters.Correlation with tidal deformability is clearly seen with L sym , K sym and P (2 ρ ). L s y m ( M e V ) K s y m ( M e V ) K s a t ( M e V ) Q s y m ( M e V ) Q s a t ( M e V ) P ( ) ( M e V / f m ) (1)(1.2)(1.4)(1.6)(1.8)(2) FIG. 5: Pearson correlation parameters for different NSmasses.
R/M = 1.31×10 (R/M) FIG. 6: Tidal deformability vs. inverse compactness for1.2, 1.4, 1.6, 1.8 solar mass NS. A list of equations whose solutions will lead to thevalue of Λ from the above structural functions will beshown without derivation. Please refer to Refs. [16, 97]for details. To begin with, an auxiliary variable y R = y ( R ) is calculated, r dy ( r ) dr + y ( r ) + y ( r ) F ( r ) + r Q ( r ) = 0 . (A2)where F ( r ) = r − πr ( E ( r ) − P ( r )) r − M ( r ) . (A3) Q ( r ) = 4 πr (5 E ( r ) + 9 P ( r ) + E + P ( r ) ∂P ( r ) /∂ E − πr r − M ( r ) − (cid:104) ( M ( r ) + 4 πr P ( r ) r (1 − M ( r ) /r ) (cid:105) . (A4)The tidal Love number k can then be calculated withthe following expression: k = 120 (cid:16) R s R (cid:17) (cid:0) − R s R (cid:17) (cid:104) − y R + ( y R − R s R (cid:105) × (cid:110) R s R (cid:16) − y R + 3 R s R (5 y R −
8) + 14 (cid:16) R s R (cid:17) × (cid:104) − y R + R s (3 y R − R + (cid:16) R S R (cid:17) (1 + y R ) (cid:105)(cid:17) + 3 (cid:16) − R s R (cid:17) (cid:104) − y R + R s ( y R − R (cid:105) × ln (cid:16) − R S R (cid:17)(cid:111) − . (A5)In the above, R S = 2 M is the Schwarzschild radius. Thevalue of Λ is then extracted with Eq. (2).0 Appendix B: Metamodeling parameters and Taylorparameters mapping
ELFc energy functional is written as a sum of kineticenergy term and potential energy term: E EF Lc ( ρ, δ ) = t F G ∗ ( ρ, δ ) + v NEF Lc ( ρ, δ ) , (B1)where ρ is the density and δ is the asymmetry parameter.The kinetic energy term t F G ∗ ( ρ, δ ) in the above is writtenas: t F G ∗ ( ρ, δ ) = t F G sat (cid:16) ρρ (cid:17) (cid:16)(cid:16) κ sat ρρ (cid:17) ((1 + δ ) +(1 − δ ) (cid:17) + κ sym ρρ δ ((1 + δ ) − (1 − δ ) )) . (B2)In the above, the parameters t F G sat = 22 . κ sym and κ sat are effective mass parameters describedin Eq. (11).The potential energy term v NEF Lc ( ρ, δ ) is written as: v NEF Lc ( ρ, δ ) = (cid:88) i =0 i ! ( v isi + v ivi δ )(1 − ( − − i ) × exp (cid:16) − . ρρ (cid:17) x i . (B3)In the above, the parameters v isi and v ivi are free parame-ters. These 10 parameters can be uniquely mapped ontoTaylor parameters using the following formulas (For adetailed derivation, please refer to Ref. [32]): v is = E sat − t F G sat (1 + κ sat ) , (B4) v is = − t F G sat (2 + 5 κ sat ) , (B5) v is = K sat − t F G sat ( − κ sat ) , (B6) v is = Q sat − t F G sat (4 − κ sat ) , (B7) v is = Z sat − t F G sat ( − κ sat ) , (B8) v iv = S − t F G sat (1 + ( κ sat + 3 κ sym )) , (B9) v iv = L − t F G sat (2 + 5( κ sat + 3 κ sym )) , (B10) v iv = K sym − t F G sat ( − κ sat + 3 κ sym )) , (B11) v iv = Q sym − t F G sat (4 − κ sat + 3 κ sym )) , (B12) v iv = Z sym − t F G sat ( − κ sat + 3 κ sym )) . (B13)When exploring the parameter space, Taylor parameterswill be translated to Metamodeling EOS using the aboveformulas and NS features will then be calculated withTOV equation. Neutron star properties will be examinedin order to search for Taylor parameter spaces flavoredby the observed tidal deformability. Appendix C: Full correlation between tidaldeformability and parameters
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