# Snowmass 2021 Cosmic Frontier White Paper:
The Dense Matter Equation of State and QCD Phase Transitions

Slavko Bogdanov  Columbia Astrophysics Laboratory, Columbia University, 550
West 120th Street, New York, NY 10027, USA  Emmanuel Fonseca  Department of
Physics and Astronomy, West Virginia University, White Hall, Box 6315,
Morgantown, WV 26506, USA  Rahul Kashyap  Institute for Gravitation and the
Cosmos, Department of Physics, Penn State University, University Park, PA
16802, USA  Aleksi Kurkela  Faculty of Science and Technology, University of
Stavanger, 4036 Stavanger, Norway  James M. Lattimer  Department of Physics
and Astronomy, State University of New York at Stony Brook, Stony Brook, NY
11794, USA  Jocelyn S. Read  Department of Physics, California State
University, Fullerton, 800 N. State College Blvd., Fullerton, CA 92831, USA
Bangalore S. Sathyaprakash  Institute for Gravitation and the Cosmos,
Department of Physics, Penn State University, University Park, PA 16802, USA
Department of Astronomy and Astrophysics, Penn State University, University
Park, PA 16802, USA  H. Thankful Cromartie  Cornell Center for Astrophysics
and Planetary Science and Department of Astronomy, Cornell University, Ithaca,
NY 14853, USA  Tim Dietrich  Max Planck Institute for Gravitational Physics
(Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
Institute for Physics and Astronomy, University of Potsdam, Potsdam 14476,
Germany  Arnab Dhani  Institute for Gravitation and the Cosmos, Department of
Physics, Penn State University, University Park, PA 16802, USA  Timothy Dolch
Department of Physics, Hillsdale College, 33 East College Street, Hillsdale,
MI 49242, USA  Eureka Scientific, 2452 Delmer Street, Suite 100, Oakland, CA
94602-3017, USA  Tyler Gorda  Technische Universität Darmstadt, Department of
Physics, 64289 Darmstadt, Germany  ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany
Sebastien Guillot  IRAP, CNRS, 9 avenue du Colonel Roche, BP 44346, F-31028
Toulouse Cedex 4, France  Université de Toulouse, CNES, UPS-OMP, F-31028
Toulouse, France  Wynn C. G. Ho  Department of Physics and Astronomy,
Haverford College, 370 Lancaster Avenue, Haverford, PA 19041, USA  Rachael
Huxford  Institute for Gravitation and the Cosmos, Department of Physics, Penn
State University, University Park, PA 16802, USA  Frederick K. Lamb  Center
for Theoretical Astrophysics and Department of Physics, University of Illinois
at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA  Department
of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green
Street, Urbana, IL 61801, USA  Philippe Landry  Canadian Institute for
Theoretical Astrophysics, University of Toronto, 60 St. George Street,
Toronto, ON M5S 3H8 Canada  Bradley W. Meyers  Department of Physics and
Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,
BC V6T 1Z1 Canada  M. Coleman Miller  Department of Astronomy and Joint Space-
Science Institute, University of Maryland, College Park, MD 20742-2421, USA
Joonas Nättilä  Columbia Astrophysics Laboratory, Columbia University, 550
West 120th Street, New York, NY 10027, USA  Physics Department, Columbia
University, 538 West 120th Street, New York, NY 10027, USA  Center for
Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY
10010, USA  Risto Paatelainen  Department of Physics and Helsinki Institute of
Physics, University of Helsinki, Helsinki, Finland  Chanda Prescod-Weinstein
Department of Physics and Astronomy, University of New Hampshire, Durham, NH
03824, USA  Saga Säppi  Department of Physics and Helsinki Institute of
Physics, University of Helsinki, Helsinki, Finland  Ingrid H. Stairs
Department of Physics and Astronomy, University of British Columbia, 6224
Agricultural Road, Vancouver, BC V6T 1Z1 Canada  Nikolaos Stergioulas
Department of Physics, Aristotle University of Thessaloniki, 54124
Thessaloniki, Greece  Ingo Tews  Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Aleksi Vuorinen  Department of Physics
and Helsinki Institute of Physics, University of Helsinki, Helsinki, Finland
Zorawar Wadiasingh  Astrophysics Science Division, NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA  Universities Space Research Association
(USRA), Columbia, MD 21046, USA  Anna L. Watts  Anton Pannekoek Institute for
Astronomy, University of Amsterdam, Science Park 904, 1090GE Amsterdam, The
Netherlands

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

###### Contents

  1. 1 Introduction
  2. 2 Theoretical Equation of State
    1. 2.1 Discovering Quark Matter Cores in Massive Neutron Stars
    2. 2.2 Perturbative QCD
    3. 2.3 Nucleon Superfluidity
    4. 2.4 Dark Matter in Neutron Stars and Exotic Compact Objects
  3. 3 Connections with Nuclear Experiment and Theory
  4. 4 Constraints from Radio Observations of Neutron Stars
    1. 4.1 Measuring Neutron Star Masses in Pulsar Binary Systems
    2. 4.2 Measurements of the Neutron Star Moment of Inertia
  5. 5 Constraints from X-ray Observations of Neutron Stars
    1. 5.1 X-ray Observational Techniques for EOS Constraints
  6. 6 Merger Transient Gravitational Wave and Electromagnetic Observations
  7. 7 Recommendations and Conclusions
    1. 7.1 Requirements for the Coming Decade

## Executive Summary

Our limited understanding of the physical properties of matter at ultra-high
density ($\rho_{s}>2.7\times 10^{14}$ g cm-3 or $n_{s}=0.16/$fm3), high
proton/neutron number asymmetry, and low temperature ($\lesssim 10^{10}~{}{\rm
K}$) is presently one of the major outstanding problems in physics, owing to a
number of challenges both in the experimental and theoretical realms [see,
e.g., 283, for a review]. A plethora of well-motivated theoretical predictions
for the state of matter in this regime have been proposed, ranging from normal
nucleonic matter, to particle exotica such as hyperons, deconfined quarks,
color superconducting phases, and Bose-Einstein condensates.
The principal macroscopic diagnostic of the state of dense matter is the
pressure-density-temperature relation of bulk matter, the so-called equation
of state (EOS). The EOS can, in principle, be employed to deduce crucial
aspects of the microphysics, such as the role of many-body interactions at
nuclear densities or the presence of deconfined quarks at high densities.
Matter in this extreme regime is known to only exist stably in the cores of
neutron stars, the collapsed compact remnants of massive stars. As such,
neutron stars can provide important insight into the properties of matter in a
regime that is otherwise inaccessible.
In the upcoming decade it is possible to arrive at definitive answers to the
following key questions: • How does matter behave in the centers of neutron
stars? • What are the physical properties of matter in the ultra-high density,
large proton/neutron number asymmetry, and low temperature region of the
quantum chromodynamics phase space? These can be achieved through
complementary measurements from electromagnetic (radio and X-ray) and
gravitational wave astrophysical observations of neutron stars, combined with
terrestrial laboratory constraints and further theoretical investigations.
This endeavor imposes the following requirements for facilities and resources
in the upcoming decade and beyond: • A next generation of gravitational wave
detectors to uncover more double neutron star and neutron star-black hole
mergers • Sensitive radio telescopes to find the most massive ($M>2$
$M_{\odot}$) and fastest spinning neutron stars • Large-area, high-time-
resolution and/or high angular resolution X-ray telescopes to constrain the
neutron star mass-radius relation • Suitable laboratory facilities for nuclear
physics experiments to constrain the dense matter equation of state • Funding
resources for theoretical studies of matter in this regime.  • The
availability of modern large-scale high performance computing infrastructure.
The same facilities and resources would also enable significant advances in
other high-profile fields of inquiry in modern physics such as the nature of
dark matter, alternative theories of gravity, nucleon superfluidity and
superconductivity, as well as a wide array of astrophysics, including but not
limited to stellar evolution, nucleosynthesis, and primordial black holes.

##  1 Introduction

Neutron stars (NSs) are the densest objects in the Universe outside of black
holes. Their density increases toward the center of the star reaching
densities around 5–10$n_{s}$, several times the nuclear saturation density
$n_{s}=0.16/$fm3. At these densities we currently do not know how matter
behaves, what the phase structure is or what the dynamic degrees of freedom
are. NSs offer a unique laboratory to study strongly interacting matter and
the underlying theory of Quantum Chromodynamics (QCD) in the most extreme
conditions. They have the potential to facilitate the discovery of novel
exotic phases of matter in their cores including the appearance of strangeness
in form hyperonic matter and ultimately the melting of nucleonic structure
giving rise to novel form of cold quark matter.

It is clear that understanding the structure of NSs and the state supranuclear
matter is of immense importance. In this White Paper, we outline the various
techniques that can be employed to this end, including electromagnetic
(primarily radio and X-ray) observations of Galactic NSs and gravitational
wave detections of merging NSs at cosmological distances, complementary
laboratory measurements of neutron-rich nuclei, and additional theoretical
investigations.

The White Paper is organized as follows. In Section 2, we summarize the
theoretical underpinnings of NSs and the dense matter EOS. In Section 3 we
describe terrestrial laboratory nuclear experiment approaches to constrain the
nuclear symmetry energy. The measurements based on radio and X-ray
observations of NSs are detailed in Sections 4 and 5, respectively, while in
Section 6, we describe the approach based on merger transient gravitational
wave and correlated electromagnetic observations. We offer conclusions and
recommendatations in Section 7.

##  2 Theoretical Equation of State

The nature of matter at ultra-high densities ($\rho_{s}>2.8\times 10^{14}$ g
cm-3), large proton/neutron number asymmetry, and low temperatures ($\lesssim
10^{10}~{}{\rm K}$) is, at present, one of the major outstanding problems in
modern physics, owing to a number of challenges both in the experimental and
theoretical realms, but a plethora of well-motivated theoretical predictions
for the state of matter in this temperature-density regime have been proposed,
ranging from normal nucleonic matter, to particle exotica such as hyperons,
deconfined quarks, color superconducting phases, and Bose-Einstein condensates
(for a review see [283]). Matter in this extreme regime is known to only exist
stably in the cores of NSs.

The structure of NSs is determined by the competition between self gravity and
pressure of strong nuclear interactions keeping the star in a hydrostatic
equilibrium. This interaction is described in the simplest case of non-
rotating NSs by the Tolman-Oppenheimer-Volkoff (TOV) equations [208, 273],
which map the equation of state (EOS) of dense nuclear matter to the
macroscopic properties of NSs, making the EOS the primary object of interest
for the nuclear physics of NSs. A large community effort has been put to
investigating the EOS using different _ab-initio_ calculations as well as
various models. The evidence from observations, in particular with the advent
of multimessenger astronomy, is more complicated than the simple static
systems described by the TOV equations and a significant push has been made in
the past years to numerically solve the combined Einstein and relativistic-
fluid-dynamic equations [244].

Figure 1 displays a schematic figure of the NS structure. The crust and the
outer core down to a depth of roughly $0.5$ km, where densities are of the
order of nuclear density, is under good theoretical control [45, 206]. Beyond
that, our understanding of the structure relies on theoretical extrapolations.
In particular, the phase of the inner core is currently unknown. Figure 2 is a
schematic view of the hypothesized phase diagram of QCD. It is a firm
prediction of QCD wherein, at sufficiently high temperatures and/or densities,
ordinary hadronic matter melts to a partonic form of matter—Quark Matter.

In the regime of high temperatures and low baryon densities, the deconfiement
transition to Quark Matter is well studied using lattice field theory [111,
26, 150] and its existence is confirmed in two decades of experimentation with
ultra-relativistic heavy-ion collisions [184] at the RHIC and the LHC. Further
experimental program runs at the LHC aim to quantify the transport properties
and the conditions of the onset of Quark Matter.

The deconfinement transition is a cross-over at low baryon densities but it
has been long hypothesized that the transition becomes stronger with
increasing baryon density. New theoretical arguments based on topological
features of QCD have been recently put forward supporting the first-order-
nature of the transition [73, 116]. The beam energy scan (BES) program at the
RHIC [10] and the future FAIR facility [7] are geared to discover the critical
point separating the crossover transition from the first-order transition. The
discovery of the critical point would have a profound impact on the physics of
NSs. Furthermore, heavy-ion collisions at beam energies up to 2 GeV per
nucleon directly constrain the EOS at the density range of 1-2 $\rho_{s}$
[145].

Figure 1: A cartoon depiction of a neutron star and its internal structure.
Credit: NASA/B. Link.

At very-high densities, owing to an attractive interaction between quarks in
QCD, it is expected that Quark Matter is in the form of a color
superconductor, fundamentally affecting the transport properties of Quark
Matter [18, 17]. Based on large-$N_{c}$ arguments, it has also been speculated
[186] that, at low temperatures, there may be a further intermediate phase
where the degrees of freedom inside the Fermi sea may be treated as quarks,
while confining forces remain important near the Fermi surface. Owing to the
similarities with both the hadronic and Quark Matter phases, this hypothetical
phase is dubbed the quarkyonic phase.

Figure 2: Schematic view of the QCD phase diagram. The figure highlights
regions probed by the RHIC, LHC, FAIR, and FRIB experiments, regions of
validity for lattice QCD and chiral EFT, and environments reached in neutron
stars, supernovae, and neutron star mergers. Abbreviations: EFT=effective
field theory, QCD=quantum chromodynamics. Figure adapted from [98].

###  2.1 Discovering Quark Matter Cores in Massive Neutron Stars

The study of the QCD phase diagram presents formidable challenges in the limit
of high baryon densities. On the theory side, lattice Monte-Carlo simulations
are hampered by the Sign Problem, and observational information is similarly
scarce due to ultrarelativistic heavy-ion experiments being limited to probing
the hot and relatively baryon-dilute Quark Matter. The possible existence of
the high-density Quark Matter inside massive NSs has been theorized a long
time ago, but due to a combination of theoretical and observational
challenges, this assertion has not been verified yet, but remains one of the
grand challenges of nuclear astrophysics. Recent years have witnessed
unprecedented progress in the astrophysical observations of NSs, ranging from
the famous observation of a gravitational-wave signal from a binary NS merger
and its electromagnetic counterparts to more traditional X-ray and radio
pulsar timing observations, providing insight into the macroscopic properties
of NSs, such as their masses and radii [211], as well as superfluidity,
superconductivity, thermal conductivity, and heat capacity of the matter in
their interiors. This progress has opened a new window to ultra-dense matter
and may even allow for a convincing detection of exotic phases of matter in
NSs in the near future. However, in order to fully exploit the recent and
upcoming observational data, a commensurate theoretical advances are also
required.

In the absence of a smoking-gun observation that could only be explained by
the presence of quark cores in NSs, a recent approach to the problem has been
to combine all available ab-initio theoretical results from nuclear and
particle theory to robust observational information on NSs in an effort to
model-independently constrain the properties of matter in the centers of NSs
of various masses. The EOS reflects the phase diagram of the matter in the
neutron star and a sufficiently accurate determination of the EOS may
facilitate the identification of transition (a phase transition or a cross-
over) from hadronic matter to deconfined Quark Matter. Certain features of the
EOS differ qualitatively between the hadronic phase characterised by the
confinement mass scale and the nearly conformal Quark Matter. Several such
features have been suggested as indicators between the hadronic and Quark
Matter phases, including the speed of sound, number of effective degrees of
freedom, as well as the interaction measure [23, 157, 115]. As shown in [23],
together with state-of-the-art observational data for the masses and tidal
deformabilities of NSs, current results for the EOS of nuclear and quark
matter suffice to provide very stringent limits to how QCD matter needs to
behave in order for quark cores to be absent even inside maximally massive
NSs. There are several physical quantities, the constraining of which would
play a major role in a future tightening of the current results, including:

  * •

EOS of nuclear matter at 1-2 nuclear saturation densities and of quark matter
at 20-40 saturation densities,

  * •

Speed of sound of dense nuclear matter beyond saturation density

  * •

Order of the deconfinement transition at small temperatures and high
densities, and the magnitude of the related latent heat,

  * •

Radii and tidal deformabilities of NSs with accurately known masses,

  * •

Resolved ringdown-phase observations of gravitational wave detections from NS
merger events, and

  * •

Electromagnetic observations of the NS merger remnants and the subsequent
kilonova explosions.

In light of recent advances in the field of nuclear astrophysics, it is
plausible that the eventual discovery of QM cores in massive NSs will not
occur via a single dramatic observation, but rather through multiple
theoretical and observational advances that work together in gradually
eliminating the possibility of all NSs being composed of nuclear matter. Given
the general importance of this question, improving the status of the
aforementioned related results is a top priority in the field, and will
hopefully receive considerable attention from all relevant research
communities.

###  2.2 Perturbative QCD

While the Sign Problem prevents non-perturbative calculations of the EOS, at
sufficiently high densities the EOS becomes accessible by _ab-initio_
calculations within perturbative QCD due the asymptotic freedom. The current
state of the art in these calculations is the partial Next-to-Next-to-Next
leading-order calculation at zero temperature, which becomes reliable at
densities around and above $n=40n_{s}$ [122]. Effects of temperature and
strange quark mass to the EoS have been discussed in [162, 163, 123]. While
the perturbative calculations become reliable only at densities which greatly
exceed the densities which are reached within neutron stars, it has been
recently demonstrated that the perturbative calculations constraint the EoS at
neutron star densities [158]. These constraints arise from the requirement
that the EoS is causal, mechanically stable, and thermodynamically consistent
at all densities, which, together with the known perturbative-QCD limit,
imposes restrictions on the possible behaviour of the EoS at intermediate
densities reached in NSs. Further improving the perturbative calculations will
make these constraints more stringent.

###  2.3 Nucleon Superfluidity

NSs are also critical testbeds for theoretical calculations of nucleon
superfluidity since there are many observable behaviors of NSs that are
understood to be due to the existence of superfluid neutrons and
superconducting protons in the NS crust and core (see Figure 1). For example,
evolution of the spin rate of pulsars gives strong evidence for superfluid
neutrons in the NS crust [22, 21, 179] and core [142]. Meanwhile, the observed
temperatures of NSs of different ages require nucleon superfluids to be
present in the core and even provide constraints on the critical temperature
for neutron superfluidity $T_{\rm cn}(\rho)$ and proton superconductivity
$T_{\rm cp}(\rho)$ [130, 216]. For example, the rapid cooling of the NS in the
Cassiopeia A supernova remnant necessitates a maximum $T_{\rm cn}=(5-10)\times
10^{8}$ K and $T_{\rm cp}>2\times 10^{9}$ K [217, 253, 252]. Furthermore, type
I versus type II superconductivity predicts decaying or long-lived magnetic
fields in NS cores [44] and thus has important implications for magnetic field
evolution in NSs, from when NSs are young, such as in magnetars, until they
are very old, like in millisecond pulsars. Therefore, continued astronomical
observations to monitor NSs will provide improved understanding of properties
of nucleon superfluidity. The interpretation of such observations would also
benefit from consistent calculations of $T_{\rm cn}$ and $T_{\rm cp}$ from EOS
models, such as those only just recently being done [20], as well as effects
such as entrainment [66, 281].

###  2.4 Dark Matter in Neutron Stars and Exotic Compact Objects

Despite extensive efforts using a wide array of direct and indirect detection
methods, the particle nature of dark matter (DM) remains elusive. Because of
their strong gravity and extreme core densities, NSs may accumulate DM from
their environment by capturing DM particles after they scatter off of nucleons
[88, 160]. Depending on the DM-nucleon interaction cross-section and the DM
particle mass, the DM could form a core [51, 74, 126] or become admixed with
the baryonic matter [174, 207]. DM could even be produced during the merger of
two NSs [94]. For certain DM models, a configuration of baryonic and non-
baryonic matter is gravitationally stable [220, 92, 227]. The DM concentration
is then sensitive to the NS’s age, temperature, mass and environment,
producing diversity in the EOS-dependent observables, like tidal
deformabilities, measured for the population. In other cases, DM may implode
the stars [58]. Alternatively, DM that is formed out of ultra-light scalar
field could agglomerate on its own, forming compact objects that mimic true
NSs and BHs [64, 249, 77, 96]. GW observations over the next decade,
especially in the 3G detector era, combined with EM searches will give access
to the full population of compact binary mergers, allowing us to search for
“smoking-gun” EOS variability due to DM. In addition, X-ray observations of
nearby NSs can be used to place constraints on the fraction of asymmetric DM
inside a NS [245]. Therefore, observational studies of NSs using the
techniques described below can, in principle, yield unique insight into the
properties of DM.

##  3 Connections with Nuclear Experiment and Theory

Constraints on the EOS from mass and radius observations of NSs can be
complemented with constraints on the nuclear symmetry energy obtained from
nuclear experiments and ab-initio neutron matter theory. It has been known for
some time that there is a high degree of correlation between NS radii and the
pressure of neutron star matter slightly above the nuclear saturation density.
The NS matter pressure at $n_{s}$ is nearly completely determined by the slope
$L$ of the symmetry energy at the same density [169], which is experimentally
probed by nuclear binding energies, dipole polarizabilities and neutron skin
thicknesses of neutron-rich nuclei, and theoretically calculated from neutron
matter studies.

The NS radius, in the absence of a strong first-order phase transition like
that possibly associated with the QCD transition, is expected to vary by just
a few tenths of a kilometer for NS masses in the range from $1.2M_{\odot}$ to
$1.8M_{\odot}$, so uncertainties in a mass measurement are relatively
unimportant. Because the NS radius is most highly correlated with the pressure
at about twice $n_{s}$, and $L$ is measured at $n_{s}$, other symmetry energy
parameters such as $S_{V}$, $K_{sym}$ and $Q_{sym}$ (which are related to the
NS matter energy and its second and third derivatives, respectively, at
$n_{s}$) can also be relevant. The symmetry parameters are extractable, to
varying degrees of precision, from the aforementioned nuclear experiments and
theory. As an example of the power of this correlation, the linear relation
$R/{\rm km}=(8.76\pm 0.37)+(L/19.6{\rm~{}MeV})$ [292] characterizes to 68%
confidence about 200 of the most commonly used Skyrme-type nuclear
interactions. It is likely that a more precise relation can be found when
higher-order symmetry parameter ($K_{sym},Q_{sym})$ information is included.
In general, nuclear data constrain linear relations among the symmetry
parameters with more precision than their individual values, but different
classes of experiments result in different correlations so that combining them
can result in more accurate constraints in the individual parameters.

Supporting current experimental estimates of the symmetry parameters are
inferences from neutron matter theory. Recently developed chiral effective
field theory allows a systematic expansion of nuclear forces at low energies
based on the symmetries of quantum chromodynamics (see [98] for a review). It
exploits the gap between the pion mass (the pseudo-Goldstone boson of chiral
symmetry-breaking) and the energy scale of short-range nuclear interactions
established from experimental phase shifts. It provides the only known
consistent framework for estimating energy uncertainties. These ab-initio
results for pure neutron matter have the least systematic uncertainties, at
present. The predicted energy and pressure can be combined with the binding
energy and saturation density of symmetric nuclear matter (well-determined
from nuclear mass fits) to find values of $S_{V}$ and $L$ that are in
remarkable agreement with existing information from fits to binding energies
of thousands of stable nuclei. The framework has also recently been used to
study the effect of finite temperatures to the EOS [156, 155].

In practice, predictions of NS radii from nuclear mass fits and neutron matter
theory are currently competitive with those from astrophysical observations.
Additional nuclear experiments, such as measurements of the skin thicknesses
and dipole polarizabilities of nuclei, can also impact this problem. Combined
with binding energy fits and neutron matter theory, they presently constrain
the symmetry parameters conservatively as $S_{V}=32\pm 2$ MeV and $L=50\pm 10$
MeV [168] (see also [103]), suggesting $R=11.3\pm 0.7$ km from the above
linear relation.

Neutron skin and dipole polarizability measurements of neutron-rich nuclei are
mostly sensitive to $L$, and until recently were largely consistent with
expectations from binding energy fits and neutron matter theory. But the small
value of neutron skins, of order 0.1–0.2 fm, have resulted in large
experimental uncertainties. Therefore, new experimental constraints from
parity-violating electron scattering measurements of the neutron skins of both
Ca48 and Pb208 from the CREX [218] and PREX [11] experiments, respectively,
have been particularly notable. Such experiments have been argued [272] to
have less systematic uncertainties than other types of measurements.
Interestingly, the PREX experiment produced a measurement about 2 standard
deviations larger than previous experimental results, while the CREX
experiment produced a measurement smaller than, but within 1 standard
deviation, of previous results. Taken together, these two experiments are
satisfied at the 90% confidence level by $L$ values in the narrow range
already predicted by binding energy fits and neutron matter theory, but the
current large experimental uncertainties demand additional efforts to resolve
this matter. In addition, a new technique using FRIB and similar facilities to
measure the proton radii of mirror nuclei such as Fe54 and Ni54 is an exciting
development. Proton radii can be measured with much higher precision than
neutron radii, and their differences, in the absence of well-understood
Coulomb effects, in mirror nuclei is equivalent to a neutron skin measurement
[15].

##  4 Constraints from Radio Observations of Neutron Stars

An important set of measurements in nuclear astrophysics comes from studying
the timing properties of radio pulsars, rotating NSs that emit highly beamed
radiation along their magnetic poles. An early example is the discovery and
analysis of the “Hulse-Taylor” radio pulsar-binary system that (indirectly)
confirmed of the existence of gravitational radiation [285, 266], and produced
mass estimates with high precision. The recent discoveries of additional
relativistic pulsar orbits and high-mass NSs, as well as the eventual
detection of nanohertz-frequency gravitational radiation through pulsar
timing, show that radio pulsars continue to serve as ideal laboratories for
fundamental physics.

###  4.1 Measuring Neutron Star Masses in Pulsar Binary Systems

Figure 3: The signature of Shapiro delay in PSR J0740+6620 as a function of
the pulsar’s binary orbital phase, with colored points denoting pulse arrival-
time residuals ($\mathcal{R}$), excess delay not accounted for by the timing
model, measured at three different radio frequencies [see 113, for details].
The top panel is the magnitude of the Shapiro delay if it is not included in
the timing model but with all other model parameters fixed at their best-fit
value. The middle panel shows the best-fit residuals obtained using an orbital
model that does not account for general-relativistic effects. The bottom panel
shows the fit with the full timing solution (including the relativistic
Shapiro delay parameters).

One of the most effective ways of obtaining NS mass measurements through
pulsar timing is by observing the relativistic Shapiro delay [250] in binary
orbits containing a pulsar. The Shapiro delay manifests as a small and
periodic delay in pulse arrival times (of order $\sim$10 $\mu$s) induced by
the spacetime curvature of a pulsar’s companion star. Measurement of the
Shapiro delay directly yields the mass of the companion star as well as the
geometry of the system; the NS mass is then determined by the Keplerian mass
function. The Shapiro delay can be measured in a pulsar-binary system of any
closed shape and size, so long as the companion mass and system orientation
lead to delay amplitudes greater than $\sim$10 $\mu$s (see Figure 3).

The Shapiro delay alone has provided some of the most impactful measurements
of high-mass NSs to date [93, 84]. While rare, observations of $>$2 M⊙ NSs
have greatly impacted our understanding of the NS interior EOS; for example,
the initial measurement of the mass of PSR J1614$-$2230 effectively ruled out
most contemporary non-baryonic EOS models (i.e., hyperons, kaons, Bose-
Einstein condensates, free-quark stars, etc.), though recent work argues that
the cores of maximally-massive NSs can be composed of quark-gluon matter under
certain conditions (e.g., see [23]). Large NS masses also constrain the
interaction between hadronic and strange-quark matter, and particular in phase
transitions between the two states [43]. On the other hand, measuring masses
and radii of the lowest-mass NSs can constrain nuclear symmetry-energy
parameters, at mean central densities close to the nuclear-saturation limit,
that have strong implications for unknown aspects of supernova physics and
other astrophysical phenomena [262].

Relativistic pulsar-binary systems, with orbital periods $\sim$ hours, are
comparatively rare but nonetheless considered as idealized astrophysical
environments due to the extreme gravitation in such systems. The discovery and
long-term radio timing of merging double-neutron-star (DNS) systems, such as
the Hulse-Taylor system, yield a variety of “post-Keplerian” (PK) parameters –
the Shapiro delay being an example of PK effects – that quantify corrections
to purely Newtonian motion beyond O($|v|^{2}/c^{2}$), where $|v|$ represents
the NS’s orbital speed. An ensemble of PK parameters measured in a single
pulsar-binary system can be used to test general relativity and directly
estimate the binary-component masses [161, 112, 183, 278].

A diverse set of tests for the presence of scalar fields as gravity mediators
[290, 202] are accessible when examining two/three-body systems contain
pulsars with white-dwarf companions and of various sizes [259, 121]. These
particular tests place stringent limits on tensor-scalar gravity theories,
where notable effects include temporal variation of Newton’s gravitational
constant, violation of Lorentz invariance through excess acceleration of
spinning, self-gravitating bodies within a preferred reference frame, and
violations of equivalence principles through differing accelerations felt by
bodies of different internal compositions and binding energies [293]. The
recent discovery and analysis of a pulsar in a three-body system with two
white dwarfs [236] have provided the strongest constraints of equivalence-
violation principles to date [28, 279]. Future discoveries and refined
measurements will continue to provide exceptional strong-field tests of
gravity.

###  4.2 Measurements of the Neutron Star Moment of Inertia

With sufficiently long timing baselines, PK effects in DNS systems at
O($|v|^{4}/c^{4}$) will become resolvable. An anticipated PK effect is apsidal
motion due to spin-orbit interaction of a pulsar with a NS companion [86];
this particular effect directly depends on the NS moment of inertia, which is
itself a quantity that depends on both the NS mass and radius. The radio-
timing measurement of the NS moment of inertia (e.g., [144]) will provide a
completely independent test for nuclear theory [195, 47, 109, 178], being able
to constrain the NS radius as well as nuclear-physics parameters at nuclear
saturation density [173, 235, 124], assuming general relativity is the correct
theory of gravity. Such constraints may also be able to probe the possible
existence of a hadron-quark phase transition at supranuclear densities, i.e.,
whether or not NSs may have a quark core [16]. An EOS-insensitive relation
subject only to the conditions of causality and the existence of a
$2M_{\odot}$ minimum to the NS maximum mass is $I=(0.87\pm
0.05)(GM/Rc^{2})^{1/2}MR^{2}$ [292].

In fact, the first meaningful constraint of an NS moment of inertia has
already been reported [168] for the $1.338M_{\odot}$ primary of the double
pulsar PSR J0737$-$3039, $I_{1.338}=64^{+65}_{-44}M_{\odot}$ km2 after
applying causality constraints. This suggests, from the above EOS-insensitive
relation, that $R_{1.338}=11.5^{+6.9}_{-6.3}$ km, consistent with expectations
but not yet of the necessary accuracy to improve existing constraints.

The exquisite timing in PSR J0737-3039 and other DNS systems permitting
resolution of PK effects also has important implications for current theories
of strong-field gravitation [161]. NS interiors are in both the strong-field
and high spacetime-curvature regime where gravity modifications may appear in
alternative theories and alter the interior structure. NS interiors thus
provide both a strong-field and high-curvature test of gravity complementary
to binary-pulsar tests. Given that the nuclear EOS is itself uncertain, this
presents a possible observational degeneracy when one allows gravity to
deviate from general relativity. A number of observable macroscopic NS
properties obey quasi-universal relations, such as the I-Love-Q relation which
is very weakly sensitive to the EOS [288], allow for testing gravity with NSs
in an EOS-independent manner, regardless of nuclear-physics uncertainties. The
I-Love-Q relation is an order of magnitude more precise than the EOS-
insensitive relation among I, M and R described above. For example, combining
independent measurements of the moment of inertia (from radio timing) and the
tidal deformability (from gravitational waves) can rule out alternative
gravity theories [254] as well as test quasi-universality [167].

##  5 Constraints from X-ray Observations of Neutron Stars

As noted previously, the mass-radius ($M$-$R$) relation of NSs has a strong
dependence on the EOS of the dense matter in their interior [see, e.g., 170,
171, 214, 238, 134]. Thus, precise mass and radius measurements of several NSs
using astrophysical observations can provide insight into the state of matter
in their cores (Figure 4).

Since it is not possible to directly sample the matter from the interior of
NSs, it is necessary to rely on indirect inference using sensitive
observations of their exteriors. This has motivated the development of a host
of observational and data modeling techniques for inferring the mass and
radius of NSs using electromagnetic observations primarily in the X-ray range
($\approx$0.1–30 keV) of the surface thermal radiation from NSs. These
approaches rely on the fact that the properties of the observed X-ray photons
are greatly affected by the immense gravity in the vicinity of the star [e.g.,
136, 189, 222, 55, and references therein], which in turn, is determined by
the stellar $M$ and $R$.

Figure 4: The particle content and their interactions in the high density –
low temperature setting at the cores of NSs is highly uncertain. Our lack of
knowledge about these microphysical aspects (lower left: uds = up down strange
quarks) is encapsulated in the EOS (top left). A sampling of plausible
theoretical EOS models is shown that includes a nucleonic star (red) [14], a
quark star (magenta) [176], a hybrid star consisting of a nucleonic outer core
and quark matter inner core (blue) [291], and a hyperon star with nucleonic
outer core and hyperonic matter inner core (black) [46]. The light blue region
represents the approximate range spanned by the set of currently viable models
[134]. The different EOS govern the global properties of the star such as $M$,
$R$ and oblateness for a given rotation rate, via their influence on stellar
structure (top right). These determine the exterior space-time properties of
the star, which measurably alter the properties of the radiation propagating
from the NS surface, encoding in it information about the EOS and the
associated microphysics. Figure adapted from [237].

###  5.1 X-ray Observational Techniques for EOS Constraints

Pulse profile modeling. For a spinning NS that emits X-rays from small regions
on its surface (commonly referred to as “hot spots”), its $R$ and $M$ can be
estimated by modeling the observed rotation-induced X-ray pulsations. This is
possible because the properties of the observed pulsed signal depend on $R$
and $M$ through the general and special relativistic effects on rotationally-
modulated emission from the hot spots. Three varieties of NSs with surface hot
spots suitable for such analyses are: rotation-powered millisecond pulsars,
accretion-powered millisecond pulsars, and thermonuclear burst oscillation
sources. The currently operating Neutron Star Interior Composition Explorer
(NICER; see [118]) NASA X-ray timing mission is focusing on producing $R$ and
$M$ measurement of a few radio millisecond pulsars that produce thermal
radiation [242, 188, 193, 243]. This pulse profile modeling technique (also
known as waveform or light curve modeling) is mature, having been studied
extensively over the last few decades [221, 190, 224, 223, 196, 34, 225, 19,
205]. Existing sophisticated models incorporate all practically important
physics (atmospheric radiation properties, gravitational light-bending,
Doppler boosting, aberration, propagation time delays, and the effects of
rotationally-induced stellar oblateness) with a very high degree of accuracy.
Interfacing these models with Baysian parameter estimation codes permits
inference of either exterior space-time parameters or EOS parameters directly
from the X-ray data [181, 191, 241, 228, 230]. Efforts for the NICER mission
have produced inference codes highly optimized specifically for this purpose.

Extremely rapid rotators. Discovering one or more NSs with spin rate
$\gtrsim$1 kHz can lead strong and clean (i.e., systematics-free) EOS
constraints. Each theoretical EOS has an associated prediction for the maximum
spin rate before break-up (e.g, [132]), so such NSs could, in principle, rule
out a swath of proposed EoS models. X-ray pulse timing offers a particularly
promising approach, since there is no evidence yet that the spin distribution
of accreting millisecond pulsars drops off at high spin rates as seen in the
radio pulsar sample [283]. Such rapidly spinning stars are likely to exhibit
short-lived, intermittent X-ray pulsations, requiring hard X-ray detectors
with high-time resolution and high instantaneous sensitivity.

Absorption line spectroscopy. Detection of photospheric absorption lines from
a NS, can, in principle, provide a measure of the stellar compactness $M/R$
[62] from their gravitational redshift. In addition, precise measurement of
the line shape may lead to constraints on $M$ and $R$ separately via line
broadening physics [215, 182, 67]. Absorption lines in NSs can arise from
either atomic transitions or magnetic effects [see, e.g. 209, for a review].
Magnetic absorption features rely on an independent determination of magnetic
field strength for redshift constraints. The most promising targets for atomic
line features are thermonuclear (type-I) X-ray bursters, which generate bright
episodic emission from the stellar surface, supply heavy elements to the
photosphere through accretion, and are not affected by broad cyclotron lines
owing to their weak magnetic field strengths [213].

Radius expansion bursts. Some thermonuclear bursts are powerful enough to
exceed the Eddington critical luminosity, inflating the photosphere above the
stellar surface [265, 175]. For sources with measured distances (e.g., those
situated in globular clusters), analysis of these photospheric radius
expansion (PRE) allows measurements of $M$ and $R$ separately [see, e.g.,
210]. However, because PRE events exhibit a decline in blackbody temperature,
most of the luminosity falls outside the X-ray energy coverage of most
previous X-ray missions, resulting in large uncertainties in the measured
Eddington luminosity [e.g., 164, 131]. In one approach, the Eddington flux is
measured at the moment when the photosphere falls back onto the stellar
surface as the burst cools [212, 261]. In the other, the evolution of the
spectrum of the cooling tail of the burst is examined [204]. The energy band
of the current NICER mission, which extends as low as 0.25 keV, is well suited
to such measurements. Future missions with similar low-energy coverage but
larger collecting area will be able to measure the photospheric evolution with
greatly increased fidelity. Aside from determining the touchdown point more
reliably, the evolution itself will provide additional determinations of $M$
and $R$, as both the Eddington limit and the gravitational redshift at each
radius depend upon these parameters [85].

Quiescent X-ray transients and cooling. For a NS radiating uniformly from the
entire surface, one can derive constraints on $M$ and $R$ from spectral fits
to its X-ray emission if the temperature, composition of the topmost layer of
the atmosphere, and its distance are known and the magnetic field is
sufficiently weak ($\ll 10^{10}$ G) so as not to affect the opacity or
temperature distribution on the NS surface. These criteria appear to be
satisfied in quiescent low-mass X-ray binaries (qLMXBs) containing NSs, in
particular, those located in globular clusters, to which the distances are
well-determined [246, 128, 54, 260]; plus additional targets in the field of
the Galaxy with reliable parallaxes obtained with GAIA. Due to source crowding
in the dense cores of globular clusters, effective studies of these targets
require soft X-ray telescopes with high-angular resolution ($\leq
1^{\prime\prime}$) imaging capabilities.

##  6 Merger Transient Gravitational Wave and Electromagnetic Observations

Motivation.

The Advanced LIGO [177] and Advanced Virgo [8] gravitational-wave (GW)
detectors’ discovery of a BNS merger in 2017, GW170817 [6], ushered in a new
era of multimessenger astronomy. Electromagnetic (EM) facilities observed a
coincident short gamma-ray burst [1, 120, 248] and subsequent kilonova [31,
27, 83, 180, 256, 264, 277], demonstrated that BNSs are prolific formation
sites for many of the heavy elements found in nature [154, 282]. through the
coordinated action of $\sim$ 70 Late-time near-infrared observations showed
BNS mergers to be sites of the rapid neutron-capture process (r-process), i.e.
nucleosynthesis producing many of the heavy elements found in nature [154,
282]. Analyses of the GW data provided limits on the size of NSs [5, 24, 40,
87, 199, 110, 233, 268, 63, 166, 95], and joint GW/EM analyses enabled an
independent measurement of cosmological parameters [2, 152, 79, 80, 95]. The
most recent LIGO-Virgo observing run produced another BNS merger [269] as well
as two NS-BH signals [269].

Figure 5: Numerical relativity simulation of a binary NS merger showing the GW
signal and the matter evolution. Top panel: The gravitational signal emitted
during the last orbits before the merger (late-inspiral phase) and during the
postmerger phase of the BNS coalescence. Bottom panels: Rest-mass density
evolution for the inspiral (left panel), the merger (middle panel) and the
postmerger phase after the formation of the black hole (right panel). Figure
adapted from [97].

The current network of LIGO, Virgo, and KAGRA [30] will continue commissioning
and observing runs through to c. 2028. The network expects to reach design
sensitivity in the coming observing runs, at which up to tens of BNS mergers
per year may be detected. Planned upgrades in c. 2025, including the addition
of a fifth observatory (LIGO-India [148]), will extend their reach for binary
NS detections from ${\sim}100$ Mpc (in 2020) to ${\sim}300$ Mpc [4]. Third
generation (3G) observatories, such as the Einstein Telescope [226, 138] and
Cosmic Explorer [185, 239], currently in the design stage, may observe BNSs
and BHNSs out to the era of peak star formation in our Universe. Proposed
dedicated high-frequency GW detectors [185, 9] may target the merger and post-
merger phases of NS coalescence.

At the same time, EM surveys such as the Vera C. Rubin Observatory’s (VRO)
Legacy Survey of Space and Time (LSST) [147] and the Dark Energy Spectroscopic
Instrument (DESI) [13], as well as the advent of 30 m class telescopes, will
push the boundaries of depth and cadence. The combined operation of GW and EM
observatories in the next decade will provide a unique opportunity to resolve
long-standing questions such as the nature of the NS EOS including possible
phase transitions to exotic matter, an independent measurement of the
expansion rate of the Universe, and a possibility to reveal the nature of dark
matter (DM). There is also the prospect of detecting long-duration GWs from
single NSs; such GWs could be produced by a quadrupole moment or fluid
oscillation and would provide unique insights into dense nuclear matter such
as its elasticity and transport properties and exotic phases like hyperons and
quarks [119, 240]

Dense matter in NS interiors. GW and EM observations constraining NS
properties are sensitive to the dense matter EOS due to the strong connections
between astrophysical observables and microphysical interactions. The presence
of a NS imprints on to gravitational-wave inspiral signal information about
cold, dense matter up to a few times nuclear saturation density. The post-
merger gravitational wave signal, on the other hand, contains information
about hot dense matter EOS at even greater densities. Analyses of GW170817
already provide stringent constraints of the radius of a typical
$1.4M_{\odot}$ NS [5, 40, 87, 199, 110, 233, 268, 63, 166, 229, 95]. Several
of these estimates included multi-messenger input, combining GW and EM
observations of BNS mergers, radio observations of massive pulsars [25, 29,
84], X-ray observations by NASA’s Neutron Star Interior Composition Explorer
(NICER) mission [188, 242], and nuclear-physics constraints [135, 268] (see
also, e.g., [32, 127, 234, 101, 232, 97, 49, 68, 102]).

Figure 6: Left: Simulated GW observations of neutron-star mergers of 5
moderately loud signals (signal to noise ratio $>13$) with LIGO and Virgo
(H1L1V1), randomly drawn from component masses between 1 and 2 $M_{\odot}$.
Merger waveform dependence produce posterior distributions encompassing 90%
probability around the assumed $M$-$\Lambda$ relation (blue line), shown here
with mass ratio ($q$) dependence suppressed. The recovery is compared to the
same signals recovered using Cosmic Explorer and the Einstein Telescope
(CE/ET). CE/ET would also detect many additional signals in the same time
period. Details will appear in [286]. Right: Mass-radius curves for three
different equations of state along with the accuracy with which neutron star
radii could be measured for 30 loudest events using Fisher matrix
approximation [56] in a network consisting of two Cosmic Explorer detectors
and an Einstein Telescope [151, 146]. Errors in mass measurement are much
smaller, hence are not visible in the plot.

In the next decade, stronger constraints from combining information from
multiple events [166, 57, 289] will enable percent-level measurements of the
NS radius and the fundamental interactions of dense neutron-rich matter [90,
71, 165, 137, 97, 38, 39, 41, 75, 76, 275, 60, 280, 99, 102, 194]. At this
stage, systematic uncertainties due to modeling, which are currently
negligible for individual events, are likely to lead to biases in the final
result. In the era of CE-ET, these systematics will be larger than the
statistical errors coming from the noise in the detectors. Hence, modeling the
systematics bias is required to constrain the precise model EOS [151]. On the
left panel, Fig. 6, improvements in the measurement of chirp mass and
effective tidal deformability are shown for five events. Such improvements
will lead to highly accurate measurements of NS radii. In the right panel of
Fig. 6, we show 30 simulated events for three EOSs (ALF2, APR3 and APR4) with
highest SNR observed in the detector combination with two cosmic explorers and
one Einstein Telescope. Both radius and mass errors are shown for each circle
which depend on the EOS used for the simulation.

It will be increasingly important to control systematic errors from prior
assumptions about the EOS and the mass distribution of NSs in merging binaries
[287], emphasizing the need to understand the limitations of EOS models and
the population of sources as a whole. EOS-insensitive (quasi-universal)
relations can be tested with GW observations [247] and applied to improve the
accuracy of source properties [65]. Nuclear physics may imprint on the GW
source mass distribution itself [106]. Precise knowledge of the EOS or the
mass distribution, particularly any sharp features therein, may enable GW
observations alone to constrain the local expansion rate of the universe [187,
267, 107, 91] and directly calibrate standard candles such as type Ia SNe
[129]. Going beyond the equilibrium EOS, future BNS signals may reveal
dynamical processes, e.g. stellar oscillations, that probe transport
properties of normal matter or exotica such as boson condensates, hyperons,
and quarks [139, 119, 141, 127].

New regions of the QCD phase space, black hole formation. Both density and
temperature increase dramatically after two NSs collide. Therefore, observing
merging NSs enables us to probe not only the low-temperature QCD phase
diagram, but provides a window on the most extreme conditions in the universe,
which will shed additional light on the existence of phase transitions or
exotic forms of matter. While these transitions or the breakdown of purely
hadronic models could be detected during the inspiral phase under certain
conditions [70, 52, 219, 102], their existence may be more apparent during the
post-merger phase when densities are higher [231, 35, 89, 133, 198, 284, 197].
The required sensitivity to detect the post-merger GW signal from a
GW170817-like event is likely to be achieved with future GW detector upgrades
[76, 275], and steadier rates are expected with next-generation
facilities[258]. Stacking data from multiple distant events may first reveal
the post-merger [57, 289]. Increasing detector sensitivity allows us to
observe post-merger GW signatures and determining the threshold mass for
prompt collapse of the merger remnant to a BH will provide stringent
constraints on the EOS and verify possible strong phase transitions in the NS
EOS [12, 36, 40, 42, 153]. In addition, knowledge of the exact threshold mass
can inform future EM counterpart searches, as systems undergoing prompt
collapse are most often not connected to bright, detectable EM signals, e.g.,
[81, 82], and can be used to distinguish between different types of compact
binary progenitors [100], possibly resolving long-standing issues associated
with the lower “mass gap” between observed NSs and BHs, as well as
uncertainties in the core-collapse supernova (CCSN) explosion mechanism (e.g.,
[114, 48]) and the nucleosythesis therein.

Additionally, the precise value of the threshold mass to prompt collapse is
found to be proportional to the maximum mass that a particular EOS supports
[251, 143]. The correlations found in numerical relativity simulations are
useful in constraining the properties of EOS such as radius of 1.4 M⊙, 1.6 M⊙
and maximum mass NS [36, 40, 159, 37, 274]. New correlations have been found
using a large collection of EOSs, where classification of events as either
prompt or delayed collapse will also help us in placing upper and lower limits
on the maximum mass of NSs as well as the radius of 1.4 $M_{\odot}$, 1.6
$M_{\odot}$ and maximum-mass NSs [153] (see Fig. 7 for GW170817 and a few
hypothetical BNS merger events with their assumed outcomes). Most importantly,
accurate modeling of tidal effects during the inspiral phase as well as
greater coverage of the parameter space (e.g., total mass, mass ratio and
longer evolution time) for accurate post-merger will put heavy demand on
computational resources. give upper limits on the radius and maximum mass of
neutron stars but also revealed new constraints on radius and tidal
deformability of a 1.4 M⊙ NS [153].

Figure 7: Constraints from GW170817 along with a few hypothetical scenarios
expected in next generation GW detectors for constraining upper and lower
limit of maximum mass and radius of maximum mass NS. Similar plots exist for
radii of 1.4 M⊙ and 1.6 M⊙ NSs. Figure adapted from [153].

Post-merger signals from BNS mergers will become visible in 3G detectors. In
addition to the properties of hot EOS, QCD phase transition leaves subtle
imprints on the post-merger signal such as a shift in the frequency of peak
frequency $f_{2}$ in the post-merger gravitational waves [52]. New
correlations have been observed between $f_{2}$, effective tidal deformability
$\tilde{\Lambda}$ and symmetric mass ratio $\eta$ [50]. QCD phase transition
could break such correlations and hence an opportunity of its detection arises
in BNS mergers that undergo QCD phase transition. However, to infer the full
nature of the post-merger signal, one must develop an accurate waveform model
that could be used in Bayesian inference. Several attempts have been made to
develop such a model [76, 69, 276, 99, 257]. More recently, new post-merger
waveform models have been developed using numerical relativity simulations in
the parameter space of total mass, $M\in[2.4,3.4]M_{\odot}$, mass ratio, $q<2$
and non-precessing dimensionless spin up to 0.2, that have faithfulness high
enough to detect the phase transition in ET [60, 59, 61].

Matter outflows and r-process nucleosynthesis. Postmerger GW and EM
observations trace the dynamic aftermath in the extreme environments created
by compact binary mergers. Folding in EM observations provides an opportunity
to independently constrain the source properties and to obtain a better
understanding of the physical processes and outcomes of mergers [78, 95]. The
successful search and detection of the _kilonova_ associated with GW170817
revealed NS mergers as a critical site of rapid neutron-capture process
(r-process) nucleosynthesis. Combined with the inferred BNS merger rate [3,
269] and detailed nucleosynthesis simulations, future multimessenger
observations of NS mergers will elucidate the processes that make the heavy
elements [172, 270]. An interdisciplinary effort implicating GW and EM
observations, numerical relativity (NR) simulations, and nuclear theory
calculations will enable detailed predictions of the abundances of individual
elements. In the future, we expect that wide field-of-view, optical survey
instruments are most likely to find fast-fading kilonovae. While today’s
searches for these counterparts are difficult due to their large sky
localizations [105, 255, 104], 3G detectors will significantly improve the
localizability. Information from the inspiral GWs together with NR simulations
can inform EM follow-up campaigns by making robust predictions about potential
matter outflows and light curves [82, 33]. Together, GW and EM observations
may reveal exotic components like primordial black holes [72, 108, 140, 117,
263].

##  7 Recommendations and Conclusions

###  7.1 Requirements for the Coming Decade

Radio Timing of Pulsars. Until recently, most radio observatories observed
only a small fraction of the known pulsar population due to limited and
competitive resources. The recent operation of wide-field interferometers,
such as the UTMOST and CHIME telescopes, are allowing for high-cadence
observations of over one thousand pulsars on a near-daily basis. These
observations allow for continued monitoring and improved measurements of
relativistic deviations in pulsar-binary orbits, yielding the potential for
additional mass and geometric measurements in the near future.

Figure 8: Simulated constraints on the EOS (pressure-density) and $M-R$ relation based on 15 expected measurements with STROBE-X. X-ray pulse profile modeling produces $M-R$ posteriors with $\pm 5$% accuracy (dashed ellipses) scattered around an underlying $M-R$ relation (black line). The corresponding EOS is shown on the left hand panel, with the dashed lines indicating the range permitted by current models. A piecewise-polytropic parameterization is assumed for the EOS with fixed transition densities. Following the procedure outlined in [125], the 1$\sigma$ constraints resulting from inference using these 15 measurements are shown by the blue band. The orange band shows the results for a choice of 6 of these stars, based on ensuring an even spread over the mass range $1.2-2.0$ M⊙, for deeper observations that result in $\pm 2$% accuracy posteriors (not shown). See [237] for further details. Technique | NS source class | Observatories
---|---|---
Pulse profile modeling | rotation-powered MSPs |  STROBE-X, Athena
| accretion-powered MSPs | STROBE-X
| bursting NS LMXBs | STROBE-X
Extremely rapid rotation | accretion-powered MSPs | STROBE-X
Radius expansion bursts | bursting NS LMXBs | STROBE-X
Absorption line spectroscopy | bursting NS LMXBs |  Athena, Lynx
Continuum spectroscopy | quiescent NS LMXBs |
Continuum spectroscopy | young cooling NSs |  AXIS, Athena, Lynx
Table 1: Summary of measurement techniques and NS source classes suitable for
dense matter EOS constraints and the proposed X-ray observatories that have
the capabilities to carry out these investigations.

Forthcoming radio interferometers will provide greater sensitivity due to
enlarged scales and frequency coverage. Recently, [113] argued that timing of
the high-mass PSR J0740+6620 with future observatories like the CHORD and
DSA-2000 interferometers will allow for order-of-magnitude improvements on the
pulsar mass precision, and likely allow for the measurement of apsidal motion
in the system with several years of timing data. Once fully operational, the
Square Kilometer Array will be able to detect the Shapiro time delay in at
least 80% of all known pulsar-binary systems with companion masses $\geq$ 0.1
M⊙. The discovery space therefore remains open for exploration in the radio
regime.

X-ray Observatories in the 2020s and Beyond. Space-borne observations at X-ray
energies offer various means for obtaining strong constraints on the allowed
dense matter EOS, providing unique insight into the low temperature-high
density region of the QCD phase diagram. While current telescopes have made
important headway, they lack the required capabilities to fully exploit the
information about the dense matter EOS encoded in the observed X-ray emission
from NSs. This important undertaking requires a new generation of X-ray
facilities with at least an order-of-magnitude improvement in sensitivity
relative to current observatories, while also offering high time resolution
required for effective studies of rapidly spinning NSs.

The Spectroscopic Time-Resolving Observatory for Broadband Energy X-rays
(STROBE-X) is a proposed probe-class X-ray (0.2–30 keV) timing and
spectroscopy mission, with a 2–5 m2 collecting area, high spectral and
temporal resolution, and rapid slew capabilities [237]. STROBE-X would enable
several observational techniques to study the EOS [see Table 1 and 283, for a
review]. The pulse profile modelling that can be done at present for a few
pulsars with NICER (sufficient to provide a proof of concept, [53]) would be
possible for $\sim$20 pulsars with STROBE-X (Figure 8), with more source
classes becoming accessible through the hard X-ray band not presently covered.

The Advanced X-ray Imaging Satellite (AXIS) is a proposed probe-class soft
X-ray (0.2–10 keV) mission, with $\sim$$1^{\prime\prime}$ angular resolution
imaging capabilities and a collecting area of $\approx 5,000$ cm2 around 1 keV
[201], with possible deployment in the late 2020s or early 2030s. These design
characteristics are optimal for spectroscopic studies of NS in crowded
regions, such as sources in Galactic globular clusters and those embedded in
supernova remnants and pulsar wind nebulae.

The Advanced Telescope for High-ENergy Astrophysics (Athena) is the European
Space Agency’s next-generation flagship X-ray observatory, with key hardware
contributions from NASA, slated for launch in the early 2030s [203]. Its high
sensitivity, high spectral resolution, and high count rate capability will
enable studies of NSs where greatly improved spectroscopic sensitivity is
desirable, such as for PRE burst systems and qLMXBs [200].

The Lynx X-ray Observatory [271] is a concept soft X-ray facility for possible
selection as a NASA Large Strategic Science Mission with a target launch date
in the mid-2030s. It will have a novel combination of $\approx$2 m2 collecting
area around 1 keV, high spectral resolution (through the use of a
microcalorimeter and dispersion gratings), and grazing-incidence focusing
optics that enable sub-arcsecond imaging. With these performance
characteristics, Lynx would be able to produce potentially strong NS EOS
constraints through high fidelity spectra of both quiescent and bursting NS
LMXBs, especially those in the dense confines of globular clusters and the
Galactic center where such systems are overabundant.

Table 1 summarizes the various techniques and NS source classes that can be
employed to provide constraints on the dense matter EOS using X-ray
observations, and the future planned observatories with the capabilities to
accomplish the desired measurements. The ability to target multiple NS
varieties is crucially important, as it enables verification of the
measurement techniques, allowing characterization and mitigation of systematic
errors. For instance, a number of NSs exhibit both accretion-powered
pulsations and thermonuclear burst oscillations, permitting pulse profile
modelling for the same source using two different types of hot spot.
Additionally, for the bursting sources, conducting spectroscopic modeling of
the burst cooling tail can offer additional cross-checks of techniques [see,
e.g. 204]. By targeting more NSs, it will be possible to sample the EOS across
a wider range of core densities. This will map the EOS more fully, probing any
potential phase transitions with finer resolution, and will move us out of the
regime where EOS model parameter inference may be prior-dominated [see for
example 125].

Double neutron star mergers. We call upon the community to build the
multidisciplinary ingredients necessary for multi-messenger astronomy of
compact binary systems. This includes the installation of highly-sensitive
observatories, such as 3G GW detectors and the next generation of EM
telescopes, and the development of an efficient, reliable, and
interdisciplinary hierarchical Bayesian framework for the interpretation of
the growing number of upcoming multi-messenger sources. Such a framework must
account for selection effects within both GW and EM observations while
considering all aspects of the source population simultaneously, including the
fundamental aspects of strong interactions. By reliably extracting source
properties (masses, radii, dynamical states, magnetic field structures, ejecta
properties, etc.), this framework will enable reliable measurements of the
dense matter EOS, the QCD phase diagram, and r-process nucleosynthesis.
Ultimately, future observations of these astronomical systems at the extremes
of gravity and density have the potential to reveal physics beyond the
Standard Model. Investment in large scale computing is required to
sufficiently cover the parameter space and accurate modelling of inspiral and
post-merger waveform from BNS mergers.

In addition to the necessity for the aforementioned multi-messenger
observational facilities, ample funding resources for further theoretical and
terrestrial laboratory studies of matter in this regime are highly desirable.
Finally, for all measurement techniques mentioned, the availability of state-
of-the-art large-scale high performance computing resources is of essence.

Collectively, electromagnetic and gravitational wave astrophysical
measurements combined with terrestrial laboratory constraints, hold the
promise to deliver definitive empirical constraints on the true nature of the
densest matter in the Universe and the high-density, low-temperature region of
the quantum chromodynamics phase space [192, 229, 149, 166], potentially
within a century of James Chadwick’s discovery of the neutron. Furthermore,
these efforts can, in principle, provide unique insight into other high-
profile topics in modern physics such as the nature of dark matter,
alternative theories of gravity, nucleon superfluidity and superconductivity,
as well as a wide array of astrophysics, including r-process nucleosynthesis,
primordial black holes, and stellar evolution.

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