nucleons and clusters that are characterized by a mass number and a charge.
Such matter is conveniently treated within the nuclear statistical equilibrium
approach [343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355,
356]. Besides light clusters, the pionization of matter at sufficiently high
temperatures [328, 357, 358, 359] and the formations of condensates of
deuterons [360, 361, 362] and alpha-particles at low temperatures [363, 364,
352, 365, 366, 367] have been studied.

In the present context, one may ask about the possible role of hyperons and
resonances in low-density matter containing clusters and condensates. As is
well known, nuclear systems show rich nucleon-$\Delta$-pion dynamics, which
were extensively explored in the context of heavy-ion physics. Similarly, the
role of the quartet of $\Delta$-resonances and the isotriplet of pions,
$\pi^{\pm,0}$, may be important in the treatment of the nucleons and clusters
[368] in warm dilute nuclear matter. In a first approximation, the full
nuclear statistical ensemble can be approximated by the light clusters with
the mass number $A\leq 4$ and a heavy nucleus such as 56Fe. Since the
hyperonic interactions are less important at low densities, the lightest
hyperon (i.e., $\Lambda$) is expected to contribute predominantly to the
strangeness content of matter. The low-density matter with hyperons in the
high-temperature regime was studied in the supernova and BNS matter [368] and
low-temperature crusts of neutron stars [369, 330]. Pions at high temperatures
were also studied in the supernova context [328, 357, 358, 359]. Furthermore,
the possibility of the formation of light hypernuclei along with non-strange
clusters was studied in Refs. [370, 371].

Figure 6.5: Mass fractions of constituents as a function of temperature at
fixed density $10^{-2}\rho_{\rm sat}$ for charge fractions $Y_{Q}=0.4$ (top
panels) and 0.1 (bottom panels) [368]. The two left panels include only
nucleons and light clusters, whereas the two right panels account for the full
composition. The full composition includes neutrons $n$, protons $p$,
deuterons $d$, triton $t$, helium $h$, $\alpha$-particles,
$\Delta$-resonances, $\Lambda$-hyperon, and isotriplet of pions $\pi^{0,\pm}$.
The increase of the mass fraction of heavy baryons with temperature is
associated with their excitation by temperature, which helps to bridge the
difference between their and nucleon rest masses which determines their
relative statistical importance. Figure 6.6: Dependence of the mass fractions
of the particles in dilute nuclear matter on density, for $T=10$ MeV [368].
The top and lower panels correspond to charge fractions $Y_{Q}=0.4$ and 0.1.
The left and right panels correspond to the cases containing only nucleons and
light clusters and the full composition, respectively. The composition
includes neutrons and protons (solid lines), deuterons (short-dashed), triton
and helium (long-dashed), $\alpha$-particles (dash-dotted),
$\Delta$-resonances (dash-double-dot), $\Lambda$-hyperon (dash-triple-dot),
and pions (double-dash-dot). In the figure on the right-hand-side, the
clusters disappear for $\rho/\rho_{\rm sat}\geq 9\times 10^{-2}$ (shaded area)
due to the Pauli-blocking of the phase space. The mass fraction of
$\isotope[56]{Fe}$ is not visible on the figure’s scale.

Let us consider how the thermodynamics of a mixture of clusters is included in
the description of low-density nuclear matter. If a nucleus (cluster) is
characterized by a mass number $A$ and a charge $Z$, its chemical potential is
expressed as

$\mu_{A,Z}=(A-Z)\mu_{n}+Z\mu_{p}.$ (6.3)

For the chemical potentials of heavy baryons, the relations (4.1a)-(4.1c)
still hold. The chemical potentials of the pions obey the following relations

$\displaystyle\mu_{\pi^{0}}=0,\qquad\mu_{\pi^{+}}=\mu_{p}-\mu_{n},\qquad\mu_{\pi^{-}}=\mu_{n}-\mu_{p}.$
(6.4)

The baryon number density $\rho$ (the subscript $B$ is dropped for simplicity)
and the charge neutrality conditions are changed to the following form,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\rho_{n}+\rho_{p}+\sum_{c}A_{c}\rho_{c}+\rho_{\Delta^{++}}+\rho_{\Delta^{+}}+\rho_{\Delta^{-}}+\rho_{\Delta^{0}}+\rho_{\Lambda},$
(6.5a) $\displaystyle\rho Y_{Q}$ $\displaystyle=$
$\displaystyle\rho_{p}+\sum_{c}Z_{c}\rho_{c}+2\rho_{\Delta^{++}}+\rho_{\Delta^{+}}-\rho_{\Delta^{-}}+\rho_{\pi^{+}}-\rho_{\pi^{-}},$
(6.5b)

where the summations sum the densities of the deuteron ($d$), triton ($t$),
$\isotope[3]{He}$ ($h$), $\alpha$-particle and $\isotope[56]{Fe}$ nucleus.
Equations (6.5a) and (6.5b) determine the two unknown chemical potentials
$\mu_{n}$ and $\mu_{p}$ at a given temperature $T$ for fixed values of $\rho$
and $Y_{Q}$.

As an illustrative example, we show in Fig. 6.5 the composition of dilute
nuclear matter at temperatures in the range of $10\leq T\leq 30$ MeV and at
fixed density of $10^{-2}\rho_{\rm sat}$ for charge fractions $Y_{Q}=0.4$
(characteristic for supernova matter) and $Y_{Q}=0.1$ (characteristic for BNS
mergers) [368]. The clusters’ abundances are characterized by a mass fraction
$X_{i}=A_{i}\rho_{i}/\rho$, where $A_{i}$ is the mass number of a constituent
and $\rho_{i}$ is its number density. At the temperatures considered here, the
mass fraction of 56Fe is not visible on the figure’s scale; it becomes
significant only at low temperatures of the order of a few MeV [363]. The main
observation is that there is a transition from light baryons and clusters to
heavy-baryon-containing matter at temperatures of about $T_{\rm tr}\sim 30$
MeV. Once heavy baryons and pions are included the isospin asymmetry in the
neutron and proton components is reduced. This affects the abundance of the
helion and triton, which are then closer together. The transition temperature
itself is dependent on the treatment of the interactions and assumed
composition, but this appears to be a generic feature. The emergence of heavy
baryons at high temperatures is due to their excitation by temperature which
increases their statistical weight compared to nucleons.

The density dependence of the particle abundances at fixed $T=10$ MeV is shown
in Fig. 6.6. At a density of $\rho/\rho_{\rm sat}\simeq 0.1$ the clusters
abruptly disappear as a consequence of the vanishing of their binding energy
with increasing density. This manifests itself in a jump in the density of
nucleons and is also reflected in the pion mass fraction. Heavy baryons do not
appear at these low temperatures. The abruptness of the transition due to the
vanishing of the cluster binding energies depends on the details of how the
Pauli-blocking effects are modeled, in particular on the dependence of the
cluster binding energies on the center-of-mass momentum of a cluster, which
appear to be smooth in some models [343].

In closing, let us mention some of the model-independent aspects of the
studies: (a) if heavy nuclei are taken into account, they act to suppress the
abundances of light clusters at low temperatures, but are negligible at high
temperatures; (b) the phase-space occupation effectively suppresses the
abundances of light clusters at sufficiently low temperatures and for
densities roughly $\geq 0.1\rho_{\rm sat}$ due to the Pauli blocking [355,
347, 356]; for experimental studies of signatures of Pauli blocking in heavy-
ion collisions see, e.g., Ref. [372]; (c) at low enough temperatures,
deuterons can cross-over from a Bose-Einstein to Bardeen–Cooper–Schrieffer
pair condensate [360, 361, 362] as the density increases.

The discussion above suggests that the composition of matter in supernovae and
BNS mergers can be much richer than studied so far. Additional degrees of
freedom, such as heavy baryons or thermally excited mesons affect the neutrino
interactions with the matter by providing additional scattering centers and
channels of neutrino emission. The presence of heavy baryons may also affect
the transport coefficients, such as the bulk viscosity, which is relevant for
the damping of the post-merger oscillations [342, 373].

## 7 Pairing in Hypernuclear Matter

### 7.1 Pairing patterns in compact stars

Bardeen–Cooper–Schrieffer (BCS) type pairing in compact stars is a complex
problem because of the lack of information about the nuclear interactions at
relevant densities and intricate many-body effects, such as the screening of
the nuclear interaction. For the nucleonic components, in a first
approximation, the bare nucleon-nucleon (NN) interaction can be used as the
pairing force in the dominant attractive partial wave channel. Assume a
single-component, isotropic, and homogeneous Fermi system with attractive
interaction $v_{0}(p,p^{\prime})$ in the $S$-wave, where $p$ and $p^{\prime}$
are the magnitudes of the relative incoming and outgoing momenta of the
particles. According to the BCS theory, the pairing gap $\Delta_{p}$ in the
quasiparticle spectrum is given, at the mean-field level, by the gap equation

$\Delta_{p}=-\frac{1}{V}\sum_{{\bm{p}}^{\prime}}v_{0}(p,p^{\prime})\frac{\Delta_{p^{\prime}}}{2E_{p^{\prime}}}(1-2f_{p^{\prime}}),$
(7.1)

where $V$ is the volume and $f_{p}=(e^{E_{p}/T}+1)^{-1}$ is the Fermi–Dirac
distribution function at temperature $T$. The quasiparticle energy in the
paired states is given by $E_{p}=\sqrt{\varepsilon_{p}^{2}+\Delta_{p}^{2}}$,
where $\varepsilon_{p}$ is the quasiparticle spectrum in the normal state. For
nucleonic pairing at low densities, the BCS calculations can be performed with
NN-interaction models that fit the scattering data with high precision, i.e.,
by the so-called “realistic” NN interactions or chiral interactions to
sufficiently high order. Going beyond the bare driving interaction entails
including such effects as screening of the pairing interaction, which sums up
the infinite series of loops that describe the screening effects in dense
matter. The self-energy effects (renormalization of the mass of the fermion
participating in the pairing) are also included in the approximation of the
quasiparticle spectrum $\varepsilon_{p}$. It is important to note that both
the pairing interaction and the (renormalized) quasiparticle spectrum are
strongly dependent on the momentum of the particle. Therefore, solutions based
on contact-type interactions cannot be applied. The dominant pairing channel
in the case of nucleons can be identified from the inspection of the phase
shifts derived from the analysis of different partial-wave channels
${}^{2S+1}L_{J}$ of the two-nucleon scattering problem. Here $S$, $L$, and $J$
refer to the total spin, orbital angular momentum, and total angular momentum
of the two-nucleon system. Because of the large disparity between the Fermi
energies of neutrons and protons, the $T=1$ isospin pairing dominates in
neutron-rich matter. The $T=0$ pairing, which competes with $T=1$ pairing in
more isospin symmetric systems (nuclei, low-density nuclear matter away from
beta-equilibrium), is not relevant to our discussion.

To summarize, in compact stars at low densities, the $T=1$ pairing is due to
the attraction in the ${}^{1}S_{0}$ partial-wave channel and may lead to the
formation of neutron–neutron and proton-proton Cooper pairs. At higher
densities, such as the nuclear saturation density, the
${}^{3}P_{2}$–${}^{3}F_{2}$ coupled partial wave is the dominant attractive
channel for neutron–neutron pairing as the ${}^{1}S_{0}$-wave interaction
becomes repulsive. The peculiarity of the pairing in this state is that Cooper
pairs have total spin $S=1$ and that there is a multitude of ground states
which differ by the values of projection of the orbital angular momentum. If
at higher densities compact star cores develop equal populations of neutrons
and protons, $S=1$ and $T=0$ pairs may be formed in the ${}^{3}D_{2}$ channel,
which applies exclusively to neutron–proton scattering. For a more detailed
discussion of the pairing patterns in neutron stars see Ref. [374].

### 7.2 Hyperonic pairing in compact stars

By analogy to the nucleonic component in compact stars, we anticipate that the
attractive component of the nuclear force between hyperons at low energies can
lead to the formation of Cooper pairs made of hyperons. It is immediately
clear that hyperon-nucleon pairing will be suppressed by the large difference
in their masses unless, under some special conditions, their Fermi energies
match due to the disparity in their populations. We first concentrate on a
low-density hyperonic component where the attraction comes from the
interaction in the ${}^{1}S_{0}$ partial-wave channel. In this case, in
complete analogy to neutrons and protons, hyperons will form spin-singlet
$S=0$ Cooper pairs. Since the problem of determining the pairing force (i.e.,
the model for $v_{0}(p,p^{\prime})$) can be decoupled in a first approximation
from the problem of determining the quasiparticle spectrum, it has been argued
that the gap can be determined from a non-relativistic BCS equation with a
chosen two-nucleon potential, where the single-particle energies and particle
composition are computed from CDFs. In the case of nucleonic matter, such an
approach was applied first in Ref. [375] and has been validated by the
computations of finite nuclei within the relativistic Hartree–Fock–Bogoliubov
theory [136, 376]. This mismatch of the ingredients parallels the many-body
computations in non-relativistic models where a similar decoupling between the
pairing interaction and self-energies have been employed.

The pairing among $\Lambda$ hyperons was studied early on in Ref. [377]. More
recent work considered particle backgrounds that were derived from CDFs [378].
The pairing among $\Xi$ hyperons has been mentioned in Ref. [379], but its
implications on physical observables remained unexplored. The $S$-wave pairing
between types of hyperons was studied within such an approach in Ref. [380],
i.e., the composition was taken from a CDF based on the Hartree approximation
with density-dependent couplings together with $\Lambda\Lambda$ pairing
interaction taken to be the configuration space parameterization of ESC00
potentials [381], as presented by Ref. [382]. In the case of the pairing among
$\Xi^{-}\Xi^{-}$ and $\Xi^{0}\Xi^{0}$ hyperons, the interaction was taken from
Ref. [383]. It is based on the Nijmegen Extended Soft Core ESC08c potential
[384]. Note that the potentials were chosen in Ref. [380] to maximize the
attraction between $\Lambda\Lambda$ and $\Xi\Xi$, therefore the obtained gaps
are likely to be upper bounds. Pairing in the $\Sigma\Sigma$ channel can, in a
first approximation, be neglected because there is no solid evidence for
attraction in this channel at low energies. This is also reflected in the fact
that the ESC08c potential for the $\Sigma\Sigma$ channel is repulsive.

Figure 7.1: Dependence of ${}^{1}S_{0}$ hyperon pairing gaps on Fermi
momentum, according to Ref. [380]. Three compositions are considered based on
the DDME2 (solid), NL3 (dashed), and GM1A (long-dashed) CDFs. The left panel
shows the $\Lambda$ hyperons gaps, the right one — $\Xi^{0,-}$ hyperon gaps.
In the case of the DDME2 CDF, the $\Xi^{0}$ hyperon is absent in matter.

For isotropic $S$-wave pairing, the gap equation for a specific hyperon ($Y$)
channel reads

$\Delta_{Y}(p)=-\frac{1}{4\pi^{2}}\int dp^{\prime}p^{\prime
2}\frac{V_{YY}(p,p^{\prime})\Delta_{Y}(p^{\prime})}{\sqrt{\left[E^{Y}(p^{\prime})-\mu_{Y}\right]^{2}+\Delta_{Y}^{2}(p^{\prime})}},$
(7.2)

where $E(p)$ denotes the single-particle energy of hyperon $Y$. For a Hartree
CDF, this energy is given by

$E^{Y}(p)=\sqrt{p^{2}+m_{Y}^{*2}}+g_{\omega Y}\omega+g_{\phi Y}\phi+g_{\rho
Y}\tau_{3Y}\rho+\Sigma_{R}.$ (7.3)

The quantity $\mu_{Y}$ is the chemical potential and $m_{Y}^{*}$ is the Dirac
effective mass of hyperon $Y$. In the case of ${}^{1}S_{0}$ pairing, the
interaction in momentum space is related to that in configuration space
$V_{YY}(r)$ via the relation

$V_{YY}(p,p^{\prime})=\langle p|V_{YY}|p^{\prime}\rangle=4\pi\int
drr^{2}j_{0}(pr)V_{YY}(r)j_{0}(p^{\prime}r),$ (7.4)

where $j_{0}(pr)=\sin(pr)/(pr)$ is the spherical Bessel function of order
zero. Solution of the gap equation (7.2) are shown in Fig. 7.1 for several
different choices of background CDFs — the DDME2 model discussed above, as
well as the non-linear models NL3 of Ref. [385] and GM1A of Ref. [34]. The use
of different background models allows us to assess the effects of different
compositions of matter on the pairing gaps, which are derived from the same
pairing interaction. The shapes of the pairing gaps resemble those of the
nucleonic gaps: the increase of the gaps at low densities is caused by the
increase in the density of states; the drop-off at high densities is caused by
the decrease of the pairing force at high densities in the $S$-waves. The
reduction of the hyperon masses at high densities also contributes to the
reduction of the gaps as the density increases. The stronger reduction of the
$\Lambda\Lambda$ pairing gap compared to the $\Xi^{-}\Xi^{-}$ and
$\Xi^{0}\Xi^{0}$ gaps can be attributed to the stronger reduction of the
effective mass of $\Lambda$s. The $\Lambda$ $S$-wave pairing gap vanishes at
densities that are relevant phenomenologically. At higher densities, $P$-wave
$\Lambda$-pairing is a possibility which will affect the cooling behavior of
hypernuclear stars [386]. The pairing of $\Lambda$s, as well as protons in the
high-density regions of neutron stars, remains however an open issue.

Figure 7.2: Critical temperature of $\Lambda$ superfluidity as a function of
radial distance inside of $1.8M_{\odot}$ and $2\,M_{\odot}$ mass stars for two
CDFs [386]. $P$-wave pairing appears in the $2\,M_{\odot}$ mass star, as shown
by the dash-dotted lines.

Qualitative estimates of the gaps of baryons in higher partial waves can be
made using the knowledge of the $P$-wave phase-shifts and gaps in neutron
matter [380, 386]. In the case of protons, the isospin invariance of nuclear
forces suggest that ${}^{3}P_{2}$-${}^{3}F_{2}$ pairing should emerge for a
sufficiently dense proton component. A simple rescaling of the density of
states allows one to obtain the proton ${}^{3}P_{2}$-${}^{3}F_{2}$ pairing
gap, which amounts to replacing the neutron Landau mass $m_{n}^{*}$ by the
proton Landau mass $m_{p}^{*}$, which results in a rescaling of the
dimensionless pairing interaction by the factor
$\alpha_{p}^{-1}=m^{*}_{p}/m^{*}_{n}$. As a result, the weak-coupling estimate
for the $P$-wave pairing gap for protons can be written as

$\displaystyle\Delta_{p}=E_{{F_{p}}}\left(\frac{\Delta_{n}}{E_{{F_{n}}}}\right)^{\alpha_{p}},$
(7.5)

where $E_{{F_{i}}}$ is the Fermi energy of baryon $i$. This procedure provides
an elementary estimate of the $P$-wave pairing gap for protons under the
assumption that the BCS weak-coupling formula
$\Delta_{i}=\epsilon_{Fi}\exp[-1/(\nu_{i}V_{i})]$ can be applied. Here
$\nu_{i}$ and $V_{i}$ denote the density of states and the pairing matrix
element of a baryon of type $i$.

Similar estimates can be made for $P$-wave pairing among $\Lambda$’s. Applying
the same argument as in the case of protons and additionally rescaling the
pairing interaction among neutrons by a factor of $2/3$ following the spirit
of SU$(6)$ spin-flavor quark model. Combining all the factors one finds that
the dimensionless pairing interaction is rescaled by the factor
$\alpha_{\Lambda}^{-1}=2m_{\Lambda}^{*}/3m_{n}^{*}$ and, consequently,

$\displaystyle\Delta_{\Lambda}=E_{{F_{\Lambda}}}\left(\frac{\Delta_{n}}{E_{F_{n}}}\right)^{\alpha_{\Lambda}}.$
(7.6)

The inclusion of the higher partial-wave pairing may have an important
influence on the cooling of compact stars, as will be discussed later in Sec.
8. However, we point out here already that for the CDF parameterizations
DDME2, NL3 and GM1A studied in Ref. [386], the ${}^{3}P_{2}$-${}^{3}F_{2}$
proton superfluidity has only a marginal effect on the cooling, since in the
innermost regions of the stars the critical temperature is below the
characteristic stellar temperature. In the case of $\Lambda$s, their
concentration could be large enough so that one finds a $P$-wave $\Lambda$
superfluid in the inner core of a compact star. Figure 7.2 illustrates the
critical temperature of $\Lambda$ pairing inside of a $2M_{\odot}$ star for
two models for the EoS.

Better models for the $P$-wave pairing of protons and $\Lambda$s are needed to
draw more definitive conclusions about the role of superfluidity in the deep
interiors of baryonic compact stars.

## 8 Cooling of Hypernuclear Stars

### 8.1 Neutrino radiation reactions

The long-term cooling of compact stars serves as a sensitive indicator of
their interior composition [10, 387, 388, 389, 13, 390]. It is characterized
by neutrino emission from the bulk of the stellar interior during the first
$\sim 10^{5}$ years after the star’s birth, which is followed by late-time
photon cooling from the surface, assuming that there is no internal heating
mechanism operating at any stage of evolution. With the advent of CDFs, which
were tuned to reproduce the available astrophysical and laboratory data, it
became possible to perform simulations of cooling of compact stars with
hyperonization in a more constrained way than was ever possible before [380,
386, 391, 392, 393].

Hyperonic matter cools via the direct Urca (dUrca) processes [394],

$\displaystyle\Lambda$ $\displaystyle\to$ $\displaystyle p+l+\bar{\nu}_{l},$
(8.1a) $\displaystyle\Sigma^{-}$ $\displaystyle\to$
$\displaystyle\left(\begin{array}[]{c}n\\\ \Lambda\\\
\Sigma^{0}\end{array}\right)+l+\bar{\nu}_{l},$ (8.1e) $\displaystyle\Xi^{-}$
$\displaystyle\to$ $\displaystyle\left(\begin{array}[]{c}\Lambda\\\ \Xi^{0}\\\
\Sigma^{0}\end{array}\right)+l+\bar{\nu}_{l},$ (8.1i) $\displaystyle\Xi^{0}$
$\displaystyle\to$ $\displaystyle\Sigma^{+}+l+\bar{\nu}_{l},$ (8.1j)

where $l$ stands for a lepton, either electron or muon, and $\bar{\nu}_{l}$ is
the associated anti-neutrino.

There exist density thresholds for these processes to operate, which are
dictated by the kinematics involved in these reactions. But these densities
are very low for hyperons, i.e., the reactions start operating at a density
that is slightly above the onset density for a hyperon participating in a
reaction. Baryon pairing is known to suppress the rates of the Urca (and other
baryonic) processes. Therefore, another unknown in the cooling simulations is
the magnitude of the gaps in the spectra of various hyperons [380, 386].

In addition to the dUrca processes, the neutrino emission via the Cooper pair-
breaking and formation (PBF) mechanism

$\displaystyle\\{YY\\}\to Y+Y+\nu+\bar{\nu},\quad
Y+Y\to\\{YY\\}+\nu+\bar{\nu},$ (8.2)

contributes to neutrino emission, where $\\{YY\\}$ stands for a hyperonic
Cooper pair. Note that this charge neutral reaction produces neutrino–anti-
neutrino pairs of all flavors. One needs to include the PBF processes also for
the neutron $S$\- and $P$-wave superfluids and proton $S$-wave superconductor.
The expression for the $S$-wave $\Lambda$ condensate emission can be easily
obtained from the analog expression for the neutrons with appropriate changes
in the condensate parameters and the weak charges, see Ref. [380]. The
emission from the $P$-wave hyperonic condensate can be obtained, in a similar
fashion, from the emission rate derived for neutrons in Ref. [395].

### 8.2 Cooling tracks

Despite the uncertainties in the pairing gaps for baryons, especially in the
high-density patterns of pairing, it is possible to obtain some generic
insights into the cooling of compact stars [380, 386]. To quantify the cooling
evolution models that are based on an alternative parameterization of the
DDME2 CDF were used with the $\sigma$ meson couplings to hyperons defined by
$R_{\sigma\Lambda}=0.6154$, $R_{\sigma\Sigma}=0.474$, and
$R_{\sigma\Xi}=0.3259$. This, alternative, choice leads to larger populations
of $\Sigma$-hyperons [380]. As a consequence, $\Sigma^{-}$ appears in larger
amounts and at lower densities than in the models discussed in Sec. 3.

There is a mass hierarchy with respect to cooling behavior. The lightest stars
that contain hyperons with $M/M_{\odot}\gtrsim 1.5$ cool via the Urca process
$\Lambda\to p+l+\bar{\nu}_{l}$. The appearance of the $\Xi^{-}$ in slightly
more massive stars opens a competing cooling channel via
$\Xi^{-}\to\Lambda+l+\bar{\nu}_{l}$, the degree of its efficacy depending on
the pairing gaps of the $\Xi^{-}$. For very massive stars with
$M/M_{\odot}\sim 2$, the $S$-wave gaps of protons and $\Lambda$’s will vanish
at high densities [because the interactions will become repulsive at high
(Fermi) energies]. There are three alternatives in this case: (a) both
components are non-superfluid and therefore the Urca process involving
$\Lambda$s and protons (8.1a) will be the dominant one; (b) both components
are superfluid, and there is a competition between processes on $\Lambda$s and
$\Xi$s; (c) the third possibility, which is the more likely case, would be
that the proton $P$-wave gap is too small to be relevant so the suppression of
neutrino emission is solely due to $\Lambda$ pairing and the process (8.1a)
dominates as in the case (a). Note that here and in all models that will be
discussed below the $(\Sigma^{-},n)$ channel (8.1e) is energetically
forbidden. This is not surprising given the large difference in their
abundances.

Cooling simulations for DDME2 CDF are shown in Fig. 8.1 to illustrate some
generic features that are observed also for other CDFs. For this CDF model,
the allowed neutrino emission channels as a function of density are as
follows:

  * •

the $(\Lambda,p)$ process (8.1a) is active in the density range
$0.34\leq\rho_{b}\leq 1.02$ fm-3 for stars with $M/M_{\odot}\geq 1.4$,

  * •

the $(\Xi^{-},\Lambda)$ channel from Eq. (8.1i) is active in the range
$0.37\leq\rho_{b}\leq 0.98$ fm-3 for stars with $M/M_{\odot}\geq 1.54$,

  * •

the $(\Sigma^{-},\Lambda)$ channel from Eq. (8.1e) is active in the range
$0.39\leq\rho_{b}\leq 0.60$ fm-3 for stars with masses in the range $1.6\leq
M/M_{\odot}\leq 2$.

The interior-to-surface temperature relation adopted in these simulations
assumes that the surface is made of iron, which predicts a lower surface
temperature for a given interior temperature than is the case for a
light–element atmosphere. However, it is expected that young compact objects
may have some light components in their atmospheres such as hydrogen, helium,
or carbon. For example, the compact central object (CCO) in Cassiopeia A with
an estimated age of $t\sim 330$ yr, or XMMU J173203.3-344518 with $t\sim
2.7\times 10^{3}$ yr, are expected to be such stellar objects. Therefore,
these simulations are likely to underestimate the actual temperatures of young
stars.

The cooling tracks corresponding to the constant mass models within the mass
range $1\leq M/M_{\odot}\leq 1.85$ cover the observed range of temperatures
well [380, 386], thus showing a clear mass hierarchy of the cooling behavior
of compact stars in the neutrino cooling era, $t\leq 10^{5}$ yr.
Interestingly, this hierarchy is inverted at the very early ($t\leq 10$ yr)
stages of thermal evolution, which is however observationally insignificant.

Let is consider some more specific features of cooling tracks. An inspection
of the simulation curves shows that switching on and off neutron $P$-wave
pairing has an important effect on the cooling of low-to-intermediate mass
stars, as this pairing significantly reduces the heat capacity of the core and
induces one of the dominant cooling processes via PBF. For the late cooling
era of $t\geq 10^{5}$ yr, one finds that the cooling tracks computed for zero
neutron $P$-wave pairing gaps are in better agreement with the data than the
cooling tracks computed for non-zero gaps.

Figure 8.1: Dependence of surface temperature of compact stars on their age
for models that are based on the DDME2 CDF EoS with masses in the range $1\leq
M/M_{\odot}\leq 2$ [386], as indicated in the plot. The left panels (a) and
(c) use the neutron ${}^{1}S_{0}$ gap from Ref.[396], proton ${}^{1}S_{0}$ gap
from Ref.[397], and we assume that the neutron $P$-wave pairing gap is zero.
The right panels use the same proton gap, but the $S$\- and $P$-wave neutron
pairing gaps are from Ref. [398]. The $\Lambda$ ${}^{1}S_{0}$-pairing gaps are
included in all panels and are based on the BCS computations of Ref. [380].
The $\Lambda$ ${}^{3}P_{2}$-gaps are included in panels (c) and (d) and are
obtained from a scaling procedure described in Ref. [386], which uses the
neutron $P$-wave results of Refs. [399] in panel (c) and those of Ref. [398]
in panel (d). For the observational data and references, see Fig. 1 of Ref.
[400]. The arrows indicate upper limits.

Let us turn to the role of the hyperonic component. As discussed above, the
hyperonic dUrca processes start to operate in stars with masses above the
canonical mass of $M/M_{\odot}\geq 1.4$. Thus, stars in the mass range
$1.4\leq M/M_{\odot}\leq 1.6$ would cool down via the $(\Lambda,p)$ channel,
which is the only active dUrca process in their interiors. For more massive
stars, $M/M_{\odot}\geq 1.7$, several reactions are triggered of which the
$(\Sigma^{-},\Lambda)$ channel is the dominant one. Nevertheless, the cooling
of stars with $M/M_{\odot}\leq 1.85$ is not that fast because of the presence
of a proton and/or $\Lambda$ superfluidity.

For stars with a larger mass, the central density exceeds the density at which
the $\Lambda$ $S$-wave gap disappears. In this case, the stars cool fast via
the $(\Sigma^{-},\Lambda)$ dUrca processes, because at high densities
$\Lambda$s cannot pair in the $S$-wave state. However, if one allows for a
higher $P$-wave pairing of $\Lambda$s, this slows down the cooling of the
$M/M_{\odot}=2$ model, as seen in Fig. 8.1. The $\Lambda$ $P$-wave pairing
scaled from the neutron $P$-wave pairing results of Ref. [398] (right panel)
is less effective than the one obtained from that of Ref. [399] (left panel).
The reason is that the latter spans a wider range of densities and its maximum
is located at a larger value.

Given the mass hierarchy, the observations of the surface temperatures of
neutron stars can be explained by the variation of their masses within their
population (the light objects are hot, and the heavier ones are cold). Note
that the massive models may not develop normal cores of hyperons due to the
possible pairing in the $P$-wave channel in high-density matter, in which case
fast cooling will not take place [386]. The pairing in the hyperonic sector
remains the main unknown for cooling simulations of hypernuclear stars. Given
this uncertainty, some studies neglect the hyperonic pairing altogether [391,
392].

There are uncertainties in the studies of the cooling of neutron stars that
are unrelated to the hyperonic component, which we list here for completeness.
These include the composition of the atmosphere [401], which substantially
affects the surface temperature of a star, and the pairing gaps of neutrons
and protons in the domains where interactions are attractive [362]. Large
magnetic fields are a contributing factor, too, as they dissipate sufficient
energy to heat up a star [402], which can counterbalance the fast cooling by
Urca processes [403, 404]. Additional heating may arise due to the frictional
effects between the superfluid and normal components inside of a neutron star
[405].

## 9 Universal Relations

### 9.1 Relations for static compact stars

The integral quantities of a compact star such as the mass, radius, moment of
inertia, quadrupole moment, etc. sensitively depend on the EoS. However,
universal relations among some integral quantities have been established [406,
407, 408, 289, 409, 410] and have been intensively studied in recent years.
The universality of the relations refers to the fact that they are highly
insensitive to the input EoS. The pair-wise universal relations between the
(dimensionless) moment of inertia, quadrupole moment, and tidal deformability
were established in Ref. [411], which raised interest in universal relations
among and beyond these quantities under various conditions such as rapid
rotation, strong magnetic fields, etc. Specifically, universal relations have
been derived for cold non-rotating stars [412, 413, 414, 415, 416, 417, 418,
65, 419], slowly and rapidly rotating stars [420, 421, 422, 423, 424, 425,
426], magnetized stars [427], finite temperature stars [65, 425], as well as
compact stars in a binary [428]. The universal relations have also been
studied in alternative theories of gravity [429, 430, 431, 432]. For a review,
see Ref. [433]. The reason for this universal behavior is not yet fully
understood, but its practical use is undisputed, as it helps us constrain
quantities that are difficult to access observationally and eliminate the
uncertainties associated with the EoS when analyzing data. For example, in the
context of gravitational-wave analysis, the universalities allow one to break
degeneracies between integral quantities, e.g., the quadrupole moment and the
neutron-star spins in binary inspiral waveforms.

The universalities are proven empirically by testing a large collection of EoS
that are already constrained by astrophysical and nuclear laboratory data. The
so-called $I$-Love-$Q$ relations for non-rotating compact stars can be
numerically fitted with a polynomial on a log scale [411]:

$\displaystyle\ln y=a_{0}+a_{1}\ln x+a_{2}\,(\ln x)^{2}+a_{3}\,(\ln
x)^{3}+a_{4}\,(\ln x)^{4},$ (9.1)

where pairs $(x,y)$ represent $(\Lambda,\bar{I})$, $(\Lambda,\bar{Q},)$ and
$(\bar{Q},\bar{I})$. The normalized quantities are defined as
$\bar{I}=I/M_{G}^{3}$, $\bar{Q}=QM_{G}/J^{2}$, where $M_{G}$ is the
gravitational mass and $J$ stands for the angular momentum. In Fig. 9.1, we
plot the results for $\bar{I}(\Lambda)$ and $\bar{Q}(\Lambda)$ for compact
stars constructed using EoS with different matter compositions – nucleonic
(labeled $N$), hypernuclear ($NY$) and hypernuclear–$\Delta$-resonance-admixed
($NY\Delta$). The EoS collection includes Hartree (based on DDME2
parameterization) and Hartree-Fock (based on PKO3 parameterization) CDF
modeling of dense stellar matter, with the radius and the maximum mass,
respectively, covering $12\lesssim R_{1.4}\lesssim 14$ km and $2.0\lesssim
M_{\rm max}\lesssim 2.5\,M_{\odot}$, where $R_{1.4}$ is the radius of the
$M=1.4M_{\odot}$ star. It is seen in Fig. 9.1 that the maximum absolute
fractional difference is about $1\%$. Clearly, the universality holds for the
third pair $\bar{I}-\bar{Q}$ as well.

Figure 9.1: The $I$-$\Lambda$ and $Q$-$\Lambda$ relations for non-rotating
cold compact stars. The top panels show universal relations for various EoS
models and matter compositions, together with fitting curves; bottom panels
show fractional errors between the fitting curve and numerical results. The
results are derived from a collection of EoS models with nucleonic ($N$),
hypernuclear ($NY$), and hypernuclear-$\Delta$-admixed ($NY\Delta$)
compositions. The Hartree models are based on the DDME2 parameterization (open
symbols), the Hartree-Fock models are based on the PKO3 parameterization
(filled symbols). The bottom panels show fractional errors between the fitting
curve and numerical results.

The search for universal relations among global parameters of compact stars
stretches back to the work of Refs. [434, 94], who established the
universality for the combination involving the moment of inertia
$\tilde{I}=I/\left(M_{G}R^{2}\right)$, specifically

$\tilde{I}\approx
0.21(1-2C)^{-1},\qquad\tilde{I}=c_{0}+c_{1}C+c_{2}C^{2}+c_{3}C^{3}+c_{4}C^{4},$
(9.2)

with the compactness of a star $C=M_{G}/R$. For the moment of inertia
$\bar{I}=I/M_{G}^{3}$ (introduced above in the context of $I$-Love-$Q$
relations) universality was also established according to the relation [421]

$\bar{I}=a_{1}C^{-1}+a_{2}C^{-2}+a_{3}C^{-3}+a_{4}C^{-4}.$ (9.3)

Reference [435] showed the universality of compactness as a function of
normalized tidal deformability according to

$C=b_{1}+b_{2}\ln{\Lambda}+b_{3}\left(\ln{\Lambda}\right)^{2}.$ (9.4)

Polynomial fits for $\bar{Q}$ as a function of inverse compactness $C^{-1}$
were found in Ref. [65],

$\bar{Q}=e_{0}+e_{1}C^{-1}+e_{2}C^{-2}+e_{3}C^{-3},$ (9.5)

which is an analog of Eq. (9.3) for $\bar{I}$.

Another interesting quantity is the compact star’s binding energy, which is
defined as the difference between the baryonic and gravitational masses,
$E_{B}=M_{B}-M_{G}$. Its insensitivity towards the input EoS model, if
normalized by the gravitational mass, was shown in Refs. [436, 421]. A
possible fit is [436]

$\frac{E_{B}}{M_{G}}=\frac{d_{1}C}{1-d_{2}C}.$ (9.6)

The values of the fitting parameters $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$,
$e_{i}$ entering Eqs. (9.2)-(9.6) can be found in tabulated from in Ref. [65].

Figure 9.2: Dependence of $\bar{Q}$ and $\bar{I}$ on the compactness $C$ at
constant $S/A=2$ and $Y_{L,e}=0.2$ for compact stars based on different
finite-temperature EoS with nucleonic only ($N$), hyperonic ($NY$), and
$\Delta$-admixed hyperonic ($NY\Delta$) compositions according to Ref. [65].
The bottom panels show fractional errors between the fitting curve and
numerical results.

Let us now turn to the discussion of universality at nonzero temperature [437,
275, 331, 65]. Reference [437] demonstrated the violation of the universality
of $I$-Love-$Q$ relations in the presence of entropy gradients that exist
during the early evolution of proto-neutron stars. Similar conclusions were
reached in Ref. [275], which suggested that these universalities are broken
when thermal effects become important. It is now firmly established that if
proto-neutron stars can be approximated by constant entropy and electron
fractions, the universalities are restored, as shown for the moment of inertia
in Ref. [331], and for all the quantities listed above in Ref. [65]. These
universalities have been established using EoS that include hyperonic
components [275, 331, 65]. The last of these references includes also
$\Delta$-resonance featuring EoS. In Fig. 9.2 we show the universality of
$\bar{I}(C)$ and $\bar{Q}(C)$ for a collection of EoS, for $N$, $NY$, or
$NY\Delta$ compositions based on DDME2 parameterization. Also shown in this
figure are the results obtained with the purely nucleonic EoS models LS220
[438] and HS(IUF) [439] as well as the hyperonic SFHoY EoS [330]. The
universality visible in Fig. 9.2 holds to high accuracy with the deviations
being at the level of a few percent. The general conclusion is that the
universality holds well for all quoted relations also at finite temperature if
the same thermodynamic conditions are considered and the precision is similar
to that of cold $\beta$-equilibrated matter. There are only a few exceptions
for the $E_{B}/M_{G}(C)$ relation (9.6), where the fits can deviate by up to
$20\%$. Clearly, the constant entropy approximation will break down to some
degree under realistic conditions. While we can be certain that there are no
physical obstacles in maintaining the universality in the finite-temperature
domain, a formulation that accounts for entropy gradients and maintains the
same thermodynamic conditions (if enforceable at all) for the finite-
temperature EoS is necessary to extend the validity of universal relations to
dynamical (time-dependent) settings of supernova and BNS mergers.

### 9.2 Relations for rapidly rotating compact stars

Figure 9.3: $\bar{Q}$-$C$ and $\bar{I}$-$\bar{Q}$ relations for rotating cold
compact stars with different values of the normalized angular momentum. The
top panels show universal relations for various EoS models and matter
compositions, together with fitting curves. The bottom panels show fractional
errors between the fitting curve and numerical results. The EoS collection is
the same as in Fig. 9.1.

The zero- and finite-temperature universalities for hyperonic models of
compact stars discussed above can be extended further to rapidly rotating
stars without loss of accuracy [425]. When considering rapidly rotating
compact stars, the quasi-universal relations between dimensionless compactness
$C$, the moment of inertia $\bar{I}$ and quadrupole moment $\bar{Q}$ can be
constructed from a stellar sequence with fixed spin parameter $j=J/M_{G}^{2}$
[421]. In Fig. 9.3, we show the $\bar{Q}$-$C$ and $\bar{I}$-$\bar{Q}$
relations for four values of $j$. The adopted EoS models are the same as in
Fig. 9.1. The fitting curves correspond to Eqs. (9.5) and (9.1), respectively.

The universalities are of practical importance in determining the maximum
masses of compact stars from the analysis of gravitational-wave events, such
as GW170817 [82]. The fact that the merger remnant apparently collapsed to a
black hole after producing the electromagnetic emission suggests a scenario in
which initially a hypermassive compact star evolves into a supramassive,
maximally fast rotating star which then collapses into a black hole. Then,
universal relations can be used to find the maximum mass of a static star from
the deduced mass of the supramassive star, see for more details Refs. [440,
441, 442, 425]. Thus, in this context, it is important to establish the
relation between the maximum masses of maximally rapidly (Keplerian) rotating
compact stars and their static counterparts. If both stars are taken at finite
temperature and at the same thermodynamic state (for example, constant entropy
and electron faction) then the relation between these masses is universal.
However, if the static counterpart is cold, then this relation breaks down.
Now, because the universality is broken between the cold and hot masses on the
Keplerian and static branches, the theory does not work all the way for
deducing the static and cold maximum masses. Rather, one is only able to
deduce the mass of the hot counterpart of the static maximum-mass
configuration [425].

## 10 Summary and Outlook

In this review article, we presented the CDF approaches to nuclear and
hypernuclear matter and their application to the astrophysics of compact
stars. As is evident from the discussion, the density functional approach
allows one to develop a flexible theoretical framework that can be adjusted to
experimental data from laboratory and astrophysical observations. Additional
degrees of freedom, such as $\Delta$-resonances and a kaon condensate, can be
included in the CDF approach. This framework provides an avenue for further
studies towards a CDF that is highly constrained by the available data. So
far, such an approach has provided one of the few avenues to study
hypernuclear stars with significant depth, including, for example, their
structure (Secs. 4.2, 4.3), rapid rotation (Sec. 5), magnetism (Sec. 4.5), and
thermal evolution (Sec. 8.2).

The central motivation to study heavy baryonic degrees of freedom comes from
the observation of massive compact stars that are in tension with the low
values of maximum masses of compact stars with hyperons in a multitude of
models. This hyperon puzzle drew attention to CDF approaches as universal
tools for fitting the input directly to the astrophysical data, rather than
taking the circuitous route of fitting to the microscopic quantities, such as
phase shifts, on which to build a many-body theory. The uncertainties in the
CDF are easily quantified in terms of observable low-order characteristics of
nuclear matter, see Sec. 3.4.

The static properties of hypernuclear stars can be constructed from several
extensions of nucleonic CDFs in a manner that allows for massive static stars
that are consistent with the currently observed heavy millisecond pulsars. The
CDFs that involve couplings consistent with the SU(6) quark model (in the
vector meson sector) predict compact stars containing a significant fraction
of hyperons in their cores, of the order of $20\%$, with $\Lambda$s becoming
the most abundant species at asymptotically high densities, see Secs. 4.2 and
4.3. However, to interpret the 2.5-2.8 $M_{\odot}$ mass objects identified in
gravitational-wave mergers, where the primary is a massive black hole, one
needs models which assume SU(3) symmetry with extreme values of parameters,
leading to very stiff hyperonic EoS. However, these extreme SU(3) models
predict a very low fraction (on the order of a few percent) of hyperons.
Therefore, one is essentially dealing with nucleonic stars. Thus, one may
conclude that the hypernuclear stars with significant (above a few percent)
hyperons must have masses close 2.1 $M_{\odot}$. Assuming rapid rotation,
close or at the Keplerian limit, increases the stellar masses by 20$\%$, which
may also be an option for the origin of highly massive compact stars, see Sec.
5.2.

The hypernuclear CDFs can be extended straightforwardly to finite temperatures
and conditions that are relevant for supernovae and BNS mergers by including
trapped neutrinos, see Sec. 6. The count rates of nearby type-II supernovae
leave open the prospect of observations of such events in the near future. The
prospects of observations of BNS mergers, in particular their post-merger
phases that is sensitive to the composition of matter, are much better and can
potentially provide independent constraints on the CDF parameters.
Interestingly, finite temperatures promote the appearance of hyperons at low
densities, making them relevant for the description of matter below nuclear
saturation density.

It has been well established that nucleonic pairing has a crucial impact on
the cooling of neutron stars by modifying the rates of neutrino emission and
the specific heat of nucleonic matter. Similar studies for hyperons in
hypernuclear stars with modern CDFs are in the early stages. The key feature
of hyperons is the rapid direct Urca cooling, which can be moderated by the
pairing gaps in the spectra of hyperons. As discussed in Sec. 8.2, the
combined effect of direct Urca processes involving hyperons and hyperon
pairing leads to a mass hierarchy in the cooling behavior of hypernuclear
stars, such that the heavier the star, the faster it cools. This feature
accounts for a wide diversity of temperature distributions of compact stars in
their temperature–age diagram. In the future, an improved understanding of
hyperonic pairing and cooling will be needed to test CDFs and their
applications to the thermal evolution of compact stars.

Universal relations among the various integral parameters of neutron stars
have been established during the last decades. They have become useful for
making EoS-independent predictions and inferences. Hypernuclear stars obey
these universalities both in the static and rapidly rotating limits with an
accuracy that is comparable to their nucleonic counterparts, see Sec. 9.
Therefore, methods and astrophysical scenarios that are based in part or
entirely on the assumption of universality can be applied to hypernuclear
stars as well. They can also be applied at finite temperatures to isentropic
stars with the same compositions. A deeper understanding of the physical
origins of universal relations will be needed to reveal their insensitivity to
the internal composition (e.g., nucleonic vs. hyperonic and $\Delta$-admixed)
and identify the astrophysical signatures of hypernuclear stars that
distinguish them from nucleonic stars. This will eventually allow one to
answer the fundamental question of the existence of hyperons in compact stars.

To conclude, the hyperonization of dense matter and the astrophysics of
compact stars featuring hyperons and $\Delta$-resonances is an enormously
vivid and fruitful field of research, with a strong relation to the
observational multimessenger astronomy of compact stars. The recent advances
on the observational front, which include the observation of gravitational
waves from BNS mergers, notably GW170817, simultaneous measurements of masses
and radii of neutron stars, and the discovery of very massive pulsars motivate
further improvements to existing theoretical models and exploration of their
implications for novel astrophysical scenarios. The ongoing and future
hypernuclear programs at CERN, BNL, JLab, GSI-FAIR, and J-PARC, and elsewhere
will eventually provide us with extremely valuable new insight into
hypernuclei. This can be used to narrow down the parameters of CDFs but also
constrain other theoretical models. The results achieved so far suggest that
the combination of theoretical work and observational progress has the
potential to reveal the detailed features of high-density matter in compact
stars in the near future.

## Acknowledgments

It is a great pleasure to thank M. Alford, G. Colucci, A. Harutyunyan, W.
Long, M. Oertel, A. Raduta, M. Sinha, and V. Thapa for collaborations from
which in part this review derives. A. S. acknowledges support by Deutsche
Forschungsgemeinschaft Grant No. SE 1836/5-2 and the Narodowe Centrum Nauki
(NCN) Grant No. 2020/37/B/ST9/01937 at Wrocław University. J. L. is supported
by the National Natural Science Foundation of China (Grant No. 12105232), the
Venture & Innovation Support Program for Chongqing Overseas Returnees (Grant
No. CX2021007), and by the Fundamental Research Funds for the Central
Universities (Grant No. SWU-020021). F. W. acknowledges support from the U.S.
National Science Foundation under Grant PHY-2012152.

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