of 23
Influence of muons, pions, and trapped neutrinos on neutron star mergers
Michael A. Pajkos
1
,*
and Elias R. Most
1,2
1
TAPIR, Mailcode 350-17,
California Institute of Technology
, Pasadena, California 91125, USA
2
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
,
Pasadena, California 91125, USA
(Received 16 September 2024; accepted 15 January 2025; published 6 February 2025)
The merger of two neutron stars probes dense matter in a hot, neutrino-trapped regime. In this work, we
investigate how fully accounting for pions (
π
), muons (
μ
), and muon-type neutrinos (
ν
μ
) in the trapped
regime may affect the outcome of the merger. By performing fully general-relativistic hydrodynamics
simulations of merging neutron stars with equations of state to which we systematically add those different
particle species, we aim to provide a detailed assessment of the impact of muons and pions on the merger
and postmerger phases. In particular, we investigate the merger thermodynamics, mass ejection, and
gravitational wave emission. Our findings are consistent with previous expectations, that the inclusion of
such microphysical degrees of freedom and finite temperature corrections leads to frequency shifts on the
order of 100
200 Hz in the postmerger gravitational wave signal, relative to a fiducial cold nucleonic
equation-of-state model.
DOI:
10.1103/PhysRevD.111.043013
I. INTRODUCTION
Neutron star mergers (NSNS) are ideal probes for
nuclear matter under extreme conditions
[1
4]
. Neutron
star interiors can reach densities beyond several times
nuclear saturation, at which physics beyond that of neu-
trons (
n
) and protons (
p
) can be probed
[3,5]
. This can
include exotic degrees of freedom such as hyperons
[6,7]
or
deconfined quarks
[8,9]
, but also mesons, including pions
(
π

)
[10
12]
. Revealing this complex and intricate inter-
play of the physics of the strong interaction is at the core of
modern astrophysical neutron star research. As such,
various ways have been proposed and investigated to reveal
matter under extreme conditions in neutron stars. These
include x-ray observations of neutron star cooling curves
and hot spots on neutron stars
[5,13]
, which have yielded
novel constraints on the dense matter equation of state
(EOS)
[14
18]
, especially when combined with advances
in chiral effective field theory descriptions of nuclear matter
around saturation
[19]
.
Within the realm of gravitational wave astrophysics,
detections of gravitational waves from merging neutron
stars
[20,21]
have the potential to directly probe and
constrain dense matter physics, including finite-temperature
effects
[22]
. The inspiral part of the gravitational waveform
can constrain neutron star radii and the cold equation of state
[23,24]
, with remarkable constraints obtained from the first
neutron star merger event GW170817 (e.g., Refs.
[25
29]
;
see also
[30
38]
). With next-generation facilities, future
detections of postmerger gravitational waves have the
potential to detect the kilohertz oscillations of the neutron
star merger remnant formed in intermediate- and low-mass
mergers
[39,40]
. These frequencies are quasi-universally
related to properties of the dense matter equation of state
[41
49]
(see also
[50]
for bounds on this universality).
The postmerger phase probes higher densities and temper-
atures than those present in the inspiral, and it could reveal
the appearance of quark matter
[51
58]
, hot dense matter
[59
67]
,hyperons
[68
70]
, and neutrinos
[71
75]
, fueling
effective chemical reactions inside the merger remnant.
While the latter could in principle affect the outcome of
the merger via an effective viscosity
[72]
, the strength of the
effect depends strongly on the neutrino conditions and
opacity
[74,75]
. In particular, it has been found that likely
the remnant will be optically thick to neutrino emission,
leading to an effective trapping of neutrinos and to a
correction of the equation of state
[74]
. This neutrino-trapped
regime can depend crucially on the microphysical inter-
actions included
[76]
, including through the appearance of
muons (
μ

) and pions,
π

[77]
(see also
[78]
). Recent
numerical studies have further been aimed at quantifying the
impact of these particles independently
[79,80]
. Similar
conclusions have also been found for core-collapse super-
novae, where the inclusion of muons and pions may critically
affect theexplosion mechanism
[81
83]
. Recent works in the
field have made substantial progress toward quantifying the
effects of additional particles. Reference
[80]
varies the mass
of the pion and quantifies the characteristics of the NSNS
dynamics and resulting GW signal. Reference
[79]
uses an
advanced postprocessing scheme to quantify the changes in
*
Contact author: mpajkos@caltech.edu
PHYSICAL REVIEW D
111,
043013 (2025)
2470-0010
=
2025
=
111(4)
=
043013(23)
043013-1
© 2025 American Physical Society
pressure in the presence of muons and trapped neutrinos,
finding changes in pressure of order 10%. More recently,
Ref.
[84]
allows for the advection of muons, paired with
a neutrino leakage scheme, while Ref.
[85]
performs
five-species neutrino transport alongside muonic reactions
in NSNS.
Building upon these works, we perform fully general-
relativistic neutron star merger simulations, which model
the influence of mesons, such as pions (
π

) in their thermal
and condensed state, and leptons, such as muons (
μ

) and
(anti)neutrinos (
ν
) in the fully trapped regime. We then
analyze the impact on the merger dynamics, gravitational
wave emission, and mass ejection from the system.
Our work is organized as follows: In Sec.
II A
, we outline
the relevant particle processes we choose to model.
Section
II B
provides the statistic description of the various
particles. In Sec.
II C
, we outline the relevant steps to update
nuclear EOSs to include these particles for the NSNS
system, with more detail in the Appendix. Section
II D
provides thermodynamic contributions from each species.
Section
II E
introduces the EOSs used in this work.
Section
II F
describes the general properties of the newly
constructed EOSs. Section
II G
quantifies the impact of
different species on isolated NSs. Section
II H
details the
numerical tools in this work. Section
III
describes our
simulation results. Lastly, Sec.
IV
concludes.
Unless otherwise noted, in this work we adopt units
G
¼
c
¼
1
, where
G
is the gravitational constant, and
c
is
the speed of light.
II. METHODS
In the following, we outline the construction of the
equation of state, as well as the setup for our numerical
simulations.
A. Particle processes
We begin by discussing the main particle interactions of
muons and pions relevant to neutron star mergers. Our
presentation largely follows that of Refs.
[79,80]
.
The principal decay channel of a charged pion (
π

, with
bare mass
m
π
140
MeV) into charged muons,
μ

,isas
follows:
π
μ
þ
̄
ν
μ
;
ð
1
Þ
where
̄
ν
μ
denotes a muon antineutrino. Charged muons
(with bare mass
m
μ
106
MeV) will further decay into
electrons
e
, electron-type antineutrinos
̄
ν
e
, and muon-type
neutrinos
ν
μ
:
μ
e
þ
̄
ν
e
þ
ν
μ
:
ð
2
Þ
The last relevant reaction is neutron (
n
, with bare mass
m
n
940
MeV) decay into a proton (
p
), electron, and
electron-type antineutrino:
n
p
þ
e
þ
̄
ν
e
:
ð
3
Þ
We now assume that neutrinos inside the neutron star
merger remnant are trapped
[74]
, and therefore they
approximate weak decays inside the hot and dense merger
remnant
[60]
as being in weak-interaction equilibrium. As a
result, we can equate the reactions in terms of their
chemical potentials,
μ
i
:
μ
π
¼
μ
μ
μ
ν
μ
¼
μ
e
μ
ν
e
¼ð
μ
n
μ
p
Þ
;
ð
4
Þ
where the different signs correspond to negatively and
positively charged current reactions. Apart from weak-
interaction equilibrium, we also need to account for charge
neutrality. As a result, the overall particle fractions,
Y
i
¼
n
i
=n
b
, where
n
i
are the particle number densities
and
n
b
is the baryon number density, obey
Y
p
¼
Y
e
þ
Y
π
þ
Y
μ
;
ð
5
Þ
where
Y
e
¼
Y
e
Y
e
þ
,
Y
π
¼
Y
π
Y
π
þ
þ
Y
c
π
, and
Y
μ
¼
Y
μ
Y
μ
þ
. Here,
Y
c
π
accounts for the presence of a
negatively charged pion condensate. Particles that follow
Bose-Einstein statistics can form multiple particles in
the same quantum state at the lowest energy level of the
system. These particles have rest mass but no kinetic
energy, and they do not contribute pressure to the surround-
ing system. As such, we call pions which do not condense
(
Y
π
;Y
π
þ
)
thermal
pions. Neutral pions (
π
0
) are added by
assuming a vanishing chemical potential,
μ
π
0
¼
0
.
Similarly to Ref.
[79]
, we define the lepton fraction for a
species
i
, described as
Y
l;i
¼ð
Y
i
Y
i
þ
Þþ
Y
ν
i
Y
̄
ν
i
j
i
f
e;
μ
g
:
ð
6
Þ
One central assumption to our hydrodynamic scheme is that
the electron-lepton number in the simulation is advected
along with the fluid in the trapped regime
[60,86]
,
u
μ
μ
Y
l;e
¼
0
;
ð
7
Þ
where
u
μ
is the fluid four-velocity, which is in good
agreement with simulations of neutron star mergers with
more advanced treatments of neutrino transport
[74]
.
Y
l;
μ
is
chosen to be a constant of 0.01 below a density threshold of
10
14
gcm
3
. Above this threshold, it is parametrized as a
function of density
Y
l;
μ
ð
ρ
Þ
. For a detailed explanation of our
choice of
Y
l;
μ
, see the Appendix. We approximate the
particle masses in this work with corresponding vacuum
rest masses
m
μ
¼
105
.
70
MeV,
m
π

¼
139
.
57
MeV, and
m
π
0
¼
134
.
98
MeV. In reality, varying masses of the pions
in matter
[80,87]
will increase, producing a less pronounced
effect of pions on the dynamics, allowing increased muon
effects. Likewise, effective muon masses can change due to
relativistic effects
[88]
.
MICHAEL A. PAJKOS and ELIAS R. MOST
PHYS. REV. D
111,
043013 (2025)
043013-2
B. Statistical descriptions of pions and muons
In order to describe muon and pion corrections to the
equation of state, we need to translate Eqs.
(4)
and
(5)
into
corrections of the energy density and pressure of the
equation of state. As an intermediate step, this involves
computing the particle fractions of
e

,
μ

,
π

,
ν
i
, and
̄
ν
i
.
We will also provide the expressions for the number
density,
n
i
, of the respective particles.
1. Muons and electrons
Leptons (
l
¼
μ
;e
) are treated as an ideal Fermi gas, whose
number density can be described as [Eq. (16) of
[79]
]
n
l

¼
K
l
θ
3
=
2
l
½
F
1
=
2
ð
η
0
l

;
θ
l
Þþ
θ
l
F
3
=
2
ð
η
0
l

;
θ
l
Þð
8
Þ
for
K
l
¼
8
ffiffiffi
2
p
π
ð
m
l
c
2
=hc
Þ
3
,
θ
l
¼
k
B
T=
ð
m
l
c
2
Þ
, and
η
0
l

¼
ð
μ
l

m
l
c
2
Þ
=
ð
k
B
T
Þ
, where
F
k
are Fermi functions of order
k
,
m
l
is the mass of the lepton, and
h
and
k
B
are Planck
s and
the Boltzmann constants, respectively. These Fermi func-
tions are defined as [Eq. (6) of
[89]
]
F
k
ð
η
;
θ
Þ¼
Z
0
dx
x
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
0
.
5
θ
x
p
e
x
η
þ
1
:
ð
9
Þ
2. Pions
Pions, by contrast, are treated as a free Bose gas. Note
that the use of
μ
in this section refers to chemical potential,
rather than muons. As a Bose gas, the pion number density
is given by
[90]
n
π
¼
1
λ
3
g
3
=
2
ð
z
Þ¼
1
λ
3
2
ffiffiffi
π
p
Z
0
dy
y
1
=
2
z
1
e
y
1
;
ð
10
Þ
where
λ
¼ð
h
2
=
2
π
mk
B
T
Þ
1
=
2
, and the fugacity is
z
¼
e
μ
=
ð
k
B
T
Þ
¼
e
ð
̃
μ
m
Þ
=
ð
k
B
T
Þ
; for the corresponding antiparticle,
z
¼
e
ð
̃
μ
m
Þ
=
ð
k
B
T
Þ
. The integral in Eq.
(10)
can be approxi-
mated with an infinite series, simplifying to
n
π
¼
1
λ
3

z
þ
z
2
2
3
=
2
þ
z
3
3
3
=
2
þ

:
ð
11
Þ
We follow the nomenclature of
[91]
, where
̃
μ
is the
chemical potential of the particle with the rest mass
included, and
μ
is the chemical potential without the
particle rest mass; concretely,
μ
i
¼
̃
μ
i
m
i
. As an impor-
tant note for pions, if
̃
μ
>m
π
, a negatively charged pion
condensate (
π
c
) will form. In regions of our simulation
where this is the case, to populate the
thermal
pions
π
, one
follows the above Bose-Einstein statistics, with the chemi-
cal potential of the thermal pion equal to the charged pion
rest mass,
̃
μ
π
¼
m
π
. To calculate the number density of
the condensate, one must follow a more detailed procedure
in the Appendix that relies on balancing charge neutrality.
3. Neutrinos
In this work, we assume that neutrinos are trapped
through the neutron star merger remnant, which is consistent
with recent simulations using full neutrino transport
[74]
.In
the neutrino-trapped regime, neutrinos behave as a massless
Fermi gas, whose number density can be described as
[79]
n
ν
¼
4
π
ð
hc
Þ
3
ð
k
B
T
Þ
3
F
2
ð
η
ν
Þ
exp
ð
ρ
lim
=
ρ
Þ
;
ð
12
Þ
where exp
ð
ρ
lim
=
ρ
Þ
isan exponential dampingfactor
[92]
to
model trapped neutrinos above
ρ
lim
¼
10
14
gcm
3
. Here,
the degeneracy parameter of the neutrinos is calculated as
η
ν
l
¼ð
μ
p
μ
n
þ
μ
l
Þ
=k
kB
T;
ð
13
Þ
and
η
ν
¼
η
ν
. In practice, these enter our calculation via
the net number fraction of neutrinos, represented by
Y
ν
Y
̄
ν
n
ν
n
̄
ν
. Applying Eq.
(12)
, we use the exact
expression provided by
[93]
Y
ν
Y
̄
ν
F
2
ð
η
ν
Þ
F
2
ð
η
ν
Þ¼
1
3
η
ν
ð
π
2
þ
η
2
ν
Þ
;
ð
14
Þ
which provides a more accurate expression of the net
number density, without numerical integration of Fermi
integrals
F
2
ð
η
ν
Þ
. Outside the trapped regime, neutrino
emission is modeled with a leakage scheme, whose details
are specified in Sec.
II H
.
C. Equation of state
Armed with the statistical descriptions for the number
density for each species, we now outline our methodology
to populate a given equation of state with each new particle.
We begin with charge conservation for the proton fraction
Y
p
¼
Y
e
þ
Y
μ
þ
Y
π
:
ð
15
Þ
Likewise, we assert lepton number conservation for
species of electrons and muons:
Y
l;e
¼
Y
e
þ
Y
ν
e
Y
̄
ν
e
ð
16
Þ
and
Y
l;
μ
¼
Y
μ
þ
Y
ν
μ
Y
̄
ν
μ
:
ð
17
Þ
Combining the previous three equations yields
ˆ
Y
p
¼
Y
l;e
ð
Y
ν
e
Y
̄
ν
e
Þþ
Y
l;
μ
ð
Y
ν
μ
Y
̄
ν
μ
Þþ
Y
π
:
ð
18
Þ
Here,
ˆ
Y
p
indicates our iteration variable of choice that will
eventually converge to
Y
p;
new
for our updated equation
of state.
INFLUENCE OF MUONS, PIONS, AND TRAPPED NEUTRINOS
...
PHYS. REV. D
111,
043013 (2025)
043013-3
Equation
(18)
provides a modular framework to add
particles to the system. In our work, we create three new
variants by modifying the
base
equation of state. The first
variant purely accounts for electron-type (anti)neutrinos,
the
þ
ν
e
case. The
þ
ν
e
case zeros out the
Y
l;
μ
,
ð
Y
ν
μ
Y
̄
ν
μ
Þ
,
and
Y
π
terms. The second variant accounts for electron-
type (anti)neutrinos and pions, the
þ
ν
e
þ
π
case. This
variant zeros out the
Y
l;
μ
and
ð
Y
ν
μ
Y
̄
ν
μ
Þ
terms. The third
variant accounts for electron-type (anti)neutrinos, pions,
muons, and muon-type (anti)neutrinos, the
þ
ν
e;
μ
þ
π
þ
μ
case. This variant includes all terms in Eq.
(18)
. In the
Appendix, we enumerate our procedure, in detail, to add
the new particle species to the EOS.
D. Calculating thermodynamic quantities
Having calculated the charge fractions (or number
densities) of pions and muons, we can calculate the
pressure,
P
, and specific internal energy,
ε
, of the pions,
muons, and trapped neutrinos. In the following sections, we
list the expressions for energy density, which has units of
energy per volume. To convert the energy density expres-
sions to specific internal energy, we simply divide by the
rest mass density
ε
μ
=
π
¼
ε
μ
=
π
=
ρ
.
1. Pions
For pions, we leverage Bose-Einstein statistics. For the
pressure,
P
Bose
¼
g
5
=
2
ð
z
ð
T
ÞÞ

2
π
m
π
h
2

3
=
2
ð
k
B
T
Þ
5
=
2
;
ð
19
Þ
where
g
5
=
2
ð
z
ð
T
ÞÞ
can be described as
g
5
=
2
ð
z
ð
T
ÞÞ¼
1
Γ
ð
5
=
2
Þ
Z
0
dy
y
3
=
2
z
1
e
y
1
:
ð
20
Þ
Note that
Γ
ð
5
=
2
Þ¼
3
ffiffiffi
π
p
=
4
, and
g
5
=
2
ð
z
¼
1
Þ¼
ζ
ð
5
=
2
Þ
1
.
342
. For the energy density
[94]
,
ε
Bose
¼
3
2
p
Bose
þ
n
π
m
π
c
2
. Note the additional contributions from the rest
mass,
n
μ
m
μ
c
2
.
2. Muons
For muons, we leverage Fermi-Dirac statistics
[79,95]
.
For pressure, we have
P
μ

¼
1
3
K
μ
m
μ
c
2
θ
5
=
2
μ
½
2
F
3
=
2
ð
η
0
μ

;
θ
μ
Þþ
θ
μ
F
5
=
2
ð
η
0
μ

;
θ
μ
Þ
:
ð
21
Þ
For the energy density,
ε
μ

¼
K
μ
m
μ
c
2
θ
5
=
2
μ
½
F
3
=
2
ð
η
0
μ

;
θ
μ
Þþ
θ
μ
F
5
=
2
ð
η
0
μ

;
θ
μ
Þ
þ
n
μ
m
μ
c
2
:
ð
22
Þ
Note that the last term in the equation accounts for
contributions from the rest mass of the muons.
3. Neutrinos
We treat neutrinos as a massless Fermi gas
[79]
. For the
energy density, we have
ε
ν
¼
4
π
ð
hc
Þ
3
ð
k
B
T
Þ
4
F
3
ð
η
ν
Þ
exp
ð
ρ
lim
=
ρ
Þ
;
ð
23
Þ
where the term exp
ð
ρ
lim
=
ρ
Þ
is used to smoothly cut off
neutrino trapping above
10
14
gcm
3
[96]
. Note that in
practice, the calculations of
F
3
ð
η
ν
Þ
can be cumbersome.
However, since we are interested in the contributions from
antineutrinos as well, the expressions for the net energy
density from neutrinos and corresponding antineutrinos
becomes
ε
ν
þ
ε
̄
ν
F
3
ð
η
ν
Þþ
F
3
ð
η
ν
Þ¼
7
π
4
60
þ
1
2
η
2

π
2
þ
1
2
η
2

;
ð
24
Þ
where in the last expression, we make use of the properties
of sums of Fermi integrals
[93]
. This expression both
simplifies computation and is more precise than performing
numerical integration. The expression for pressure simply
follows from that of an ultrarelativistic gas,
P
ν
¼
ε
ν
=
3
:
ð
25
Þ
E. Equation-of-state models
In order to apply pion and muon corrections, we need to
adopt an underlying equation-of-state framework. We here
adopt two EOS models, SFHo
[97]
and DD2
[98
100]
.
The calculations used in both EOSs are based on a relativistic
mean field model for nucleons. Both unmodified tables
tabulate basic thermodynamic quantities (e.g., pressure or
speed of sound) against three independent variables
ð
ρ
;T;Y
p
Þ
. These tables assume nuclear statistical equilib-
rium (NSE) among the constituent particles: nuclei, nucle-
ons, electrons, positrons, and photons. For these unmodified
tables, only the protons, electrons, and positrons are assumed
to contribute to the charge fraction, or
Y
p
¼
Y
e
Y
e
þ
.As
outlined in the Appendix, we detail our procedure to create
tabulated EOSs against
ð
ρ
;T;Y
l;e
Þ
.
Corrections for pions and muons are applied to the
entirety of the new EOS table, whereas EOS contributions
from trapped neutrinos are only added in the regime roughly
above the neutrino trapping limit of
10
14
gcm
3
due to the
exponential damping factor of exp
ð
ρ
=
ρ
lim
Þ
in Eq.
(24)
.
Ideally, this approximation to neutrino trapping should not
MICHAEL A. PAJKOS and ELIAS R. MOST
PHYS. REV. D
111,
043013 (2025)
043013-4
be used during the inspiral phase of the merger, because
there are no expected neutrino contributions from isolated
NS companions. Because we do not see major neutrino
fractions in the isolated companions, and for the simplicity
of a single EOS table during the entire evolution, we employ
this approximation.
F. General equation-of-state properties
In this section, we present general properties of the
ν
e
,
μ
, and
π
-augmented equations of state, including particle
fractions, pressure profiles, and specific internal energy
profiles. We begin our analysis to look at general properties
of the EOS by investigating estimates for particle fractions
Y
i
under typical thermodynamic conditions in the merger.
These plots are generated for typical conditions found in a
neutron star merger remnant
[60,101]
:
ρ
¼
1
.
5
n
sat
¼
3
.
9
×
10
14
gcm
3
,
Y
l;e
¼
0
.
06
, and
0
.
01
T
80
MeV.
These calculations are presented in Figs.
1
and
2
for the
SFHo and DD2 EOSs, respectively.
We begin by examining the top-left panel of Fig.
1
.We
see the particle fractions for protons
ð
Y
p
Þ
and electron-type
neutrinos
Y
ν
e
. Recall that in this work, we define
Y
ν
e
(and
similarly,
Y
ν
μ
) as the difference of the particle fraction of
the neutrinos from the particle fraction of the corresponding
antineutrinos [Eq.
(17)
]. This definition implies that neg-
ative values of
Y
ν
e
signify the presence of more antineu-
trinos than neutrinos. At low temperatures, there is no
noticeable imbalance between neutrinos and antineutrinos.
With increasing temperature, there is an increasing imbal-
ance, indicated by
Y
ν
e
∼−
0
.
07
at
T
¼
80
MeV. The
Y
p
FIG. 1. Physical quantities for the SFHo EOS, taken at rest mass density
ρ
¼
3
.
9
×
10
14
gcm
3
and lepton fraction
Y
l;e
¼
0
.
06
. The
left column represents only electron-type (anti)neutrinos
ð
ν
e
Þ
. The middle column represents
ν
e
and pions (
π
) (including condensed
pions,
π
c
). The right column includes
ν
e
,
π
, muons
μ
, and muon (anti)neutrinos
ð
ν
μ
Þ
. Top row: particle fractions for various species.
Middle row: pressure,
P
, as a function of temperature,
T
, for the unmodified SFHo equation of state and SFHo with additional
particles. The shaded region represents where the pressure
P
SFHo
>P
þ
ν
e
þ
π
. Bottom row: specific internal energy,
ε
, as a function of
temperature.
INFLUENCE OF MUONS, PIONS, AND TRAPPED NEUTRINOS
...
PHYS. REV. D
111,
043013 (2025)
043013-5
behavior is characterized by a steady increase with temper-
ature from
Y
p
¼
0
.
06
to
Y
p
0
.
13
. This behavior is
defined by Eq.
(16)
. Since the table is defined at a fixed
Y
l;e
¼
0
.
06
,as
Y
ν
e
steadily decreases,
Y
e
and thus,
Y
p
must steadily increase.
In the middle panel of the first row, we additionally
examine pion-related quantities for the
þ
ν
e
þ
π
case. The
newly labeled quantities are condensed pions
ð
Y
π
c
Þ
, ther-
mal and condensed charged pions (
Y
π
), and thermal neutral
pions
Y
π
0
. At lower temperature values, the dashed orange
line displays values around 0.04 for
Y
π
. This is due to pion
condensate forming. As temperatures increase, the thermal
population of pions increases. By contrast, the condensate
does not increase as rapidly. Thus,
Y
π
c
=Y
π
is decreasing as
temperature increases. This behavior represents pions
moving from the lowest available energy state into
higher energy levels due to increased temperature. At
T
¼
80
MeV, we see
Y
π
c
0
.
08
and
Y
π
0
.
18
. There is
a moderate increase in
Y
π
0
due to thermal effects. Note the
drastically different behavior of
Y
ν
e
. In contrast with the
þ
ν
e
case,
Y
ν
e
remains close to 0. This is a direct
consequence of pion condensation via Eq.
(4)
. Because
μ
π
is bounded by
m
π
, and the pions are in strong
equilibrium with the baryons,
μ
n
μ
p
is also limited by
m
π
. This in turn limits
η
ν
e
, limiting the imbalance between
electron neutrinos and antineutrinos. As a consequence of
these low
Y
ν
e
values, notice the similar shapes of
Y
p
(solid
black line) and
Y
π
. These two curves are shifted by
Y
e
Y
l;e
0
.
06
, as a result of Eq.
(5)
.
In the right panel of the first row, we additionally
examine muon-related quantities for the
þ
ν
e;
μ
þ
π
þ
μ
case. At low temperatures, in the dash-dotted red line,
we see that
Y
μ
0
.
02
and at temperatures near 80 MeV, it
climbs to
Y
μ
0
.
06
. This behavior is justified, as higher
temperatures increase the total available amount of thermal
energy to produce new particles. At lower temperatures,
there is no noticeable imbalance between muon-type
neutrinos and muon-type antineutrinos. However, at higher
temperatures, we see
Y
ν
μ
∼−
0
.
04
. Similarly to before,
Y
μ
and
Y
ν
μ
mirroring each other is a direct consequence of
FIG. 2. Same as Fig.
1
, but for the DD2 EOS.
MICHAEL A. PAJKOS and ELIAS R. MOST
PHYS. REV. D
111,
043013 (2025)
043013-6