# Bayesian inference of thermal effects in dense matter within the covariant
density functional theory

Adriana R. Raduta<EMAIL_ADDRESS>Mikhail V. Beznogov
<EMAIL_ADDRESS>Micaela Oertel<EMAIL_ADDRESS>National
Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest,
Romania  LUTH, Observatoire de Paris, Université PSL, CNRS, Université de
Paris Cité, 92190 Meudon, France

###### Abstract

The high temperatures reached in a proto-neutron star or during the post-
merger phase of a binary neutron star coalescence lead to non-negligible
thermal effects on the equation of state (EOS) of dense nuclear matter. Here
we study these effects within the covariant density functional theory
employing the posteriors of a Bayesian inference, which encompasses a large
sample of EOS models. Different densities and temperatures are considered. We
find that for a number of quantities thermal effects are strongly correlated
with the Dirac effective mass ($m^{*}$) of the nucleons and/or its logarithmic
derivative as a function of density. These results can be explained within the
low temperature approximation though they survive beyond this limit.

###### keywords:

Equation of state , Hot and dense matter , Thermal effects , Covariant density
functional

##  1 Introduction

Numerical simulations of core-collapse supernovae and the subsequent evolution
of the proto-neutron star into a mature cold neutron star (NS), simulations of
the formation of stellar black holes and of binary neutron star (BNS) mergers
require detailed knowledge of the equation of state (EOS) over wide domains of
baryon number density $10^{-14}~{}\mathrm{fm}^{-3}\lesssim
n_{\mathrm{B}}\lesssim 2~{}\mathrm{fm}^{-3}$, temperature $0\leq T\lesssim
100~{}\mathrm{MeV}$ and electron fraction $0\leq Y_{\mathrm{e}}\lesssim 0.6$
[1, 2].

More than a hundred different EOS models considering non-zero temperatures
have been developed so far, see, e.g., those available in tabulated form on
CompOSE [3, 4]111https://compose.obspm.fr/. The huge majority of them rely
either on a non-relativistic potential model or on the covariant density
functional (CDF) theory to describe dense nuclear matter. They assume that the
effective interactions have the same functional form at zero and finite
temperature.

In default of constraints from nuclear experiments, and awaiting third
generation gravitational wave detectors which might be able to constrain
thermal effects in the post-merger phase of a BNS, see, e.g., [5], insight
into the thermal behavior of nuclear matter can be gained by confronting the
predictions of available models against each other and by comparison with ab
initio calculations. Pioneering works of Constantinou et al. [6, 7] have
proved that, in non-relativistic potential models, the thermal response
functions are controlled by the Landau effective masses of the nucleons and
their derivative with respect to density. This feature has, for instance, been
used to assess the importance of thermal effects in the post-merger remnant in
Ref. [8].

Models with finite range interactions have been further shown to present
traits distinct from those obtained by models with zero-range interactions [7,
9]. Non-relativistic models with finite- and zero-range interactions differ in
the predictions for the specific heat at constant volume ($C_{\mathrm{V}}$)
and the thermal index (${\Gamma}_{\mathrm{th}}$) as functions of density.
While Skyrme models predict for $C_{\mathrm{V}}$ a behavior similar to the one
of free Fermi gases, where the maximum value of 1.5 is obtained for vanishing
densities, non-relativistic models with more involved momentum dependencies
lead to a non-monotonic behavior of $C_{\mathrm{V}}$ as a function of density,
with a maximum, in excess of 1.5, occurring at finite density. Besides, in
standard Skyrme models ${\Gamma}_{\mathrm{th}}$ has a strong and monotonic
increase with the density, while in models with finite range interactions it
is almost flat. All these differences originate from the differences in the
density dependence of effective masses which, for finite-range interactions,
also depend on temperature [9]. Qualitative differences among non-relativistic
and CDF-based models are obvious in the density dependence of thermal pressure
(${p}_{\mathrm{th}}$), thermal chemical potential (${\mu}_{\mathrm{th}}$) and
thermal index. Non-relativistic models predict for the first two quantities
above a steep (zero-range interactions) or gentle (finite-range interactions)
increase with the density, while in CDF models ${p}_{\mathrm{th}}$ and
${\mu}_{\mathrm{th}}$ reach their maxima at finite densities and then
saturate. The latter feature entails a strong decrease of
${\Gamma}_{\mathrm{th}}$ at high densities. Analytic formulas derived at next-
to-leading order in an expansion in temperature have further shown that, for
all these types of models, temperature corrections to thermal state variables
can be expressed in terms of the level density parameter, Fermi momentum,
effective mass and its derivative with respect to density [9]. The
sophistication degree of the model gets reflected into the complexity of these
formulas. The simplest expressions correspond to Skyrme interactions.

Ref. [10] systematically confronted the thermal behavior of a selection of
mean field models frequently used in astrophysical numerical simulations. It
came out that constraining a Skyrme potential to reproduce the density
dependence of effective mass in ab initio models results in a thermal behavior
deviating from the one of standard models, see, e.g., the high density
increase of ${e}_{\mathrm{th}}$ and related decrease of
${\Gamma}_{\mathrm{th}}$ of NRAPR [11] in Ref. [10]. On the other hand, the
thermal response of CDF models has been shown to strongly depend on the
mesonic couplings, with ${p}_{\mathrm{th}}$ being the most affected quantity.
The role of non-nucleonic degrees of freedom has been further investigated in
Ref. [12], whose most preeminent results are the increase (decrease) of
thermal energy density (pressure) with the number of species and the fact
that, under certain conditions, thermal pressure (${p}_{\mathrm{th}}$) can
become negative. The analytic intractability of CDF models and the temperature
dependence of Dirac effective mass along with the usage of models with
different types of couplings, i.e., density-dependent, non-linear, mixed,
nevertheless prevented Ref. [10] from easily tracking the “ingredients” which
govern the thermal behavior of nuclear matter in CDF models as has been done
for the non-relativistic ones [6, 7].

The aim of this letter is to bridge this gap and verify numerically some of
the assumptions done in the analytical calculations in Ref. [9]. The thermal
behavior is studied for a large sample ($10^{5}$) of EOS models belonging to
run 1 in Ref. [13]. These models rely on a modified version of the CDF model
with simplified density dependent couplings [14] and have been generated
within a Bayesian approach. The latter aspect is essential for a thorough
sampling of the huge parameter space of this kind of model. Compliance with
most current constraints available for cold matter at various densities and
isospin asymmetries is fulfilled by accounting for information from nuclear
physics, ab initio calculations of pure neutron matter and observations of
NSs, for details see Ref. [13]. Throughout this paper natural units will be
used ($\hslash=k_{\mathrm{B}}=c=1$).

##  2 The model

###  2.1 CDF

The theoretical framework implemented in this work is the CDF theory with
density dependent couplings [15, 16]. Nucleons are treated as fundamental
particles and the interaction is described by the exchange of $\sigma$-,
$\omega$\- and $\rho$-“mesons”. The name of the meson thereby determines the
quantum numbers for the interaction channel. For the sake of brevity in the
following we provide only the most relevant information. Note that, in spite
of being omitted in the equations, the contribution of anti-particles is
accounted for in numerical calculations.

At finite temperature, $T$, the scalar and number densities of nucleons
($i=\mathrm{n,p}$) are given by

$\displaystyle n_{i}^{\mathrm{s}}$
$\displaystyle=\frac{1}{\pi^{2}}\int_{o}^{\infty}dkk^{2}\frac{m^{*}_{i}}{E_{i}(k)}f_{\mathrm{FD}}\left(E_{i}(k)-\mu^{*}_{i}\right),$
(1) $\displaystyle n_{i}$
$\displaystyle=\frac{1}{\pi^{2}}\int_{o}^{\infty}dkk^{2}f_{\mathrm{FD}}\left(E_{i}(k)-\mu^{*}_{i}\right),$
(2)

where $k$ and $E_{i}(k)=\sqrt{k^{2}+m^{*2}_{i}}$ stand for the wave number and
kinetic part of the single particle energy, respectively. The Dirac effective
masses $m^{*}_{i}$ and effective chemical potentials $\mu^{*}_{i}$ are related
to the nucleon mass $m_{\mathrm{N}}$ and the chemical potential $\mu_{i}$ via
$m^{*}_{i}=m_{\mathrm{N}}-g_{\sigma i}\bar{\sigma}$ and
$\mu^{*}_{i}=\mu_{i}-g_{\omega i}\bar{\omega}-g_{\rho
i}t_{3i}\bar{\rho}-\Sigma_{R}$, where $\bar{M}$ and $g_{Mi}$ represent the
mean field expectation value of the corresponding field $M=\sigma,\omega,\rho$
and its coupling to the nucleon $i$; $t_{3i}$ represents the third component
of isospin of species $i$ with the convention that $t_{3{\mathrm{p}}}=1/2$;
$\Sigma_{R}$ is the “rearrangement” term;
$f_{\mathrm{FD}}\left(x\right)=1/\left[1+\exp\left(x/T\right)\right]$
represents the Fermi-Dirac distribution function.

The energy density and pressure can be cast as sums of a kinetic term, an
interaction term and, in the case of pressure, also a “rearrangement” term,

$\displaystyle e$ $\displaystyle=$ $\displaystyle
e_{\mathrm{kin}}+e_{\mathrm{int}},$ (3) $\displaystyle p$ $\displaystyle=$
$\displaystyle p_{\mathrm{kin}}+p_{\mathrm{int}}+p_{\mathrm{rearrang}}.$ (4)

The kinetic terms in eqs. (3) and (4) account for the kinetic contributions of
all species,

$\displaystyle e_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle\sum_{i={\mathrm{n},p}}\frac{1}{\pi^{2}}\int_{0}^{\infty}dkk^{2}E_{i}(k)f_{\mathrm{FD}}\left(E_{i}(k)-\mu^{*}_{i}\right),$
(5) $\displaystyle p_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{i={\mathrm{n},p}}\frac{1}{\pi^{2}}\int_{0}^{\infty}\frac{dkk^{4}}{E_{i}(k)}f_{\mathrm{FD}}\left(E_{i}(k)-\mu^{*}_{i}\right).$
(6)

The interaction terms exclusively depend on mean field expectation values of
the mesonic fields

$\displaystyle e_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle\frac{m_{\sigma}^{2}}{2}\bar{\sigma}^{2}+\frac{m_{\omega}^{2}}{2}\bar{\omega}^{2}+\frac{m_{\rho}^{2}}{2}\bar{\rho}^{2},$
(7) $\displaystyle p_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle-\frac{m_{\sigma}^{2}}{2}\bar{\sigma}^{2}+\frac{m_{\omega}^{2}}{2}\bar{\omega}^{2}+\frac{m_{\rho}^{2}}{2}\bar{\rho}^{2},$
(8)

with $m_{M}$ representing the mass of meson $M$.

The “rearrangement” term in eq. (4) is given by

$p_{\mathrm{rearrang}}=n\Sigma_{R}=n\sum_{\mathrm{i=n,p}}\left(\frac{\partial
g_{\omega i}}{\partial n_{i}}\bar{\omega}n_{i}+t_{3i}\frac{\partial g_{\rho
i}}{\partial n_{i}}\bar{\rho}n_{i}-\frac{\partial g_{\sigma i}}{\partial
n_{i}}\bar{\sigma}n_{i}^{s}\right)$

and arises from the density dependence of the couplings and is essential for
thermodynamic consistency.

The mean-field expectation values of the meson fields are given by

$\displaystyle m_{\sigma}^{2}\bar{\sigma}=\sum_{i={\mathrm{n},p}}g_{\sigma
i}n_{i}^{s},~{}~{}m_{\omega}^{2}\bar{\omega}=\sum_{i={\mathrm{n},p}}g_{\omega
i}n_{i},~{}~{}m_{\rho}^{2}\bar{\rho}=\sum_{i={\mathrm{n},p}}g_{\rho
i}t_{3i}n_{i}.$

Computed from the thermodynamic identity $Ts=e+p-\sum_{i}\mu_{i}n_{i}$, the
entropy density takes the form:

$s=\frac{e_{kin}+p_{kin}-\sum_{i={\mathrm{n},p}}\mu_{i}^{*}n_{i}}{T}~{}.$ (9)

To be specific for the following discussion, thermal contributions to the
different quantities are defined as
${X}_{\mathrm{th}}=X(T,n,Y_{\mathrm{p}})-X(0,n,Y_{\mathrm{p}})$, where $n$ is
the density and $Y_{\mathrm{p}}=Y_{\mathrm{e}}$ is the proton fraction.

###  2.2 EOS models

Table 1: Values of selected nuclear matter parameters corresponding to the EOS models used in this work. Provided are the saturation density (${n}_{\mathrm{sat}}$); energy per nucleon (${E}_{\mathrm{sat}}$), compression modulus (${K}_{\mathrm{sat}}$), skewness (${Q}_{\mathrm{sat}}$) and kurtosis (${Z}_{\mathrm{sat}}$); the symmetry energy (${J}_{\mathrm{sym}}$), its slope (${L}_{\mathrm{sym}}$), compressibility (${K}_{\mathrm{sym}}$), skewness (${Q}_{\mathrm{sym}}$) and kurtosis (${Z}_{\mathrm{sym}}$); Dirac effective mass of the nucleons (${m}_{\mathrm{sat}}^{*}$), all values given for symmetric nuclear matter at ${n}_{\mathrm{sat}}$. Columns 3 and 4 list the medians and the 68% confidence interval (CI) of $10^{5}$ EOS models in the DDB∗ set, which correspond to run 1 in Ref. [13]. Characteristics of models with extreme values of ${m}_{\mathrm{sat}}^{*}$ are itemized on columns 5 and 7, respectively. Column 6 corresponds to a model with an intermediate value of ${m}_{\mathrm{sat}}^{*}$.  Par. | Unit | Median | 68% CI | DDB∗min | DDB∗med | DDB∗max
---|---|---|---|---|---|---
${n}_{\mathrm{sat}}$ | $\mathrm{fm}^{-3}$ | $0.153$ | ${}^{+0.0049}_{-0.0049}$ | $0.147$ | $0.152$ | $0.160$
${E}_{\mathrm{sat}}$ | MeV | $-16.1$ | ${}^{+0.2}_{-0.2}$ | $-15.7$ | $-15.9$ | $-15.7$
${K}_{\mathrm{sat}}$ | MeV | $247$ | ${}^{+33}_{-28}$ | $344.5$ | $228.7$ | $195.9$
${Q}_{\mathrm{sat}}$ | MeV | $-39.9$ | ${}^{+160}_{-130}$ | $153.9$ | $-124.5$ | $-268.2$
${Z}_{\mathrm{sat}}$ | MeV | $1360$ | ${}^{+410}_{-830}$ | $-3756.3$ | $1258.5$ | $1985.2$
${J}_{\mathrm{sym}}$ | MeV | $32.1$ | ${}^{+1.8}_{-1.8}$ | $32.6$ | $33.9$ | $33.7$
${L}_{\mathrm{sym}}$ | MeV | $42.3$ | ${}^{+15}_{-13}$ | $47.8$ | $59.8$ | $87.8$
${K}_{\mathrm{sym}}$ | MeV | $-105$ | ${}^{+27}_{-24}$ | $-86.1$ | $-129.5$ | $-34.3$
${Q}_{\mathrm{sym}}$ | MeV | $932$ | ${}^{+360}_{-420}$ | $605.6$ | $556.1$ | $74.8$
${Z}_{\mathrm{sym}}$ | MeV | $-6440$ | ${}^{+3100}_{-3800}$ | $-8225.5$ | $-3087.0$ | $-765.5$
${m}_{\mathrm{sat}}^{*}$ | $m_{\mathrm{N}}$ | $0.657$ | ${}^{+0.041}_{-0.045}$ | $0.516$ | $0.648$ | $0.751$

For our analysis, we investigate dense matter at finite temperature within
each of the $10^{5}$ EOS models in run 1 of Ref. [13], hereafter dubbed DDB∗,
and which has been obtained within a Bayesian approach.

For information, the values of some selected nuclear matter (NM) parameters
corresponding to the set DDB∗ of EOS models are provided in Table 1. The large
dispersion in the values of the higher order parameters $Q_{i}$, $Z_{i}$ are
representative of current uncertainties in the behavior of the EOS at high
densities, see, e.g., Refs. [17, 18, 19]. Values of NM parameters for three
particular EOS models within the set are listed, too. Two of them correspond
to EOS models with extreme values of the nucleon Dirac effective mass at
${n}_{\mathrm{sat}}$; the third model corresponds to a EOS model with an
intermediate value of ${m}_{\mathrm{sat}}^{*}$. The criterion by which the
first of these models have been selected explains why their values of
${K}_{\mathrm{sat}}$ and ${L}_{\mathrm{sym}}$ fall outside the domains
recommended in Refs. [1, 17] even if this is of no relevance for the present
work.

###  2.3 Sommerfeld pseudo-expansion

Figure 1: Correlations between the nucleon effective mass ($m^{*}$) and
effective chemical potential ($\mu^{*}$) in symmetric nuclear matter (SNM) for
various thermodynamic conditions. Models correspond to the DDB∗ set of
effective interactions.  Figure 2:  Test of validity of the low-$T$
approximation. Thermal kinetic energy density ($e_{\mathrm{kin;th}}$), thermal
kinetic pressure ($p_{\mathrm{kin;th}}$) and thermal interaction energy
($e_{\mathrm{int;th}}$) are represented as a function of temperature in
bottom, middle and top panels, respectively. Solid lines are used to
illustrate the predictions of eqs. (5), (6) and (7) with $n^{\mathrm{s}}_{i}$
computed according to eq. (1); predictions of eqs. (18), (19) and (7) with
$n^{\mathrm{s}}_{i}$ computed according to eq. (17) are depicted with dashed
lines. Left and right panels correspond to SNM with
$n=0.15~{}\mathrm{fm}^{-3}$ and $n=0.45~{}\mathrm{fm}^{-3}$. EOS models under
consideration are DDB∗min, DDB∗med and DDB∗max.

In non-relativistic potential models [6], for a given density and composition,
the temperature dependence and thus thermal effects arise exclusively from the
kinetic contributions. In CDF models, the only explicit $T$-dependence arises
from the kinetic contributions, too, but there is an additional implicit
temperature dependence via the effective mass and the scalar energy density in
the interaction terms. The latter thus contribute to the thermal effects, too.

In order to understand these effects in the different EOS related quantities,
we will now start the discussion by considering a low temperature expansion of
the EOS. At low temperatures ($T\ll\mu^{*}-m^{*}$) integrals involving the
Fermi-Dirac distribution function can be cast into a series of even powers of
$T$ using the so-called Sommerfeld expansion. In the case of CDF models, this
expansion has the generic form:

$\displaystyle I$
$\displaystyle=\int_{0}^{\infty}G(e)f_{\mathrm{FD}}\left(e-\mu^{*}\right)de$
$\displaystyle\approx\int_{0}^{\mu^{*}}G(e)de+\frac{\pi^{2}}{6}T^{2}\left.\frac{dG}{de}\right|_{e=\mu^{*}}+\frac{7\pi^{4}}{360}T^{4}\left.\frac{d^{3}G}{de^{3}}\right|_{e=\mu^{*}}+...~{},$

where $G$ depends on $m^{*}$ and both $m^{*}$ and $\mu^{*}$ depend on
temperature. This means that in this limit all quantities whose structure fits
the one in the equation above can be expressed in terms of $m^{*}$, $\mu^{*}$
and $T$ only. We emphasize that due to the $T$-dependence of $m^{*}$ and
$\mu^{*}$ our following discussion regarding the terms of different orders in
temperature is not strictly accurate as the $T$-dependence of $m^{*}$ and
$\mu^{*}$ is not considered in the expansion (hence, pseudo-expansion).

The expressions for the resulting low $T$ expansions for particle number
density, scalar density, density of kinetic energy, kinetic pressure, entropy
and specific heat within CDF models are provided in Appendix A.

Considering now that $m^{*}$ and $\mu^{*}$ are strongly correlated, see Fig.
1, thermal effects are governed by only one quantity in addition to $T$
itself, which is conveniently chosen to be $m^{*}$. This figure also shows
that $m^{*}$ increases with $T$ whereas $\mu^{*}$ decreases with $T$;
$T$-effects on $\mu^{*}$ and $m^{*}$ are equally important.

The validity of the low-$T$ approximation is investigated in Fig. 2, where
thermal contributions to kinetic energy, kinetic pressure and interaction
energy are plotted as a function of $T$. The results of eqs. (18) and (19) are
compared with those of eqs. (5) and (6). In the interaction energy, for a
given density and composition thermal effects arise due to dependence on the
scalar density. We thus show results for $e_{\mathrm{int;th}}$, with
$n_{i}^{\mathrm{s}}$ provided by eq. (17), and those obtained when instead eq.
(1) is used. For this comparison, we have considered symmetric nuclear matter
(SNM) with $n=0.15~{}\mathrm{fm}^{-3}$ and $n=0.45~{}\mathrm{fm}^{-3}$, and
show results for the three models in Table 1, with minimum, maximum and
intermediate values of ${m}_{\mathrm{sat}}^{*}$.

The following aspects are worth noticing: i) for fixed $T$, thermal effects
depend on $m^{*}$, ii) the validity domain of the low-$T$ approximation
shrinks with $m^{*}$, iii) $e_{\mathrm{kin;~{}th}}$ and
$p_{\mathrm{kin;~{}th}}$ are positive and increase with $T$, iv)
$e_{\mathrm{int;~{}th}}=-p_{\mathrm{int;~{}th}}$ is negative and its absolute
value increases with $T$, v) $|e_{\mathrm{int;th}}|<e_{\mathrm{kin;th}}$ and
the higher the temperature the less important the contribution of
interactions, vi) $p_{\mathrm{int;th}}\approx p_{\mathrm{kin;th}}$.

These results can be understood considering that the predictions of eqs.
(17)–(19) are the result of the interplay among the different terms entering
the low-$T$ expressions. For the thermodynamic conditions considered here the
terms without explicit $T$-dependence in $n^{\mathrm{s}}$, $e_{\mathrm{kin}}$
and $p_{\mathrm{kin}}$ decrease with $T$ while those $\propto T^{2}$ increase
with $T$. The terms $\propto T^{4}$ do not present a universal behavior. In
particular, the contribution to $p_{\mathrm{kin}}$ can be increasing or
decreasing. This means that even if each of these terms increases with
$m^{*}$, since they are alternating, their sum does not necessarily increase.
The opposite $T$-dependence of terms without explicit $T$-dependence and
$\propto T^{2}$ in $n^{\mathrm{s}}$ explains why the $m^{*}$-dependence of
$e_{\mathrm{int;~{}th}}$ is not the same at low and high densities. The
reduced validity domain of the low-$T$ approximation in models with large
$m^{*}$ is merely due to the much steeper increase of the terms $\propto
T^{2}$, especially for $n^{\mathrm{s}}$ and $e_{\mathrm{kin;th}}$. For DDB∗med
and DDB∗max the high temperatures bending of $p_{\mathrm{kin;~{}th}}$ computed
according to eq. (19) is caused by the decrease with $T$ of the term $\propto
T^{4}$.

##  3 Thermal effects on state variables, thermal coefficients and speed of
sound

Figure 3: Joint probability density plots for ${e}_{\mathrm{th}}$,
${p}_{\mathrm{th}}$, $\mu_{\mathrm{N;th}}$, $S/A$, $C_{\mathrm{V}}$,
$C_{\mathrm{P}}$, ${\Gamma}_{\mathrm{th}}$, $\Gamma_{\mathrm{S}}$ and
$c_{\mathrm{S}}^{2}$ vs $m^{*}$ and $q$ for the case of SNM with
$n=0.15~{}\mathrm{fm}^{-3}$ and $T=20~{}\mathrm{MeV}$, condition I. DDB∗ set
of EOSs.

Figure 4: The same as in Fig. 3 but at $n=0.15~{}\mathrm{fm}^{-3}$ and
$T=50~{}\mathrm{MeV}$, condition II.

Figure 5: The same as in Fig. 3 but at $n=0.6~{}\mathrm{fm}^{-3}$ and
$T=20~{}\mathrm{MeV}$, condition III.

Figure 6: The same as in Fig. 3 but at $n=0.6~{}\mathrm{fm}^{-3}$ and
$T=50~{}\mathrm{MeV}$, condition IV.

Let us now consider the full set of EOS models and investigate thermal effects
for different quantities. As above, only SNM will be considered.
Generalization to nuclear matter with various degrees of isospin asymmetry,
i.e., neutron to proton ratios, is straightforward.

Figs. 3-6 show different thermal quantities as a function of $m^{*}$ with the
aim of identifying potential correlations. Correlations with

$q=\frac{m^{*2}}{\mu^{*2}}\left(1-\frac{3n}{m^{*}}\frac{dm^{*}}{dn}\right),$
(10)

that appear in the Constantinou’s et al. next to leading order corrections to
thermal state variables [9], e.g., ${p}_{\mathrm{th}}$, ${\mu}_{\mathrm{th}}$,
$C_{\mathrm{P}}$, are examined as well. Please note that at variance with Ref.
[9], we consider here the finite-$T$ value of $m^{*}$ and $\mu^{*}$. This
choice takes into account the small increase of $m^{*}$ with temperature of
$\sim$ 2% to 8% (at $T=50$ MeV), depending on the model. In particular, we
noticed that stronger increase of $m^{*}$ with temperature corresponds to the
models with low values of $m^{*}(T=0)$. While not illustrated here,
correlations with zero temperature values of $m^{*}$ and $q$ have been
analyzed, too, and found to be qualitatively quite similar and of comparable
strength as those depicted here. This result is not surprising given the
magnitude of the above-mentioned increase of $m^{*}$ with temperature. Largely
different thermodynamic conditions are considered: I:
($n=0.15~{}\mathrm{fm}^{-3}$, $T=20~{}\mathrm{MeV}$), II:
($n=0.15~{}\mathrm{fm}^{-3}$, $T=50~{}\mathrm{MeV}$),
III:($n=0.6~{}\mathrm{fm}^{-3}$, $T=20~{}\mathrm{MeV}$) and IV:
($n=0.6~{}\mathrm{fm}^{-3}$, $T=50~{}\mathrm{MeV}$). The third set of
conditions (III) thereby lies within the validity domain of the low-$T$
approximation for all EOS models, whereas the others do not.

The different quantities we are studying here are the thermal energy density
(${e}_{\mathrm{th}}$), thermal pressure (${p}_{\mathrm{th}}$), thermal
chemical potential ($\mu_{\mathrm{th;N}}$), entropy per baryon ($S/A$),
thermal index,

${\Gamma}_{\mathrm{th}}=1+\frac{{p}_{\mathrm{th}}}{{e}_{\mathrm{th}}};$ (11)

heat capacities per nucleon at constant volume

$C_{\mathrm{V}}=T\left.\left(\frac{\partial\left(S/A\right)}{\partial
T}\right)\right|_{V,\\{N_{i}\\}},$ (12)

and at constant pressure

$C_{p}=T\left.\left(\frac{\partial\left(S/A\right)}{\partial
T}\right)\right|_{P,\\{N_{i}\\}}=C_{\mathrm{V}}+\frac{T}{n^{2}}\frac{\left.\left(\frac{\partial
P}{\partial T}\right|_{n}\right)^{2}}{\left.\frac{\partial P}{\partial
n}\right|_{T}}~{};$ (13)

adiabatic index,

$\Gamma_{\mathrm{S}}=\left.\frac{\partial\ln P}{\partial\ln
n}\right|_{S}=\frac{C_{\mathrm{P}}}{C_{\mathrm{V}}}\frac{n}{P}\left.\frac{\partial
P}{\partial n}\right|_{T}~{};$ (14)

and speed of sound squared (in units of squared speed of light)

$c_{\mathrm{S}}^{2}=\left.\frac{dP}{de}\right|_{S,A,Y_{\mathrm{p}}}=\Gamma_{\mathrm{S}}\frac{P}{e+P}.$
(15)

In eqs. (12) – (15) $S$, $V$, $N_{i}$ and $A$ denote the total entropy,
volume, total number of $i$-particles and total number of particles
$A=\sum_{i}N_{i}$.

For all thermodynamic conditions ${e}_{\mathrm{th}}$, $S/A$ and
$C_{\mathrm{V}}$ perfectly scale with $m^{*}$. ${e}_{\mathrm{th}}$ and $S/A$
feature strong, positive and almost linear correlations with $m^{*}$. Within
the validity domain of the low-$T$ approximation their behavior is perfectly
understandable. The results corresponding to $n=0.15~{}\mathrm{fm}^{-3}$ show
that these correlations survive even beyond the domain of validity of that
approximation. The situation of $C_{\mathrm{V}}$ is particularly interesting
as the shape and sign of the correlation changes. This behavior is
attributable to the opposite $m^{*}$-dependencies of the kinetic and
interaction terms of the energy density at densities lower or equal to
${n}_{\mathrm{sat}}$, see Fig. 2. For condition II, $C_{\mathrm{V}}$ has
values in excess of 1.5. While the same happens in the case of non-
relativistic models with finite-range interactions [7], here the explanation
might be different. Condition II approaches the limit of a relativistic ideal
gas, for it $C_{\mathrm{V}}$ can exceed the value of 1.5. Whenever the
degenerate gas limit is approached, i.e., at low temperatures, $S/A\simeq
C_{\mathrm{V}}\simeq C_{\mathrm{P}}$, which is confirmed for condition III,
see Fig. 5. For large $n$-values strong and positive correlations between
$C_{\mathrm{P}}$ and $m^{*}$ occur also at high temperatures. The fact that
the correlations between ${p}_{\mathrm{th}}$ and $m^{*}$ are weaker than the
correlations between ${e}_{\mathrm{th}}$ and $m^{*}$ is attributable to the
“rearrangement” term. Indeed, extra dependencies introduced by the couplings
and their density derivatives result in a ”scattering” of the
${p}_{\mathrm{th}}$ dependence on $m^{*}$. We note that correlations between
${p}_{\mathrm{th}}$ and $m^{*}$ manifest only at high densities. A sizable
dependence of $\mu_{\mathrm{n;th}}$ on $m^{*}$ manifests as well, though it
qualitatively changes with $n$ and $T$.

A number of strong correlations with $q$ occur as well. Those involving
${p}_{\mathrm{th}}$ and $\mu_{\mathrm{th;N}}$ at ($n=0.6~{}\mathrm{fm}^{-3}$,
$T=20~{}\mathrm{MeV}$) are expected given that in this case the system is
close to degenerate [9]. In this limit non-negligible positive correlations
among $q$ and ${e}_{\mathrm{th}}$; $S/A$; $C_{\mathrm{V}}$; $C_{\mathrm{P}}$
are to be noticed as well. The remaining figures show that some of those
correlations, e.g., $q-\mu_{\mathrm{th;N}}$, persist even outside the validity
domain of the low-$T$ approximation. Particularly interesting is the case of
${\Gamma}_{\mathrm{th}}$, for which a strong positive linear correlation with
$q$ is obtained under all circumstances. With the exception of
($n=0.15~{}\mathrm{fm}^{-3}$, $T=20~{}\mathrm{MeV}$) this result can be
ascribed to the good correlations between $q$ and ${e}_{\mathrm{th}}$;
${p}_{\mathrm{th}}$. At $n=0.15~{}\mathrm{fm}^{-3}$ a clear correlation exists
between $q$ and $c_{\mathrm{S}}^{2}$, which means that, whenever the
temperatures is high, the EOS stiffness is regulated by the density dependence
of $m^{*}$.

##  4 Summary

In this work we have addressed the EOS dependence of finite-$T$ effects in hot
and dense matter built upon a CDF approach with density dependent couplings.
$10^{5}$ EOS models previously generated within a Bayesian approach [13] to
cold NM have been employed. They comply with current constraints from nuclear
physics, ab initio calculations of pure neutron matter and NS observations and
ascertain a thorough sampling of the parameter space of this type of EOS
models.

We found that thermal contributions to state variables and thermal
coefficients depend mainly on the Dirac effective mass of the nucleons, with
the thermal energy density and entropy per baryon showing very strong
correlations. This behavior can be understood within the low-$T$ approximation
but extend beyond its validity domain. While specific heat at constant volume
also shows strong correlations with $m^{*}$, the sign of this correlation
changes with thermodynamic conditions. Correlations between
$\mu_{\mathrm{N;th}}$, ${\Gamma}_{\mathrm{th}}$, $\Gamma_{\mathrm{S}}$ and $q$
have been identified as well. All these correlations consider the values that
$m^{*}$ and $q$ take at finite-$T$. Correlations of similar shape and
strengths have been found with the $T=0$ values of $m^{*}$ and $q$. Such
correlations have been previously discussed by Constantinou et al. [9] who
carried out a full Sommerfeld expansion, i.e., an expansion in actual powers
of $T$ with $m^{*}$ and $\mu^{*}$ taken at $T=0$. Note, however, that to
obtain such elegant expressions, Constantinou et al. [9] have assumed that the
variation of $m^{*}$ with temperature can be neglected compared to the
variation of $\mu^{*}$. We have explicitly verified that these variations are,
in fact, comparable. Nevertheless, we observed that the correlation patterns
are robust to switching between zero and finite temperature values of $m^{*}$
and $q$. This behavior may be seen as the trivial consequence of relatively
small temperature effects on $m^{*}$. From this perspective our numerical
study complements the work done in [9] and checks its validity under wider
thermodynamic conditions. We expect all the correlations found here to
manifest also in other versions of CDF models. It remains nevertheless to
check what exactly determines the magnitude of finite temperature effects and
whether effects stronger than those seen here are realistic.

Numerical simulations of the dynamics of core-collapse supernovae performed
with non-relativistic Skyrme-like EOS models showed that the Landau effective
mass governing the thermal effects impacts the contraction of the proto-NS and
the neutrino and gravitational wave signals [20, 21, 22] as well as the
evolution of proto-NS in failed core-collapse supernova and subsequent
formation of black holes [23]. Numerical simulations of the post-merger
dynamics of a BNS have been carried out too, establishing links between
$m^{*}$, the properties of the merger and those of the GW signal [8, 24].
These BNS simulations have employed either Skyrme-like EOS models or a
phenomenological model of the density dependence of the effective mass [25].
The common feature of all these simulations is that they use parameterized
effective interactions that allow independent tuning of various nuclear matter
parameters and disregard the $T$-dependence of $m^{*}$. We conjecture that a
similar sensitivity to the Dirac effective mass and its density dependence
will manifest should simulations with EOS derived within the CDF approach be
done. What remains to be tested is whether the $T$-dependence of $m^{*}$
enhances the $m^{*}$ sensitivity of dynamical astrophysics phenomena.
Confrontation with results produced by using EOSs derived within dissimilar
theoretical frameworks, e.g., relativistic vs non-relativistic mean field,
will be beneficial in many respects. First, this will help to verify the
robustness of the finite-$T$ imprints on observational signatures of EOS.
Then, it will contribute to assess the uncertainties related to hot matter
properties and their importance with respect to uncertainties related to cold
matter, which is an important question to answer in connection to ongoing
planning for future GW detectors.

## Acknowledgements

A.R.R. and M.V.B. acknowledge financial support from the Ministry of Research,
Innovation and Digitization under contracts PN-III-P4-ID-PCE-2020-0293 and PN
23 21 01 02. M.O. acknowledges financial support from the Agence Nationale de
la Recherche (ANR) under contract ANR-22-CE31-0001-01.

##  Appendix A The low temperature expansion for CDF

In this limit the CDF expressions for particle number density, scalar density,
density of kinetic energy, kinetic pressure and entropy density take the
forms:

$\displaystyle\pi^{2}n\left(T\right)$
$\displaystyle=\int_{0}^{\infty}de\sqrt{e^{2}+2em^{*}}\left(e+m^{*}\right)f_{\mathrm{FD}}\left(e-(\mu^{*}-m^{*})\right)$
$\displaystyle\approx\frac{1}{3}\left(\mu^{*2}-m^{*2}\right)^{3/2}+\frac{\pi^{2}T^{2}}{6}\frac{2\mu^{*2}-m^{*2}}{\sqrt{\mu^{*2}-m^{*2}}}$
(16)
$\displaystyle+\frac{7\pi^{4}T^{4}}{360}\frac{3m^{*4}}{\left(\mu^{*2}-m^{*2}\right)^{5/2}}~{},$

$\displaystyle\frac{\pi^{2}}{m^{*}}n^{\mathrm{s}}\left(T\right)=\int_{0}^{\infty}de\sqrt{e^{2}+2em^{*}}f_{\mathrm{FD}}\left(e-(\mu^{*}-m^{*})\right)$
$\displaystyle\approx\frac{\mu^{*}}{2}\sqrt{\mu^{*2}-m^{*2}}-\frac{m^{*2}}{2}\ln\left(\frac{\sqrt{\mu^{*2}-m^{*2}}}{m^{*}}+\frac{\mu^{*}}{m^{*}}\right)$
(17)
$\displaystyle+\frac{\pi^{2}T^{2}}{6}\frac{\mu^{*}}{\left({\mu^{*2}-m^{*2}}\right)^{1/2}}+\frac{7\pi^{4}T^{4}}{360}\frac{3m^{*2}\mu^{*}}{\left(\mu^{*2}-m^{*2}\right)^{5/2}}~{},$

$\displaystyle e_{\mathrm{kin}}\left(T\right)$
$\displaystyle=\frac{1}{\pi^{2}}\int_{0}^{\infty}de\left(e+m^{*}\right)^{2}\sqrt{e^{2}+2em^{*}}f_{\mathrm{FD}}\left(e-(\mu^{*}-m^{*})\right)$
$\displaystyle\approx\frac{\mu^{*}}{8\pi^{2}}\left(2\mu^{*2}-m^{*2}\right)\sqrt{\mu^{*2}-m^{*2}}$
$\displaystyle-\frac{m^{*4}}{8\pi^{2}}\ln\left(\frac{\sqrt{\mu^{*2}-m^{*2}}}{m^{*}}+\frac{\mu^{*}}{m^{*}}\right)$
(18)
$\displaystyle+\frac{1}{6}T^{2}\frac{\mu^{*}\left(3\mu^{*2}-2m^{*2}\right)}{\left(\mu^{*2}-m^{*2}\right)^{1/2}}$
$\displaystyle+\frac{7\pi^{2}}{120}T^{4}\mu^{*}\frac{2\mu^{*4}-5m^{*2}\mu^{*2}+4m^{*4}}{\left(\mu^{*2}-m^{*2}\right)^{5/2}},$

$\displaystyle p_{\mathrm{kin}}\left(T\right)$
$\displaystyle=\frac{1}{3\pi^{2}}\int_{0}^{\infty}de\left(e^{2}+2em^{*}\right)^{3/2}f_{\mathrm{FD}}\left(e-(\mu^{*}-m^{*})\right)$
$\displaystyle\approx\frac{\mu^{*}}{24\pi^{2}}\left(2\mu^{*2}-5m^{*2}\right)\sqrt{\mu^{*2}-m^{*2}}$
$\displaystyle+\frac{1}{8\pi^{2}}m^{*4}\ln\left(\frac{\sqrt{\mu^{*2}-m^{*2}}}{m^{*}}+\frac{\mu^{*}}{m^{*}}\right)$
(19) $\displaystyle+\frac{1}{6}T^{2}\mu^{*}\left(\mu^{*2}-m^{*2}\right)^{1/2}$
$\displaystyle+\frac{7\pi^{2}}{360}T^{4}\mu^{*}\frac{2\mu^{*2}-3m^{*2}}{\left(\mu^{*2}-m^{*2}\right)^{3/2}}~{},$

$s=\frac{T}{3}\mu^{*}\sqrt{\mu^{*2}-m^{*2}}+\frac{7\pi^{2}T^{3}}{90}\mu^{*}\frac{2\mu^{*2}-3m^{*2}}{\left(\mu^{*2}-m^{*2}\right)^{3/2}}~{}.$
(20)

The first terms in eqs. (16) and (17) along with the sums of the first two
terms in eqs. (18) and (19) have the same functional forms as at zero
temperature. The values of $\mu^{*}$ and $m^{*}$ are nevertheless different.
Indeed, the $T$\- and $n$-dependencies of these two quantities have been
omitted for notational simplicity.

From eq. (20) one can compute the specific heat at constant volume,

$\displaystyle c_{\mathrm{V}}$ $\displaystyle=T\left.\frac{\partial
s}{\partial T}\right|_{\\{N_{i}\\},V}$ (21)
$\displaystyle=\frac{T}{3}\mu^{*}\sqrt{\mu^{*2}-m^{*2}}+\frac{T^{2}}{3}\left.\frac{\partial\mu^{*}}{\partial
T}\right|_{\\{n_{i}\\}}\sqrt{\mu^{*2}-m^{*2}}$
$\displaystyle+\frac{T^{3}\pi^{2}}{90}\mu^{*}\frac{22\mu^{*2}-53m^{*2}}{\left(\mu^{*2}-m^{*2}\right)^{3/2}}$
(22)

Here $V$, $N_{i}$ stand for the total volume and total number of particles
belonging to the $i$-species. We see that, up to the lowest order in $T$,
$c_{\mathrm{V}}=s$; this result is identical to the degenerate limit of a
Fermi gas. To cast eq. (22) we have used the relation

$\displaystyle m^{*}\frac{\partial m^{*}}{\partial
T}\left[\frac{\mu^{*2}-m^{*2}}{\pi^{2}}-\frac{m^{*2}}{6\left(\mu^{*2}-m^{*2}\right)}T^{2}\right]=$
$\displaystyle\mu^{*}\frac{\partial\mu^{*}}{\partial
T}\left[\frac{\mu^{*2}-m^{*2}}{\pi^{2}}+\frac{2\mu^{*2}-3m^{*2}}{6\left(\mu^{*2}-m^{*2}\right)}T^{2}\right]$
(23)
$\displaystyle+\frac{T}{3}\left(2\mu^{*2}-m^{*2}\right)+\frac{7\pi^{2}}{30}T^{3}\frac{m^{*4}}{\left(\mu^{*2}-m^{*2}\right)^{2}}~{},$

which was obtained from eq. (16) by requiring that $\partial n/\partial T=0$
and keeping terms up to the third order in $T$.

## References

  * [1] M. Oertel, M. Hempel, T. Klähn, S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89 (2017) 015007.  doi:10.1103/RevModPhys.89.015007.
  * [2] G. Fiorella Burgio, A. F. Fantina, Nuclear Equation of state for Compact Stars and Supernovae, Astrophys. Space Sci. Libr. 457 (2018) 255–335.  arXiv:1804.03020, doi:10.1007/978-3-319-97616-7_6.
  * [3] S. Typel, M. Oertel, T. Klaehn, CompOSE CompStar online supernova equations of state harmonising the concert of nuclear physics and astrophysics compose.obspm.fr, Phys. Part. Nucl. 46 (4) (2015) 633–664.  doi:10.1134/S1063779615040061.
  * [4] S. Typel, et al., CompOSE Reference Manual, Eur. Phys. J. A 58 (11) (2022) 221\.  arXiv:2203.03209, doi:10.1140/epja/s10050-022-00847-y.
  * [5] C. A. Raithel, V. Paschalidis, Detectability of Finite-Temperature Effects From Neutron Star Mergers with Next-Generation Gravitational Wave Detectors (12 2023).  arXiv:2312.14046.
  * [6] C. Constantinou, B. Muccioli, M. Prakash, J. M. Lattimer, Thermal properties of supernova matter: The bulk homogeneous phase, Phys. Rev. C 89 (2014) 065802.  doi:10.1103/PhysRevC.89.065802.
  * [7] C. Constantinou, B. Muccioli, M. Prakash, J. M. Lattimer, Thermal properties of hot and dense matter with finite range interactions, Phys. Rev. C 92 (2015) 025801\.  doi:10.1103/PhysRevC.92.025801.
  * [8] J. Fields, A. Prakash, M. Breschi, D. Radice, S. Bernuzzi, A. da Silva Schneider, Thermal Effects in Binary Neutron Star Mergers, Astrophys. J. Lett. 952 (2) (2023) L36.  doi:10.3847/2041-8213/ace5b2.
  * [9] C. Constantinou, B. Muccioli, M. Prakash, J. M. Lattimer, Degenerate limit thermodynamics beyond leading order for models of dense matter, Annals of Physics 363 (2015) 533–555.  doi:https://doi.org/10.1016/j.aop.2015.10.003.
URL https://www.sciencedirect.com/science/article/pii/S000349161500367X

  * [10] A. R. Raduta, F. Nacu, M. Oertel, Equations of state for hot neutron stars, Eur. Phys. J. A 57 (12) (2021) 329.  arXiv:2109.00251, doi:10.1140/epja/s10050-021-00628-z.
  * [11] A. Steiner, M. Prakash, J. Lattimer, P. Ellis, Isospin asymmetry in nuclei and neutron stars, Physics Reports 411 (6) (2005) 325–375.  doi:https://doi.org/10.1016/j.physrep.2005.02.004.
URL https://www.sciencedirect.com/science/article/pii/S0370157305001043

  * [12] A. R. Raduta, Equations of state for hot neutron stars-II. The role of exotic particle degrees of freedom, Eur. Phys. J. A 58 (6) (2022) 115.  arXiv:2205.03177, doi:10.1140/epja/s10050-022-00772-0.
  * [13] M. V. Beznogov, A. R. Raduta, Bayesian inference of the dense matter equation of state built upon covariant density functionals, Phys. Rev. C 107 (2023) 045803\.  doi:10.1103/PhysRevC.107.045803.
  * [14] T. Malik, M. Ferreira, B. K. Agrawal, C. Providência, Relativistic Description of Dense Matter Equation of State and Compatibility with Neutron Star Observables: A Bayesian Approach, ApJ 930 (1) (2022) 17.  doi:10.3847/1538-4357/ac5d3c.
  * [15] S. Typel, H. Wolter, Relativistic mean field calculations with density-dependent meson-nucleon coupling, Nuclear Physics A 656 (3) (1999) 331–364.  doi:https://doi.org/10.1016/S0375-9474(99)00310-3.
  * [16] F. Hofmann, C. M. Keil, H. Lenske, Density dependent hadron field theory for asymmetric nuclear matter and exotic nuclei, Phys. Rev. C 64 (2001) 034314.  doi:10.1103/PhysRevC.64.034314.
  * [17] J. Margueron, R. Hoffmann Casali, F. Gulminelli, Equation of state for dense nucleonic matter from metamodeling. i. foundational aspects, Phys. Rev. C 97 (2018) 025805.  doi:10.1103/PhysRevC.97.025805.
  * [18] H. D. Thi, C. Mondal, F. Gulminelli, The Nuclear Matter Density Functional under the Nucleonic Hypothesis, Universe 7 (10) (2021) 373.  arXiv:2109.09675, doi:10.3390/universe7100373.
  * [19] P. Char, C. Mondal, F. Gulminelli, M. Oertel, Generalized description of neutron star matter with a nucleonic relativistic density functional, Phys. Rev. D 108 (10) (2023) 103045.  arXiv:2307.12364, doi:10.1103/PhysRevD.108.103045.
  * [20] A. S. Schneider, L. F. Roberts, C. D. Ott, E. O’Connor, Equation of state effects in the core collapse of a $20\text{$-$}{M}_{\bigodot}$ star, Phys. Rev. C 100 (2019) 055802.  doi:10.1103/PhysRevC.100.055802.
  * [21] H. Yasin, S. Schäfer, A. Arcones, A. Schwenk, Equation of state effects in core-collapse supernovae, Phys. Rev. Lett. 124 (2020) 092701.  doi:10.1103/PhysRevLett.124.092701.
  * [22] O. E. Andersen, S. Zha, A. da Silva Schneider, A. Betranhandy, S. M. Couch, E. P. O’Connor, Equation-of-state dependence of gravitational waves in core-collapse supernovae, Astrophys. J. 923 (2) (2021) 201.  doi:10.3847/1538-4357/ac294c.
  * [23] A. Schneider, E. O’Connor, E. Granqvist, A. Betranhandy, S. Couch, Equation of State and Progenitor Dependence of Stellar-Mass Black-Hole Formation, Astrophys. J. 894 (1) (2020) 4.  doi:10.3847/1538-4357/ab8308.
  * [24] C. A. Raithel, V. Paschalidis, Influence of stellar compactness on finite-temperature effects in neutron star merger simulations, Phys. Rev. D 108 (2023) 083029.  doi:10.1103/PhysRevD.108.083029.
  * [25] C. A. Raithel, F. Özel, D. Psaltis, Finite-temperature extension for cold neutron star equations of state, The Astrophysical Journal 875 (1) (2019) 12.  doi:10.3847/1538-4357/ab08ea.