# Quasi-universal relations in the context of future neutron star detections

Lami Suleiman<EMAIL_ADDRESS>Nicholas and Lee Begovich Center for
Gravitational Wave Physics and Astronomy, California State University
Fullerton, Fullerton, California 92831, USA.  Jocelyn Read  Nicholas and Lee
Begovich Center for Gravitational Wave Physics and Astronomy, California State
University Fullerton, Fullerton, California 92831, USA.

###### Abstract

The equation of state dependence of neutron star’s astrophysical features
modeling is key to our understanding of isospin asymmetric and dense matter.
However, there exists a series of almost equation-of-state independent
relations reported in the literature, called quasi-universal relations, that
are used to determine neutron star radii and moments of inertia from X-ray and
gravitational wave signals. Using sets of equations of state constrained by
multi-messenger astronomy measurements and nuclear-physics theory, we discuss
quasi-universal relations in the context of future gravitational-wave
detectors Cosmic Explorer and Einstein Telescope, and X-ray detector STROBE-X.
We focus on relations that involve the moment of inertia $I$, the tidal
deformability $\Lambda$ and the compactness $C$: $C(\Lambda)$, $I(\Lambda)$
and $I(C)$. The quasi-universal fits and their associated errors are
constructed with three different microphysics approaches which include state
of the art nuclear physics theory and astrophysical constraints.
Gravitational-wave and X-ray signals are simulated with the sensitivity of the
next generation of detectors. Equation of state inference on those simulated
signals is performed to assess if quasi-universal relations will offer a
better precision on the extraction of neutron star’s macroscopic parameters
than equation of state dependent relations. We confirm that the relation
$I(\Lambda)$ offers a more pronounced universality than relations involving
the compactness regardless of the equation of state set. We show that
detections with the 3rd generation of gravitational wave detectors and the
X-ray detector STROBE-X will be sensitive to the fit error marginalization
technique. We also find that the sensitivity of those detectors will be
sufficient that using full equation of state distributions will offer better
precision on extracted parameters than quasi-universal relations. We also note
that nuclear physics theory offers a more pronounced equation of state
invariance of quasi-universal relations than current astrophysical
constraints.

††preprint: APS/123-QED††preprint: APS/123-QED

##  I Introduction

As the densest stars in the Universe, neutron stars are particularly well
suited to investigate ultra-dense matter. Our understanding of the innermost
layers of neutron stars remains uncertain due to limitations of nuclear
physics laboratories to reach equivalent regimes of temperature and density.
Neutron star astrophysical features strongly depend on the equation of state
of ultra-dense matter, thus offering the opportunity to probe the neutron star
interior with multi-messenger astronomy. On the other hand, a series of
relations between various neutron star observables, referred to as quasi (or
almost) universal relations, were empirically found to depend weakly on the
exact equation of state [1].

While the perfectly holding universality (as per the no-hair theorem) of
isolated and stationary black holes in the gravitional theory of General
Relativity is in principle not applicable to compact stars, their external
gravitational field presents features that are almost independent of the
neutron star’s interior. This (almost) universality applies to non-magnetized
neutron stars on a static metric in General Relativity and holds for
magnetized [2] and spinning neutron stars [3], as well as in modified gravity
theories [4]. The underlying physics of this universality has been connected
to approximate no-hair relations, where high order multiple moments of compact
stars are approximately determined by low order multipole moments, and to the
self-similarity of isodensities in compact star. For a detailed introduction
and history of universality in compact stars, as well as the derivation of the
approximate no-hair relations, details on the I-Love-Q relations and also on
the black-hole limit of this almost universality, we refer to the extensive
review and work of Ref. [5] and references therein.

The era of multi-messenger astronomy has allowed for the detection of various
neutron star’s astrophysical features and is expected to provide increasingly
more precise observational data in the future. Almost equation of state
independent relations between macroscopic parameters of neutron stars are well
suited to extract the astrophysical detection of one parameter from the
measure of another (or others). The measurement of post-Keplerian parameters
in binary systems via pulsar timing has provided the most precise measurements
of neutron star’s gravitational mass $M$. Various wavelength of the
electromagnetic spectrum (radio, X-ray and optics) as well as gravitational
wave signals from binary neutron star mergers have provided a large number of
mass measurements111For a list of neutron star masses measurements with
reported precision, see https://compose.obspm.fr/resources.. The Neutron star
Interior Composition Explorer (NICER) telescope, which relies on the effects
of General Relativity on the detection of hot spots on the surface of a
rotating neutron star, has provided the simultaneous measurement of both the
gravitational mass and the radius $R$ for two sources (J0030$+$0451 and
J0740$+$6620) [6, 7, 8, 9]. Although the precision of the NICER telescope has
made some constraint on the equation of state of dense matter, the future
Spectroscopic Time-Resolving Observatory for Broadband Energy X-rays
(STROBE-X) [10] is expected to offer measurements of mass-radius contours two
to three times tighter. The tidal deformability $\Lambda$ of a neutron star
was for the first time from the gravitational wave measurement of the double
neutron star binary merger source GW170817 [11]. The significant increase in
sensitivity brought forth by the next generation of gravitational wave
detectors, such as the Cosmic Explorer [12] and the Einstein Telescope [13,
14], is expected to increase the number and precision of detections by several
orders of magnitude. Double pulsar binaries are the most promising systems to
detect the star’s moment of inertia $I$; the famous double pulsar PSR
J$0737$-$3039$ for which more than 15 years of data was gathered [15] has not
permitted the direct measurement of the moment of inertia yet. However, Ref.
[16] have been able to use a quasi-universal relation that involves $I$, $M$
and $\Lambda$ to extract constraints on the moment of inertia of
J0737$-$3039A.

In this paper, we assess the quasi-universality of three relations and discuss
their use in the context of next generation of detectors. In Section II, the
quasi-universal relations used in this paper and the equation of state sets on
which they are based are discussed. We then give details on quasi-universal
relations fits and how to introduce the equation of state variability as the
fit’s error. We show how next-generation detections are simulated. In Section
III, we compare the quasi-universal relations designed with different equation
of state sets. We then discuss the impact of different marginalizations of the
fit error on the extraction of neutron star’s macroscopic parameters. We
contrast the detector sensitivity with the equation of state variability of
quasi-universal relations. Finally, we use equation of state inference to
compare the accuracy of parameter extraction with quasi-universal relations
and equation of state dependent relations. In the Appendix, we present the
mass, radius, tidal deformability and moment of inertia modeling, as well as
parameters for the quasi-universal relation fits.

##  II Methods

In this paper, we explore astrophysical features of a neutron star within the
theory of General Relativity, assuming the line element of a spherically
symmetric, static and isotropic space-time. The mass $M$ and radius $R$ are
solved using the Tolman-Oppenheimer-Volkov [17, 18] differential equations.
The moment of inertia $I$ is determined in the slow-rotation approximation of
Ref. [19] and the tidal deformability follows the Ref. [20] derivation. For
details on the modeling of macroscopic parameters, see Appendix A.

###  II.1 Quasi-universal relations

####  II.1.1 $C(\Lambda)$, $I(\Lambda)$ and $I(C)$

In this paper, we study the following relations:

  1. 1.

The relation between the compactness and the dimensionless tidal deformability
$C(\Lambda)$, with ${C=GM/(Rc^{2})}$. In the assumption that this relation is
equation of state independent, it can be used to extract the radius of the
neutron star from the gravitational wave measurement of its mass and tidal
deformability, see e.g., Ref. [21].

  2. 2.

The relation between the dimensionless moment of inertia
$\bar{I}=Ic^{4}/(G^{2}M^{3})$ and the dimensionless tidal deformability
$\bar{I}(\Lambda)$. In the assumption that this relation is equation of state
independent, it can be used to extract the moment of inertia from the
gravitational wave measurement of the mass and tidal deformability, see e.g.,
Ref. [16].

  3. 3.

The relation between the dimensionless moment of inertia and the compactness
$\bar{I}(C)$. In the assumption that this relation is equation of state
independent, it can be used to extract the moment of inertia from the
simultaneous measurement of the mass and radius by telescope such as NICER or
Strobe-X, see e.g., Ref. [22].

Not all of the above mentioned relations are equal in the face of equation of
state independence (see Ref. [5, 23]), but they are all considered in the
literature to be quasi-universal. If the relations hold through equation of
state variation, it is also possible to parameterize them with what is
referred to as fits, see, e.g., Refs. [24, 1, 5, 25, 26, 27] or Section II.3.2
of the present paper.

####  II.1.2 Quasi-universal or data driven relations

The universality qualifier attributed to the relations studied in this paper
should relate astrophysical features of neutron star’s in spite of the
description of the neutron star interior, or in other words the Equation of
State (EoS). It should be possible to use minimally constrained EoSs (only
causal and themodynamically consistent) and still retain the quasi-
universality in the relations. Such relations would reflect the EoS
independence driven by General Relativity, not a decreased EoS variability
related to our increasing knowledge on the behavior of ultra-dense matter, or
our bias in choosing certain EoSs. The introduction of astrophysical and
nuclear constraints on the EoS can appear as stronger universality but would
be better described as a data driven narrowing of the EoS dependence.

In this paper, we do not use quasi-universal relations established from
minimally constrained EoS sets, but data-driven relations from informed sets.
We consider both up to date astrophysical data and nuclear physics data.

###  II.2 Equations of state sets

####  II.2.1 Nuclear physics based set

The first set comprises sixty one EoSs established from full nuclear physics
calculations. The EoSs are gathered from Ref. [28] to which we add seven EoSs
presented in Ref. [29]. They describe cold and catalyzed $\beta$-equilibrated
matter as such thermodynamic conditions are relevant for isolated ”adult” (not
proto) neutron stars and for the inspiral phase of a neutron star merger222We
acknowledge that a global thermodynamic equilibrium (catalyzed) neutron star
crust may not be appropriate to describe some of the potential STROBE-X
sources located in accreting binaries, but nevertheless we reasonably neglect
the impact of an accreted crust on the modeling of macroscopic parameters. .

The EoSs of this set were computed with two different nuclear physics
approaches: relativistic mean field theory (x39), Skyrme force energy density
functionals (x22). Among the relativistic mean field models are included ten
EoSs with a hyperonic core (for a recent review on hyperonic compact stars,
see Ref. [30]) and fourteen hybrid models with a core quark phase transition
(see, e.g., Refs. [31, 32]); the rest have nucleonic cores.

All EoSs of this set follow thermodynamic consistency; note that the Skyrme
density functional are based on a non-relativistic approach that does not
guarantee sound speed causality. They are unified, meaning that the crust (low
density) and the core (high density) have been calculated with the same
nuclear physics model; we refer to Ref. [33] for a discussion on the role of
non-unified EoSs in neutron star modeling. A large majority of them also
permit the Direct Urca process, a neutrino emission reaction which X-ray
observations indicate exists in neutron stars [34]. They all meet at least the
mass constraint imposed by the 1$\sigma$ pulsar timing measurement of the
millisecond pulsar J0740$+$6620 with a mass of $2.08\pm 0.07$ M⊙ [35]333In
Ref. [28], the mass constraint follows J1614$-$2230 measured at $1.908\pm
0.016$ M⊙ [36], which is why in our set we did not include H3, hyperonic DD2
and FSU2H, BSk19, KDE0v1, SKOp and BCPM. In overall, this set comprises
various core compositions and nuclear approaches, with reasonable microphysics
parameters when compared to modern nuclear physics laboratory (see Ref. [33])
and astrophysics measurements while keeping a certain variability in their
nuclear physics features.

####  II.2.2 Agnostic sets

The two other sets are based on an agnostic approach, that is to say that the
EoSs are not constructed with specific nuclear physics calculations. The point
of such sets is to explore the parameter space of pressure and density and to
go beyond our usual EoS constructions. In the following, we discuss two
agnostic sets: one parametric which we will refer to as the ”meta-model” set,
and one non parametric, later on referred to as the ”Gaussian process” set.

The first of the agnostic sets is based on Ref. [37]. To construct one EoS,
the nuclear empirical parameters that are the saturation density, the energy
of symmetric matter at saturation density, the symmetry energy, the isoscalar
and isovector incompressibility, skewness and kurtosis, the nucleon effective
mass and the effective mass isosplit are randomly thrown in intervals
determined by nuclear physics laboratory experiments. Taking advantage of
chiral-effective-field theory calculations presented in Ref. [38] for pure
neutron matter, the $\beta$-equilibrated EoSs can be constrained. For a given
collection of nuclear empirical parameters, the low density part of the EoS is
reconstructed according to the meta-modeling approach discussed in Ref. [39],
with a compressible liquid drop model for the inhomogeneous crust. The high
density part is constructed with five polytropes for which the adiabatic index
and the polytropic constant are randomly thrown. The meta-model set, respects
thermodynamic consistency, sound speed causality and provides EoSs in
accordance with nuclear physics laboratory experiments while keeping the
freedom permitted by the unknown core behavior and the error bars of nuclear
experiments at low density. It is a parametric approach, that is to say it
follows a specific functional, which implements bias. This set is constrained
by the pulsar timing mass measurement of the millisecond pulsar J0740$+$6620,
it is composed of $5\times 10^{4}$ EoSs, and is denoted MM+$\chi$+PSR.

The second agnostic set is publicly available and constructed with the
Gaussian process approach discussed in Ref. [40]. Contrary to a parametric
approach, the Gaussian process EoSs are not bound by the functional chosen to
parameterize, and therefore avoid the sort of bias inherent to parametrized
constructions, e.g., the discontinuous sound speed of piecewise polytropes in
the meta-model set. The Gaussian process set is trained on fifty nuclear
physics based EoSs; the selection of EoSs constitutes a bias, but the training
is believed to be sufficiently loose that this bias is tamed. The low density
part of the EoS is conditioned on the DH crust [41]. This set does not offer a
well constrained and well constructed crust and core-crust transition,
contrary to the meta-model set and the nuclear physics based set. It follows
thermodynamic consistency and sound speed causality. This set of EoSs is
constrained by the mass of J0740$+$6620, NICER measurements and by the tidal
deformability measurement of GW170817, we refer to it as GP+astro.

###  II.3 Parameter extraction

####  II.3.1 Parameter extraction from future astrophysical detections

We refer to parameter extraction as the technique that consists in determining
the value of an astrophysical parameter from the measurement of one or several
others. To do so, one can either use EoS dependent relations, or quasi EoS
independent relations: for example, using a mass measurement we can estimate
the radius from the EoS dependent relation $M(R)$, or using a mass and tidal
deformability measurement we can estimate the radius from the almost EoS
independent relation $C(\Lambda)$.

The reason why the relations presented in Section II.1.1 are preferred in
parameter extraction is related to the idea that we can count on the
universality to retain a tight error bar on the extracted parameter. But in
the context of increasing detector precision, we have to assess whether the
EoS variability of the relations discussed in this paper overcomes the
detector precision. In other words, we shall assess if the quasi-universality
or our current state of the art of astrophysics and nuclear physics
constrained relations are sufficient for a precise parameter extraction, or if
we will have to keep the relations up to date with upcoming EoS constraints.

If the detector precision imposes the use of increasingly data driven
relations in parameter extraction, it is also reasonable to question if
Section II.1.1 relations are still more relevant for parameter extraction than
EoS dependent relations. It is important to note that Section II.1.1 relations
require the detection of at least two parameters to estimate a third, which
implies at least two detection errors. EoS dependent relations may involve the
detection of only one parameter. Therefore, in the context of increasing
detection precision, we have to assess if EoS dependent relations that are
sufficiently informed by astrophysical measurements might provide tighter
error bars on the extracted parameter than source or data driven relations.

To assess those two points, we simulate neutron-star gravitational and X-ray
detections and estimate their associated detection errors:

  * •

A double neutron star binary merger emitting gravitational waves, for which
individual masses and tidal deformabilities are recovered with the public
parameter estimation software Bilby [42] with ROQ likelihoods [43] suited for
third generation telescope analysis [44]. Sky localization parameters of the
binary (right ascension and declination) are fixed, or in other words we
assume that an electromagnetic counterpart has been found. The spins are
aligned, equal and low ($\chi\leq 0.015$) and we consider a binary with a
neutron star of mass ${M_{1}=1.5{\;\rm M}_{\odot}}$ and a neutron star of mass
${M_{2}=1.3{\;\rm M}_{\odot}}$. Tidal deformabilities are calculated from the
relation $\Lambda(M)$ determined by the unified equation of state model DD2
[45] as presented in Ref. [33]: ${\Lambda_{1}\simeq 470}$ and ${\Lambda\simeq
1055}$. Simulations are performed with different detector sensitivities:
projected O4 sensitivity of LIGO and Virgo facilities (denoted GW-O4), and
third generation telescopes Cosmic Explorer and Einstein Telescope sensitivity
(denoted GW-3G)444The sensitivity curves for CE and ET (ET-D design [46]) used
to simulate the signal in this paper can be found at
https://dcc.ligo.org/LIGO-T1500293/public. Updated sensitivity ET-D curves
were provided in Ref. [14] recently, but conclusions of our paper remain
robust.. The signal to noise ratio of this event is quite high ($\simeq 1800$)
although not as high as the source presented in Ref. [44].

  * •

An X-ray source with a neutron star mass ${M_{\rm xray}=1.4{\rm\;M}_{\odot}}$
and a radius established from the equation of state model DD2 $R_{\rm
xray}\simeq 13.16$ km. We use the precision reported in Ref. [7] for the
detector sensitivity denoted NICER and the projected sensitivity reported in
Ref. [10] denoted STROBE-X, to simulate a distribution of masses and radii
using a bi-variate normal distribution peaked on Mxray and $R_{\rm xray}$.

####  II.3.2 Performing fits on agnostic equation of state sets

The usual approach for parameter extraction is to use fits. They correspond to
a parametrized function (usually a polynomial) fitted to macroscopic
parameters computed with a certain EoS set, and reported with a precision that
attests to the largest difference between the fit and the set’s macroscopic
parameters.

The pertinence of fits based on nuclear physics EoSs to represent the EoS
variability in neutron stars must be questioned. First, the different core
compositions have an impact on the modeling of macroscopic parameters. The
first fits available were based on a limited number of EoS models. For
example, the fits presented in Ref. [24] used three EoSs with purely nucleonic
cores; the reported precision of $2\%$ for the relation $C(\Lambda)$ was shown
in Ref. [33] to be too small when compared to several EoSs of various
stiffness. Recognizing that the so called ”universal” relations were only
quasi-independent of the description of neutron star’s interior, efforts were
made to fit to larger sets of nuclear physics based EoSs, including various
core compositions thus increasing the reported error of the fits; for example,
in Ref. [5], the fits were performed using twenty EoSs with nucleonic cores,
seven EoSs with hyperonic or kaon cores and three quark stars; a reported
precision of 6.5% is given for neutron stars and 15% for quark stars. Neutron
star’s core composition remains unknown, there is therefore no reason to
perform the fit on sets containing more nucleonic models than hybrid or
hyperonic ones. This is however what is done in practice, systematically with
fits based on nuclear physics EoS, likely because nucleonic cores are easily
accessible in the literature. A similar reasoning applies to the nuclear
physics approach used to establish the set of EoSs on which the fits are based
on: relativistic mean field based models tend to be stiffer than e.g., Skyrme
energy density functional models. Secondly, there is no reason to reduce the
set of EoSs to known hypothesis of the core composition nor our approaches of
nuclear physics. To that end, the more relevant set of EoSs to perform the
fits should be agnostic, as presented e.g. in Ref. [25, 23].

In this paper, we perform fits of the three relations presented in Section
II.1.1 with the different EoS sets presented in this paper. For each relation
and each set, a non-linear least square method is used to determine the
parameter $a_{k}$ with $k\in[0,5]$ in

$\displaystyle C_{\rm fit}$
$\displaystyle=\sum_{k=0}^{5}a_{k}(\ln\Lambda)^{k}\;,$ (1)
$\displaystyle\ln\bar{I}_{\rm fit}$
$\displaystyle=\sum_{k=0}^{5}a_{k}\ln(\Lambda)^{k}\;,$ (2)
$\displaystyle\bar{I}_{\rm fit}$ $\displaystyle=\sum_{k=0}^{5}a_{k}C^{-k}\;.$
(3)

An error is associated to each fit and denoted ${(\Delta X_{\rm fit})^{\rm
S}}$ with $X$ the fitted quantity and $S$ the set of EoSs. The fits are
performed using macroscopic parameters of neutron star with at least a mass of
1.0 M⊙ up to the last stable mass configuration. The error is assessed from
the full range of the (finite) nuclear set, and the 99% percentiles of the
GP+astro set and MM+$\chi$+PSR set. The fit parametrizations for $C(\Lambda)$,
$\bar{I}(\Lambda)$ and $\bar{I}(C)$ presented in Ref. [5] (later denoted Yagi
& Yunes) are also used as a comparison.

####  II.3.3 Marginalization of the error

To include the fit error ${(\Delta X_{\rm fit})^{\rm S}}$ in the extraction of
a parameter, we must specify a marginalization technique. This can be done by
defining an extra variable $\delta X$ with a random distribution reflecting
the fit uncertainty, such that each sample fitted parameter is multiplied by a
factor $(1+\delta X_{\rm fit}^{S})$. In this paper, we describe this
distribution in relation to the error on the fit for a given relation
${(\Delta X_{\rm fit})^{\rm S}}$. For example, in Refs. [11, 47], the
marginalization consisted of drawing the fit error parameter $\delta X$ from a
Gaussian function peaked on the fitted relation with a ${(\Delta C_{\rm
fit})^{\rm yy}}$ variance; with the fit’s error assigned to a 3-$\sigma$
precision, the standard deviation for the Gaussian distribution is ${(\Delta
X_{\rm fit})^{\rm S}/3}$. We refer to this technique as the Gaussian
marginalization.

In the absence of information on the distribution of macroscopic parameters
computed from the EoS set used to perform the fit, we propose that users
choose a uniform distribution instead of a Gaussian one: ${\delta X_{\rm
fit}^{S}\in[-(\Delta X_{\rm fit})^{\rm S};(\Delta X_{\rm fit})^{\rm S}]}$ is
randomly drawn from a uniform distribution; this approach is referred to later
on as the uniform marginalization. It leads to a larger systematic error on
the extraction of the macroscopic parameter but avoids the fit line bias. To
illustrate that the distribution of points is not necessarily a Gaussian for a
given set of EoS, we present the distribution of compactness and dimensionless
moment of inertia around the fit for equation of state sets in Appendix B.

One can also assess the full extent of the fit error. We take the two extremes
of the relation ${X_{\rm fit\pm}^{S}(Z)=X_{\rm fit}^{S}(Z)\pm(\Delta X_{\rm
fit})^{\rm S}}$. We then perform the parameter extraction and obtain two
distributions which represent the two extremes of the fit error; we refer to
this approach as the fit limits.

###  II.4 Equation of state inference

(a) Relation between the compactness $C$ and the dimensionless tidal
deformability $\Lambda$.

(b) Relation between the compactness $C$ and the dimensionless moment of
inertia $\bar{I}$.

(c) Relation between the dimensionless tidal deformability $\Lambda$ and the
dimensionless moment of inertia $\bar{I}$.

Figure 1: Contours (99 percentile) presented for the nuclear set,
MM+$\chi$+PSR set and the GP+astro set. In black we present the Yagi & Yunes
fit [5].

Astrophysical sources inform the EoS: sources such as the simulated ones
presented Section II.3.1 therefore have the potential to constrain the sets of
EoSs presented in Section II.2. We use the following EoS inference approaches.

For the X-ray simulation, we use a simple likelihood estimation on the
1-$\sigma$ STROBE-X sensitivity data and the MM+$\chi$+PSR and GP+astro sets.
For the gravitational wave simulated data, we use the EoS inference technique
discussed in Refs. [48, 40, 49, 50] with the software Likelihood Weighting
Protocol (LWP555https://git.ligo.org/reed.essick/lwp) to infer on the
MM+$\chi$+PSR and GP+astro set with the two neutron stars of the binary
coalescence in Cosmic Explorer and Einstein Telescope sensitivity.

With this newly informed sets of EoSs, we can extract parameters from the EoS
$M(R)$ and $M(I)$ and compare the EoS dependent parameter extraction to data-
driven relation technique.

(a) MM+$\chi$+PSR set

(b) GP+astro set

Figure 2: $M(R)$ extracted from the $C(\Lambda)$ relation and a simulated GW
signal in the sensitivity of O4 (orange) and Cosmic Explorer and Einstein
telescope (green); contours are shown at 1-$\sigma$. Results are presented for
the fit line (plain lines), with the Gaussian marginalization of the error
(dashed line), and with the uniform marginalization of the error (dotted
line); the limits of the error are presented in shades of colors.

##  III Results

###  III.1 Comparison of the different equation of state sets

In Fig. 1 we present the relations of Section II.1.1 established from
different sets of EoS: the nuclear set, the GP+astro and the MM+$\chi$+PSR
set. In black is also presented the fits of Yagi & Yunes.

The agnostic GP+astro set presents the largest EoS variability. It explores a
larger astrophysics parameter space than the nuclear set and the agnostic
MM+$\chi$+PSR set, and overlaps them both, thus showing that the chiral
effective field theory constraints used for the meta-modeling approach offer
stronger constraints than current astrophysical measurements from NICER and
GW170817. We note also that the GP+astro set does not encompass the meta-model
(nor the nuclear) set in a symmetric way, particularly at low compactness: the
meta model set favors higher tidal deformabilities and moment of inertia, in
other words, stiffer EoSs.

The nuclear set and the meta model do not perfectly overlap, as some of the
EoSs used in the nuclear set are not in accordance with chiral effective field
theory constraints. The nuclear set is in good accordance with the fit of Yagi
& Yunes, which was also based on complete nuclear physics calculation EoSs.
The nuclear set was established using 61 EoSs with various core compositions
(including deconfined quarks), however it presents a stronger EoS invariance
than the Yagi & Yunes fit based on $\sim 30$ EoSs. For example, the error
associated to the Yagi & Yunes fit of $C(\Lambda)$ is of 6.5% and 15%
excluding and including quark stars respectively. Our fit for this relation
(see Appendix for parameters) gives a 6% maximum error (see Table 1 in the
Appendix) even though it includes a significant portion of hybrid models. The
selection of modern EoS which are calibrated to astrophysical and nuclear
physics data plays a role in the smaller error (although not significant) of
our fit: for example, the set used to establish the Yagi & Yunes fit includes
models presenting low radii at fixed mass (i.e. high tidal deformability and
moment of inertia), which microphysics parameters have been reproved by
nuclear physics laboratory experiments e.g. WFF1 and WFF2 [51].

The relation $C(\Lambda)$ in Fig. 1(a) and $\bar{I}(C)$ in Fig. 1(b) present a
clear EoS variability. The relation $\bar{I}(\Lambda)$ in Fig. 1(c) is EoS
invariant to the point that we cannot distinguish differences between the sets
of EoSs: this relation is sufficiently strongly universal to overcome the EoS
variability emerging from the use of different EoS sets. This result is in
accordance with Ref. [23] which quantifies the degree of universality of
various relations: we also find sub-percent errors on the fits (see Table 1 in
the Appendix). Contrary to the compactness involved relations in which the
different sets and their various EoS constraints can be distinguished, the
$\bar{I}(\Lambda)$ universality cannot be attributed to our use of data driven
EoS sets.

The particular EoS invariance of the relation $\bar{I}(\Lambda)$ was discussed
in Ref. [5] and reference therein in the larger context of I-Love-Q relations.
One of the hypothesis put forward to explain the EoS invariance of I-Love-Q
relations was the lack of variability far from the core (low densities). The
GP+astro set presents a unique crust while the meta model offers crust
variability yet the former presents a slightly larger error on the fit of
$\bar{I}(\Lambda)$ than the latter: we did not observe an obvious loss of
invariance when including the nuclear physics constrained crust variability of
the meta model.

Finally, from this hierarchy of universality between the different relations,
we can anticipate that there exists a detector sensitivity for which the
radius extraction from the measurement of the tidal deformability in a
gravitational wave signal could be overcome by the EoS variability, while the
extraction of the moment of inertia from the same signal would not.

###  III.2 Error marginalization

####  III.2.1 Radius from binary neutron star mergers

In Fig. 2, we present the extraction of the radius from the simulated
gravitational wave signal of a binary neutron star merger using the relation
$C(\Lambda)$; the O4 sensitivity is presented in orange and the Cosmic
Explorer and Einstein telescope sensitivity is presented in green. Fig. 2(a)
used the fit based on the meta-modeling set of EoS while Fig. 2(b) used the
Gaussian process set.

The radius extraction using the Gaussian or the uniform marginalization of the
error, or simply the fit line and without including any error, are (roughly)
equivalent at O4 sensitivity. This fit limits (in shaded colors) are larger at
that sensitivity but not significantly relative to the size of the contour,
although it is more pronounced for the Gaussian process fit.

In the case of 3rd generation sensitivity, we first see that there is a
significant difference between the Gaussian and uniform marginalization, as
well as with the fit line without inclusion of the error: the Gaussian
marginalization has overestimated the precision on the radius by around a
factor of two compared to uniform marginalization. We can conclude that the
technique used to include the error has an impact on the results in this
sensitivity of the detectors. We also see on that figure that the extraction
of the radius from the fit line is significantly thinner than the error
marginalization: this indicates that the parameter extraction contours at that
sensitivity are related to the fit error and not the the tidal deformability
recovery from the gravitational wave signal. This is confirmed by the limits
of the error presented in shades of green: they are at the opposite ends of
the uniform marginalization and do not overlap at all.

Finally, the meta-model (MM+$\chi$+PSR) fit offer a better precision on the
parameter extraction than the Gaussian process (GP+astro) fit, as the maximum
error presented in Table 1 indicated. This indicates that in parameter
extraction, current astrophysical constraints are not equivalent to nuclear
theory calculations in data-driven relations. We also note that higher radii
are extracted for a given mass with the Gaussian process fit, in accordance
with our discussion of Fig. 1(a) in Section III.1.

Figure 3: $I(M)$ extracted from the $\bar{I}(\Lambda)$ relation and a
simulated GW signal in the sensitivity of O4 (orange) and Cosmic Explorer and
Einstein telescope (green); contours are shown at 1-$\sigma$. Results are
presented for the fit line (plain lines), with the Gaussian marginalization of
the error (dashed line), and with the uniform marginalization of the error
(dotted line); the limits of the error are presented in shades of colors. Fits
used are that of GP+astro.

####  III.2.2 Moment of inertia from binary neutron star mergers

(a) MM+$\chi$+PSR set

(b) GP+astro set

Figure 4: $I(M)$ extracted from the $\bar{I}(C)$ relation and the simulated
X-ray signal in the sensitivity of NICER (orange) and Strobe-X (green);
contours are shown at 1-$\sigma$. Results are presented for the fit line
(plain lines), with the Gaussian marginalization of the error (dashed line),
and with the uniform marginalization of the error (dotted line); the limits of
the error are presented in shades of colors.

In Fig. 3, we present the extraction of the moment of inertia using the
relation $\bar{I}(\Lambda)$ and the fit based on the Gaussian process set
(which produces the largest fit error, see Table 1). In that case, we can see
that the two marginalization techniques, the limit of the error and the
extraction from the fit line all overlap, both for O4 sensitivity and CE+ET
sensitivity. This is not explained by the parameter estimation of the tidal
deformability, as we use the same source and same set of EoS as for Fig. 2 but
by the nature of the quasi-universality of the relation $\bar{I}(\Lambda)$. As
can be seen in Fig. 1(c), this relation presents a significant universality,
which makes considerations of the marginalization technique, or even purely of
the error, irrelevant. The error associated to the parameter estimation
dominates whatever the sensitivity of the detector. We conclude that the
universality of the relation $\bar{I}(\Lambda)$ will strongly hold with the
3rd generation of gravitational wave detectors.

####  III.2.3 Moment of inertia from X-ray detections

In Fig. 4, we present the extraction of the moment of inertia using the
relation $\bar{I}(C)$. For the meta-model fits presented in Fig 4(a), the
marginalization technique does not impact the results. For the fits based on
the Gaussian process set presented in Fig. 4(b), the results are impacted at
high sensitivity (STROBE-X) as the Gaussian marginalization underestimates the
error by almost a third compared to the uniform marginalization. The
constraints brought forth by nuclear physics in the meta-modeling allow us to
consider that STROBE-X sensitivity still dominates (or is at least equivalent)
to the EoS variability of the meta-model based data-driven relation
$\bar{I}(C)$.

###  III.3 Equation of state inference vs data-driven relations

####  III.3.1 Moment of inertia extraction

(a) From STROBE-X X-ray data.

(b) From CE+ET gravitational wave data.

Figure 5: Moment of inertia distribution from parameter extraction operated
with data-driven relations and EoS inference.

We show the moment of inertia distribution extracted with the EoS dependent
relation $M(I)$ from the GP+astro and MM+$\chi$+PSR EoSs inferred on by the
simulated STROBE-X source in Fig. 5(a). We compare it to the moment of inertia
distribution obtained with the data-driven relation fits which parameters are
presented in Table 1. The EoS inference leads to a better determination of the
moment of inertia than the use of data-driven relation fits. In the case of
the GP+astro set, the improvement is significant, while in the case of the
meta-model it is almost equivalent: this is in accordance with results
presented in Fig 4. The peaks of the EoS inference distributions are not the
same for the GP+astro and the meta-model set. The GP+astro set is informed by
GW170817 which softens the EoS distribution in that set while the meta-model
set can explore stiffer models. The distribution for $I$ determined by the
data-driven relation of the GP+astro however explores higher values of the
moment of inertia than its EoS inference counterpart. The fit established from
a set of EoS informed by data (astrophysics or nuclear physics) does not keep
memory distribution of that set: the fit only cares about the boundary points,
not the distribution of points itself.

The moment of inertia distribution extracted with gravitational wave data
simulated with Cosmic Explorer and Einstein Telescope sensitivity is presented
in Fig. 5(b). The extraction of $I$ with the data-driven relation
$\bar{I}(\Lambda)$ are very similar: as discussed in Fig. 1(c), this relation
is very universal, although we can better distinguish between the two
different EoS sets. The difference of the peak’s values in the EoS inference
and data-driven relation distributions is also visible even though the
distributions overlap well: the use of fits which loose prior information on
the sets leads again to higher values of the moment of inertia, suggesting a
slightly stiffer EoS. Finally, we also note that the 3G gravitational wave
detection offer a better constraint on the moment of inertia than STROBE-X
does with EoS inference.

####  III.3.2 Radius extraction

Figure 6: Radius distribution from parameter extraction operated with data-
driven relations and EoS inference.

We show the radius distribution extracted with the EoS dependent relation
$C(\Lambda)$ from the GP+astro and MM+$\chi$+PSR EoSs inferred on by the
simulated gravitational wave data source in Fig. 6. Results are similar to
Fig. 5(b), except that the distribution offered by data-driven relations is
much larger. We also note that the meta model inference peak on the radius in
Fig 6 is much more pronounced than the peak in moment of inertia of Fig. 5(b).
This is because the radius is strongly related to the description of the crust
while the moment of inertia is mainly sensitive to the core. While the core
treatment of the MM+$\chi$+PSR and the GP+astro seem to lead to similar
distributions for the moment of inertia inference, this is not the case for
the radius inference. The power of the low density treatment of the meta model
explicitly shows in this case.

We also note that the distributions presented in Fig. 5 and Fig. 6 are not
always very well resolved. In this paper, we have used pre-computed EoS sets.
They will be an issue for future detectors and a generator of EoS set
operating with the inference will be necessary to offer well resolved
distributions.

##  IV Conclusion

In this paper, we discuss three macroscopic parameter relations :
$C(\Lambda)$, $\bar{I}(C)$ and $\bar{I}(\Lambda)$. We have studied the use of
so called quasi-universal relations in moment of inertia and radius extraction
from gravitational wave and X-ray signals with current and future detector
sensitivity.

We have used three different sets of EoSs to design the relations and their
associated fits. We have shown that for compactness related relations
discussed in this paper, the EoS variability lets us distinguish between the
different sets of EoS while the $\bar{I}(\Lambda)$ presents a more pronounced
universality. To accurately express the EoS set dependence of the relations,
we refer to data-driven relation instead of quasi-universal relations.

We have discussed the fit error marginalization in parameter extraction and
have shown that different approaches do not impact the result for O4 or NICER
sensitivity, but do impact the result for Einstein Telescope and Cosmic
Explorer and STROBE-X sensitivity.

Finally, we have shown that using data-driven relations in parameter
extraction with the future detector sensitivity will lead to much larger
uncertainties on the extracted parameter than direct EoS inference. In
overall, when the quasi-universality of data-driven relations is overcome by
the sensitivity of the detector and its consequent information on the EoS,
using data-driven relations is sub-optimal.

In previous usage, data driven relations were broadly equivalent in parameter
extraction when compared to direct EoS inference (for example, in GW170817
[21]). We show here that even if future data-driven relations are conditioned
on EoS knowledge with precision measurement, direct usage of those EoSs will
be more effective in extracting parameters.

###### Acknowledgements.

The authors thank Rory Smith, Sunny Ng, Philippe Landry, Carl-Johan Haster,
Isaac Legred and Emily Wuchner for useful discussion about numerical
applications. The authors thank collaborators of the Nuclear Physics from
Multi-Messenger Mergers (NP3M) Focused Research Hub for useful discussions.
The authors acknowledge the financial support of the National Science
Foundation grant Number PHY 21-16686 and PHY 21-10441. The authors are
grateful for computational resources provided by the LIGO Laboratory and
supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.
Software: This work makes use of scipy [52], numpy [53] and matplotlib [54].

##  Appendix A Mass, radius, moment of inertia and tidal deformability
modeling

We study the astrophysical features of a non-rotating neutron star within the
gravity theory of General Relativity, assuming the line element ${\rm d}s$ of
a spherically symmetric, static and isotropic space-time determined by the
metric $g_{\mu\nu}$ as

$\displaystyle{\rm d}s^{2}$ $\displaystyle=g_{\mu\nu}{\rm d}x^{\mu}{\rm
d}x^{\nu}$ (4) $\displaystyle=-e^{2\phi(r)}{\rm d}t^{2}+e^{2\lambda(r)}{\rm
d}r^{2}+r^{2}{\rm d}\theta+r^{2}\sin(\theta){\rm d}\phi^{2}\;,$

with $r$ the Schwarzschild like radial coordinate. Considering vacuum outside
the neutron star, this metric reduces to the Schwarzschild one. Inside the
star, the functions $\phi$ (gravitational redshift) and $\lambda$ (radial
gravitational distortion) are solved using hydrostatic equilibrium equations
and the Equation of State (EoS) ${P=P(\epsilon)}$, with $P$ the pressure and
$\epsilon$ the energy density. In the assumption that the matter inside the
neutron star is a perfect fluid, we obtain the Tolman-Oppenheimer-Volkov (TOV)
differential equations

$\displaystyle\frac{{\rm d}m}{{\rm d}r}=\frac{4\pi
r^{2}}{c^{2}}\epsilon(r)\;,$ (5) $\displaystyle\frac{{\rm d}\phi}{{\rm
d}r}=\frac{Gm(r)}{r^{2}}\bigg{(}1+\frac{4\pi
r^{3}P(r)}{m(r)c^{2}}\bigg{)}\bigg{(}1-\frac{2Gm(r)}{rc^{2}}\bigg{)}^{-1}\;,$
(6) $\displaystyle\frac{{\rm d}P}{{\rm
d}r}=-\bigg{(}\frac{\epsilon(r)+P(r)}{c^{2}}\bigg{)}\frac{{\rm d}\phi}{{\rm
d}r}\;,$ (7)

with $G$ the gravitational constant, $c$ the speed of light and $m(r)$ the
gravitational mass radial profile. The central energy density or central
pressure, used as a boundary condition to solve the TOV equations, determines
the total gravitational mass $M$ and the total radius $R$ of the neutron star.

The dimensionless tidal deformability expresses the quadruple moment response
induced on the neutron star by an external gravitational field. Its value at
the surface of the star denoted $\Lambda=\lambda_{2}(R)$ is given by the tidal
deformability radial profile

$\lambda_{2}(r)=\frac{2}{3}k_{2}(r)C(r)^{-5}\,,$ (8)

with $C(r)=Gm(r)/(rc^{2})$ the compactness of a star of radius $r$. The tidal
Love number $k_{2}$ is given by

$\displaystyle k_{2}(r)=$
$\displaystyle\,\frac{8C(r)^{5}}{5}\big{(}1-2C(r)\big{)}^{2}\big{[}2+2C(r)\big{(}y(r)-1\big{)}-y(r)\big{]}$
$\displaystyle\times\bigg{(}2C(r)\big{[}6-3y(r)+3C(r)\big{(}5y(r)-8\big{)}\big{]}$
$\displaystyle+4C(r)^{3}\big{[}13-11y(r)+C(r)\big{(}3y(r)-2\big{)}$
$\displaystyle\hskip 42.67912pt+2C(r)^{2}\big{(}1+y(r)\big{)}\big{]}$
$\displaystyle+3\big{(}1-2C(r)\big{)}^{2}\big{[}2-y(r)+2C(r)\big{(}y(r)-1\big{)}\big{]}$
$\displaystyle\hskip 71.13188pt\ln\big(1-2C(r)\big{missing})\bigg{)}^{-1}\;,$
(9)

see, e.g., Ref. [20] and reference therein. The function $y(r)$ is to be
solved simultaneously with the TOV equations using the additional differential
equation [20, 55]

$r\dfrac{{\rm d}y}{{\rm d}r}+y(r)^{2}+F(r)y(r)+Q(r)=0\,,$ (10)

with the boundary condition $y(0)=2$, see Section IV.A of Ref. [56]. The
functions $F(r)$ and $Q(r)$ are given by

$\displaystyle F(r)=$ $\displaystyle\bigg{(}1-\frac{4\pi
G}{c^{2}}r^{2}\frac{\epsilon(r)-P(r)}{c^{2}}\bigg{)}\bigg{(}1-\frac{2Gm(r)}{rc^{2}}\bigg{)}^{-1}\,,$
(11) $\displaystyle Q(r)=$ $\displaystyle\frac{4\pi
G}{c^{2}}r^{2}\bigg{(}1-\frac{2Gm(r)}{rc^{2}}\bigg{)}^{-1}\Biggl{[}\frac{5\epsilon(r)+9P(r)}{c^{2}}$
$\displaystyle\hskip
8.5359pt+\frac{\epsilon(r)+P(r)}{c_{s}(r)^{2}}-\frac{6\,c^{2}}{4\pi
Gr^{2}}\Biggr{]}\;$ (12) $\displaystyle\hskip
8.5359pt-4\bigg{(}\frac{Gm(r)}{rc^{2}}+\frac{4\pi
G}{c^{4}}r^{2}P(r)\bigg{)}^{2}\bigg{(}1-\frac{2Gm(r)}{rc^{2}}\bigg{)}^{-2}$

with $c_{s}$ the sound speed that should be treated with caution around a
discontinuous density, see, e.g., Ref. [57].

To model the moment of inertia, we use the slow and rigid rotation
approximation detailed in Ref. [19]666Modeling the moment of inertia outside
of this approximation requires solving the Einstein’s equations with a metric
describing a stationary and axisymmetric space-time, see, e.g., the LORENE
library.. When considering rigid rotation, the uniform angular frequency is
contributed to by the local spin frequency $\omega(r)$ and the angular
momentum $j(r)$ of a sphere of radius $r$. We denote $\Omega$ the uniform
angular frequency of the star, defined at the surface (${r=R}$) as

$\Omega=\omega(R)+\frac{2Gj(R)}{c^{2}R^{3}}\;.$ (13)

The uniform angular frequency, the angular momentum and the moment of inertia
are solution of the differential equations

$\displaystyle\frac{{\rm d}I}{{\rm
d}r}=\frac{8\pi}{3}\,\frac{r^{4}}{e^{\phi(r)}}\frac{\omega(r)}{\Omega}\frac{\epsilon(r)+P(r)}{c^{2}}\bigg{(}1-\frac{2Gm(r)}{c^{2}r}\bigg{)}^{-1/2}\;,$
(14) $\displaystyle\frac{{\rm d}\omega(r)}{{\rm
d}r}=\frac{6G}{c^{2}}\frac{e^{\Phi(r)}}{r^{4}}j(r)\bigg{(}1-\frac{2Gm(r)}{c^{2}r}\bigg{)}^{-1/2}\;,$
(15) $\displaystyle\frac{{\rm d}j(r)}{{\rm
d}r}=\frac{8\pi}{3}\frac{r^{4}}{e^{\phi(r)}}\omega(r)\frac{\epsilon(r)+P(r)}{c^{2}}\bigg{(}1-\frac{2Gm(r)}{c^{2}r}\bigg{)}^{-1/2}\;,$
(16)

to be solved simultaneously with the TOV equations.

##  Appendix B Distribution of EoSs in the sets

We present in Fig. 7 the distribution of compactness and dimensionless moment
of inertia points for GP+atro and MM+$\chi$+PSR set.

Figure 7: Distribution of points of compactness (top) and dimensionless moment
of inertia (bottom) for a few values of the tidal deformability and
compactness respectively, in MM+$\chi$+PSR and GP+astro sets. Vertical lines
correspond to the 99 percentile limits of the distributions.

##  Appendix C Fit parameters and errors associated to the fits

We present in Table 1 the parameters of the fit for the relations presented in
Section II.3.2 and the maximum error associated to the fits with the different
EoS sets presented in this paper.

Relation | Set | $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ |  $\Delta_{\rm max}$ (in %)
---|---|---|---|---|---|---|---|---
$C(\Lambda)$ | GP+astro | 3.7678 $\times 10^{-1}$ | -5.1851 $\times 10^{-2}$ | 8.3659 $\times 10^{-3}$ | -1.6529 $\times 10^{-3}$ | 1.5470 $\times 10^{-4}$ | -5.0440 $\times 10^{-6}$ | 12.6129
Nuclear | 3.5636 $\times 10^{-1}$ | -3.6950 $\times 10^{-2}$ | 4.5372 $\times 10^{-3}$ | -1.1316 $\times 10^{-3}$ | 1.1773 $\times 10^{-4}$ | -4.0028 $\times 10^{-6}$ | 5.9967
MM+$\chi$+PSR | 3.3888 $\times 10^{-1}$ | -1.0262 $\times 10^{-2}$ | -9.8856 $\times 10^{-3}$ | 2.0656 $\times 10^{-3}$ | -1.9511 $\times 10^{-4}$ | 7.2814 $\times 10^{-6}$ | 5.2443
$\bar{I}(\Lambda)$ | GP+astro | 1.5011 $\times 10^{0}$ | 5.2134 $\times 10^{-2}$ | 2.4508 $\times 10^{-2}$ | -9.9877 $\times 10^{-4}$ | 2.7064 $\times 10^{-5}$ | -3.3703 $\times 10^{-7}$ | 0.9699
Nuclear | 1.4874 $\times 10^{0}$ | 6.9698 $\times 10^{-2}$ | 1.9206 $\times 10^{-2}$ | -2.5145 $\times 10^{-4}$ | -2.0918 $\times 10^{-5}$ | 7.4502 $\times 10^{-7}$ | 0.6774
MM+$\chi$+PSR | 1.4857 $\times 10^{0}$ | 6.7440 $\times 10^{-2}$ | 1.8129 $\times 10^{-2}$ | 2.2972 $\times 10^{-4}$ | -7.7256 $\times 10^{-5}$ | 2.8044 $\times 10^{-6}$ | 0.5033
$\bar{I}(C)$ | GP+astro | 1.5333 $\times 10^{1}$ | -1.1303 $\times 10^{1}$ | 3.8536 $\times 10^{0}$ | -5.0621 $\times 10^{-1}$ | 3.2323 $\times 10^{-2}$ | -7.5871 $\times 10^{-4}$ | 22.2608
Nuclear | 6.1602 $\times 10^{0}$ | -4.7746 $\times 10^{0}$ | 2.1004 $\times 10^{0}$ | -2.8280 $\times 10^{-1}$ | 1.9466 $\times 10^{-2}$ | -4.9244 $\times 10^{-4}$ | 9.6337
MM+$\chi$+PSR | 2.3584 $\times 10^{0}$ | 1.9423 $\times 10^{-1}$ | -4.5225 $\times 10^{-2}$ | 1.1677 $\times 10^{-1}$ | -1.4423 $\times 10^{-2}$ | 5.7714 $\times 10^{-4}$ | 9.1280
Table 1: Fit parameters of the relation $C(\Lambda)$, $\bar{I}(C)$ and
$\bar{I}(\Lambda)$ based on the parametrization presented in Eqs. (1), (2),
(3), and maximum error associated to the fit $\Delta_{\rm max}$. Results are
presented for the EoS sets discussed in this paper, GP+astro, the nuclear set
and the MM+$\chi$+PSR.

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