∎

11institutetext: Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon,
UMR 5822, F-69622, Villeurbanne, France 22institutetext: Institute for Nuclear
Theory, University of Washington, Seattle, WA 98195-1550, USA 33institutetext:
JINA-CEE, Michigan State University, East Lansing, MI, 48823, USA

# Confronting a set of Skyrme and $\chi_{EFT}$ predictions for the crust of
neutron stars

On the origin of uncertainties in model predictions

Guilherme Gramsaddr1  Jérôme Margueronaddr1  Rahul Somasundaramaddr1  Sanjay
Reddyaddr2,addr3

(Received: date / Accepted: date)

###### Abstract

With the improved accuracy of neutron star observational data, it is necessary
to derive new equation of state where the crust and the core are consistently
calculated within a unified approach. For this purpose we describe non-uniform
matter in the crust of neutron stars employing a compressible liquid-drop
model, where the bulk and the neutron fluid terms are given from the same
model as the one describing uniform matter present in the core. We then
generate a set of fifteen unified equations of state for cold catalyzed
neutron stars built on realistic modelings of the nuclear interaction, which
belongs to two main groups: the first one derives from the phenomenological
Skyrme interaction and the second one from $\mathrm{\chi EFT}$ Hamiltonians.
The confrontation of these model predictions allows us to investigate the
model dependence for the crust properties, and in particular the effect of
neutron matter at low density. The new set of unified equations of state is
available at the CompOSE repository.

###### Keywords:

neutron star equation of state crust compressible liquid-drop model Skyrme and
chiral EFT interactions

††journal: Eur. Phys. J. A

##  1 Introduction

The description of the neutron star (NS) equation of state (EoS) from the
crust to the core represents a challenge for modern nuclear and particle
physics, as well as for astrophysics Lattimer:2004 ; LATTIMER2016127 , see
also Ref. Burgio2021  for a review. A typical density at the transition
between the crust and the core is about half saturation energy-density
($\rho_{\mathrm{sat}}\approx 2.6~{}10^{14}$ g cm-3) while in the core, it
reaches up to several times this energy-density. These densities determine NS
global equilibrium properties, such as their masses, radii, moment of inertia
or tidal deformabilities Haensel:2007yy . Recently, NICER X-ray observatory
has released the measurement of two NS masses and radii with an unprecedented
accuracy: PSR J0030 has been estimated to have a mass M$=1.4\pm 0.05$ M⊙ and a
radius R$=13.02^{+1.24}_{-1.06}$ km MillerNICER2019  or
R$=12.71^{+1.14}_{-1.19}$ km Riley19 , and PSR J0740 with a mass M$=2.07\pm
0.05$ M⊙ and a radius R$=12.35\pm 0.75$ km Miller21  or
R$=12.39^{+1.30}_{-0.98}$ km NICER2021 . Now measurable with LIGO-Virgo
interferometers, the tidal deformability $\tilde{\Lambda}$ has been estimated
for the first time to be $\tilde{\Lambda}=280^{+300}_{-200}$ from GW170817
Abbott2017 , the gravitational waveforms emitted during the last inspiral
orbits of the binary NS mergers. The tidal deformability is in turn strongly
correlated to the mass and radius Tews:2018kmu , converting the uncertainty on
$\tilde{\Lambda}$ into an uncertainty on the radius of a 1.4M⊙ NS of about
$\pm 1$ km. The confrontation of theoretical modeling against accurate
observational data, especially the very recent ones from NICER and the LIGO-
Virgo-Kagra collaboration (LVKC), is now possible and represents a major
challenge requiring more and more precise modeling of dense matter properties.

The new precision era, opened by NICER and LVKC for the measurement of radii
and tidal deformabilities, requires to control more systematically the sources
of theoretical uncertainties in the modeling of NS EoS. One of them is the
modeling of NS crust, which may be inconsistent with the one used for the
core. While the crust represents a small fraction of the NS, 10% in terms of
radius and less than 1% of the mass, it was estimated that the method used to
connect the crust and the core could influence the theoretical prediction of
the NS radius by 3-5%, or in other words, by a few hundred of meters
Fortin2016 . See also Ref. Suleiman2021  for a recent study on the impact of
non consistent treatment of crust and core EoS on NS macroscopic properties.
Presently, the experimental uncertainties in the measurement of NS radii are
still larger – about 1-2 km – than this theoretical one. However, anticipating
future observational improvements, it is preferable to resolve this source of
uncertainty by employing unified models from the crust to the core of NS, as
it was already suggested by several teams DouchinHaensel2001 ; Fantina12 ;
Fortin2016 .

Even when NS EoS are unified from the crust to the core, they still lead to
different predictions depending on the nuclear interaction on which they are
built on. The scope of this paper is therefore to confront a set of unified
EoS for cold catalyzed NS matter in order to evaluate the model dependence of
the crust predictions. To do so, we employ a compressible liquid-drop model
(CLDM) originally suggested by Baym, Bethe and Pethick bbp1971 , and which we
have recently applied to include state-of-the-art nuclear physics constraints,
e.g. $\mathrm{\chi EFT}$ interaction used in Many-Body Perturbation Theory
(MBPT) framework, as well as nuclear physics data such as the AME2016 database
AME2016 . In the CLDM, the EoS used in the core is unified with the crust,
through the bulk and neutron fluid contributions, see for instance the recent
Refs. Carreau2019a ; Grams:2021a  and references therein. The effects of non-
uniformities in the crust are implemented through finite-size (FS) terms. In
Ref. Grams:2021b , we have shown that these terms can be sorted according to
the leptodermous expansion Myers1973 . We have also presented in detail the
use of the meta-model to capture the predictions of the $\mathrm{\chi EFT}$
Hamiltonians. A first comparison of the $\mathrm{\chi EFT}$ predictions
against a representative phenomenological model, SLy4 Chabanat1997 , was
performed in Ref. Grams:2021a . In this paper, we perform a more extensive
comparison considering a set of seven Skyrme parametrisations which are used
to determine the properties of finite nuclei, and perform pretty well. These
Skyrme models are also chosen such that they span over the present
uncertainties in the EoS properties, e.g. slope of the symmetry energy
$L_{\mathrm{sym}}$, etc. In addition, we provide the new EoS within the format
of the CompOSE catalog compose , which offers an access to a large set of EoS
ready to be implemented in astrophysical modeling. It also allows us to
analyze the recent progresses in the field of dense matter physics.

The present paper is ordered as follow: in Section 2 we compare the Skyrme and
$\mathrm{\chi EFT}$ predictions in uniform matter and analyze their
systematical differences, especially for low-density neutron matter. Then in
Sec. 3 we compare the various crust EoS provided by the CLDM based on
different nuclear interactions. Finally, we show our results in Sec. 4 using
the same files and units as the ones uploaded on the CompOSE catalog.

Table 1: Nuclear empirical parameters for the Skryme models and the Hamiltonians derived from $\mathrm{\chi EFT}$ used in the present work. The lowest order NEP are defined at $n_{\mathrm{sat}}$ ($E$, $L$/$n$, $K$), while the higher order ones ($Q$, $Z$) are defined from a fit of SM and NM up to 10$n_{\mathrm{sat}}$ for Skyrme, and a fix to reach 2M⊙ for the $\mathrm{\chi EFT}$ models. The last four columns show the low-density correction parameters $b_{\mathrm{sat}}$ and $b_{\mathrm{sym}}$, and the effective mass $m^{*}_{\mathrm{sat}}$ at saturation in symmetric matter and the effective mass splitting $\Delta m^{*}_{\mathrm{sat}}$, see text for more details.  Model | $E_{\mathrm{sat}}$ | $n_{\mathrm{sat}}$ | $K_{\mathrm{sat}}$ | $Q_{\mathrm{sat}}$ | $Z_{\mathrm{sat}}$ | $E_{\mathrm{sym}}$ | $L_{\mathrm{sym}}$ | $K_{\mathrm{sym}}$ | $Q_{\mathrm{sym}}$ | $Z_{\mathrm{sym}}$ | $b_{\mathrm{sat}}$ | $b_{\mathrm{sym}}$ | $m^{*}_{\mathrm{sat}}$ | $\Delta m^{*}_{\mathrm{sat}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (MeV) | (fm-3) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) |  |  | ($m_{N}$) | ($m_{N}$)
H1MM | -17.0 | 0.186 | 261 | -220 | -200 | 33.8 | 46.8 | -154 | 700 | 500 | 11.37 | 10.46 | 0.59 | 0.43
H2MM | -15.8 | 0.176 | 237 | -220 | -200 | 32.0 | 43.9 | -144 | 700 | 500 | 9.59 | 9.34 | 0.61 | 0.41
H3MM | -15.3 | 0.173 | 232 | -220 | -200 | 31.8 | 50.6 | -96 | 700 | 500 | 10.35 | 20.56 | 0.61 | 0.34
H4MM | -15.0 | 0.169 | 223 | -220 | -200 | 31.0 | 42.1 | -138 | 700 | 500 | 8.37 | 8.70 | 0.63 | 0.38
H5MM | -13.9 | 0.159 | 207 | -220 | -200 | 29.4 | 40.2 | -128 | 700 | 500 | 6.41 | 8.24 | 0.66 | 0.33
H7MM | -13.2 | 0.139 | 201 | -220 | -200 | 28.1 | 36.5 | -150 | 700 | 500 | 9.40 | -1.46 | 0.67 | 0.41
DHS${}^{L59}_{\mathrm{MM}}$ | -14.0 | 0.168 | 200 | -220 | -200 | 31.4 | 58.9 | -30 | 700 | 500 | 9.00 | 10.00 | 1.0 | 0.0
DHS${}^{L69}_{\mathrm{MM}}$ | -14.6 | 0.173 | 216 | -220 | -200 | 33.7 | 69.0 | -20 | 700 | 500 | 9.00 | 10.00 | 1.0 | 0.0
BSK14MM | -15.9 | 0.159 | 239 | -88 | -896 | 30.00 | 43.9 | -152 | 213 | -676 | 1.22 | 0.01 | 0.80 | 0.03
BSK16MM | -16.1 | 0.159 | 242 | -91 | -895 | 30.00 | 34.9 | -187 | 245 | -696 | 1.22 | 0.01 | 0.80 | 0.04
F0MM | -16.0 | 0.162 | 230 | -124 | -749 | 32.00 | 42.4 | -113 | 294 | -625 | 1.23 | 0.06 | 0.69 | -0.19
LNS5MM | -15.6 | 0.160 | 240 | -113 | -574 | 29.15 | 50.9 | -119 | 195 | -853 | 1.26 | 0.00 | 0.60 | 0.23
RATPMM | -16.0 | 0.160 | 240 | -109 | -697 | 29.26 | 32.4 | -191 | 266 | -923 | 1.25 | -0.00 | 0.67 | 0.26
SGIIMM | -15.6 | 0.158 | 215 | -90 | -874 | 26.83 | 37.6 | -146 | 221 | -915 | 1.25 | 0.00 | 0.79 | 0.28
SLY5MM | -16.0 | 0.160 | 230 | -104 | -749 | 32.03 | 48.3 | -112 | 195 | -326 | 1.26 | -0.01 | 0.70 | -0.18

Figure 1: Energy per particle in uniform matter (left), for NM (top) and for
SM (bottom), normalized by the Free Fermi Gas energy $e_{\mathrm{FFG}}$
(right). Plots are drawn as function of the densities, $n_{n}$ in NM and
$n_{B}$ in SM, as well as function of the Fermi energy, $k_{Fn}$ in NM and
$k_{FB}$ in SM. Note that in SM $n_{n}=n_{B}/2$ and $k_{Fn}=k_{FB}$. We plot
the 8 $\mathrm{\chi EFT}$ Hamiltonians (dashed lines), a gray band for H1-H4,
and 7 Skyrme interactions (solid lines), see the legend for more details. Note
that for the clarity of the legend in all the figures shown in this paper, we
drop the index MM to all models. It is implicit that in all this work, we use
the meta-model adjusted to the predictions of the microscopic interactions.
For $\mathrm{\chi EFT}$ Hamiltonians however, the extrapolation at high
density stands for only one among all possible realizations of the Hamiltonian
guided by the condition to reproduce the radio observations of massive
pulsars, as detailed in the text.

Figure 2: Symmetry energy w.r.t to baryon density (left) and normalized by the
FFG symmetry energy w.r.t Fermi momentum (right). Inset on left plot shows
$e_{sym}$ at high densities. Continuous (dashed) lines shows the meta-model
fitted to Skyrme ($\mathrm{\chi EFT}$) models. Dark purple and magenta dots
show predictions from Brussels-Montreal microscopic models BSk22 and BSk25
Pearson18 . Light (darl) blue bands show constraints from isobaric analog
state IAS (IAS + neutron skin) Danielewicz2014 . The recent constraint from
neutron skin experiments by PREX-II is shown in yellow band Brendan2021 .
Figure 3: Uniform matter energy in $\beta$-equilibrium.

##  2 Uniform matter in the core of neutron stars

The core of NS is composed of uniform matter at $\beta$-equilibrium, whose
properties are strongly related to the symmetry energy originating from the
energy difference between neutron matter (NM) and symmetric matter (SM). We
therefore first discuss uniform matter properties for the set of models
considered in this analysis. Our description of uniform matter is based on the
meta-model (MM) Margueron2018a  which is calibrated on existing models, such
as the Skyrme model or the $\mathrm{\chi EFT}$ Hamiltonian, by the use of the
nuclear empirical parameters (NEPs). These NEPs are obtained from the
derivatives of the energy per particle in SM and the symmetry energy, as
detailed in Ref. Margueron2018a  for instance.

The reproduction of the Skyrme predictions for uniform matter is presented in
Ref. Margueron2018a  and in Ref. Grams:2021b  we have presented the
calibration of the MM to the many-body perturbation theory predictions based
on the $\mathrm{\chi EFT}$ Hamiltonians: H1-H7 Drischler2016  and the
DHSL59-DHSL69 Drischler2021 . Note that in the present work, the Skyrme MM
adjustment is slightly different from the one shown in Ref. Margueron2018a
where it was analytically fixed to reproduce Skyrme’s predictions ($E/A$ and
$P$) at 4$n_{\mathrm{sat}}$. We replaced this – somehow arbitrary –
prescription by a fit over the densities in SM and NM and from
$n_{\mathrm{sat}}$ up to $10n_{\mathrm{sat}}$ taking the parameters
$Q_{\mathrm{sat}/\mathrm{sym}}$ and $Z_{\mathrm{sat}/\mathrm{sym}}$ are free
parameters, the other NEPs being taken at their predicted values. This new
prescription avoids extrapolating the MM in dense region where it may deviate
from the original Skyrme’s predictions.

In the following, we investigate the following nucleonic models:

  * •

Skyrme forces: we select several Skyrme forces which have been adjusted to the
ground-state properties of finite nuclei, and therefore are expected to
reproduce well SM around $n_{\mathrm{sat}}$. Their predictions in NM however
differ largely and they represent the actual uncertainty for NM EOS based on
phenomenological approaches. In the following, we select BSK14Goriely07 ,
BSK16Chamel08 , F0Lesinski06 , LNS5Cao06 , RATPRayet82 , SGIINguyen81 ,
SLy5Chabanat98 . Anticipating further results, the wide predictions in NM from
these approaches mostly reflect their different predictions for the parameter
$L_{\mathrm{sym}}$, see Table 1.

  * •

$\mathrm{\chi EFT}$: this approach takes as experimental constraints the NN
scattering properties in vacuum complemented with the binding energy in $d$
and 3He. We have selected several of the latest predictions in SM and NM which
are H1-H7 Drischler2016  (except H6), and DHSL59-DHSL69 Drischler2021 . These
eight Hamiltonians explore uncertainties in the chiral NN and 3N interactions.
Since the MM needs constraints at high density, beyond the break-down density
of these approaches, we have additionally fixed the high order empirical
parameters $Q_{\mathrm{sat}/\mathrm{sym}}$ and $Z_{\mathrm{sat}/\mathrm{sym}}$
such that these EOS predict a maximal mass $M_{\mathrm{TOV}}$ above the
largest observed mass at about $2M_{\odot}$. Since this prescription is added
on top of the original $\mathrm{\chi EFT}$ prediction, we decided to fix the
high-order NEPs to the same values for all EOS.

The NEPs predicted by these models are given in Table 1. Note that the third
and forth order parameters ($Q$ and $Z$) are not fixed at saturation density,
but rather imposed by a fit over the densities (for Skyrme) or by the
requirement to reach 2M⊙ (for $\mathrm{\chi EFT}$).

The predictions in SM and NM are shown in Fig. 1, see caption for details. A
gray band captures the uncertainties originating from $\mathrm{\chi EFT}$
Hamiltonians H1-H4, which reproduce experimental nuclear masses at best, see
table 3. In NM the gray $\mathrm{\chi EFT}$ band is much narrower than the
dispersion among the Skyrme models, see top panels of Fig. 1. The reason is
related to the fact that $\mathrm{\chi EFT}$ theory is well-suited to describe
low-density NM, which is directly constrained by the nucleon-nucleon phase
shifts and three-body forces. Skyrme interactions are calibrated using the
properties of finite nuclei, which reflect more directly the properties of SM,
as it can be seen from Fig. 1(bottom panels), where the $\mathrm{\chi EFT}$
band is now larger than the dispersion among the Skyrme models. At first
sight, the predictions from $\mathrm{\chi EFT}$ and Skyrme models are
complementary: the former describes NM better, while the latter is better for
SM.

The low-density energy per particle is shown in the right panels of Fig. 1, as
function of the Fermi momentum, $k_{Fn}=(3\pi^{2}n_{n})^{1/3}$ (NM) and
$k_{FB}=(3\pi^{2}n_{B}/2)^{1/3}$ (SM). The Free Fermi Gas (FFG) energy per
particle $e_{\mathrm{FFG}}^{\mathrm{NM}}=3\hbar^{2}k_{F}^{2}/(10m_{N})$, where
$m_{N}$ is the nucleon mass, $k_{F}=k_{Fn}$ in NM and $k_{F}=k_{FB}$ in SM,
scales the energies shown in the left panels of Fig. 1. In this
representation, it is clear than the predictions from Skyrme models appear to
be almost unconstrained in NM, as already suggested in Ref. Roggero:2015 ,
while the ones based on $\mathrm{\chi EFT}$ are very consistent among each
other. In a previous analysis Grams:2021a , we have compared in detail the
$\mathrm{\chi EFT}$ predictions and the SLy4 Skyrme model, which is almost
identical to the SLy5 presented here. By investigating other Skyrme models,
which perform rather well for the ground state of finite nuclei, we now
explore more widely the Skyrme’s uncertainties in the predictions for the
properties of low-density NM. These properties are interesting since low-
density NM is close to the unitary gas limit Carlson:2008 ; Bulgac:2012 ;
Vidana:2021 , which is universal from nuclear systems to cold atom gas
Navon:2010 . From our previous analysis, we have concluded that the crust
composition ($A_{\mathrm{cl}}$, $Z_{\mathrm{cl}}$) depends mainly on the
properties of SM, while NM influences mostly the energy per particle, the
pressure and the sound-speed in non-uniform matter. We will use these results
in the discussion of the crust prediction in the next section.

The symmetry energy, defined as,

$e_{\mathrm{sym}}(n)=e_{\mathrm{NM}}(n)-e_{\mathrm{SM}}(n)\,,$ (1)

where $e=E/A$, is shown in Fig. 2. It represents the cost in energy per
particle to convert SM into NM. In dense matter at $\beta$-equilibrium, this
energy is provided by electrons at their Fermi energy. Despite the dispersion
of $\mathrm{\chi EFT}$ predictions in SM, these models predict symmetry energy
in a narrower band compared to the Skyrme’s one, since Skyrme models are
penalized by their poor reproduction of NM. We additionally display
constraints from nuclear experiments: the light (dark) blue band are shown the
constraints from isobaric analog state IAS (IAS + $\Delta r_{np}$, i.e., IAS +
neutron skin) Danielewicz2014 , and the yellow band shows the PREX-II
predictions from the neutron skin in Pb Brendan2021 , leading to
$E_{\mathrm{sym}}=38.1\pm 4.7$ MeV and $L_{\mathrm{sym}}=106\pm 37$ MeV. We
note that all models predict symmetry energy lower than PREX-II band when
$n_{B}\approx n_{\mathrm{sat}}$ and inside the very lower band for
$n_{B}\approx 0.1$ fm-3, which shows a disagreement between PREX-II measure
and the model predictions. Regarding the constraints from IAS Danielewicz2014
all $\mathrm{\chi EFT}$ models predict symmetry energy located inside both
blue bands while most Skyrme models predict $e_{\mathrm{sym}}$ inside the
light blue band, except SGII that is always softer and LNS5 which crosses the
constraint band just around saturation. For the more stringent constraint, IAS
+ neutron skin, we note that SGII and LNS5 predict low values for the symmetry
energy. RATP while compatible with the symmetry energy band around saturation
density tends to be as soft as SGII and LNS5 above saturation density.
Similarly to Fig. 1, we show in Fig. 2 (right panel) the symmetry energy
normalized by the FFG prediction as function of the nucleon Fermi momentum. It
is interesting to remark the very large deviation from one Skyrme model to the
other on the vertical axis, which reflects the influence of the interaction to
the symmetry energy (potential and effective mass contributions), see
discussion in Ref. Somasundaram2021  for instance. At low density, for
instance $k_{FB}=0.4~{}\mathrm{fm}^{-1}$ ($n_{B}\approx
0.004~{}\mathrm{fm}^{-3}$), the potential term contributes to double the
symmetry energy. The potential term in $\mathrm{\chi EFT}$ models increases
the normalized symmetry energy up to $k_{FB}=0.6$-$0.7~{}\mathrm{fm}^{-1}$
($n_{B}\approx 0.015~{}\mathrm{fm}^{-3}$), reaching $2.5$-$2.7$, and then it
decreases slowly as the Fermi momentum $k_{FB}$ increases. This low-density
behavior is however not characteristic of all Skyrme models: Some Skyrme
interactions predict the symmetry energy to be lower than the $\mathrm{\chi
EFT}$ predictions (BSK14, LNS5, SGII) while some others predict it to be
larger than the $\mathrm{\chi EFT}$ predictions (F0, SLy5). In Fig. 2 we also
show symmetry energy predictions from BSk22 and BSk25 Skyrme interactions
Pearson18 . These two models represent the two extreme cases investigated in
Ref. Pearson18 : BSk22 is ASYsoft (BSk25 is ASYstiff) below saturation
density, as shown in Fig. 2. The symmetry energy from BSk22 is similar to the
$\mathrm{\chi EFT}$ band, while BSk25 predicts a large value for the symmetry
energy below saturation density. In Ref. Pearson18 , these two models are used
in Hartree-Fock-Bologiubov modeling of the NS crust, as we will show in the
following.

At high density, as shown in the left panel of Fig. 2, several Skyrme models
also predict a bending down of the symmetry energy (BSK14, BSK16, RATP, SGII,
LNS5) that the $\mathrm{\chi EFT}$ do not predict. The inset in Fig. 2 shows
that the bending down of the symmetry energy around saturation density of
these Skyrme models leads to a crossing of the zero axis at high density. This
is a well known feature of phenomenological nuclear interactions such as
Skyrme, see for instance Ref. Danielewicz:2009  and references therein.

To conclude this section, we show in Fig. 3 the energy per particle at
$\beta$-equilibrium in uniform matter. The condition to fulfill beta-
equilibrium (without neutrinos and without muons) is the following:
$\mu_{n}-\mu_{p}=\mu_{e}$. In cold catalyzed neutron stars, neutrinos do not
contribute to the chemical equilibrium. Muons can however appear when
$\mu_{\mu}\geq m_{\mu}c^{2}$ and they contribute to $\beta$-equilibrium
through the thermodynamical relation: $\mu_{e}=\mu_{\mu}$. Charge neutrality
is also imposed: $n_{e}+n_{\mu}=n_{p}$. These three conditions complemented by
the baryon conservation number $n_{B}=n_{n}+n_{p}$ lead to uniquely determine
the composition of npe$\mu$ matter. Below saturation density, the dispersion
among the Skyrme predictions observed in Fig. 3 is very large, much larger
than the predictions from $\mathrm{\chi EFT}$. In the following, we will
investigate the impact of this large dispersion among the Skyrme predictions
on the crust properties in non-uniform matter. This will be performed in the
next section based on the CLDM approach.

Above saturation density and as the density increases, the $\mathrm{\chi EFT}$
band for the symmetry energy gets larger and larger. While some Skyrme models
are close but out of the band, e.g., SLy5 (above) and RATP (bellow), the
$\mathrm{\chi EFT}$ band represents a far estimation of the current
uncertainties up to twice saturation density.

##  3 The crust of neutron stars

In a recent paper Grams:2021b , we have presented the CLDM used in the present
study for non-uniform matter and we have detailed how the bulk terms are
connected to uniform matter by using the meta-model approach Margueron2018a ,
which reproduces uniform matter from $\mathrm{\chi EFT}$ and Skyrme
approaches. The CLDM employed here describes isolated finite nuclei as well as
nuclear clusters in the crust of NS. At variance with some other approaches
where the experimental nuclear masses, when they exist, are included directly
in the description of nuclear cluster energetic, we use in our approach the
comparison to the experimental data to quantify the goodness of the models.
This approach satisfies the unified prescription where finite systems and
infinite nuclear matter are based on the same model. For this reason, we first
discuss in this section the constraints from experimental nuclear data, and
then we present our crust EoS.

###  3.1 Constraints from experimental nuclear data

The CLDM defines the finite nuclei energy as
$E_{\mathrm{nuc}}=E_{\mathrm{bulk}}+E_{\mathrm{FS}}$, which sums the bulk and
finite-size (FS) contributions. The present notations are consistent with Ref.
Grams:2021b , where more details are given.

Table 2: Standard parameters the surface and curvature terms employed in the
CLDM approach considered in this work. These values are obtained from
averaging over the parameters given in Ref. Carreau2019b .

$\sigma_{\mathrm{surf,sat}}^{\mathrm{std}}$ | $\sigma_{\mathrm{surf,sym}}^{\mathrm{std}}$ | $p_{\mathrm{surf}}^{\mathrm{std}}$ | $\sigma_{\mathrm{curv,sat}}^{\mathrm{std}}$ | $\beta_{\mathrm{curv}}^{\mathrm{std}}$
---|---|---|---|---
MeV fm-2 | MeV fm-2 |  | MeV fm-1 |
1.1 | 2.31 | 3.0 | 0.1 | 0.7

1 Note the associated value $b_{\mathrm{surf}}=29.9$.

For each nucleus defined by its mass $A$ and its charge $Z$ localizing it over
the nuclide chart, the mechanical equilibrium is imposed:
$P_{\mathrm{nuc}}=n_{\mathrm{cl}}^{2}\partial e_{\mathrm{nuc}}/\partial
n_{\mathrm{cl}}=0$, where $n_{\mathrm{cl}}$ is the nucleus density. This
variational condition fixes the nuclear cluster density $n_{\mathrm{cl}}(A,Z)$
at equilibrium: each nuclei have a different density, and they overall explore
the density dependence of the CLDM model around saturation density, at
variance with the liquid-drop model where the model coefficients are kept
constant.

The bulk term, assuming the local density approximation, originates from
uniform matter calculations presented in Sec. 2. The FS term is composed of
the Coulomb, surface and curvature contributions, as described in the FS4
prescription in Ref. Grams:2021b . These terms are moreover fine-tuned to
better reproduce the experimental binding energies for $A\geq 12$ and $Z\geq
6$, as detailed in Ref. Grams:2021b . The standard parameters for the surface
and curvature terms are given in Tab. 2. In the practical implementation of
the CLDM, each of these values are multiplied by the parameters
$\mathcal{C}_{i}$ ($\mathcal{C}_{\mathrm{surf},\mathrm{sat}}$,
$\mathcal{C}_{\mathrm{surf},\mathrm{sym}}$,
$\mathcal{C}_{\mathrm{curv},\mathrm{sat}}$, $\mathcal{C}_{\beta}$) as,

$\displaystyle\sigma_{\mathrm{surf},\mathrm{sat}}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\mathrm{surf},\mathrm{sat}}\sigma_{\mathrm{surf},\mathrm{sat}}^{\mathrm{std}}\,,$
(2) $\displaystyle\sigma_{\mathrm{surf},\mathrm{sym}}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\mathrm{surf},\mathrm{sym}}\sigma_{\mathrm{surf},\mathrm{sym}}^{\mathrm{std}}\,,$
(3) $\displaystyle\sigma_{\mathrm{curv},\mathrm{sat}}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\mathrm{curv},\mathrm{sat}}\sigma_{\mathrm{curv},\mathrm{sat}}^{\mathrm{std}}\,,$
(4) $\displaystyle\beta_{\mathrm{curv}}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\beta}\beta_{\mathrm{curv}}^{\mathrm{std}}\,,$ (5)

and the Coulomb energy is corrected with the coefficient
$\mathcal{C}_{\mathrm{Coul}}$. Since these coefficients incorporate in an
effective way the slight correction due to the actual density profile in
finite nuclei, which we approximate by a hard sphere in the CLDM, the values
for these variational parameters $\mathcal{C}_{i}$ are expected to be close to
1 as shown in Table 3.

Table 3: Optimization of the parameters $\mathcal{C}_{i}$ over the nuclide chart, for $A\geq 12$ and $Z\geq 6$, and considering the experimental energies from the 2016 Atomic Mass Evaluation (AME) AME2016  for the interactions considered in this work (Skyrme and $\mathrm{\chi EFT}$). Odd-even mass staggering parameters $\Delta_{\mathrm{sat}}$ and $\Delta_{\mathrm{sym}}$, $\chi_{E}$ for the optimization with (without) odd-even mass staggering. In the two last columns, the minimal energy $e_{\mathrm{cl}}$ for each model is given together with its position in the nuclide chart $(A,Z)_{\min}$. Experimental binding energy are $e(^{56}$Fe$)$ = -8.79 MeV and $e(^{68}$Zn$)$ = -8.76 MeV. Model | $\mathcal{C}_{\mathrm{Coul}}$ | $\mathcal{C}_{\mathrm{surf},\mathrm{sat}}$ | $\mathcal{C}_{\mathrm{surf},\mathrm{sym}}$ | $\mathcal{C}_{\mathrm{curv},\mathrm{sat}}$ | $\mathcal{C}_{\beta}$ | $\Delta_{\mathrm{sat}}$ | $\Delta_{\mathrm{sym}}$ | $\chi_{E}$ | $\min(e_{\mathrm{nuc}})$ | (A,Z)min
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | (MeV) | (MeV) | (MeV) | (MeV) |
BSK14MM | 0.965 | 1.002 | 0.890 | 0.770 | 1.100 | 5.75 | -7.35 | 2.61 (2.74) | -8.72 | (68,30)
BSK16MM | 0.973 | 1.076 | 0.876 | 0.777 | 0.817 | 5.68 | -4.88 | 2.66 (2.79) | -8.70 | (68,30)
LNS5MM | 0.949 | 0.916 | 0.668 | 0.531 | 1.646 | 5.69 | -4.84 | 2.66 (2.79) | -8.75 | (68,30)
RATPMM | 0.970 | 1.095 | 0.676 | 0.583 | 0.694 | 5.54 | 0.22 | 2.81 (2.93) | -8.70 | (68,30)
SGIIMM | 0.952 | 0.942 | 0.406 | 0.262 | 1.981 | 5.42 | 4.97 | 2.95 (3.07) | -8.75 | (68,30)
F0MM | 0.966 | 1.064 | 1.264 | 1.144 | 0.922 | 5.82 | -10.47 | 2.60 (2.73) | -8.65 | (68,30)
SLy5MM | 0.967 | 1.039 | 1.257 | 1.116 | 0.980 | 5.85 | -11.26 | 2.59 (2.72) | -8.67 | (68,30)
H1MM | 0.966 | 1.582 | 1.307 | 0.942 | -0.255 | 5.53 | 0.22 | 3.07 (3.18) | -8.17 | (68,30)
H2MM | 0.927 | 1.037 | 1.123 | 1.053 | 1.141 | 5.79 | -9.46 | 2.58 (2.71) | -8.37 | (70,30)
H3MM | 0.913 | 0.847 | 1.095 | 1.127 | 1.560 | 5.94 | -14.83 | 2.65 (2.77) | -8.61 | (68,30)
H4MM | 0.903 | 0.716 | 0.967 | 1.144 | 1.833 | 5.99 | -16.85 | 2.77 (2.89) | -8.84 | (68,30)
H5MM | 0.868 | 0.325 | 0.646 | 1.215 | 2.616 | 6.32 | -29.26 | 3.78 (3.86) | -9.83 | (58,26)
H7MM | 0.858 | 0.110 | 0.339 | 1.086 | 3.235 | 6.61 | -43.69 | 5.32 (4.95) | -10.23 | (56,26)
DHSL59,MM | 0.853 | 0.370 | 1.038 | 1.686 | 2.089 | 6.41 | -33.40 | 4.34 (4.41) | -9.66 | (58,26)
DHSL69,MM | 0.871 | 0.567 | 1.571 | 1.971 | 1.649 | 6.28 | -28.51 | 4.02 (4.10) | -9.10 | (60,26)
Figure 4: Binding energy for the Iron group using the CLDM with SLy5, LNS5 and
H2. A black square marks the lowest energy for each model.

A small improvement of the fit to the experimental nuclear masses has been
implemented in the present work in order the reduce the effect of the odd-even
mass staggering in the data: we have corrected the experimental masses as,
$\tilde{E}_{\mathrm{exp}}^{i}=E_{\mathrm{exp}}^{i}-\Delta
E_{\mathrm{exp}}^{i}$, with

$\Delta
E_{\mathrm{exp}}^{i}=\left[\Delta_{\mathrm{sat}}+\Delta_{\mathrm{sym}}\left(\frac{N_{i}-Z_{i}}{A_{i}}\right)^{2}\right]\,A_{i}^{-1/3}\,\delta(N,Z)\,.$
(6)

where $\delta(N,Z)=1$ if $N$ and $Z$ are odd, $0$ if either $N$ or $Z$ is odd,
and $-1$ if both $N$ and $Z$ are even BohrMottelson1969 . The parameters
$\Delta_{\mathrm{sat}}$ and $\Delta_{\mathrm{sym}}$ are varied together with
the CLDM parameters $\mathcal{C}_{i}$ in the fit to the experimental masses.
The loss function $\chi_{E}$, required for the fit, is defined as,

$\displaystyle\chi_{E}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{N}\sum_{i=1}^{N}(\tilde{E}_{\mathrm{exp}}^{i}-E_{\mathrm{nuc}}^{i})^{2}\right]^{1/2}\,,$
(7)

where $N=3375$ (2443 without extrapolated data) is the number of considered
nuclei from the experimental nuclide chart and $E_{\mathrm{nuc}}^{i}$ the
energy from the CLDM model for the nucleus $i$. By comparing $\chi_{E}$ and
$\chi_{E/A}$ (defined as function of $E/A$ instead of $E$), the impact of the
loss function has been shown to be non-negligible in general, but smaller than
other uncertainties Grams:2021b  such as for instance the one coming from
varying the model for the bulk. We will thus employ only $\chi_{E}$ for the
fit, without considering the uncertainties originating from a different
measure of the goodness of the model to reproduce experimental nuclear masses.
Note that we also checked that our predictions are weakly impacted by
incorporating the extrapolated data for the experimental masses, compared to
the case where we would restrict the nuclear chat only to the measured masses.

The results of the fit to finite nuclei are shown in Table 3. The coefficients
$\mathcal{C}_{i}$ are in general of the order of 1, as expected. The odd-even
mass staggering parameters $\Delta_{\mathrm{sat}}$ and $\Delta_{\mathrm{sym}}$
are also shown. The isoscalar parameter $\Delta_{\mathrm{sat}}$ is similar to
the value suggested in Ref. Vogel:1984  (7.2 MeV). The isovector parameter
$\Delta_{\mathrm{sym}}$ however is much smaller than the one suggested in Ref.
Vogel:1984  (-44 MeV), where the fit is done by considering only $A>40$
nuclei. We indeed found that this quantity is very sensitive to the nuclei
which we fit it. When we select only $A>40$ nuclei $\Delta_{\mathrm{sym}}$
change from -7.35 (-11.26) to $\approx$-16.83 (-16.60) MeV for BSK14 (SLy5).
The value of $\Delta_{\mathrm{sym}}$ therefore varies from one region of the
nuclear chat to another. It is difficult to determine a constant value over
the nuclide chart, but its influence is small since nuclei do not explore
large isospin asymmetries ($N<2Z$ for heavy nuclei).

With a loss function $\chi_{E}\lesssim 3$ MeV, the $\mathrm{\chi EFT}$ models
H1 to H4 are the more accurate ones to reproduce the experimental masses. They
are comparable with the Skyrme models considered here. Note that our
conclusion remains valid even when we do not correct for the odd-even mass
staggering in the nuclear data, see number is parenthesis in the $\chi_{E}$
column on Table 3. The nuclear masses however disfavor the $\mathrm{\chi EFT}$
models: H5, H7, DHSL59 and DHSL69. In the following, Hamiltonians H1-H4 will
be considered as our best models and will be marked with a gray band, since
they reproduce nuclear masses at best.

In the two last columns of table 3 are shown the minimal energy per particle
obtained for each model. This quantity is sometimes used to define the offset
of the energy per particle in the crust, see next sub-section. Experimentally,
the nucleus 56Fe minimizes the energy, but it is a bit model dependent due to
the approximation scheme. The search for the minimal energy per particle
configuration is illustrated in Fig. 4 for a few models (LNS5, SLy5 and H2)
and for iron group isotopic chains (Fe, Co, Ni, Zn). For all the Skyrme
models, the lowest energy configuration is 68Ni. This is also the lowest
energy configuration for H1, H3, H4, while H2 prefers 70Ni and the other
$\mathrm{\chi EFT}$ models prefer 56-60Fe nuclei.

###  3.2 Crust equation of state

The non-uniform matter in the crust is modeled according to the CLDM, which
sums the nuclear cluster energy $E_{\mathrm{cl}}$, with bulk, FS and electron
interaction contributions, the neutron fluid contribution $E_{n}$ as well as
the kinetic electron gas $E_{e}$:

$E_{\mathrm{tot}}=E_{\mathrm{cl}}+E_{n}+E_{e}\,.$ (8)

Note that as the electron density goes to zero, $n_{e}\rightarrow 0$, then
$E_{\mathrm{cl}}\rightarrow E_{\mathrm{nuc}}$, previously defined. Details on
these different terms are given in Ref. Grams:2021b .

Figure 5: Energy per particle, pressure, sound speed and volume fraction at
beta equilibrium in the NS crust. Dark gray band shows the prediction for the
outer-inner crust transition for the models that better reproduce nuclear
masses (all Skyrme and H1-H4). Light gray band shows the uncertainty including
all models.

The energy, pressure, sound speed and volume fraction,
$u=V_{\mathrm{cl}}/V_{\rm WS}$ where $V_{\mathrm{cl}}$ is the cluster volume
and $V_{\rm WS}$ the Wigner-Seitz cell volume, at $\beta$-equilibrium are
shown in Fig. 5 for a range of densities spanning over the outer (partially)
and inner crust. As already suggest in Ref. Grams:2021a , these quantities are
sensitive to the properties of low-density NM, as shown in Fig. 1. The models
LNS5 and SGII (F0 and SLy5), which predict binding energies in low-density NM
lower (higher) than the $\mathrm{\chi EFT}$ band, also predict that the
binding energy at $\beta$-equilibrium is lower (higher) in the density range
going from $3~{}10^{-3}$ to $10^{-1}$ $\mathrm{fm}^{-3}$. Same remarks can be
done for the pressure, the sound speed and the volume fraction at
$\beta$-equilibrium. Vertical dark gray bands in Fig. 5 shows the prediction
for the outer-inner crust transition for the models that better reproduce
nuclear masses, i.e., all Skyrme models and H1-H4, while light gray band shows
the prediction including all models used in this work. Note the importance of
reproducing experimental nuclear masses to better constraint the location of
the neutron drip.

It can also be remarked that in the outer crust the $\mathrm{\chi EFT}$
Hamiltonians are more spread than the Skyrme models. Since the Skyrme models
reproduce better the experimental nuclear binding, they lead to tighter
predictions in the outer crust. Note that the gray band localizing the
predictions from H1-H4 $\mathrm{\chi EFT}$ Hamiltonians are in full agreement
with the Skyrme’s ones. One can thus conclude that the experimental nuclear
masses are important constraints to accurately predict the EoS in NS outer
crust. In the inner crust however, since a large amount of neutrons drip off
clusters forming a neutron fluid, some properties such as the binding energy,
the pressure, the sound speed and the volume fraction are largely impacted by
the properties of uniform NM.

The transition from non-homogeneous matter (crust) to homogeneous matter
(core) is obtained by computing the energy per particle for the two phases and
comparing their values with the increase of the density. The lower energy is
the favorable state. In the bottom-right panel of Fig. 5 the end of the volume
fraction curves shows this transition. The spread on the end of curves shows
the uncertainty on the crust-core transition for the 15 models investigated in
this work. In Tab. 4 we add the numerical values for the outer-inner crust
transition together with the crust-core for the different models.

##  4 Tables of equation of state

The fifteen EoS presented in this paper are available on the CompOSE catalog
compose  and can be interfaced with the Lorene
library111https://lorene.obspm.fr/. We briefly describe here a few quantities
given in these tables and we start with the enthalpy.

The enthalpy per particle is defined as

$h(n_{B})=\langle m\rangle c^{2}+e_{\mathrm{tot}}+p_{\mathrm{tot}}/n_{B}\,,$
(9)

where the mass $\langle m\rangle=(n_{n}m_{n}+n_{p}m_{p})/n_{B}$ for the baryon
density $n_{B}=n_{n}+n_{p}$ contributes to the rest mass term in Eq. (9). From
Eq. (9) one could define the enthalpy per unit mass $m_{\mathrm{ref}}$ as
$h/m_{\mathrm{ref}}$, where the unit mass $m_{\mathrm{ref}}$ is an arbitrary
quantity. It could be taken to be the atomic unit mass $m_{u}$
222$m_{u}=931.494028(23)$ MeV c-2, or the lowest measured mass of 56Fe
333$m(^{56}$Fe$)=930.411790$ MeV c-2, and sometimes, it is just taken to be
the neutron mass 444$m_{n}=939.565413(6)$ MeV c-2. As long as the definition
of the rest mass is well defined, all these choices are equivalent.

The equilibrium of static neutron stars is given by the solution of the TOV
equations Tolman1939 ; Oppenheimer1939 . In some numerical resolution of these
equations, see for instance Ref. Lindblom1992  or the Lorene library, the EOSs
are sampled according to the log enthalpy per unit mass defined as,

$H\equiv\log\left(\frac{h}{m_{\mathrm{ref}}c^{2}}-1\right)\,.$ (10)

In this case, it is important to fix $m_{\mathrm{ref}}$ to be below the lowest
possible mass such that the ratio $h/m_{\mathrm{ref}}$ is always above $1$,
and the quantity given by Eq. (10) always well-defined. A natural choice would
be to take $m(^{56}$Fe$)$ since it is the most stable system in nature.
However, the NS EOS is not experimentally measured but instead is based on a
modeling of the experimental nuclear masses. Due to the approximation scheme
in the modeling, it is not given that the model mass reproduces the
experimental quantity. It shall then be checked that the choice for
$m_{\mathrm{ref}}$ ensures the ratio $h/m_{\mathrm{ref}}$ is above 1. To
address this question in a more quantitative way, we have investigated in Fig.
4 the location of the minimal energy nucleus, and we found that it is not
necessarily 56Fe for all EOSs. For each EOS, we have then reported the value
of the binding energy corresponding to the minimal energy nucleus in Tab. 3.
The reference mass shall thus be fixed to be the minimum energy per particle
predicted by the considered model. From Tab. 3 we can check that
min$(e_{\mathrm{nuc}})/m(^{56}$Fe$)c^{2}<1$ for all Skyrme model we
considered, but some of the $\mathrm{\chi EFT}$ models, while not being our
preferred ones, do not respect this condition. Employing these models in TOV
solver using (10) should then be performed carefully, and it could be
necessary to lower down the value of the reference mass for these cases.

Figure 6: Particle fractions. Top: leptons, $Y_{e}$ left axis ($Y_{\mu}$ right
axis) in continuous (dotted) lines. Bottom: baryons, $X_{n}$ left axis
($X_{p}$ right axis) in continuous (dotted) lines.

The neutron, proton, electron and muon particle fractions are shown in Fig. 6.
For densities lower than $10^{-3}~{}\mathrm{fm}^{-3}$ all models agree well
together, except H7, which slightly overestimates $Y_{e}$. At higher densities
however, a dispersion among the model predictions appears. Here also the low-
density energy per particle in NM plays an important role: a reduction of the
energy per particle in NM would ease the production of neutrons
$\beta$-processes from electrons and protons, and thus would reduce the
electron fraction $Y_{e}$. This is exactly what happens: the stiffest models,
e.g. F0 and SLy5, predict the largest particle fractions in the density range
going from $10^{-3}-10^{-1}~{}\mathrm{fm}^{-3}$, while the softest ones, e.g.
LNS5 and SGII, predict the lowest electron fractions.

Muons appear at around $0.1$-$0.2~{}\mathrm{fm}^{-3}$, and this onset density
is rather model independent (at least for density scale of the figure). The
muon fraction increases for all models. At a few times saturation density,
some models such as BSK14, BSK16, RATP, SGII and LNS5 predict however a
bending down of the electron and muon density. This behavior results from the
symmetry energy, see Fig. 2, which bends down to zero for these models as the
density exceeds saturation density. As the symmetry energy reduces, it is more
and more easy to produce neutrons and then the lepton fraction goes down to
zero.

Figure 7: Neutron chemical potentials in the inner crust with respect to the
baryon density.

The neutron chemical potential is shown in Fig. 7. In the outer crust
$\nu_{n}<0$, since all neutrons are bound to nuclei. The neutron drip is
defined as the density at which the first neutron drips out of nuclei or
equivalently by the condition that the neutron chemical potential $\nu_{n}>0$.
Fig. 7 only shows this positive case. The small dispersion of the curves in
the lower left side of Fig. 7 reflects the uncertainties in the position the
outer-inner crust transition. As we already commented in the discussion of
Fig. 5, the position of the inner-outer crust transition is better defined
with Skyrme models than with $\mathrm{\chi EFT}$ ones. This indicates that the
inner-outer crust transition is mainly determined by the experimental nuclear
masses. As the density increases above the drip point, the neutron fluid
contribution becomes more important. As a consequence a convergence of the
$\mathrm{\chi EFT}$ models is observed, while the uncertainties on the Skyrme
models predictions are getting larger and larger. As seen in the previous
figures, we note that F0 and SLy5 (LNS5 and SGII) predicts higher (lower)
values than the other models, while RATP, BSK14 and BSK16 predicts $\nu_{n}$
compatible with the $\mathrm{\chi EFT}$ models.

Figure 8: Cluster composition, $A_{\mathrm{cl}}$ (top) and $Z\mathrm{cl}$
(bottom), for the eight $\mathrm{\chi EFT}$ Hamiltonians and the seven Skyrme
models. The silver band shows our best results including H1-H4.

The cluster composition in the inner and outer crust is shown in Fig. 8. Our
best predictions from the $\mathrm{\chi EFT}$ models H1-H4 are bounded by the
gray band. The Asy-Stiff models such as H5, H7, DHSL59 and DHSL69 predict
lower $Z_{\mathrm{cl}}$ than the gray band, and the Asy-Soft models such as
RATP, LNS5, SGII, BSK16 predict larger $Z_{\mathrm{cl}}$ than the gray band.
Other models are compatible with the gray band. Note however that H5 and H7
predict much lower $A$ and $Z$ than the other $\mathrm{\chi EFT}$ ones. This
is most probably due to the fact that these two hamiltonians predict the
lowest values for $n_{\mathrm{sat}}$ compared to other $\mathrm{\chi EFT}$
predictions, see Tab. 1. Since the cluster radius is mostly controlled by the
virial condition, a lower value for $n_{\mathrm{sat}}$ induces a lower value
for $A$. In addition, the cluster asymmetry parameter $I_{\mathrm{cl}}$ is
predicted to be quite model independent, see our study in Ref. Grams:2021b
for instance, a lower value for $A$ induces a lower value for $Z$.

An interesting feature appears close to the core-crust transition density: the
behavior of $A_{\mathrm{cl}}$ and $Z_{\mathrm{cl}}$ can be quite different
from one model to another. This is reflecting the important role played by the
isospin asymmetry parameter $p_{\mathrm{surf}}$, which is fixed to be
$p_{\mathrm{surf}}=3$ in our present study. The role of $p_{\mathrm{surf}}$ on
the core-crust transition density has been discussed in Refs. Carreau2019a ;
Grams:2021b . It is indeed a parameter which is difficult to determine, since
finite nuclei do not explore isospin asymmetries large enough to be impacted
by the value of $p_{\mathrm{surf}}$. This parameter could be determined from a
slab configuration calculation exploring large asymmetries. In the present
study, we however prefer to fix $p_{\mathrm{surf}}$ for simplicity, but a more
accurate estimation of the core-crust transition density shall include a
proper adjustment of the parameter $p_{\mathrm{surf}}$.

Figure 9: Neutron star MR diagram, for the eight $\mathrm{\chi EFT}$
Hamiltonians and the seven Skyrme models. The squares indicate the density at
which causality is violated. The light crosses mark the central density in
units of $n_{\mathrm{sat}}$: the numbers given in the diagram indicate the
value of the central density in units of $n_{\mathrm{sat}}$. Gray contours
show the NICER observations for the pulsars J0030 (obtained from Ref. Riley19
) and J0740 (obtained as an average of the analyses of Refs. Miller21  and
NICER2021 ) together with LIGO/Virgo observation GW170817
LIGOScientific:2018cki .

We also compare our results with other predictions in Fig. 8, such as the
Bruxelles-Montreal Hartree-Fock Bogoliubov calculation (BSk22 and BSk25)
Pearson18 , the Negele-Vautherin Hartree-Fock calculation Negele1973 , the
Douchin-Haensel CLDM calculation DouchinHaensel2000 ; DouchinHaensel2001 ;
HaenselPichon94 , and finally the original BBP model bbp1971 . The oldest
calculations, BBP, Negele-Vautherin and Douchin-Haensel, do not always overlap
with our gray band, since they were performed before the recent $\mathrm{\chi
EFT}$ achievement. The BSk22 model overlaps pretty well with our gray band,
reflecting the good agreement already noticed for the symmetry energy. The
model BSk25, which is stiffer than the $\mathrm{\chi EFT}$ H1-H4 models
predicts larger $Z_{\mathrm{cl}}$ than us. Note also that while shell effects
are absent from our calculation, our best predictions (gray band) is
compatible with models which have them, e.g. BSk22 or Negele-Vautherin. This
shows that while shell effects are important to get accurate
$Z_{\mathrm{cl}}$, the actual value for $Z_{\mathrm{cl}}$ is still largely
influenced by the contribution from the bulk term in the CLDM, in particular
by the symmetry energy at low-density. The leptodermous expansion, which has
been shown to provide a good ordering of the contribution of the different
terms in the mass formula in NS crust, suggests that shell effects are
comparable with curvature terms. In other words, shell effects are certainly
important for accurate predictions of the crust properties, but they are not
the main ingredient to understand the origin of the main uncertainties in
these properties. The main source of uncertainties are originating from the
bulk term, and in more detail by the low-density NM properties.

Finally, the mass-radius relations for all the considered models are given in
Fig. 9. The crosses on the curves mark the central density in units of
$n_{\mathrm{sat}}$. Note that the central density at the maximum (causal) mass
for the Skyrme models is of the order 7-8$n_{\mathrm{sat}}$. Most of the
Skyrme models considered here are softer and explore higher central densities
than the $\mathrm{\chi EFT}$ ones (5-6$n_{\mathrm{sat}}$). Note that the MR
relations require extrapolations above the $\mathrm{\chi EFT}$ breakdown
density. We indeed recall that we have fixed the values of the parameters $Q$
and $Z$ in such a way that these models reach $2M_{\odot}$. Therefore the high
density behavior of the $\mathrm{\chi EFT}$ EoS is mostly controlled by this
prescription and is only mildly impacted by the $\mathrm{\chi EFT}$ properties
at low density. It is interesting to remark that all our EoSs satisfy the
GW170817 constraint obtained from Ref. LIGOScientific:2018cki  and shown as a
dark-gray contour (90 $\%$ confidence level). The two light-gray contours in
Fig. 9 represent the NICER measurements for the pulsars J0030 Riley19  and
J0740 Miller21 ; NICER2021 . We see that all $\mathrm{\chi EFT}$ models
together with the Skyme LNS5, F0 and SLy5 respect GW and NICER constraints,
while BSK14 respects GW and only J0030. The other Skyrme models BSK16, SGII
and RATP are outside the NICER contours and respect, but very marginally, the
GW contour. In addition, these three Skyrme models are very soft and fail to
reproduce the observed lower limit of MTOV. They are however stiff enough to
predict M${}_{\mathrm{TOV}}\geq 1.7$ M⊙, above the canonical mass NS. Despite
the fact that they do not reach the observational constraint at around
$2M_{\odot}$, we keep these EoSs since the densities at which they fail are
well above the densities where nucleonic models can be trusted
(3-4$n_{\mathrm{sat}}$). These EoSs certainly undergo a phase transition in
their densest regions that we leave open for future studies.

Figure 10: Pressure versus baryon density (in fm-3) at $\beta$-equilibrium, for the eight $\mathrm{\chi EFT}$ Hamiltonians and the seven Skyrme models. The error bar correspond to the inference for the pressure at twice saturation density from the LIGO/Virgo observation GW170817 LIGOScientific:2018cki . Table 4: Neutron drip density $n_{\rm drip}$, crust-core transition density $n_{\rm cc}$, density at which the symmetry energy becomes negative $n_{\mathrm{sym}}$ and causal density $n_{\rm causal}$ above which causality is broken. For some Skyrme models (F0, LNS5 and SLy5) the maximum mass is reached before causality is broken. For some models (BSK16, LNS5, RATP and SGII) the softening of the EoS is explained from the low value of $n_{\rm sym}$. Model | $n_{\rm drip}$ | $n_{\rm cc}$ | $n_{\rm sym}$ | $n_{\rm causal}$
---|---|---|---|---
| ($\times 10^{-4}$ fm-3) | ($\times 10^{-2}$ fm-3) | (fm-3) | (fm-3)
H1MM | 2.60 | 9.60 | - | 0.68
H2MM | 2.94 | 8.67 | - | 0.65
H3MM | 2.44 | 8.45 | - | 0.65
H4MM | 2.52 | 9.06 | - | 0.65
H5MM | 2.85 | 8.70 | - | 0.64
H7MM | 3.22 | 8.31 | - | 0.57
DHS${}^{L59}_{\mathrm{MM}}$ | 2.60 | 7.21 | - | 0.62
DHS${}^{L69}_{\mathrm{MM}}$ | 2.37 | 7.32 | - | 0.64
BSK14MM | 2.52 | 7.62 | - | 1.21
BSK16MM | 2.52 | 8.46 | 0.68 | 1.20
F0MM | 2.52 | 8.33 | - | -
LNS5MM | 2.60 | 7.04 | 1.18 | -
RATPMM | 2.52 | 8.60 | 0.63 | 1.34
SGIIMM | 2.60 | 6.53 | 0.78 | 1.35
SLY5MM | 2.44 | 7.63 | - | 1.03

The Skyrme models explored here are all quite soft at high density, see Fig.
10. The soft Skyrme interactions which are marginal for the GW170817 contour
shown in Fig. 9 (BSK16, SGII and RATP) are also marginal for the prediction of
the pressure at $2n_{\mathrm{sat}}$ extracted from the analysis of GW based on
agnostic EoS LIGOScientific:2018cki . It shall however be noticed that these
interactions overlap with the GW contour for densities in the range $4$ to
$5n_{\mathrm{sat}}$, so quite larger than the constraint inferred at
$2n_{\mathrm{sat}}$. Even the stiffer models shown in Fig. 9 pass through the
GW contour for densities above $2n_{\mathrm{sat}}$. The GW inference for the
pressure at $2n_{\mathrm{sat}}$ shall therefore be considered cautiously.

The softening of BSK16, SGII and RATP EOSs is related to the bending down of
the symmetry energy, see inset in Fig. 2. Note that in Fig. 9 the EOS for
these three cases are replaced by the NM EOS for densities
$n>n_{\mathrm{sym}}$, where $n_{\mathrm{sym}}$ is the density at which
$e_{\mathrm{sym}}=0$, see Tab. 4. The other Skyrme EOSs, BSK14, LNS5, SLy5 and
F0, are softer than the $\mathrm{\chi EFT}$ ones for densities $n_{B}\gtrsim
0.3$-$0.4$ fm-3 as shown in Fig. 10. It is then interesting to study the
consequence of such soft EoSs on the properties of neutron stars. The first
two columns of Tab. 4 summarize the boundaries for each EoS. The first column
shows the density transition between the outer and inner crust $n_{\rm drip}$
while the second shows the crust-core transition $n_{\mathrm{cc}}$. In the two
last columns we show the limit of validity for the EoS that either present
negative values for the symmetry energy at high densities ($n_{\mathrm{sym}}$)
or break causality ($n_{\rm causal}$).

Finally, we observe in Fig. 10 a change of slope of the pressure for densities
of about 0.4 to 0.6 fm-3, similar to the one observed in Ref. Annala:2019 .
The authors of Ref. Annala:2019  interpreted this softening as a sign of the
presence of quark matter in massive neutron stars. This cannot be our case
since we describe purely nucleonic matter. This apparent contradiction is
discussed in Ref. Somasundaram2021b , where it is shown that the bending down
of the pressure at these densities is not necessarily correlated with the
reduction of the sound speed, as it should be for a phase transition to quark
matter. In the present approach, we indeed illustrate that the change of slope
of the pressure does not necessarily reflect the onset of a phase transition
to quark matter. The bending of the pressure is compatible with matter
composed of nucleons only, where the many-body correlations at high density
contribute to soften the EoS.

##  5 Conclusions

In the present analysis, we have explored the predictions of fifteen models,
including seven Skyrme and eight $\mathrm{\chi EFT}$ Hamiltonians. We have
focused our study on the properties of the crust, for which we have employed a
CLDM approach. This approach is adequate for the understanding of the origin
of the model dependence in the prediction for the NS crust properties. We have
indeed found that there are two important features which govern, at first
order, our predictions: i) the ability of the models to reproduce the nuclear
experimental masses over the nuclear table, and ii) the low-density energy per
particle in NM. For the models which satisfy feature (i), the condition on the
low-density energy per particle in NM is largely equivalent to the one on the
low-density properties of the symmetry energy.

Crust properties have been analyzed in terms of bulk and FS contributions,
where the bulk term explore the uncertainties in the uniform matter
prediction, e.g. the low-density NM, and the FS terms are optimized to
reproduce experimental nuclear masses. The two main sources of uncertainties
for the NS crust properties are then well identified. We have observed that
some quantities are more sensitive to the feature (i), e.g. cluster
composition $A_{\mathrm{cl}}$ and $Z_{\mathrm{cl}}$, while others are more
impacted by the feature (ii), e.g. energy per particle, pressure, sound speed,
volume fraction, electron fraction $Y_{e}$, neutron chemical potential,
mentioning only the observables that we have analyzed. In a previous analysis,
we have also illustrated the role of FS terms, in particular the effect of the
parameter $p_{\mathrm{surf}}$ controlling the isospin dependence of the
surface energy for large values of the isospin parameter.

For all the fifteen unified EoS presented here, we have generated tables in
the CompOSE format. Many nucleonic models presented here predict MTOV above
the limit set by radio-observations of about 2M⊙, except a few Skyrme EOSs.
The central densities explored by these soft EoS are however so large that the
nucleon model presented here shall be replaced by a model with exotic degrees
of freedom. The description of phase transitions is out of the scope of the
present study, but it could be performed on top of the EOS we provide. Models
with phase transition will be studied by the authors in future works.

In summary, our study illustrates the importance of constructing unified
models for NS EoS. Further extensions are currently considered, investigating
contributions from smaller terms in the leptodermous expansion, such as for
instance the pairing term or the shell effects.

###### Acknowledgements.

We thank N. Chamel for providing the nuclear configurations below saturation
predicted by BSk22 to BSK25. G.G., J.M. and R.S. are supported by CNRS grant
PICS-08294 VIPER (Nuclear Physics for Violent Phenomena in the Universe), the
CNRS IEA-303083 BEOS project, the CNRS/IN2P3 NewMAC project, and benefit from
PHAROS COST Action MP16214. S.R. is supported by Grant No. DE-FG02-00ER41132
from the Department of Energy , and the Grant No. PHY-1430152 (JINA Center for
the Evolution of the Elements), and PHY-1630782 (Network for Neutrinos,
Nuclear Astrophysics, and Symmetries (N3AS)) from the National Science
Foundation. This work is supported by the LABEX Lyon Institute of Origins
(ANR-10-LABX-0066) of the Université de Lyon for its financial support within
the program Investissements d’Avenir (ANR-11-IDEX-0007) of the French
government operated by the National Research Agency (ANR).

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###  4.7 Cohomological smoothness for stacky maps

Having completed the proof of Theorem 1.4, we now switch notation and write
$Rg_{!}$ for the functor constructed therein, and $Rg^{!}$ for its right
adjoint, in agreement with the notation in [1, 11].

Recall that in Definition 1.3.ii we have defined $\ell$-cohomologically smooth
morphisms between decent v-stacks, and that our definition agrees with [1,
Definition IV.1.11] by the argument in Remark 4.6. Both of these definitions
are extrinsic, formulated in terms of the existence of charts with various
properties. However, we can now give a purely intrinsic definition, parallel
to the definition in the 0-truncated case.

######  Proposition 4.14.

Let $f:X\to Y$ be a fine map of decent v-stacks. Then the following conditions
are equivalent.

  1. 1.

The map $f$ is $\ell$-cohomologically smooth in the sense of Definition
1.3.ii.

  2. 2.

The complex $Rf^{!}\mathbf{F}_{\ell}$ is invertible, the natural map
$Rf^{!}\mathbf{F}_{\ell}\otimes f^{\ast}(-)\to Rf^{!}(-)$ is an isomorphism,
and these statements hold after any decent base change on $Y$.

  3. 3.

The complex $Rf^{!}\mathbf{F}_{\ell}$ is invertible and its formation commutes
with any decent base change on $Y$.

###### Proof.

For (1)$\implies$(2), assume we have charts $a\colon V\to Y$, $b\colon W\to
X$, and an $\ell$-cohomologically smooth morphism $g\colon W\to V$ as in the
commutative diagram (1.1)(ii). Using the $\ell$-cohomological smoothness of
$a$, $b$, and $g$, we have $b^{*}Rf^{!}\mathbf{F}_{\ell}\cong
Rb^{!}\mathbf{F}_{\ell}^{-1}\otimes Rb^{!}Rf^{!}\mathbf{F}_{\ell}\cong
Rb^{!}\mathbf{F}_{\ell}^{-1}\otimes Rg^{!}Ra^{!}\mathbf{F}_{\ell}$ is
invertible, which implies invertibility of $Rf^{!}\mathbf{F}_{\ell}$ since
invertibility can be detected v-locally. For the second part of the claim,
suppose $A\in D_{\text{\rm\'{e}t}}(Y,\mathbf{F}_{\ell})$. It is enough to
check that $Rf^{!}\mathbf{F}_{\ell}\otimes f^{*}A\to Rf^{!}A$ is an
isomorphism after pullback through $b$, and one sees after twisting by
$Rb^{!}\mathbf{F}_{\ell}$ that this is equivalent to the statement that
$Rg^{!}Ra^{!}\mathbf{F}_{\ell}\otimes g^{*}a^{*}A\to Rg^{!}Ra^{!}A$ is an
isomorphism, which is true because $a\circ g$ is $\ell$-cohomologically
smooth. Since the property (1) is stable under decent base change on $Y$, so
are the properties we have just verified.

Now we prove (2)$\implies$(1): Let $a\colon V\to Y$ be a chart, and let
$b\colon W\to X$ be the base change of $a$ through $f$, so that once again we
have a diagram as in (1.1). By hypothesis, $Rg^{!}\Lambda$ is invertible, and
$Rg^{!}\mathbf{F}_{\ell}\otimes g^{*}(-)\to Rg^{!}(-)$ is an isomorphism. Let
$h\colon U\to W$ be a chart. Applying $h^{!}\cong
h^{!}\mathbf{F}_{\ell}\otimes h^{*}$, we find that $R(g\circ
h)^{!}\mathbf{F}_{\ell}\otimes(g\circ h)^{*}(-)\to R(g\circ h)^{!}(-)$ is an
isomorphism. This means that $g\circ h$ is an $\ell$-cohomologically smooth
morphism between charts for $X$ and $Y$ respectively, and so $f$ is
$\ell$-cohomologically smooth in the sense of Definition 1.3.ii.

By the discussion immediately following [1, Proposition IV.2.33], a
compactifiable morphism $g$ between locally spatial diamonds with finite dim.
trg is $\ell$-cohomologically smooth (in the sense of [11]) if and only if
$Rg^{!}\mathbf{F}_{\ell}$ is invertible and its formation commutes with any
base change. The above arguments can now be adapted to give the equivalence
between (1) and (3).

∎

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