# Astrophysics in the XXI Century with Compact Stars

Andreas Bauswein1 David B. Blaschke2 and Tobias Fischer2 1 GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 2 Institute
of Theoretical Physics, University of Wroclaw, Poland

##  Chapter 1 Effects of a strong phase transition on supernova explosions,
compact stars and their mergers

### Abstract

We outline a theoretical approach supporting strong phase transitions from
normal nuclear matter to the deconfined quark-gluon plasma, in the equation of
state (EOS) for compact star matter. Implications of this hypothesis are
discussed for astrophysical applications. Special emphasis is devoted to
potentially detectable signatures, which can be directly related with the
occurrence of a sufficiently strong phase transition. Therefore, simulations
of core-collapse supernovae and binary compact star mergers are considered,
including the subsequent emission of gravitational waves and, in the case of
supernova, in addition the neutrinos play the role of messengers.

###  1 Introduction

The state of matter at the interior of core-collapse supernovae and binary
neutron star mergers can reach extreme conditions, in terms of temperatures up
to several $10^{11}$ K 111$1.16\times 10^{10}~{}\rm K\simeq 1$ MeV, baryon
densities in excess of normal nuclear saturation density and large isospin
asymmetry. The latter is given by the proton fraction (in general charge
fraction) $Y_{p}$, where $Y_{p}=0$ for neutron matter (no net charges are
present) and $Y_{p}=0.5$ for isospin symmetric matter. The EOS for simulations
of supernovae and binary neutron star mergers must hence cover such an
extended three-dimensional domain, where presently first-principle nuclear
matter calculations are not available. Instead, model EOS are being developed
for supernova studies or simulations of neutron star mergers (for a
comprehensive review about the EOS in supernova studies, c.f. Fischer et al.,
2017). These combine several domains with different degrees of freedom, e.g.,
heavy nuclei at low temperatures, inhomogeneous nuclear matter composed of
light and heavy nuclei together with unbound nucleons, and homogeneous matter
at high temperatures and densities. In this chapter we reflect on the role of
the EOS in core-collapse supernovae and during binary neutron star mergers,
with particular focus on the recent finding about the presence of a phase
transition from normal hadronic matter to the quark-gluon plasma.

In the case of core-collapse supernova studies, accurate neutrino transport is
essential for the prediction of the neutrino signal from these explosive
events, as was observed from SN1987A (cf. Bionta et al., 1987; Hirata et al.,
1988). This observation is until now a benchmark. It confirmed that neutrinos
from the next galactic event will become the observable signal, from which we
will be able to learn not only details about these stellar explosions but also
about the state of matter at the supernova interior. In addition to neutrinos,
also gravitational waves are emitted from the interior of core-collapse
supernovae. The classical emission is associated with signal that originates
from the core bounce of moderately up to rapidly rotating stellar cores
collapse (c.f. Ott et al., 2004, 2006; Kotake et al., 2006; Dimmelmeier et
al., 2007, 2008; Scheidegger et al., 2008), which, however, seems difficult to
be detectable. In general, the SN GW signal does not obey any template
character. For example, it depends sensitively on the details of the radiation
hydrodynamics models used to provide GW signals form simulations and on the
EOS, as was demonstrated by Scheidegger et al. (2010). An alternative idea for
the detection prospects of gravitational wave from the long-term post-bounce
evolution has been proposed by Murphy et al. (2009). The foundation of this is
the excitation of a few dominant fluid modes, which, in turn, give rise to the
emission of GWs. This idea has been further explored and a connection could be
established between the neutrino and GW emission characteristics using
sophisticated general relativistic three-dimensional SN simulations by Kuroda
et al. (2017). Nevertheless, the lack of any SN GW measurement, for any of the
observing runs of aLIGO/VIRGO, demonstrate the complication of the problem and
point to the need of improvements, both, of the GW detectors in the kHZ range
as well as on the data analysis.

Gravitational waves are also key observable from binary neutron star mergers.
Since the equation of state of neutron star matter determines the stellar
structure and the dynamics of a neutron star merger, gravitational waves carry
information about the properties of high-density matter. This in particular
includes the prospect to learn about the presence of deconfined quark matter
in compact stars from the observations of gravitational waves produced by
neutron star mergers.

While galactic supernova explosions and neutron star mergers provide the
ultimate sites for testing the location and characteristics of the
deconfinement transition in the QCD phase diagram at high baryon densities and
low temperatures, inaccessible to heavy-ion collision experiments, they are
very rare events. Since SN1987A occurred we have not observed a supernova
neutrino signal and since GW170817, there was no binary neutron star merger
detected which would allow the extraction of the neutron star compactness from
the measurement of its tidal deformability. Up to now, we have no detection of
the postmerger gravitational wave signal. Therefore, compact stars which are
observed as pulsars in different astrophysical environments, remain excellent
tools to calibrate our knowledge of the dense matter equation of state albeit
at zero temperature.

We note that the ab-initio simulations of lattice QCD cannot make at present
any predictions for the thermodynamics in the QCD phase diagram at finite and
even high baryon density, due to the sign problem; see Muroya et al. (2003)
for an introductory review on this issue. Present techniques of lattice
quantum field theory for evaluating the QCD partition function are applicable
only for low baryochemical potentials $\mu_{B}/T\lesssim 2$. It is found that
the QCD transition is a chiral crossover with a pseudo-critical temperature
$T_{c}(0)=156.5\pm 1.5$ MeV at $\mu_{B}=0$ Bazavov et al. (2019). There is no
indication for a critical endpoint (CEP) for $\mu_{B}/T\lesssim 2$ and
$T>0.9~{}T_{c}(0)$ Bazavov et al. (2017). From recent lattice QCD simulations
at very low quark masses and their extrapolation to the chiral limit it has
been concluded that the chiral phase transition temperature is
$T_{c}^{(0)}=132^{+3}_{-6}$ MeV. For the temperature of the CEP (if that
exists at all) has to hold $T_{\rm CEP}<T_{c}^{(0)}$ Ding et al. (2019).
At low baryon chemical potentials and high temperatures the QCD EOS measured
in lattice QCD simulations is well confirmed by a Bayesian analysis of high-
statistics data obtained by heavy-ion collision (HIC) experiments at the
Relativistic Heavy-Ion Collider and at the Large Hadron Collider Pratt et al.
(2015). It is the hope that in systematic beam energy scan programs of HIC
experiments at lower collision energies ($\sqrt{s_{NN}}=3\dots 11$ GeV being
the relevant energy range in the nucleon-nucleon center of mass system of the
collision), similar analyses of the high-statistics data would reveal
indications of the CEP as a landmark in the QCD phase diagram. This hope,
however, has not been fulfilled yet. For a recent review, see An et al.
(2022). The problem has been clearly elucidated in the work by Senger (2021),
where he shows that when the temperature of the chemical freezeout (dilepton
spectrum) should fulfill the above lattice QCD constraint for $T_{\rm CEP}$,
the collision energy has to be below $\sqrt{s_{NN}}=6(3)$ GeV. In order to
reach the mixed phase of a first-order phase transition with the dynamical
trajectories of HIC simulations in the QCD phase diagram, a collision energy
of at least $\sqrt{s_{NN}}=5$ GeV is required Arsene et al. (2007). Following
this line of argument, even if a CEP exists, its temperature is too low to be
reached in HIC that could probe the QCD phase transition.

Therefore, in order to investigate the hadron-to-quark matter phase transition
at low temperatures, one has to rely on observations of compact stars in their
multi-messenger astrophysical context: as proto-neutron stars in supernova
explosions, as pulsars in isolation and in binaries as well as in their final
stages of their life, in the inspiral, merger and postmerger phases of binary
neutron star mergers.

In this contribution, we review several, recent aspects of the role that a
strong QCD phase transition can play in these three. We discuss the birth of
compact stars in core-collapse supernova events as well as the astrophysics of
binary neutron star merger events.

###  2 Hybrid neutron star formation in core-collapse supernova explosions

Proto-neutron stars (PNS) are the central, compact objects of core-collapse
supernovae. They are the transitional state of the initial collapse of the
stellar core of massive stars and the remnant neutron star. The stellar core
collapse is triggered due to the photodisintegration of heavy, iron-group
nuclei and due to electron captures on protons bound in these nuclei.
Consequently, the density and the temperature increase continuously during the
collapsing stellar core, which simultaneously deleptonises. When nuclear
saturation density is reached in the center, the short range repulsive nuclear
force balances gravity such that the core bounces back with the formation of a
hydrodynamics shock wave. The latter propagates quickly out of the stellar
core and stalls at a radius of around 100–200 km due to the continuous
photodisintegration of infalling heavy nuclei from above and the launch of the
$\nu_{e}$-burst associated with the shock propagation across the neutrino
sphere of last scattering. As a result, the dynamic shock turns into an
accretion front.

The supernova explosions are related with the revival of the stalled bounce
shock wave and the subsequent ejection of the stellar mantle that surrounds
the PNS. It has long been considered via the liberation of energy from the PNS
interior to a thin layer of accumulated material at the PNS surface (for
reviews about core collapse supernovae, see Janka et al., 2007; Mirizzi et
al., 2016). Several scenarios have been explored in the literature. The
magneto-rotational mechanism of LeBlanc & Wilson (1970) (for recent works,
c.f., Takiwaki et al., 2009; Winteler et al., 2012; Mösta et al., 2014b; Mösta
et al., 2015b), considers the required energy through the winding up of the
magentic field stress in the presence of initially high magnetic field and
rapidly rotating stellar cores. Another scenario is the neutrino-heating
mechanism of Bethe & Wilson (1985). With angle and energy dependent neutrino
transport methods once can demonstrate the development of a gain region, below
which neutrinos dominantly decouple from matter and hence establish net
cooling, while above neutrinos are still dominantly trapped and hence net
heating is established. This mechanism has been demonstrated to lead to
supernova explosions for a variety of massive progenitor stars (cf. Müller et
al., 2012; Takiwaki et al., 2012; Bruenn et al., 2013; Melson et al., 2015;
Lentz et al., 2015). However, in the framework of multi-dimensional
simulations accurate Boltzmann neutrino transport cannot be employed due to
the current computational limitations. Instead, approximate neutrino transport
schemes are commonly used, whose range of applicability is being debated
currently (cf. Nagakura et al., 2018).

####  2.1 Simulations of core-collapse supernovae with hadron-quark matter
hybrid EOS

Besides the aforementioned scenarios for the onset of the supernova explosion,
another mechanism was discovered by Sagert et al. (2009), which is due to the
occurance of a phase transition at high densities from hadronic matter to the
quark gluon plasma. The latter was being treated within the simple but
powerful bag model. During the phase transition a large amount of latent heat
is released in a highly dynamical fashion, which in turn triggers the onset of
the supernova explosion even in spherically symmetric simulations. Moreover,
it leaves an observable millisecond burst in the neutrino signal, as was first
demonstrated by Dasgupta et al. (2010) for the currently operating Super-
Kamiokande neutrino detector. These milestones demonstrate the sensitivity of
EOS uncertainties related to our understanding of core-collapse supernovae and
underline the need of a more elaborate understanding of the EOS at high
densities including better constraints in particular.

(a) $T=0$

(b) $s=3~{}k_{\rm B}$

Figure 1: (Figure adopted from Fischer, 2021) Equation of state assuming
$\beta$-equilibrium, showing the total pressure, $P$, the charge fraction,
$Y_{p}$ and the temperature $T$, as a function of the baryon density, $\rho$,
as well as the corresponding mass-radius relations, $M$–$R$, comparing the RDF
1.2 hadron-quark hybrid model (solid grey lines) and the DD2F reference
hadronic model (red dashed lines).  Figure 2:  (Figure adopted from Fischer,
2021) Supernova temperature–density region occupied in simulations that
feature a hadron-quark phase transition, where the color-coding is due to the
electron abundance. Shown are also the hadron-quark matter phase boundaries
for the onset of quark matter, $\rho_{\rm onset}$, and for reaching the pure
quark-matter phase, $\rho_{\rm final}$, for the DD2F-RDF 1.2 hybrid EOS, as
well as curves of constant entropy (green solid lines) together with the
central supernova trajectory (thick solid black line).  Figure 3:  (Figure
adopted from Fischer et al., 2018) Post-bounce evolution of the supernova
launched from a 50 M⊙ progenitor where the DD2F-RDF 1.2 hybrid EOS has been
employed with the phase transition at about 1.2 s, where the entropy per
particle being the color-code. Shown are the PNS radius, the bounce shock and
the second shock, which forms as a direct consequence of the first-order
hadron-quark phase transition. Figure 4:  (Figure adopted from Fischer et al.,
2018) Post-bounce supernova evolution of a 50 M⊙ progenitor star that takes
into account the hadron-quark matter phase transition, which occurs in this
model after at about 1.2 s. The figure shows the explosion energy, $E_{\rm
expl}$, the mass enclosed inside the PNS, $M_{\rm PNS}$, as well as the quark-
matter core, $M_{\rm quark}$, and the central density, $\rho_{\rm central}$.

For example, there is one problem concerning the hybrid EOS used in Sagert et
al. (2009), namely that it does not fulfill the constraint which arises from
the observations of pulsars with masses as high as 2 M⊙, demonstrated, e.g.,
by Antoniadis et al. (2013); Fonseca et al. (2016); Cromartie et al. (2020);
Fonseca et al. (2021). This well-known problem of bag-model like hybrid EOS
has been overcome by the inclusion of linear repulsive vector interaction,
such as the VBAG model of Klähn & Fischer (2015), similar to effective nuclear
mean-field models. Higher-order repulsive interactions were considered in
Benić et al. (2015) and Kaltenborn et al. (2017), in light of the existence of
a third family of compact stars as well as the so-called twin phenomenon where
two families of compact stars exist with nearly the same mass but different
radii. The corresponding RDF 1.2 hybrid EOS is shown in Fig. 1, for
$\beta$-equilibrium and zero temperature conditions in Fig. 1(a), together
with the mass-radius relation where the horizontal blue band marks the present
maximum neutron star constraint of $2.08\pm 0.07$M⊙ from the measurement of
the pulsar PSR J0740+6620 (Antoniadis et al., 2013; Fonseca et al., 2021). The
vertical lines in Fig. 1 indicate the onset of the phase transition (solid)
and reaching the pure quark matter phase (dash-dotted), which is due to
Maxwell’s condition for the first-order phase transition construction. In
between is the region of thermodynamic instability. The need of data in
between the onset of quark matter and reaching the pure quark matter phase,
for hydrodynamics simulations, results in a hadron-quark mixed phase where the
pressure gradient is flat in the zero-temperature case (see fig. 1(a)). It is
interesting to note that at finite entropy, examined at the example of
$s=3~{}k_{\rm B}$, which is a representative value for the PNS interior, the
phase boundaries are located at lower densities and the phase transition
region widens (see Fig. 1(b)). This has important consequences for the
stability of PNS, in connection with cold hybrid stars, for which the onset
mass increases but the maximum mass decreases (a detailed discussion about the
subject can be found in Hempel et al., 2016).

To support the temperature dependence of the phase boundaries, $\rho_{\rm
onset}$ and $\rho_{\rm final}$, Fig. 2 shows qualitative results for the
DD2F-RDF 1.2 hadron-quark model EOS, together with curves of constant entropy
per particle of $s=1-7~{}k_{\rm B}$. There is another important aspect related
to the finite temperature dependence on the phase transition construction, the
EOS softens substantially in the hadron-quark coexistence region (see Fig.
1(b)). During the post-bounce evolution, the central PNS grows not only in
mass but also the central density increases continuously, due to the
continuous mass accretion. Once the central density reaches $\rho_{\rm onset}$
the PNS becomes unstable. The initial contraction proceeds into an adiabatic
collapse, which halts when a sufficient amount of the PNS interior reaches the
pure quark matter phase, where the EOS stiffens again. In the RDF EOS class
this is additionally supported due to the vector interactions. As a
consequence, an additional strong hydrodynamics shock wave is formed at the
phase boundaries, which propagates quickly to increasingly larger radii (see
the thick solid black line in Fig. 3). This second shock wave takes over the
bounce shock, which triggers the onset of the supernova explosion. Afterwards,
on the timescale of several hundreds of milliseconds, the central PNS, now
featuring a massive core composed of deconfined quark matter (see Fig. 4),
settles into a quasi-stationary state, i.e. the central density saturates at
about five times the saturation density (see Fig. 4).

The classical picture of core-collapse supernova phenomenology, originally
proposed by Woosley et al. (2002), considers the high-mass progenitor
contributions to the failed explosion branch, which yield the formation of
black holes instead. It becomes evident that this picture might be incomplete
when taking into account a supernova explosion scenario, driven by the hadron-
quark phase transition. However, this cannot account for the bulk part of all
massive star explosions. It affects mainly the high-mass end of massive stars,
due to the rather high central densities required for the phase transition to
occur, which are not reached for low- and intermediate-mass progenitors on the
order of $\leq$25 M⊙. This picture of core-collapse supernova phenomenology
has been proposed recently by Kuroda et al. (2022). It also takes into account
that the class of core-collapse supernova explosions of very massive
progenitors, in the mass range of $M>30~{}M_{\odot}$, driven by the hadron-
quark phase transition can also only be small fraction. Otherwise there would
be a conflict with stellar population synthesis analyses, concerning the
number of black holes in this mass range. Furthermore, the role of metallicity
has yet incompletely been studied. From Fischer (2021) is becomes clear that
the situation might be completely different at (extremely) low metallicity,
usually connected with the early evolution of the universe.

####  2.2 Observational signals of the hadron-quark phase transition

One question remains, namely, is there an observable signature from the
supernova explosions driven by the hadron-quark first-order phase transition?
The present simulations based on multi-purpose hybrid EOS, applicable for
core-collapse supernova studies, reveal that there are potentially observable
signatures in both, the neutrino and the gravitational wave signals. These
cannot only directly related with the formation of a quark matter phase.

#####  2.2.1 Neutrino emission

The emission of a burst-like signals is related with the dynamics of the
formation and propagation of the second shock wave. This millisecond neutrino
burst is released when the second shock wave, which forms a a direct
consequence of the hadron-quark phase transition, propagates across the
neutrino spheres of last inelastic scattering. This neutrino burst is
detectable for a galactic event, as illustrated in Fig. 5 examined at the
Super-Kamiokande detector. This burst is absent in standard neutrino-driven
core-collapse supernova explosions (c.f. Wu et al., 2015). Hence, if observed
for a future galactic event, it represents a smoking gun signature for the
appearance of quark matter. If not observed, one might be able to constrain
the conditions, in terms of baryon density and temperature, up to which the
hadron-quark phase transition can be excluded. Similar results have been
obtained previously by Sagert et al. (2009) and Fischer et al. (2011),
however, employing a simplistic bag-model EOS for quark matter which is
incompatible with the constrain deduced from the existence of massive pulsars
of $\sim$2 M⊙.

Figure 5:  (Figure adopted from Fischer et al., 2018) Neutrino signal from a
core-collapse supernova explosion launched from a 50 M⊙ progenitor that takes
into account the hadron-quark phase transition using the DD2F-RDF 1.2 hybrid
EOS, evaluated at the Super-Kamiokande water Cherenkov detector at a fiducal
distance of 10 kpc from the Earth.

#####  2.2.2 Emission of gravitational waves

In general, core-collapse supernovae GWs stem from various different phases of
the evolution. During the early post-bounce phase, high-amplitude GWs are
emitted from convective motion, which lasts ups to several tens of
milliseconds. After that, during the proceeding post-bounce mass accretion
phase prior to the explosion onset, the GW emission is determined from a
variety of aspects, e.g., neutrino-driven convection, the presence of
hydrodynamics instabilities such as the standing accretion shock instability.
One feature seen in all present core-collapse SN simulations is the
continuously increasing GW frequency with time (c.f. Murphy et al., 2009;
Müller et al., 2013; Kuroda et al., 2016; Vartanyan et al., 2019; Mezzacappa
et al., 2020; Shibagaki et al., 2021, and references therein). This feature is
considered to be associated with PNS oscillations as well as convection at the
PNS interior at high density, which, in turn, excites the dominant post-bounce
GW $f$-mode (Morozova et al., 2018; Torres-Forné et al., 2019; Sotani &
Takiwaki, 2020). It is also present in the axially-symmetric SN simulations of
Zha et al. (2020) and Kuroda et al. (2022) employing a hadron-quark hybrid
EOS, however, prior to the phase transition when the post-bounce evolution is
determined by the hadronic phase.

Figure 6:  (Figure adopted from Kuroda et al., 2022) Gravitational waveform
$A_{+}$ (top panels) and spectrogram (bottom panels) for an axially-symmetric
core-collapse SN simulation with hadron-quark hybrid EOS, launched from a 50
M⊙ progenitor. The inlay at the top panel shows a zoom in of $A_{+}$
corresponding to the evolution shortly after the phase transition.  Figure 7:
(Figure adopted from Kuroda et al., 2022) Gravitational wave signal, spectral
amplitudes with respect to the frequency, for a sample of axially-symmetric
core-collapse SN simulations launched from different progenitors: 9.6 M⊙
(Z9.6), 11.2 M⊙ (S11.2), 50 M⊙ (S50) and 70 M⊙ (Z70). All models are of solar
metallicity except Z9.6 and Z70, which correspond to the class of extremely
metal poor stars.

Figure 6 shows the post-bounce evolution of the GW amplitude (top panel) for
an axially-symmetric SN simulation of Kuroda et al. (2022), launched from a 50
M⊙ progenitor employing the hadron-quark hybrid EOS of Fischer et al. (2018).
Shown is only the non-vanishing component in axial symmetry observed along the
equatorial plane, i.e., perpendicular to the symmetric axis. Note that since
the SN simulations employ an effective treatment of General Relativity, the GW
amplitude $A_{+}$, more precisely the GW strain, is obtained from the standard
quadrupole formula (for details, see Shibata & Sekiguchi, 2003; Kuroda et al.,
2014, and references therein). The bottom panel of Fig. 6 shows the
spectrogram of the GW emission, which is obtained using a short time Fourier
analysis. It shows the aforementioned continuously rising dominant $f$-mode
frequency, starting at around 50–100 Hz rising to several hundreds of Hz. The
sudden rise of the GW amplitude and frequency, at around 400 ms post bounce,
is the occurence of the hadron-to-quark matter phase transition and the
associated PNS collapse, second bounce and formation of the second shock wave.
It lasts for only few milliseconds. After that, the PNS oscillations continue
for several tens of milliseconds, which are responsible for the GW emission
between about 400–450 ms, shown in Fig. 6.

In order to address the question about the detectability of such GW signal, in
particular the possibility to distinguish it from SN that are not associated
with the hadron-quark phase transition, Kuroda et al. (2022) compared the GW
emission with those of canonical, neutrino-driven SN explosions as well as one
failed SN model in which case a black hole forms instead. The latter is
launched from a low-metallicity 70 M⊙ progenitor (Z70). In comparison to this
one, as well as two other neutrino-driven SN models launched from a low-
metallicity 9.6 M⊙ (Z9.6) and a solar-metallicity 11.2 M⊙ progenitor (S11.2),
the high-frequency kHz GW signal is enhanced by nearly one order of magnitude
for the 50 M⊙ SN simulation with hadron-to-quark matter phase transition (see
Fig. 7). The origin of these high amplitudes is PNS oscillations and PNS
convection after the phase transition took place. It is absent for Z70 because
a black hole forms about 1 ms after the phase transition. The future
observation of such a high GW amplitude component at kHz frequencies, which is
likely to be absent in all other SN, represents a smoking gun signature for
the presence of a hadron-to-quark matter phase transition and a resulting
stable remnant compact star with quark matter at the interior. This might be
used as constraint for the presence of quark matter at high baryon densities,
complementary to lattice QCD and experimental heavy-ion collision programs.

One caveat in predicting the GW emission from axially symmetric simulations
might arise in the overestimation of convective motion due to the presence of
a fixed symmetry axis. This aspect tends to overestimate the GW amplitude.
Another, yet incompletely understood feature is the presence of a quiescent
phase of GW emission during the post-bounce evolution (present between 200–350
ms post bounce in Fig. 6 prior to the phase transition induced sudden rise of
the amplitude). It has been speculated whether it corresponds to the period
when the shock wave gradually recedes. Only with simulations based on three
spatial dimensional setup will enable us to resolve this issue.

#####  2.2.3 Nucleosynthesis yields of heavy elements

The final phase of a supernova evolution after the supernova explosion onset,
is characterized by the deleptonization of the nascent PNS, i.e. the diffusion
of the trapped neutrinos. It lasts for several tens of seconds and is
associated with the ejection of a low-mass outflow known as the neutrino-
driven wind, which has long been studied as a possible site for the
nucleosynthesis of heavy elements in the universe (c.f. Thielemann et al.,
2011; Cowan et al., 2021, and references therein). It depends sensitively on
the nucleosynthesis conditions of the neutrino-driven wind, i.e., entropy per
particle and the neutron excess, as well as on their evolution (Woosley et
al., 1994). Recent investigations have confirmed that canonical, neutrino-
driven core-collapse SN explosions cannot yield a strong $r$-process
nucleosynthesis component (c.f. Martínez-Pinedo et al., 2014; Fischer et al.,
2020a). The reason for this are the generally too low entropies per particle,
on the order of 50–100 $k_{\rm B}$, and insufficient neutron excess or even
generally proton-rich conditions if PNS convection is considered (Roberts et
al., 2012; Mirizzi et al., 2016). No elements heavier than molybdenum, with
atomic number $Z=42$, can be produced. The key input physics in the SN
simulations is three-flavor Boltzmann neutrino transport, for the accurate
prediction of the neutrino fluxes and spectra as well as their evolution,
which determine nearly entirely these aforementioned nucleosynthesis
conditions (Qian & Woosley, 1996). Contrary, studies that are lacking neutrino
transport have to assume the nucleosynthesis conditions. Furthermore, recent
multi-dimensional SN simulations point to the presence of late-time mass
accretion, i.e., after the successful SN explosion onset, which lasts for
seconds (c.f. Müller et al., 2017; Bollig et al., 2021; Burrows & Vartanyan,
2021; Nagakura et al., 2021, and references therein). The exception might be
the class of neutrino-driven explosions of low-mass stellar progenitor in the
mass of 8–9 M⊙ (Radice et al., 2017; Stockinger et al., 2020) as well as those
which originate from binary stellar evolution leading to stripped envelope SN
explosions (Müller et al., 2019). If large down flows of matter persist after
the SN explosion onset, then the neutrino-driven wind is likely to be absent
and the nucleosynthesis results of these model might be very different from
those of the neutrino-driven wind Witt et al. (2021).

Figure 8:  (Figure adopted from Fischer et al., 2020b) Integrated elemental
yields from three QCD driven core-collapse supernova explosions, launched from
35, 40 and 50 M⊙ progenitors. For comparison, observations are shown for two
metal-poor stars, HD122564 from Roederer et al. (2012) and CS22892 from Sneden
et al. (2003).

The conditions found in SN explosions that are triggered through the hadron-
to-quark matter phase transition by Fischer et al. (2020b) are very different
from those of neutrino-driven SN explosions. One finds:

  1. (a)

an early, low-entropy and neutron-rich component with electron fraction of
$Y_{e}\simeq 0.2-0.25$, which is ejected dynamically together with the
expanding explosion shock wave,

  2. (b)

an intermediate component characterized by rising entropy and $Y_{e}$, and

  3. (c)

a late-time component.

The latter is the high-entropy neutrino-driven wind, featuring an entropy per
baryon on the order of several 100 $k_{\rm B}$, however, being subject of
large uncertainties as pointed out before. The integrated nucleosynthesis
results of three SN explosions driven by the QCD phase transition are shown in
Fig. 8, for three different progenitors of 35, 40 and 50 M⊙, showing a robust
strong $r$-process component with the production of elements not only
corresponding to the third $r$-process peak at mass number $A\simeq 195$ but
even transactinides such as plutonium. However, the abundances of the rare-
Earth elements as well as those of the third $r$-process peak are suppressed
by several orders of magnitude in comparison to the second $r$-process peak at
nuclear mass number $A\simeq 125$. Hence, this class of massive star
explosions cannot account for the bulk part of $r$-process nucleosynthesis in
the galaxy. Instead, it is as a rare $r$-process site, as was the considered
in Farouqi et al. (2021), together with the class of magneto-rotationally
driven core-collapse SN (LeBlanc & Wilson, 1970; Winteler et al., 2012; Mösta
et al., 2014a; Mösta et al., 2015a). The latter requires a high magnetic field
strength of the progenitor in combination with rapid rotation, which is likely
to be realized only for a subset of all core-collapse SN.

###  3 Quark matter in binary neutron star mergers

For the astrophysics community the very first unambiguous detection of a
neutron star coalescence in August 2017 was a seminal event as the
observations revealed a ’proto-type’ neutron star merger, which settled many
questions that had been open or tentative before (Abbott et al., 2017).
Furthermore, the detection highlights the potential to learn about the
presence of deconfined quark matter in neutron stars, i.e. at densities of a
few time nuclear saturation density. Merging neutron stars lead to a number of
observable phenomena, which in principle can carry in imprint of quark
deconfinement. This includes foremost the gravitational-wave signal but also
electromagnetic counterparts, where various processes generate emission in
different parts of the spectrum (LIGO Scientific Collaboration et al., 2017).

Figure 9:  (Figure from Bauswein et al., 2019b) Density evolution of a neutron
star merger with a hybrid equation of state with a phase transition to
deconfined quark matter. Lines display density contours of
$\rho=10^{14}~{}\mathrm{g/cm^{3}}$ (dashed) and $8.7\times
10^{14}\mathrm{g/cm^{3}}$ (solid). The latter density roughly corresponds to
the onset density of the phase transition at zero temperature.

The merger of two neutron stars proceeds through different stages, starting
with a long-lasting “inspiral” phase, where the binary looses angular momentum
and energy as a result of gravitational-wave emission, which leads to a
decrease of the orbital separation. The inspiral continuously accelerates and
finally leads to the coalescence of the binary components as a highly
dynamical process with typical time scales of milliseconds (see Fig. 9). For
higher total binary masses the merger forms a black hole-torus system because
the merger remnant cannot be stabilized against the gravitational collapse.
Recall that typical neutron star masses are about 1.3 to 1.4 $M_{\odot}$ such
that the mass of the merged object will usually exceed the maximum mass of
nonrotating neutron stars. If the total binary mass is below a certain
threshold mass, the merging binary components create a neutron star merger
remnant. Since most neutron star binaries may have a total mass of about 2.7
$M_{\odot}$ and since the threshold mass for black hole formation is likely in
the range between 2.8 and 3.4 $M_{\odot}$, one may expect that the typical
outcome of a merger is the formation of a neutron star remnant. This object is
rapidly rotating and initially highly deformed and strongly oscillates
reaching a quasi-equilibrium on time scales of several 10 milliseconds to
enter a phase of secular evolution.

During the merging process the densities increase as a more massive and more
compact object forms. Therefore, the prospects for the occurrence of
deconfined quark matter in the “postmerger” stage are generally better
compared to the inspiral. Moreover, the temperature in the merger remnant
increases and can reach several 10 MeV and in the shock-heated contact
interface even about 100 MeV. This further enhances the likelihood for the
hadron-quark phase transition to take place. Whether deconfined quark matter
is present in the postmerger remnant or even before merging during the
inspiral, is currently open and depends on the unknown temperature-dependent
onset density of the hadron-quark phase transition. This motivates to identify
specific signatures of quark deconfimenent in neutron star merger observables.

####  3.1 Gravitational waves

The gravitational wave signal of a neutron star merger reflects the stages of
the coalescence, i.e. the inspiral and the postmerger evolution, and encodes
the dynamics of the system, which are dominantly determined by the binary
masses and the equation of state. In turn, the detailed analysis of the
gravitational wave signal reveals the binary masses and information on the
equation of state (see Abbott et al., 2019). The impact of the binary masses
is described by two parameters. The chirp mass, defined as follows,

$\mathcal{M}_{\mathrm{chirp}}=(M_{1}M_{2})^{3/5}/(M_{1}+M_{2})^{1/5}~{},$ (1)

with $M_{1}$ and $M_{2}$ being the masses of the individual binary components,
has the strongest influence on the inspiral gravitational wave emission and is
thus measured with high precision. The binary mass ratio $q=M_{1}/M_{2}$
affects the signal only during the last orbits before merging and can thus be
inferred only with less precision. Combining $\mathcal{M}_{\mathrm{chirp}}$
and $q$ yields the physical masses of the system.

Equation-of-state effects during the inspiral are described by the so-called
combined tidal deformability, which is defined as follows (c.f. Flanagan &
Hinderer, 2008; Hinderer, 2008),

$\tilde{\Lambda}=\frac{16}{13}\frac{(M_{1}+12M_{2})M_{1}^{4}\Lambda_{1}+(M_{2}+12M_{1})M_{2}^{4}\Lambda_{2}}{(M_{1}+M_{2})^{5}}~{}.$
(2)

Here $\Lambda_{1}$ and $\Lambda_{2}$ are the tidal deformabilities of the
individual stars given by
$\Lambda=\frac{2}{3}k_{2}\left(\frac{c^{2}\,R}{G\,M}\right)^{5}$ with the
stellar radius $R$ and the tidal Love number $k_{2}$. $k_{2}$ is a stellar
structure parameter, which is as the radius uniquely determined by the
equation of state. A larger $\tilde{\Lambda}$ accelerates the inspiral since
systems with a finite tidal deformability lead to stronger gravitational-wave
emission compared to point-particle binaries of the same masses. Its
definition illustrates that $\Lambda$ scales tightly with the stellar radius,
and the constraint which GW170817 put on the tidal deformability excludes
neutron stars in the given mass range with radii larger than about 13.5 km
(Abbott et al., 2018).

As the stellar radius, the tidal deformability is similarly affected by the
presence of a phase transition to deconfined quark matter. Specifically, the
hadron-quark phase transition can lead to a kink in the relation between tidal
deformability and mass. The kink occurs at the neutron star mass
$M_{\mathrm{onset}}$, where quark matter starts to appear. Hence, the presence
of quark matter may be identified by detecting systems with different masses
and measuring the respective tidal deformability (detailed can be found in
Chatziioannou & Han, 2020; Chen et al., 2020; Pang et al., 2020, and
references therein). However, even for models which predict a rather strong
impact of the phase transition on the stellar structure, the kink in
$\Lambda(M)$ may not be very pronounced (c.f. Han & Steiner, 2019; Bauswein et
al., 2019b; Christian et al., 2019). Therefore, identifying quark matter in
neutron stars during the gravitational-wave inspiral would require very
precise measurements of the tidal deformability, which seems challenging in
regard to statistical and systematic uncertainties in the gravitational-wave
data analysis.

The definition of the tidal deformability also shows that $\Lambda$ strongly
decreases with the stellar mass since more massive stars become more compact
and thus more similar to point particles. Hence, it is more challenging to
measure finite-size effects in high-mass binaries. This implies that the
identification of quark matter through the inspiral may be more intricate in
particular if the onset density of the hadron-quark phase transition is
relatively high. Although the currently very low number of neutron star merger
detections does not yield good statistics on the population properties of
neutron star binaries, one may expects that there will be fewer systems with
high total binary masses.

Figure 10:  (Figure taken from Bauswein et al., 2019a) Gravitational-wave
spectrum of two neutron star merger simulations with a total mass of 2.7
$M_{\odot}$. The black curve adopts a purely hadronic equation of state, while
the green curve displays the spectrum for a model undergoing a phase
transition to deconfined quark matter.

In this regard the postmerger phase may be more promising since the remnant
probes a density regime which is higher than that of the initial stars, and,
thus, low-mass binaries may already reveal signals of quark matter. The
equation of state affects the frequencies of the postmerger oscillations and
therefore the frequencies of the postmerger gravitational-wave emission.
Typical gravitational-wave spectra are shown in Fig. 10, where one can clearly
identify different peaks. The most prominent feature is a pronounced peak in
the kHz range, which corresponds to the dominant oscillation mode of the
remnant and is generated by the fundamental quadrupolar fluid mode. The
frequency of this oscillation is determined by the compactness of the remnant,
where more compact stellar objects oscillate at higher frequencies. This
implies a the strong sensitivity of the frequencies on the equation of state
since the properties of the equation of state shape the stellar structure of
the remnant.

Figure 10 shows simulation results for an equation of state model with phase
transition to quark matter (green) and one model without phase transition
(black) for a merger of two stars with 1.35 $M_{\odot}$. Both models are based
on the same hadronic equation of state at lower densities and only differ at
higher densities, where the model with phase transition features a significant
softening. This softening leads to a more compact remnant and thus to a higher
postmerger oscillation frequency, while the general morphology of the spectra
is similar. In the particular case displayed in Fig. 10 the phase transition
takes place right after merging (see Fig. 9) such that the inspiral proceeds
identically.

While a frequency shift by a phase transition is intuitive, it is not obvious
that a high postmerger frequency alone is indicative of the presence of
deconfined quark matter because also purely hadronic equation of state models
can reach frequencies in the range up to 4 kHz for this binary configuration.
However, an increased postmerger frequency is in fact characteristic of a
phase transition if one compares $f_{\mathrm{peak}}$ to the tidal
deformability inferred from the inspiral phase (details can be found in
Bauswein et al., 2019a, and references therein). This comparison is displayed
in Fig. 11, which compiles simulation results for many calculations with the
same binary mass configuration but with different equation of state models
with and without phase transition to deconfined quark matter.

Figure 11:  (Figure from Bauswein & Blacker, 2020) Postmerger gravitational-
wave frequency as function of the tidal deformability for 1.35-1.35
$M_{\odot}$ binaries with many equation of state models. Purely hadronic
models (crosses) follow a tight relation fitted by the black curve. Models
undergoing a phase transition before merging (dots) or after merging (plus
signs) lead to a frequency shift relative to the models without phase
transition.

All purely hadronic models follow a tight correlation between
$f_{\mathrm{peak}}$ and $\Lambda_{1.35}$ (note that for equal-mass binaries
$\tilde{\Lambda}=\Lambda_{1}=\Lambda_{2}$). This scaling is not fully
unexpected because $\Lambda_{1.35}$ is a good proxy for the stellar structure
of the inspiralling stars and consequently also for the postmerger remnant.
Although densities increase during merging, purely hadronic equations of state
do not change very abruptly such that $\Lambda_{1.35}$ does also describe the
stellar structure of objects with densities exceeding those in 1.35
$M_{\odot}$ neutron stars.

In contrast, the occurrence of deconfined quark matter can change the equation
of state strongly beyond the onset density of the phase transition.
Consequently, the softening of the equation of state and the resulting
frequency shift of the postmerger gravitational-wave emission is not captured
by stellar parameters which are only sensitive to the density regime below the
onset density (such as $\Lambda_{1.35}$ in Fig. 11). This explains the
characteristic offset of equation of state models with quark matter in Fig.
11. Hence, a significant deviation of the postmerger gravitational wave
frequency $f_{\mathrm{peak}}$ from the empirical relation
$f_{\mathrm{peak}}(\Lambda_{1.35})$ for purely hadronic models (black line)
does provide very strong evidence for the occurrence of deconfined quark
matter in neutron stars (see Bauswein et al., 2019a).

Some of the purely hadronic models in Fig. 11 feature a phase transition to
hyperonic matter, which apparently does not soften the equation of state
sufficiently to yield a strong frequency deviation. Hence, a characteristic
frequency shift relative to $\Lambda_{1.35}$ does indeed represent an
unambiguous signature of quark matter.

It is conceivable that the frequency shift is a signature which also occurs
for other hadronic equations of state describing the density regime below the
phase transition (see Bauswein et al., 2019a; Liebling et al., 2021; Blacker
et al., 2020; Weih et al., 2020; Hanauske et al., 2021; Prakash et al., 2021,
and references therein). The models depicted by plus signs in Fig. 11 are
based on the same hadronic equation of state but make different assumptions
about the properties of the hadron-quark phase transition and the properties
of deconfined quark matter in the non-perturbative regime beyond the phase
transition. Therefore, the models occur at the same tidal deformability
$\Lambda_{1.35}$ but result in differently strong frequency shifts222One model
with a slightly higher $\Lambda_{1.35}$ features a small modification of the
purely hadronic equation of state.. In fact, the magnitude of the frequency
increase encodes the “strength” of the phase transition. For instance, for a
roughly similar onset density and stiffness of the quark matter phase, the
frequency shift scales with the density jump across the phase transition (see
Bauswein et al., 2019a). Generally, it is clear that the signature of quark
matter will be more pronounced for models which feature a stronger deviation
from the corresponding hadronic reference model. This may also imply that for
certain hybrid models the aforementioned signature of a relative frequency
increase may not be very pronounced or even absent if the equation of state
closely resembles the behavior of a purely hadronic model. See also the model
in Most et al. (2019), where quark matter exhibits only a very weak impact on
the gravitational-wave frequency but influences the collapse time scale of the
remnant.

Moreover, the frequency shift may not be very strong if only a small quark
matter core forms in the center of the remnant for binary configurations which
are barely massive enough to reach conditions in the postmerger remnant to
undergo a phase transition (see Blacker et al., 2020). In this case, however,
a more massive binary with a sizeable quark matter core may yield a very
strong signature. Blacker et al. (2020) provides an extensive study of the
impact of quark matter in neutron star mergers for different binary mass
configuration including unequal-mass systems, which can similarly yield
characteristic frequency shifts. Depending on the specific properties of the
phase transition and the quark matter equation of state, the occurrence of
deconfined quark matter during the postmerger evolution may be delayed, as was
discused in Weih et al. (2020).

The dots in Fig. 11 demonstrate that a frequency increase relative to the
empirical $f_{\mathrm{peak}}(\Lambda_{1.35})$ relation may also occur for
mergers of hybrid stars, i.e. systems which contain quark matter cores already
before the coalescence during the inspiral (Bauswein & Blacker, 2020).

Figure 11 illustrates that a relatively coarse determination of the tidal
deformability of about $\Delta\Lambda\approx 100$ may be sufficient to
identify the occurrence of the hadron-quark phase transition through a
postmerger frequency shift. The dominant postmerger gravitational-wave
frequency was not detected in GW170817. However, simulations show that
relatively minor upgrades of the sensitivity of existing gravitational-wave
instruments in the kHz range are sufficient to detect the postmerger peak of a
signal similar to GW170817 (details can be found in Chatziioannou et al.,
2017; Torres-Rivas et al., 2019; Wijngaarden et al., 2022). These
gravitational-wave data analysis studies reveal that $f_{\mathrm{peak}}$ can
be measured within a few 10 Hz, which is sufficiently precise to detect
sizeable deviations from the $f_{\mathrm{peak}}(\Lambda)$ relation for purely
hadronic models.

These considerations imply that any sufficiently precise measurement of
$f_{\mathrm{peak}}$ and the tidal deformability will be informative about
properties of the hadron-quark phase transition. While a significant
postmerger frequency shift will provide very strong evidence for the presence
of deconfined quark matter, data compatible with the
$f_{\mathrm{peak}}(\Lambda)$ relation for purely hadronic models rules out the
presence of a strong phase transition in the density regime of the merger
remnant. This would thus place a constraint on the onset of quark
deconfinement assuming that the presence of quark matter would lead to a
strong signature as described above.

The probed density regime by the postmerger remnant and thus the concrete
constraint that can be put on the onset density of the hadron-quark phase
transition, is also accessible by gravitational wave parameters. For purely
hadronic equation of state models there exist tight empirical relations
between the tidal deformability or the postmerger gravitational wave frequency
$f_{\mathrm{peak}}$, respectively, and the maximum density in the merger
remnant (see Bauswein et al., 2019a; Blacker et al., 2020). Those universal
relations hold for a large range of binary masses and can be employed to infer
an upper or lower limit on the onset density of the hadron-quark phase
transition depending on whether the comparison between $f_{\mathrm{peak}}$ and
$\Lambda$ for the given binary masses reveals evidence for the presence or
absence of quark matter.

Two complications may arise, which have been stressed in detail in Blacker et
al. (2020). If the quark matter core is too small, the impact on the stellar
structure and thus postmerger gravitational wave emission will not be very
pronounced but only occur for more massive binary systems. Moreover, the
presence of quark matter may not in any case lead to a sizeable effect if for
instance the properties of the quark matter equation of state are mimicking
those of hadronic matter as in Alford et al. (2005)333If values of
$f_{\mathrm{peak}}$ and $\Lambda$ are compatible with
$f_{\mathrm{peak}}(\Lambda)$ of hadronic matter, either no phase transition
occurred or quark matter properties are very similar to hadronic matter.
Hence, the sketched procedures can only rule out the presence of a strong
phase transition up to a given density and thus constrain the onset density of
a strong phase transition (Blacker et al., 2020).. The latter issue requires
further investigations to reveal possible other signatures of quark matter
that can inform about the presence of quark matter in the case its impact on
the stellar structure is relatively weak.

####  3.2 Mass ejection and electromagnetic counterparts

Gravitational waves are not the only messenger from neutron star mergers.
Apart from gamma-ray bursts and associated electromagnetic emission in
different parts of the spectrum, mass ejection during the merger process can
lead to electromagnetic radiation in the infrared, optical and ultraviolet.
Material ejected from a neutron star merger expands with high velocities (up
to a few 10 per cent of the speed of light) and undergoes the rapid neutron
capture process forming heavy elements Cowan et al. (2021). The freshly
synthesized elements are radioactive and the subsequent decays deposit heat in
the initially opaque expanding ejecta cloud. The thermal emission of this
matter produces an electromagnetic counterpart commonly called kilonova, which
evolves on time scales of hours and days and becomes bright enough to be
observable with telescopes Metzger (2019). In the case of GW170817 the
gravitational wave signal provided an estimate of the sky localization, and
the corresponding kilonova was found several hours after the initial alert
circulated after the detection by the gravitational wave instruments (LIGO
Scientific Collaboration et al., 2017).

The properties of the electromagnetic emission yield a coarse estimate of the
ejecta parameters like their mass and outflow velocities. The brightness of
the kilonova for instance scales approximately with the amount of ejecta,
although the quantitative interpretation is very challenging because of
various uncertainties, e.g. about atomic data of heavy nuclei, and different
ejection mechanisms playing a role on different time scales and leading to
different ejecta components. The ejecta characteristics depend on the merger
dynamics and therefore on the binary masses and the equation of state (c.f.
Hotokezaka et al., 2013; Bauswein et al., 2013, and references therein).
Hence, kilonova properties in principle contain information on the equation of
state and thus possibly on the presence of quark matter in neutron stars.

Figure 12:  (Figure taken from Bauswein et al., 2019b) Amount of unbound
matter as function of time in simulations of 1.35-1.35 $M_{\odot}$ binaries
for different hybrid equations of state. The black curve shows the purely
hadronic reference model without phase transition.

Figure 12 displays the amount of unbound matter as function of time for
1.35-1.35 $M_{\odot}$ mergers. The calculations run only for a few 10 ms after
merging and thus do only provide the initial dynamical ejecta component, while
matter becoming unbound during the subsequent secular evolution is not
captured. The black curve shows the result for a purely hadronic equation of
state model, while all other curves exhibit the ejecta mass for hybrid
equations of state undergoing a phase transition to deconfined quark matter.
This set of models demonstrates that there is no very distinct and unique
impact of the hadron-quark phase transition on mass ejection in neutron star
mergers, as some models lead to enhanced ejecta masses while others yield a
comparable or even reduced amount of unbound material in comparison to the
purely hadronic reference equation of state (black line).

The scatter among the different models visible in Fig. 12 is to be compared to
ejecta mass variations of various purely hadronic equations of state. Neutron
star merger simulations for a larger sample of purely hadronic equation of
states show roughly comparable variations of the ejecta mass even for models
which result in similar stellar parameters (see Bauswein et al., 2013). This
implies that it is very difficult to unambiguously deduce the occurrence of
quark matter from kilonova observations alone since bulk ejecta properties
apparently do not carry unambiguous signatures of quark matter Bauswein et al.
(2019b); Prakash et al. (2021). Moreover, nuclear network calculations
postprocessing merger simulations do not reveal significant differences in the
resulting elemental abundance patterns comparing models with and without phase
transition (Prakash et al., 2021), although much more work is required to
properly address the full complexity of this point.

####  3.3 Black-hole formation

Albeit kilonova observations may not be immediately informative about the
occurrence of deconfined quark matter, they do contain information about the
merger outcome. All models displayed in Fig. 12 yield a neutron star merger
remnant since their total binary mass of 2.7 $M_{\odot}$ is below the
respective threshold mass $M_{\mathrm{thres}}$ for prompt black-hole
formation. If the total binary mass exceed the threshold for prompt black-hole
formation, the ejecta mass and thus the kilonova brightness is strongly
reduced Bauswein et al. (2013). Hence, combining the binary masses measured by
the gravitational wave inspiral and the interpretation of the kilonova
emission can yield an estimate of the threshold mass $M_{\mathrm{thres}}$ for
direct black hole-formation. Along these lines, it has been argued that
GW170817 did not result in a prompt collapse. Also, the presence or absence of
postmerger gravitational-wave emission informs about a gravitational collapse
after merging. In essence, it is conceivable that $M_{\mathrm{thres}}$ can be
determined or at least constrained from future observations.

This poses the question whether the occurrence of deconfined quark matter
affects the threshold mass $M_{\mathrm{thres}}$ for black-hole formation. It
is intuitively clear that the stability of a neutron star system in general
and of $M_{\mathrm{thres}}$ in particular will strongly depend on the
properties of the equation of state. Numerically $M_{\mathrm{thres}}$ can be
determined by performing several simulations for different binary masses for a
given equation of state and by checking the respective outcome. Such estimates
for a large sample of different purely hadronic equation of states yield
$M_{\mathrm{thres}}$ in a relatively broad range between roughly 2.8 and 3.8
$M_{\odot}$ depending on the specific properties of the microphysical model,
as was discussed in Bauswein et al. (2020).

Figure 13:  Collapse behavior of purely hadronic models (black) and hybrid
equations of state with phase transition to deconfined quark matter (green).
$M_{\mathrm{thres}}$ is the threshold mass for prompt black-hole formation.
$\Lambda_{\mathrm{thres}}$ is the tidal deformability of the binary with a
total mass of $M_{\mathrm{thres}}$, i.e. the tidal deformability of the system
with a marginally stable remnant for the given equation of state. Purely
hadronic equations of state lead to threshold properties below the black
dashed line.

A $M_{\mathrm{thres}}$ estimate may yield signatures of quark matter in
neutron stars since the occurrence of a phase transition can but does not
necessarily need to result in a destabilization of the merger remnant Bauswein
et al. (2020). This is summarized in Fig. 13, which displays
$M_{\mathrm{thres}}$ versus $\Lambda_{\mathrm{thres}}$ for different equations
of state. $\Lambda_{\mathrm{thres}}$ is the tidal deformability measured in a
system with a total mass of $M_{\mathrm{tot}}=M_{\mathrm{thres}}$ and can be
inferred from the gravitational-wave inspiral signal. Hence, future
observations will narrow down the area where the true equation of state lies
in Fig. 13.

It is striking that in Fig. 13 purely hadronic models are located below the
dashed line, whereas some hybrid equations of state with phase transition to
deconfined quark matter occur in the upper left corner of the diagram. Hence,
some hybrid models feature a relatively low $M_{\mathrm{thres}}$ in comparison
to the tidal deformability. This may not be surprising because in these cases
the tidal deformability is determined by the hadronic regime of the equation
of state below the onset of the phase transition, whereas the appearance of
quark matter during the merging softens the equation of state and destabilizes
the remnant, i.e. reduces $M_{\mathrm{thres}}$. The softening of the equation
of state and the associated reduction of $M_{\mathrm{thres}}$ is not captured
by the tidal deformability since it is only sensitive to the equation of state
at lower densities. This effect is somewhat similar to the shift of the
postmerger gravitational-wave frequency as result of a strong compactification
of the remnant. Such sudden changes of the equation of state do not occur for
purely hadronic models, which is why they cannot occur much beyond the dashed
line in Fig. 13.

Identifying the presence of quark matter through the collapse behavior in Fig.
13 requires only a moderate precision of a tidal deformability measurement in
comparison to resolving directly the kink induced by a phase transition in
$\Lambda(M)$. Moreover, a single measurement may already provide evidence for
quark matter if $\Lambda_{\mathrm{thres}}$ and $M_{\mathrm{thres}}$ are
correspondingly constrained. As apparent from Fig. 13 the comparison between
$\Lambda_{\mathrm{thres}}$ and $M_{\mathrm{thres}}$ may not in any case reveal
the presence of quark matter as some hybrid models can mimic the behavior of
purely hadronic equations of state.

###  4 Summary

In this contribution, we have illustrated that there exist pronounced and
unambiguous features which reveal the occurrence of deconfined quark matter in
supernova explosions and in neutron star mergers.
In supernova simulations, the quark matter deconfinement phase transition
leads to an explosion mechanism for massive blue supergiant stars which
without the phase transition would evolve into black holes and represent
failed supernovae. We have demonstrated that as a result of the deconfinement-
driven explosion mechanism neutron stars with high masses $\gtrsim
2~{}M_{\odot}$ are born, while the population of this mass segment within the
standard neutrino-driven supernova explosion is quite unlikely, see Müller et
al. (2016). As observational signatures we have discussed a second neutrino
signal which appears about 1.2 ms after bounce and would be detectable by
Super-Kamiokande if the supernova goes off in our galaxy, as well as a
gravitational wave signal in the spectrogram of an axially-symmetric core-
collapse supernova about 400 ms post bounce. It has also been found that a
supernova exploding by the quark deconfinement mechanism must be a relatively
rare event because it cannot account for the bulk part of the r-process
nucleosynthesis in the galaxy due to a strong suppression of the synthesis of
rare-earth elements and those of the third r-process peak.
Most of the signals of quark deconfinement in binary neutron star mergers are
connected to the postmerger evolution, which may not be unexpected since the
merger remnant probes higher densities than the inspiralling stars and thus
naturally carries a stronger imprint of the equation of state at higher
densities. We have focussed on the discussion of two deconfinement signals:
the shift of the peak frequency of postmerger gravitational wave signal
(occurring for both merging hadronic and merging hybrid stars) and the
correlation between binary mass and tidal deformability at the threshold for
prompt black-hole formation. The strength of these signatures depend on
details of the phase transition and the equation of state of quark matter.
This implies that under certain circumstances it may be challenging to reveal
or exclude the occurrence of quark deconfinement. This highlights the
necessity of further research investigating more hybrid models and details of
the various signals from neutron star mergers.

The prospects to identify signals of quark confinement through the various
features described above emphasizes the outstanding importance of future
measurements and sensitivity upgrades of existing instruments. This includes
in particular the ability to detect postmerger gravitational-wave emission in
the kHz range but also electromagnetic follow-up observations to infer
kilonova properties.

###  5 Epilogue

We want to take up the discussion of the Introduction that the simulations of
core-collapse SN explosions and BNS mergers with model EOS are a unique tool
to investigate the QCD phase diagram in the region of low temperatures and
high baryon densities ($T\lesssim 60$ MeV at $1\lesssim n/n_{0}\lesssim 5$)
which is inaccessible to lattice QCD simulations and HIC experiments. Not only
that the detection of the above-described signals of a strong first-order
phase transition in BNS mergers and/or supernovae would provide support for
the existence of a CEP in the phase diagram that has so far been
unsuccessfully been sought for in HIC experiments, there is theoretical
evidence for a crossover transition at very low temperatures that suggests the
existence of a second CEP or even a crossover-all-over situation. This arises
from the observation Hatsuda et al. (2006); Abuki et al. (2010) that a
coexistence of chiral symmetry breaking and diquark condensation occurs at low
temperatures due to the $U_{A}(1)$ anomaly-generating triangle diagram which,
after Fierz transformation mixes diquark and meson condensates. This effect
realizes the concept of quark-hadron continuity Schäfer & Wilczek (1999) in a
crossover transition. Support for such a picture comes also from recent
progress in NS phenomenology.
We are currently witnessing a paradigm change in the interpretation of mass
and radius measurements of pulsars that is induced by the observation that
from the multi-messenger analysis of typical-mass neutron star radii with
$R_{1.4~{}M_{\odot}}=11.7^{+0.86}_{-0.81}$ km Dietrich et al. (2020) (see also
Capano et al. (2020)) and the recent NICER radius measurement
$R_{2.0~{}M_{\odot}}=13.7^{+2.6}_{-1.5}$ km Miller et al. (2021) (see also
Riley et al. (2021)) follows that $R_{2.0~{}M_{\odot}}\gtrsim
R_{1.4~{}M_{\odot}}$. The description of such a behaviour as solution of the
Tolman-Oppenheimer-Volkoff equations requires a soft-stiff transition in the
EOS at densities $n\lesssim 2n_{0}$, just before the hyperon onset. This
transition could be the hadron-to-quark matter transition. In recent
descriptions one joins a standard nuclear EOS with a constant speed of sound
(CSS) model for the high-density phase either by a first-order phase
transition (with a vanishing speed of sound in the mixed phase Somasundaram &
Margueron (2021)) or by directly matching the nuclear and quark matter squared
speed of sound $c_{s}^{2}$ at a certain transition density $n_{\rm tr}$
without a density jump, thus mimicking a crossover transition Drischler et al.
(2021). The best phenomenological description fulfilling simultaneously the
constraints on both radii $R_{2.0~{}M_{\odot}}$ and $R_{1.4~{}M_{\odot}}$ is
obtained in this simple picture by $n_{\rm tr}\sim 0.5~{}n_{0}$ and
$c_{s}^{2}\sim 0.5$. We would like to remark that a CSS model with
$c_{s}^{2}=0.45\dots 0.54$ provides an excellent fit to a microscopic nonlocal
chiral quark model of the Nambu–Jona-Lasinio (NJL) type with diquark
condensation (color superconductivity) and repulsive vector meson mean field
Antić et al. (2021); Contrera et al. (2022). A direct, one-zone interpolation
scheme between the safely known soft nuclear matter EoS (up to about
$1.1~{}n_{0}$ as in (Hebeler et al., 2013)) and the suitably chosen stiff
quark matter EoS (e.g., from a NJL model with coupling to a repulsive vector
meson mean field) was pioneered in the works of Masuda et al. (2013a, b). Such
a phase transition construction can be understood as a shortcut for three
physical effects as ingredients:

  1. (i)

a stiffening of the nuclear matter EoS $P_{H}(\mu)$ due to the repulsive quark
Pauli blocking effect between nucleons Röpke et al. (1986); Blaschke et al.
(2020) which can be effectively accounted for with a nucleonic excluded volume
(see, e.g., Alvarez-Castillo et al. (2016)),

  2. (ii)

a strong reduction of the quark matter pressure $P_{Q}(\mu)$ at low chemical
potentials due to confining forces (which result then in a good crossing of
curves at $\mu=\mu_{c}$ allowing for the Maxwell construction
$P_{H}(\mu_{c})=P_{Q}(\mu_{c})$ of a first-order phase transition), and

  3. (iii)

a mixed phase construction (e.g., by a parabolic interpolation Ayriyan et al.
(2018)) that mimics the effects of finite-size structures (pasta phases) in
the quark-hadron coexistence region.

For more details on the physics background, (see Ayriyan et al., 2021; Baym et
al., 2018, and references therein). With this microphysical basis behind the
interpolation approach, a two-zone interpolation construction for the hadron-
to-quark matter has been developed in Ayriyan et al. (2021), where at the
matching point $\mu_{c}$ situated between $\mu_{H}$ and $\mu_{Q}$, one can
choose the condition of continuous density ($\Delta n=0$, crossover) or a
finite density jump ($\Delta n\neq 0$, first-order transition). In concluding
this epilogue, we want to give an outlook to the generalization of this two-
zone interpolation scheme to finite temperatures (and arbitrary isospin
densities) as a necessary prerequisite for investigating the consequences of
these recent developments in the interpretation of neutron star phenomenology
at zero temperature to simulations of supernova explosions and of binary
neutron star merger events. The goal is to model the general class of hybrid
EoS that corresponds to a phase diagram which has not only one critical
endpoint (CEP) at high temperatures which marks the change from a first-order
to a crossover transition regime, but also second CEP at low temperatures that
arises from the competition and mixing between dynamical chiral symmetry
breaking and color superconductivity. Within the finite-temperature
generalization of the two-zone interpolation scheme, this can be achieved by
defining the function $\Delta n[\mu_{c}(T)]$ along the matching line
$\mu_{c}(T)$ between the hadron-like and the quark-like interpolation zone in
the phase diagram. The function $\Delta n[\mu_{c}(T)]$ encodes the position of
the CEPs $T_{\rm CEP,1}$ and $T_{\rm CEP,2}$ where $\Delta n=0$ as well as the
strength of thee first-order transition between these points where $\Delta
n\neq 0$.
It is the aim of our ongoing research to investigate the dependence of the
above described signals of a strong phase transition in supernova explosions
and binary neutron star mergers on the detailed structure of the QCD phase
diagram at low temperatures and high baryon densities and thus to be prepared
for interpreting the possible observation of signals from such events in the
near future.

#### Acknowledgements

This work was supported by the Polish National Science Centre (NCN) under
grant No. 2019/33/B/ST9/03059 (D.B.) and No. 2020/37/B/ST9/00691 (T.F.). A.B.
acknowledges support by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No. 759253, by Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Project-ID 279384907 - SFB 1245, by DFG - Project-ID 138713538 -
SFB 881 (”The Milky Way System”, subproject A10) and by the State of Hesse
within the Cluster Project ELEMENTS. The work was performed within a project
that has received funding from the European Union’s Horizon 2020 research and
innovation program under grant agreement STRONG - 2020 - No. 824093.

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