Predicting the moment of inertia of pulsar J0737-3039A from Bayesian modeling of the nuclear equation of state

# Predicting the moment of inertia of pulsar J0737-3039A from Bayesian modeling of the nuclear equation of state

Yeunhwan Lim Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA    Jeremy W. Holt Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA    Robert J. Stahulak Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
July 19, 2019
###### Abstract

We investigate neutron star moments of inertia from Bayesian posterior probability distributions of the nuclear equation of state that incorporate information from microscopic many-body theory and empirical data of finite nuclei. We focus on PSR J0737-3039A and predict that for this 1.338  neutron star the moment of inertia lies in the range  g cm g cm at the 95% credibility level, while the most probable value for the moment of inertia is  g cm. Assuming a measurement of the PSR J0737-3039A moment of inertia to 10% precision, we study the implications for neutron star radii and tidal deformabilities. We also determine the crustal component of the moment of inertia and find that for typical neutron star masses the crust contributes % of the total moment of inertia, below what is needed to explain large pulsar glitches in the scenario of strong neutron entrainment.

###### pacs:
21.30.-x, 21.65.Ef,
\definechangesauthor

[name=Yeunhwan, color=magenta]YH

## I Introduction

Neutron star observations are a promising tool Lattimer and Prakash (2004) to infer the properties of matter at extraordinarily high densities on the order of several times that of atomic nuclei. Shortly after the discovery Burgay et al. (2003); Lyne et al. (2004) of the double-pulsar system J0737-3039, it was suggested Lyne et al. (2004); Lattimer and Schutz (2005) that radio timing observations of star A in the pair could lead to a measurement of periastron advance sufficiently precise to resolve the effects of relativistic spin-orbit coupling Wex (1995), which enters at second order in a post-Newtonian expansion of the orbital motion. This in turn would place constraints on neutron star moments of inertia Damour and Schäfer (1988), complementary to ongoing LIGO/VIRGO gravitational wave observations Abbott et al. (2017, 2018) for neutron star tidal deformabilities and X-ray pulsar timing measurements Watts et al. (2016) for neutron star radii. All of these efforts aim to shed light on the properties of ultra-dense matter, its equation of state, and the possible existence of novel phases of matter Baym (1973); Au and Baym (1974); Glendenning (1982); Glendenning and Moszkowski (1991); Thorsson et al. (1994); Glendenning and Schaffner-Bielich (1998); Bunta and Gmuca (2004); Weber (2005); Alford et al. (2005); Brown et al. (2007, 2008); Weissenborn et al. (2011, 2012a, 2012b); Lim et al. (2014, 2015, 2018) conjectured to exist in the cores of neutron stars.

Several studies Mølnvik and Østgaard (1984); Morrison et al. (2004); Bejger et al. (2005); Piekarewicz et al. (2014); Carreau et al. (2018) have already provided a range of predictions for neutron star moments of inertia from different theoretical descriptions of the dense matter equation of state based on (i) nonrelativistic many-body calculations with realistic two- and three-nucleon forces, (ii) nonrelativistic Skyrme effective interactions, (iii) relativistic mean field models, (iv) self-bound strange quarks, and (v) meta-modeling that includes only empirical and microscopic constraints on the phenomenological parameters entering in the equation of state. While measurement of a neutron star’s moment of inertia to 10% precision may be sufficient Morrison et al. (2004) to distinguish among several of the qualitatively different models above, a more detailed understanding of the correlations among the moment of inertia, equation of state, neutron star radius, and other bulk neutron star properties can be achieved within a Bayesian statistical framework Raithel et al. (2016, 2017); Carreau et al. (2018). In previous work Lim and Holt (2018), we have constructed a model of the dense matter equation of state based on a Taylor series expansion in powers of the Fermi momentum. Bayesian posterior probability distributions for the model parameters were then constructed using microscopic predictions Coraggio et al. (2013, 2014); Sammarruca et al. (2015); Holt and Kaiser (2017) from chiral effective field theory Weinberg (1979); Entem and Machleidt (2003); Epelbaum et al. (2009); Machleidt and Entem (2011); Epelbaum et al. (2015); Entem et al. (2017) to define prior probability distribution functions together with empirical data Dutra et al. (2012); Holt and Lim (2018) for finite nuclei to define likelihood functions. As a first application, we compared the Bayesian modeling of neutron star tidal deformabilities with the first observational data from GW170817 Abbott et al. (2017).

In the present work we employ the same framework to study the probability distributions for neutron star moments of inertia as a function of mass, focusing on the distribution of values for the 1.338 neutron star J0737-3039A. We then make predictions for the crustal fraction of the moment of inertia. This quantity is central to the ongoing debate Chamel (2012); Andersson et al. (2012); Chamel (2013); Piekarewicz et al. (2014); Watanabe and Pethick (2017); Carreau et al. (2018) whether the superfluid angular momentum reservoir in a neutron star inner crust is sufficient to produce the largest pulsar glitches, such as those observed in the Vela pulsar Cordes et al. (1988). Previous studies Andersson et al. (2012); Chamel (2013) that included for the first time a treatment of the neutron band structure in the inner crust have found that the ratio of the crustal moment of inertia to the total moment of inertia may need to be as large as 7-9% in order to account for observed pulsar glitches. In the present work, we find that such large crustal moments of inertia are statistically unlikely in the neutron star mass range 1.2-1.5 . Due to competing effects, we find only a minor correlation between the fractional crustal moment of inertia and the core-crust transition density, in contrast to previous works Link et al. (1999); Fattoyev and Piekarewicz (2010) that suggested a strong correlation with the transition pressure at the crust-core interface.

The paper is organized as follows. In Section II we describe the model used to construct the dense matter equation of state and the resulting neutron star composition and structure, including a consistent treatment of the inner crust. In Section III we solve the simultaneous equations for hydrostatic equilibrium together with the additional equation for the neutron star moment of inertia in the slow-rotation approximation. In Section IV we present our predictions for the probability distribution of neutron star moments of inertia as a function of mass together with correlations among the neutron star radius, tidal deformability, nuclear symmetry energy slope parameter, as well as the core-crust transition density and pressure. We also compute the crustal fraction of the neutron star moment of inertia and the implications for the standard model of pulsar glitches. We end with a summary and conclusions.

## Ii Parameterized nuclear energy density functional

The nuclear equation of state (EOS), which relates the energy density and pressure at a given baryon number density, is essential for understanding the phenomenology of compact stars. Purely microscopic approaches for computing the cold dense matter equation of state start from realistic nuclear forces fitted to nucleon-nucleon scattering data, the deuteron binding energy, and also the properties of few-nucleon systems when three-body forces are included. Recently there has been much interest Hebeler and Schwenk (2010); Hebeler et al. (2011); Gezerlis et al. (2013); Tews et al. (2013); Coraggio et al. (2014); Roggero et al. (2014); Carbone et al. (2014); Drischler et al. (2014); Hagen et al. (2014); Wlazłowski et al. (2014); Wellenhofer et al. (2015); Tews et al. (2016); Drischler et al. (2016); Holt and Kaiser (2017); Sammarruca et al. (2018) in deriving constraints on the equation of state from chiral effective field theory, a framework for constructing the nuclear force that allows for the quantification of theoretical uncertainties through variations in the low-energy constants of the theory, the order in the chiral expansion, and the choice of resolution scale. However, since the typical momentum-space cutoffs used to regularize ultraviolet-divergent loop integrals are on the order of  MeV, chiral nuclear potentials are not expected to provide a good description of nuclear matter for densities larger than about twice the saturation density , which corresponds to a Fermi momentum of  MeV. Extensions to higher densities are strongly model dependent, but previous works have employed general polytrope extrapolations Hebeler et al. (2010, 2013) or speed of sound parametrization Tews et al. (2018), allowing for the inclusion of phase transitions and general conformal bounds Bedaque and Steiner (2015) for strongly interacting matter at very high energy densities.

In the present work, we employ a minimal model for the nuclear energy density functional beyond in which we fit predictions from chiral effective field theory to a power series expansion in the Fermi momentum up to and then extrapolate this functional without modification to higher densities. One of the primary aims of neutron star observations is to search for indications of novel phases of strongly interacting matter in neutron star cores, and the minimal model employed in the present study provides a useful baseline scenario without exotic degrees of freedom or phase transitions. Specifically we write the energy density as

 E(n,δ)=12mτn+12mτp+[1−δ2]fs(n)+δ2fn(n), (1)

where is the isospin asymmetry, and are the proton and neutron kinetic densities, and and are the potential energy contributions for symmetric nuclear matter and pure neutron matter of the form

 fs(n)=3∑i=0ain(2+i/3),fn(n)=3∑i=0bin(2+i/3). (2)

The coefficients and are fitted to individual symmetric nuclear matter and pure neutron matter equations of state computed in many-body perturbation theory using chiral nuclear forces up to . We include ten equations of state obtained by varying the chiral order of the nucleon-nucleon potential from next-to-next-to-leading order (N2LO) to N3LO, the order in many-body perturbation theory from second to third order, and finally the momentum-space cutoff from  MeV. The mean and covariance matrices for and then define our prior Gaussian probability distributions.

Since none of the adjustable parameters in chiral nuclear potentials are explicitly fitted to the properties of nuclear matter, we then incorporate empirical constraints on the nuclear matter saturation density , saturation energy , incompressibility , and skewness . The values for these quantities are typically obtained from fitting energy density functionals to the binding energies and charge radii of finite nuclei. We consider 205 high quality models Dutra et al. (2012), which give the average and standard deviations  fm,  MeV,  MeV,  MeV. The parameters are then uniquely determined in terms of the empirical nuclear matter parameters , which provides a method to derive Bayesian likelihood functions involving the . For pure neutron matter there are no strong empirical constraints on the equation of state. However, the nuclear symmetry energy is closely related to the isospin asymmetry energy , which can be extracted from the binding energies of nuclei, giant dipole resonances, neutron skin thicknesses, and heavy-ion collision flow data. Recent analyses Lattimer and Lim (2013); Tews et al. (2017) give  MeV. From correlations Holt and Lim (2018); Margueron and Gulminelli (2018) among the isospin asymmetry energy , its slope parameter and curvature , likelihood functions involving the were obtained in Ref. Lim and Holt (2018) to derive the final Bayesian posterior probability distributions.

Randomly sampling from the and joint probability distributions, we construct 10,000 equations of state for nuclear matter in beta equilibrium. For the crust EOS, we utilize a liquid drop model with the same nuclear model used in the bulk matter equation of state Lim et al. (2017). This provides a unified approach, since the structure is constructed by a single nuclear model, without the need to stitch together various bulk matter EOSs with one specific crust EOS. Since we use the same nuclear force model for both the core and crust, the core-crust density is treated consistently, and therefore the moment of inertia from the crust can be calculated correctly.

In the liquid drop model at  MeV, the total energy density is given by

 ε= unifi+σ(xi)udrN+2π(nixierN)2ufd(u) (3) +(1−u)nnofno,

where is the volume fraction of a heavy nucleus in the Wigner-Seitz cell, is the baryon number density of the heavy nucleus, is the energy density of the heavy nucleus, is the surface tension as a function of proton fraction () of the heavy nucleus, is the dimension of the nuclear phase (e.g., spherical, cylindrical, or slab), is the radius of the heavy nucleus, is the geometric function that reflects the nuclear pasta phase, is the neutron density outside of the heavy nucleus, and is the energy density of the neutron gas. The nuclear configuration is determined by minimizing the energy density for a given total baryon number density and proton fraction . In beta equilibrium matter, we include electrons and find the proton (or electron) fraction that minimizes the total energy density. Note that we use the same energy density functional for and as used in the bulk matter EOS. The core-crust transition density is found by comparing the energy density of inhomogeneous matter and uniform nuclear matter. Near the phase boundaries, we employ a Maxwell construction to find the exact density and pressure,

 pi=po,μni=μno (4)

where () stands for inhomogeneous (homogeneous) matter.

## Iii Moment of inertia

The neutron star mass and radius are obtained by solving the Tolman-Oppenheimer-Volkov (TOV) equation, which is the hydrostatic equilibrium equation for spherically symmetric neutron stars, given by

 dpdr =−(ε+p)(m+4πr3p)r(r−2m), (5a) dεdr =4πr2ε, (5b)

where is the pressure, is the energy density (including rest mass), and is the enclosed mass within the distance from the center. The neutron star moment of inertia is then calculated by solving the conventional TOV equation with an additional equation including the rotational frequency. In a slowly rotating neutron star, the moment of inertia is given by Hartle (1967); Lattimer and Prakash (2001)

 I=8π3∫R0r4(ε+p)e(λ−ν)/2¯ωΩdr, (6)

where and are metric functions defined by

 e−λ = (1−2mr)−1, (7) dνdr = −2ε+pdpdr, (8)

is the angular velocity of a uniformly rotating neutron star, and is the rotational drag function. The unitless frequency satisfies

 ddr(r4jd~ωdr)=−4r3~ωdjdr, (9)

where . This rotational drag meets the boundary conditions

 ~ω(r=R)=1−2IR3andd~ωdr∣∣∣r=0=0 (10)

at the surface and center of the neutron star. The moment of inertia can be integrated from

 I =23∫R0r3~ωdjdrdr=16∫R0ddr(r4jd~ωdr)dr (11) =R46d~ωdr∣∣∣r=R.

The second-order differential equation (9) can be translated to a first-order differential equation by introducing , giving

 dϕdr =−ϕr(ϕ+3)−(4+ϕ)dlnjdr (12) =−ϕr(ϕ+3)+(4+ϕ)4πr2(ε+p)(r−2m),

with the boundary condition

 ϕ(r=0)=0. (13)

The total moment of inertia of a neutron star is then given as

 I=R36ϕR~ωR=ϕR6(R3−2I),I=R3ϕR6+2R, (14)

with the boundary condition in Eq. (10).

The moment of inertia of the core is given by integrating Eq. (11) up to the core radius :

 Ic=R4t6d~ωdr∣∣∣r=Rt=R3t6ϕt~ωt. (15)

From the relation between and , we have

 ~ωt=~ωRExp[−∫RRtϕ(r)rdr]. (16)

Thus, the moment of inertia of the crust is given as

 ΔI =I−Ic=R36ϕR~ωR[1−(RtR)3ϕtϕR~ωt~ωR] (17) =I{1−(RtR)3ϕtϕRExp[−∫RRtϕ(r)rdr]}.

## Iv Results

In the present section we analyze 10,000 equations of state randomly sampled from the Bayesian posterior distributions for the and parameters entering into the nuclear energy density functional of Eq. (1). For each equation of state we consider up to 110 representative neutron stars with masses in the range with spacing . In the case that the particular equation of state yields a maximum mass such that , which occurs for the softest equations of state generated, we use the same mass spacing but with a modified range . In total we therefore consider more than 1,000,000 neutron stars for analysis, each constructed with a realistic crust equation of state.

In Fig. 1 we show in red the resulting probability distribution for the ratio of the moment of inertia to as a function of . In the previous work Lattimer and Schutz (2005) it was shown that in the absence of phase transitions and other effects that strongly soften the equation of state beyond a few times normal nuclear densities, there is a nearly unique relation between the quantity and . This relation is shown as the blue band in Fig. 1 which we have generated from the empirical formula obtained in Ref. Lattimer and Schutz (2005). Observational evidence suggests that neutron star masses lie in the range while radii lie in the range . Therefore, we expect only the region to be physically relevant. For this range of neutron star compactness parameters our results are completely consistent with the empirical relation in Ref. Lattimer and Schutz (2005). We find that over the entire range of neutron star compactnesses, the following formula holds:

 IMR2=M/R+a(M/R)4b+c(M/R), (18)

where , , and . This formula is shown as the green dotted curve in Fig. 1 and should be accurate for most neutron star configurations.

In Fig. 2 we show the neutron star moment of inertia as a function of mass, plotted as a probability distribution based on the 10,000 equations of state sampled from our Bayesian posterior distribution. Naturally the moment of inertia increases approximately linearly with the mass. The uncertainty also generically increases with the neutron star mass up to about . Beyond this value, the fraction of equations of state capable of producing such massive neutron stars decreases as does the range of allowed radii for a given mass. This results in a narrowing of the moment of inertia probability distribution for the largest-mass neutron stars in our sample.

From Fig. 1 we see that a simultaneous measurement of neutron star mass and moment of inertia will indeed strongly constrain the radius. This is demonstrated more explicitly in Fig. 3, where we plot the probability distribution for the neutron star moment of inertia vs. neutron star radius for a fixed mass of corresponding to that of PSR J0737-3039A. A total of 10,000 samples are considered, one for each of the generated equations of state. We see that the moment of inertia lies in the range , where for convenience we have defined , while the radius lies between . In addition we observe an approximate linear correlation between the moment of inertia and the radius in this regime of the form

 I=(−0.881+0.1853Rkm)×1045gcm2 (19)

with correlation coefficient . From Fig. 3 we observe that the moment of inertia probability distribution is asymmetric and peaks around and  km. We present in Fig. 4 the full probability distribution function for the moment of inertia of a neutron star from the 10,000 EOSs constructed in this work. The probability function appears as an asymmetric Gaussian function and therefore the average value of the total moment of inertia does not match the most probable moment of inertia, which we denote by . In this work, we find with one- and two-sigma credibility ranges given by , , , and in units of . In comparison the most likely value of the moment of inertia is found to be .

A precise moment of inertia measurement to 10% precision is expected to be competitive with gravitational wave constraints on neutron star radii Abbott et al. (2018) as well as direct measurements of radii from X-ray observations. We now discuss the implications for such a moment of inertia measurement and how it can be implemented in the current Bayesian modeling of the equation of state. In the top left panel of Fig. 5 we show the neutron star mass and radius distribution resulting from our Bayesian analysis of the nuclear energy density functional including constraints from microscopic many-body theory and nuclear experiments. The softest equations of state generated from the statistical sampling do not reach a maximum neutron star mass of , but this is due primarily to our smooth continuation of the equation of state beyond twice saturation density. In particular we cannot rule out the existence of higher-power repulsive contributions to the nuclear energy density functional beyond that might sufficiently stiffen the equations of state to support a neutron star. In the present work we do not focus on the properties of the heaviest neutron stars and therefore we keep the softest equations of state in our subsequent analysis.

As the value of the neutron star moment of inertia increases, so too does the statistical average of the neutron star radius. Our current EOS distribution results in km, km, km, km, km for a neutron star, and the most probable radius is km. We apply Bayesian analysis to see how the credibility interval varies for a given moment of inertia measurement. Bayesian statistics gives for the posterior probability

 P(Mi|D)=P(D|Mi)P(Mi)∑jP(D|Mj)P(Mj), (20)

where stands for the nuclear model parameters, represents the data set, is the posterior probability, is the likelihood function, and is the prior distribution function, which in this case we take to be that arising from the inclusion of EOS constraints from nuclear theory and experiment (that is, our previous posterior distribution function). From measurements of the moment of inertia and the corresponding uncertainties , we define the likelihood function for a specific nuclear model as

 (21)

where is the moment of inertia from the specific nuclear model.

In the top-right, bottom-left, and bottom-right panels of Fig. 5 we show the resulting posterior probability distribution for the neutron star mass-radius relation assuming the following moment of inertia measurements for the neutron star in PSR J0737-3039: . The resulting and credibility intervals for the radius of a neutron star are shown in Table 1.

For instance, in the case that , the maximum radius for a neutron star would be shifted down to approximately  km. Likewise, under the scenario where the minimum radius would be shifted up to about  km. Even a moment of inertia measurement nearly consistent with our most probable value from the prior distribution will further constrain the equation of state. This is shown in the lower-left panel of Fig. 5, where a measured value of increases the lower bound on the radius of neutron star up to roughly  km, even though the measured value of would be only a few percent below that of the most probable value from the prior distribution.

Even more strongly constraining is the relation between the neutron star moment of inertia and dimensionless tidal deformability as pointed out in Ref. Yagi and Yunes (2013). In Fig. 6 we show the correlation between the tidal deformability and the moment of inertia of a neutron star with mass . We find the empirical formula,

 Λ=−38.520+181.912(I45)3.5, forM=1.338M⊙. (22)

Note that this formula may support different values for the coefficients when neutron stars with different masses are considered, due to the fact that the most probable values of and vary with the mass. As seen in Fig. 6 precise measurements of neutron star moments of inertia can be used to very tightly constrain the tidal deformability, and vice versa.

In Fig. 7 we demonstrate the strong correlation between and for all of the neutron stars that may be constructed from our Bayesian posterior probability distribution for the nuclear equation of state. The red solid line in Fig. 7 results from plotting individually the scaled moments of inertia vs. tidal deformabilities . The log-log functional relationship between and can be well approximated by

 Log10(¯I)=0.65967+0.097354[Log10(Λ)]1.56. (23)

This is shown as the blue dashed line in Fig. 7. In comparison we also show as the green dashed line in Fig. 7, the original correlation derived in Ref. Yagi and Yunes (2013). While both functions fit the theoretical results very well, we find that our functional form has a smaller value with fewer parameters.

Finally, we investigate the fraction of the neutron star moment of inertia contained in the crust. This quantity is related to the ratio of the superfluid angular momentum to the total angular momentum in the neutron star, which must be sufficiently large in order to support the observed glitch activity of the Vela pulsar. One can derive Link et al. (1999) that in the standard hydrodynamic two-fluid model, the ratio of the crustal moment of inertia to the total moment of inertia must satisfy

 ΔII≥0.014. (24)

However, strong entrainment of otherwise free neutrons by the inner crust can be studied within band theory calculations and has been shown Chamel (2012); Andersson et al. (2012); Chamel (2013) to reduce the neutron superfluid angular momentum reservoir. The key quantity is the neutron effective mass, defined as Chamel (2013)

 m∗n≡mnnfn/ncn, (25)

where is the bare neutron mass, is the density of conduction neutrons, and is the density of unbound neutrons. The neutron effective mass strongly depends on the density and peaks at a value of around  fm. Averaging over typical densities in the crust leads to a decrease in the superfluid angular momentum reservoir such that is required to explain observed glitch activity. Such large crustal moments of inertia are not favored in most theoretical modeling of neutron star structure, especially for the moderately soft equations of state produced by chiral effective field theory.

In Fig. 8 we show the fraction of the crustal moment of inertia to the total moment of inertia as a function of the neutron star mass. The crust moment of inertia is obtained numerically from the Eq. (17). The blue (green) dashed curves in Fig. 8 indicate () credibilities, while the black dashed curve represents the central value of the momentum fraction. Low-mass neutron stars tend to have thicker crusts than high-mass neutron stars, leading to fractional crustal moments of inertia that are systematically larger. Although the mass of the Vela pulsar is not precisely known, recent work Ho et al. (2015) has estimated a value . From Fig. 8 we see that the present EOS modeling within the scenario of strong neutron entrainment would be insufficient to explain large pulsar glitch activity. However, such large entrainment effects have recently been called into question, and in particular the inclusion of neutron pairing in band theory calculations may significantly reduce neutron entrainment and increase the angular momentum reservoir Watanabe and Pethick (2017).

Connecting the crustal fraction of the moment of inertia to specific features of the nuclear equation of state is challenging. It has been suggested Link et al. (1999); Fattoyev and Piekarewicz (2010) that the pressure of beta-equilibrium matter at the neutron star core-crust interface is strongly correlated with the crustal moment of inertia:

 ΔI∼R6tpt, (26)

where is the radius of the neutron star core and is the pressure at the core-crust boundary. Eq. (26) can be derived under the approximation of slow rotation and thin, low-density crusts. In Fig. 9 we plot the probability contour plot of the crustal density and corresponding pressure at the core-crust interface as a function of the nuclear symmetry energy slope parameter . We note here the negative correlation between and , as well as a weaker negative correlation between and . This negative correlation results in a related negative correlation between and and almost no correlation between and as shown in the right panels of Fig. 10, plotted for a neutron star with mass . That is, a low transition density at the core crust boundary results in the crustal moment of inertia making up a higher fraction of the total.

In the left and center panels of Fig. 10 we show also the probability distributions for the total moment of inertia and the crustal moment of inertia as a function of the transition density and pressure for a neutron star. We find negative correlations with the transition density and relatively weak correlations with the transition pressure. Physically, as the transition density increases, decreases and the neutron star becomes more compact. Thus the moment of inertia decreases. On the other hand, is anti-correlated with since is anti-correlated with and correlates with , the radius of the neutron star.

Since we find no strong link between the crustal moment of inertia and the core-crust transition pressure (in contrast to previous works), we examine in more detail the correlation between the transition pressure and the core radius , which was not considered in Refs. Link et al. (1999); Fattoyev and Piekarewicz (2010). In Fig. 11 we plot the probability distribution of the core radius vs. the core transition pressure. We find a statistically significant anticorrelation between the two quantities, which from Eq. (26) reduces the dependence of the crustal moment of inertia on the transition pressure. Indeed soft equations of state (with low values of ) are correlated with higher transition pressures as seen in Fig. 9 and also give rise to more compact neutron stars with smaller core radii. The combined result is almost no correlation between the crustal moment of inertia and the transition pressure, as seen in the middle panel of Fig. 10.

## V Summary

We have computed the moment of inertia of neutron stars based on a Bayesian analysis of the nuclear energy density functional constrained by chiral effective field theory and nuclear matter properties deduced from finite nuclei. We predict that for pulsar PSR J0737-3039A, with a well measured mass of , the moment of inertial lies in the range  g cm g cm at the 95% credibility level, while the most probable value for the moment of inertia is  g cm. We have also shown that a pulsar timing measurement of the PSR J0737-3039A moment of inertia to 10% precision will result in meaningful constraints on the current Bayesian modeling of the equation of state by imposing the likelihood function for the posterior probability. Three scenarios were considered and the resulting effect on the neutron star mass-radius relation were analyzed. In particular, we find that the credibility interval for the radius of a neutron star decreases from km to , and  km depending on the moment of inertia measurement.

We have studied as well correlations among the neutron star moment of inertia, radius, and tidal deformability. A strong correlation is demonstrated between the moment of inertia and tidal deformability, indicating that one of the two quantities will strongly constrain the other. From our large sample of realistic equations of state, we derived a new functional model for the vs. universal relation.

Finally, we have employed realistic modeling of the crust equation of state to determine the fraction of the neutron star moment of inertia contained in the crust. We find that for typical neutron star masses of 1.2-1.5 , the fractional crustal moment of inertia is less than . In the strong neutron entrainment scenario, our small values of the crustal moment of inertia would be unable to account for the observed large glitch activity in the Vela pulsar. We have also shown that the crustal moment of inertia is weakly correlated with the core-crust transition density. Low transition densities at the core-crust boundary allow a large ratio between the moment of inertia of the neutron star crust to the total moment of inertia, since the high transition pressure is responsible for smaller core radii in neutron stars. In contrast to previous works, we find no correlation between the crustal moment of inertia and the transition pressure. It is understood that the symmetry energy slope parameter is anti-correlated with but positively correlated with . Thus, their effects counteract each other, and no visible correlation is seen.

In the future we plan to incorporate as well neutron star radius measurements from the NICER mission and neutron star tidal deformability measurements the LIGO/VIRGO collaborations within the present Bayesian statistical modeling of the equation of state.

###### Acknowledgements.
Work supported by the National Science Foundation under Grant No. PHY1652199. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing.

## References

• Lattimer and Prakash (2004) J. M. Lattimer and M. Prakash, Science 304, 536 (2004).
• Burgay et al. (2003) M. Burgay, N. D’Amico, A. Possenti, R. N. Manchester, A. G. Lyne, B. C. Joshi, M. A. McLaughlin, M. Kramer, J. M. Sarkissian, F. Camilo, V. Kalogera, C. Kim,  and D. R. Lorimer, Nature 426, 531 (2003).
• Lyne et al. (2004) A. G. Lyne, M. Burgay, M. Kramer, A. Possenti, R. Manchester, F. Camilo, M. A. McLaughlin, D. R. Lorimer, N. D’Amico, B. C. Joshi, J. Reynolds,  and P. C. C. Freire, Science 303, 1153 (2004).
• Lattimer and Schutz (2005) J. M. Lattimer and B. F. Schutz, Astrophys. J. 629, 979 (2005).
• Wex (1995) N. Wex, Class. Quantum Grav. 12, 983 (1995).
• Damour and Schäfer (1988) T. Damour and G. Schäfer, Nuovo Cimento 101, 127 (1988).
• Abbott et al. (2017) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017).
• Abbott et al. (2018) B. P. Abbott et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), Phys. Rev. Lett. 121, 161101 (2018).
• Watts et al. (2016) A. L. Watts et al., Rev. Mod. Phys. 88, 021001 (2016).
• Baym (1973) G. Baym, Phys. Rev. Lett. 30, 1340 (1973).
• Au and Baym (1974) C.-K. Au and G. Baym, Nucl. Phys. A236, 500 (1974).
• Glendenning (1982) N. K. Glendenning, Phys. Lett. B 114, 392 (1982).
• Glendenning and Moszkowski (1991) N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991).
• Thorsson et al. (1994) V. Thorsson, M. Prakash,  and J. M. Lattimer, Nucl. Phys. A572, 693 (1994).
• Glendenning and Schaffner-Bielich (1998) N. K. Glendenning and J. Schaffner-Bielich, Phys. Rev. Lett. 81, 4564 (1998).
• Bunta and Gmuca (2004) J. K. Bunta and S. Gmuca, Phys. Rev. C 70, 054309 (2004).
• Weber (2005) F. Weber, Prog. Part. Nucl. Phys. 54, 193 (2005).
• Alford et al. (2005) M. Alford, M. Braby, M. Paris,  and S. Reddy, Astrophys. J. 629, 969 (2005).
• Brown et al. (2007) G. E. Brown, J. W. Holt, C.-H. Lee,  and M. Rho, Phys. Rept. 439, 161 (2007).
• Brown et al. (2008) G. E. Brown, C.-H. Lee,  and M. Rho, Phys. Rept. 462, 1 (2008).
• Weissenborn et al. (2011) S. Weissenborn, I. Sagert, G. Pagliara, M. Hempel,  and J. Schaffner-Bielich, Astrophys. J. 740, L14 (2011).
• Weissenborn et al. (2012a) S. Weissenborn, D. Chatterjee,  and J. Schaffner-Bielich, Nucl. Phys. A881, 62 (2012a).
• Weissenborn et al. (2012b) S. Weissenborn, D. Chatterjee,  and J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012b).
• Lim et al. (2014) Y. Lim, K. Kwak, C. H. Hyun,  and C.-H. Lee, Phys. Rev. C 89, 055804 (2014).
• Lim et al. (2015) Y. Lim, C. H. Hyun, K. Kwak,  and C.-H. Lee, Int. J. Mod. Phys. E 24, 1550100 (2015).
• Lim et al. (2018) Y. Lim, C.-H. Lee,  and Y. Oh, Phys. Rev. D 97, 023010 (2018).
• Mølnvik and Østgaard (1984) T. Mølnvik and E. Østgaard, Nucl. Phys. A437, 239 (1984).
• Morrison et al. (2004) I. A. Morrison, T. W. Baumgarte, S. L. Shapiro,  and V. R. Pandharipande, Astrophys. J. 617, L135 (2004).
• Bejger et al. (2005) M. Bejger, T. Bulik,  and P. Haensel, Mon. Not. R. Astron. Soc. 364, 635 (2005).
• Piekarewicz et al. (2014) J. Piekarewicz, F. J. Fattoyev,  and C. J. Horowitz, Phys. Rev. C 90, 015803 (2014).
• Carreau et al. (2018) T. Carreau, F. Gulminelli,  and J. Margueron, arXiv:1810.00719  (2018).
• Raithel et al. (2016) C. A. Raithel, F. Özel,  and D. Psaltis, Astrophys. J. 831, 44 (2016).
• Raithel et al. (2017) C. A. Raithel, F. Özel,  and D. Psaltis, Astrophys. J. 844, 156 (2017).
• Lim and Holt (2018) Y. Lim and J. W. Holt, Phys. Rev. Lett. 121, 062701 (2018).
• Coraggio et al. (2013) L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt,  and F. Sammarruca, Phys. Rev. C 87, 014322 (2013).
• Coraggio et al. (2014) L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, L. E. Marcucci,  and F. Sammarruca, Phys. Rev. C 89, 044321 (2014).
• Sammarruca et al. (2015) F. Sammarruca, L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt,  and L. E. Marcucci, Phys. Rev. C 91, 054311 (2015).
• Holt and Kaiser (2017) J. W. Holt and N. Kaiser, Phys. Rev. C 95, 034326 (2017).
• Weinberg (1979) S. Weinberg, Physica A 96, 327 (1979).
• Entem and Machleidt (2003) D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003).
• Epelbaum et al. (2009) E. Epelbaum, H.-W. Hammer,  and U.-G. Meißner, Rev. Mod. Phys. 81, 1773 (2009).
• Machleidt and Entem (2011) R. Machleidt and D. R. Entem, Phys. Rept. 503, 1 (2011).
• Epelbaum et al. (2015) E. Epelbaum, H. Krebs,  and U.-G. Meißner, Phys. Rev. Lett. 115, 122301 (2015).
• Entem et al. (2017) D. R. Entem, R. Machleidt,  and Y. Nosyk, Phys. Rev. C 96, 024004 (2017).
• Dutra et al. (2012) M. Dutra, O. Lourenco, J. S. Sá Martins, A. Delfino, J. R. Stone,  and P. D. Stevenson, Phys. Rev. C 85, 035201 (2012).
• Holt and Lim (2018) J. W. Holt and Y. Lim, Phys. Lett. B 784, 77 (2018).
• Chamel (2012) N. Chamel, Phys. Rev. C 85, 035801 (2012).
• Andersson et al. (2012) N. Andersson, K. Glampedakis, W. C. G. Ho,  and C. M. Espinoza, Phys. Rev. Lett. 109, 241103 (2012).
• Chamel (2013) N. Chamel, Phys. Rev. Lett. 110, 011101 (2013).
• Watanabe and Pethick (2017) G. Watanabe and C. J. Pethick, Phys. Rev. Lett. 119, 062701 (2017).
• Cordes et al. (1988) J. M. Cordes, G. S. Downs,  and J. Krause-Polstorff, Astrophys. J. 330, 847 (1988).
• Link et al. (1999) B. Link, R. I. Epstein,  and J. M. Lattimer, Phys. Rev. Lett. 83, 3362 (1999).
• Fattoyev and Piekarewicz (2010) F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C 82, 025810 (2010).
• Hebeler and Schwenk (2010) K. Hebeler and A. Schwenk, Phys. Rev. C 82, 014314 (2010).
• Hebeler et al. (2011) K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga,  and A. Schwenk, Phys. Rev. C 83, 031301 (2011).
• Gezerlis et al. (2013) A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler, A. Nogga,  and A. Schwenk, Phys. Rev. Lett. 111, 032501 (2013).
• Tews et al. (2013) I. Tews, T. Krüger, K. Hebeler,  and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013).
• Roggero et al. (2014) A. Roggero, A. Mukherjee,  and F. Pederiva, Phys. Rev. Lett. 112, 221103 (2014).
• Carbone et al. (2014) A. Carbone, A. Rios,  and A. Polls, Phys. Rev. C 90, 054322 (2014).
• Drischler et al. (2014) C. Drischler, V. Somà,  and A. Schwenk, Phys. Rev. C 89, 025806 (2014).
• Hagen et al. (2014) G. Hagen, T. Papenbrock, A. Ekström, K. Wendt, G. Baardsen, S. Gandolfi, M. Hjorth-Jensen,  and C. Horowitz, Phys. Rev. C 89, 014319 (2014).
• Wlazłowski et al. (2014) G. Wlazłowski, J. W. Holt, S. Moroz, A. Bulgac,  and K. J. Roche, Phys. Rev. Lett. 113, 182503 (2014).
• Wellenhofer et al. (2015) C. Wellenhofer, J. W. Holt,  and N. Kaiser, Phys. Rev. C 92, 015801 (2015).
• Tews et al. (2016) I. Tews, S. Gandolfi, A. Gezerlis,  and A. Schwenk, Phys. Rev. C 93, 024305 (2016).
• Drischler et al. (2016) C. Drischler, A. Carbone, K. Hebeler,  and A. Schwenk, Phys. Rev. C 94, 054307 (2016).
• Sammarruca et al. (2018) F. Sammarruca, L. E. Marcucci, L. Coraggio, J. W. Holt, N. Itaco,  and R. Machleidt, arXiv:1807.06640  (2018).
• Hebeler et al. (2010) K. Hebeler, J. M. Lattimer, C. J. Pethick,  and A. Schwenk, Phys. Rev. Lett. 105, 161102 (2010).
• Hebeler et al. (2013) K. Hebeler, J. M. Lattimer, C. J. Pethick,  and A. Schwenk, Astrophys. J. 773, 11 (2013).
• Tews et al. (2018) I. Tews, J. Carlson, S. Gandolfi,  and S. Reddy, Astrophys. J. 860, 149 (2018).
• Bedaque and Steiner (2015) P. Bedaque and A. W. Steiner, Phys. Rev. Lett. 114, 031103 (2015).
• Lattimer and Lim (2013) J. M. Lattimer and Y. Lim, Astrophys. J. 771, 51 (2013).
• Tews et al. (2017) I. Tews, J. M. Lattimer, A. Ohnishi,  and E. E. Kolomeitsev, Astrophys. J. 848, 105 (2017).
• Margueron and Gulminelli (2018) J. Margueron and F. Gulminelli, arXiv:1807.01729  (2018).
• Lim et al. (2017) Y. Lim, C. H. Hyun,  and C.-H. Lee, Int. J. Mod. Phys. E 26, 1750015 (2017).
• Hartle (1967) J. B. Hartle, Astrophy. J. 150, 1005 (1967).
• Lattimer and Prakash (2001) J. M. Lattimer and M. Prakash, Astrophys. J. 550, 426 (2001).
• Yagi and Yunes (2013) K. Yagi and N. Yunes, Phys. Rev. D 88, 023009 (2013).
• Ho et al. (2015) W. C. G. Ho, C. M. Espinoza, D. Antonopoulou,  and N. Andersson, Sci. Adv. 1 (2015).
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