Farrell et al

*

# Differential Rotation in Compact Objects with Hyperons and Delta Isobars

Delaney Farrell  Fridolin Weber  Jia Jie Li  Armen Sedrakian  Department of
Physics, San Diego State University, San Diego, CA 92115, United States
Center for Astrophysics and Space Sciences, University of California at San
Diego, San Diego, CA 92093, United States  School of Physical Science and
Technology, Southwest University, Chongqing 400715, China  Frankfurt Institute
for Advanced Studies, D-60438 Frankfurt am Main, Germany  Institute of
Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Poland
<EMAIL_ADDRESS>
(xx November 2023; xx November 2023; xx November 2023)

###### Abstract

Neutron stars may experience differential rotation on short, dynamical
timescales following extreme astrophysical events like binary neutron star
mergers. In this work, the masses and radii of differentially rotating neutron
star models are computed. We employ a set of equations of states for dense
hypernuclear and $\Delta$-admixed-hypernuclear matter obtained within the
framework of CDF theory in the relativistic Hartree-Fock (RHF) approximation.
Results are shown for varying meson-$\Delta$ couplings, or equivalently the
$\Delta$-potential in nuclear matter. A comparison of our results with those
obtained for non-rotating stars shows that the maximum mass difference between
differentially rotating and static stars is independent of the underlying
particle composition of the star. We further find that the decrease in the
radii and increase in the maximum masses of stellar models when
$\Delta$-isobars are added to hyperonuclear matter (as initially observed for
static and uniformly rotating stars) persist also in the case of
differentially rotating neutron stars.

###### keywords:

Neutron Star, Differential Rotation, Relativistic Hartree-Fock, Covariant
Density Function Theory

††articletype: Article Type

##  1 Introduction

As the densest observed stellar objects, neutron stars provide a unique,
naturally occurring laboratory for studying matter at extreme pressures and
densities not reproducible by experiments. The matter within the core of a
massive neutron star can reach densities up to an order of magnitude higher
than nuclear saturation density. One physical mechanism for supporting massive
neutron stars is rotation, as rotating neutron stars can sustain a higher rest
mass than their non-rotating counterparts. While we assume the majority of
rotating neutron stars are rotating uniformly, some neutron stars may form
with considerable differential rotation following extreme astrophysical events
like core-collapse supernovas and binary neutron star mergers (Morrison .,
2004). The state of differential rotation is short-lived, as stars will relax
to uniform rotation due to the shear viscosity of matter on timescales that
are much shorter than secular timescales.

Studies of differentially rotating neutron stars are of particular interest in
the context of binary neutron star mergers following the recent first
observation of a binary neutron star merger, GW170817 (Abbott ., 2017). Binary
neutron star mergers are likely to form differentially rotating remnants as
evidenced by general relativistic numerical simulations and expected from
complicated hydrodynamic motions during the coalescence. A differentially
rotating remnant star formed following a merger event has been confirmed
through numerical simulation in the past (Shibata Kiuchi, 2017; Shibata Uryū,
2000; Fujibayashi ., 2018). As we move into a new era of multi-messenger
astronomy following the GW170817 event, understanding not only the inspiral
phase but also the post-merger phase has the potential to provide further
information on the equation of state of dense matter, neutron star properties,
and the remnant’s evolution.

Following a binary neutron star merger, the remnant star can take the shape of
a hypermassive or supermassive neutron star (Sarin Lasky, 2021; Hotokezaka .,
2013), or promptly collapse into a black hole. Differential rotation is one
mechanism that allows these stars to sustain total masses considerably higher
than both non-rotating stars and uniformly rotating stars. These massive,
differentially rotating neutron stars may also deviate from spherical or axial
symmetry by exhibiting extreme triaxial deformations. Both the structural
deformation and differential rotation allow the compact remnant to remain
stable on short, dynamical timescales, for masses that would render
configurations unstable in the static and uniform-rotation cases.

The observable properties of a neutron star, like its mass, are not only
dependent on rotation. Additionally, the underlying microphysics, given by the
equation of state (EOS), greatly influences the overall structure of these
objects. The cores of neutron stars can reach densities up to a few times
nuclear saturation density, covering a density and temperature regime not
reproducible in laboratories and not fully understood. The high-density regime
is expected to not only contain nucleons, but also other exotic degrees of
freedom like deconfined quark matter, hyperons, and delta isobars ($\Delta$).
These additional degrees of freedom have been shown previously in numerical
simulations to greatly impact properties like the mass and radius of neutron
stars. In Li . (2018), the presence of $\Delta$’s was shown to soften the EOS
in the low to intermediate-density region and stiffen it at high densities,
resulting in a slightly increased maximum mass and considerably decreased
radius for non-rotating compact objects. In this work, we extend the work done
in Li . (2018) and Li . (2023) to explore differential rotation’s impact on
both stars containing hyperonic matter and $\Delta$ populations. The paper is
organized as follows: Section 2 describes the theoretical framework for
modeling differential rotation in neutron stars and for constructing the dense
matter EOS in covariant density functional (CDF) theory. Section 3 presents
calculated mass-radius relations for differentially rotating stars containing
both hyperon and $\Delta$ isobar populations, and density maps indicating at
what depths different particle species appear within these stars. Section 4
gives a summary of the work presented.

##  2 Background

###  2.1 Modeling Differential Rotation in Neutron Stars

To calculate the bulk properties of neutron stars, the stellar structure
equations must be solved in the framework of Einstein’s theory of general
relativity. These equations are derived from Einstein’s field equation and
depend on the stellar matter’s EOS, i.e., the underlying relationship between
pressure $P$ and energy density $\epsilon$. In general, modeling uniformly or
differentially rotating neutron stars is more complicated than modeling their
non-rotating counterparts. Rapid rotation of either kind can deform the
structure of the star, resulting in a flattening at the pole and expansion
along the radius in the equatorial direction. To account for this deviation
from spherical symmetry, the stellar structure equations for rotating stars
must be dependent on both the radial coordinate $r$ and the polar coordinate
$\theta$, assuming axisymmetry. Rotation also results in more massive objects,
as it is a physical mechanism that stabilizes massive stars against collapse;
as a result, rotating neutron stars are able to sustain roughly up to 20% more
mass than their non-rotating counterparts, depending on the underlying
equation of state (Kalogera Baym, 1996). The increased mass of rotating stars
alters the geometry of spacetime by introducing a dependence on the star’s
rotational frequency to the line element and a self-consistency condition to
the stellar structure equations to account for the dragging of local inertial
frames (Weber, 1999).

Modeling differential rotation in neutron stars begins with the same metric as
uniformly rotating stars, which is dependent on both the radial coordinate $r$
and polar angle $\theta$:

$ds^{2}=-e^{\gamma-\rho}dt^{2}+e^{2\alpha}(dr^{2}+r^{2}d\theta)+e^{\gamma-\rho}r^{2}\sin^{2}\theta(d\phi-\omega
dt)^{2},$ (1)

where $\gamma$, $\rho$, $\alpha$, and $\omega$ are metric functions, where
dragging of local inertial frames is accounted for by $\omega$. These
functions are dependent on $r$ and $\theta$, as well as the star’s angular
velocity $\Omega$. The metric functions are computed from Einstein’s field
equation

$R^{\kappa\sigma}-\frac{1}{2}Rg^{\kappa\sigma}=8\pi T^{\kappa\sigma},$ (2)

where $R^{\kappa\sigma}$ is the Ricci tensor, $R$ is the curvature scalar, and
$g^{\kappa\sigma}$ is the metric tensor. The energy momentum tensor,
$T^{\kappa\sigma}$, is given by

$T^{\kappa\sigma}=(\epsilon+P)\,u^{\kappa}u^{\sigma}+g^{\kappa\sigma}P,$ (3)

where $\epsilon$ and $P$ are given by the underlying EOS.

Once the four metric functions are solved for, they are used to solve the
equation of hydrostatic equilibrium for a barotropic fluid:

$h(P)-h_{p}=\frac{1}{2}[\gamma_{p}+\rho_{p}-\gamma-\rho-\text{ln}(1-v^{2})+F(\Omega)]$
(4)

where $h(P)$ is the enthalpy, $\gamma_{p}$ and $\rho_{p}$ are the value of the
metric potentials at the pole $p$, and $v$ is defined as

$v=(\Omega-\omega)\,r\,\sin\,\theta e^{-\rho}.$ (5)

The final term in Eqn. 4, $F(\Omega)$, defines the rotational law for the
matter. Previous work (Komatsu ., 1989; Cook ., 1992, 1994) defines the
rotation law as the linear function

$F(\Omega)=A^{2}(\Omega_{c}-\Omega),$ (6)

where $\Omega_{c}$ is the central value for the angular velocity. The
parameter $A$ is used to determine the length scale over which the frequency
changes, and acts as a scaling factor of the degree of differential rotation
(Morrison ., 2004).

Substituting Eqn. 6 into Eqn. 4 gives

$(\Omega_{c}-\Omega)=\frac{1}{\hat{A}^{2}}\left[\frac{(\Omega-\omega)\,s^{2}(1-\mu^{2})\,e^{-2\rho}}{(1-s)^{2}-(\Omega-\omega)^{2}\,s^{2}\,(1-\mu^{2})\,e^{-2\rho}}\right],$
(7)

where the rotational frequency $\Omega$ can be then isolated and solved
numerically using a root-finding algorithm like Newton-Rhapson. The parameters
$s$ and $\mu$ in Eqn. (7) are dimensionless representations of the radial and
polar coordinates. Additionally, Eqn. (7) uses a modified version of the
rotation parameter, $\hat{A}$, which is scaled by the equatorial radius
$r_{e}$: $\hat{A}\equiv A/r_{e}$. Uniform rotation is achieved in the limit
$\hat{A}^{-1}\rightarrow 0$. Bulk properties of the star, like mass, angular
momentum, and rotational kinetic energy, are calculated once a set of self-
consistent solutions to the above equations is found.

###  2.2 Equation of State

The models of dense matter EOS used in this work are based on the framework of
CDF theory, where meson-$\Delta$ coupling values are varied and the
calculations are carried out with the relativistic Hartree-Fock (RHF)
approximation. The Lagrangian of the model is given as

$\mathscr{L}=\mathscr{L}_{\text{B}}+\mathscr{L}_{\text{m}}+\mathscr{L}_{\text{int}}+\mathscr{L}_{\text{l}},$
(8)

where $\mathscr{L}_{\text{B}}$ represents free baryonic fields
$\psi_{\text{B}}$, $\mathscr{L}_{\text{m}}$ represents free meson fields
$\phi_{\text{m}}$, $\mathscr{L}_{\text{int}}$ describes the interaction
between baryons and mesons, and $\mathscr{L}_{\text{l}}$ represents the
contribution from free leptons. The baryons accounted for include the spin-1/2
octet of nucleons $N\in\\{n,p\\}$ and hyperons
$Y\in\\{\Lambda,\Xi^{0,-},\Sigma^{+,0,-}\\}$ and the spin-3/2 zero-strangeness
quartet $\Delta\in\\{\Delta^{++,+,0,-}\\}$. The mesons accounted for include
those regularly encountered in the RHF approximation including the isoscalar-
scalar meson $\sigma$, the isoscalar-vector meson $\omega$, the isovector-
vector meson $\rho$, and the pseudo-vector meson $\pi$, as well as two hidden-
strangeness mesons, $\sigma^{*}$ and $\phi$, which describe the interaction
between hyperons. The leptons accounted for are electrons $e^{-}$ and muons
$\mu^{-}$.

The standard procedure for obtaining the density functional begins with
finding equations of motion from the Euler-Lagrange equations for each
particle species. These take the form of the Dirac equations for the baryon
octet and leptons, the Rarita-Schwinger equations for the $\Delta$’s, and the
Klein-Gordon equations for the mesons. The solutions of these equations are
used to generate the energy density functional by evaluating each of the
baryon self energies, $\Sigma$, in the RHF approximation. In Dirac space, the
self-energy can be defined as

$\Sigma(k)=\Sigma_{S}(k)+\gamma_{0}\Sigma_{0}(k)+\boldsymbol{\gamma}\cdot\boldsymbol{\hat{k}}\Sigma_{V}(k),$
(9)

where $\Sigma_{S}$, $\Sigma_{0}$, and $\Sigma_{V}$ denote the scalar self-
energy, the time and space components of vector self-energy, respectively,
$\gamma_{\mu}$ ($\mu$ = 0-3) are the Dirac matrices, and
$\boldsymbol{\hat{k}}$ is a unit vector along the 3-momentum $\boldsymbol{k}$.
The self-energies, field equations, and the required charge neutrality
condition are then used to determine EOS at zero temperature. The full self-
consistent procedure is outlined in greater detail in Li . (2018); Sedrakian .
(2023); Weber (1999).

The interaction between mesons and baryons, described by
$\mathscr{L}_{\text{int}}$ in Eqn. (8), is parametrized by meson-baryon
coupling constants $g_{mB}$. The values for each of the coupling constants are
dependent on the type of baryon and must be fitted to empirical data of
nuclear and hypernuclear systems. For the hypernuclear sector, the coupling
constant values are given by the SU(3) flavor symmetry quark model for vector
mesons; for scalar mesons, the coupling constants are fitted to empirical
hypernuclear potentials. The meson-$\Delta$ coupling is parametrized by
$\Delta$ isoscalar potential VΔ, which is measured in units of the nucleon
isoscalar potential
V${}_{\text{N}}=\Sigma_{0,\omega}(\omega)+\Sigma_{S,\sigma}(\sigma)$. Heavy-
ion collision and scattering experiments provide approximate upper and lower
bounds on VΔ relating to V${}_{\text{N}}$, but there are no exact values
defined for the isoscalar potential. Following Li . (2018), three potential
depths VΔ are chosen at nuclear saturation density $\rho_{0}$ as VΔ = 2/3
V${}_{\text{N}}$, V${}_{\text{N}}$, and 4/3 V${}_{\text{N}}$. These are
consistent with inferences of $\Delta$ potentials from terrestrial experiments
and are compatible with the lower limit on the maximum mass of static compact
stars $M_{\rm TOV}\geq 2M_{\odot}$. For comparison, we also include a
hypernuclear EOS constructed from the framework of CDF theory which does not
include $\Delta$ particles, labeled "npY" throughout the text. The three mixed
$\Delta$-hyperon EOS models parametrized by VΔ and the purely hypernuclear EOS
model npY are shown in Fig. 1.

Figure 1: Pressure $P$ vs. energy density $\epsilon$ for four EOS models. The
black curve, denoted "npY", describes hypernuclear matter that does not
contain $\Delta$ isobars. The other three curves contain both hyperons and
$\Delta$’s, parametrized by different values of the potential depth VΔ (see
text for more details).

##  3 Results

The EOS models described in Section 2.2 were used to calculate the properties
of differentially rotating neutron stars. As demonstrated in Li . (2018),
accounting for $\Delta$’s reduced radii of non-rotating stars when compared to
the hypernuclear only EOS, npY. Additionally, including $\Delta$’s increased
slightly the masses of non-rotating stellar sequences, where the larger by
absolute value depth of the potential VΔ resulted in a higher maximum mass. In
the following sections, stellar sequences under different degrees of
differential rotation are calculated to compare to the non-rotating mass and
radius values shown in Li . (2018) to explore if the same trends are
replicated. Furthermore, we calculate density maps of high-mass,
differentially rotating stars to demonstrate at what depth specific particle
species (hyperons and/or $\Delta$’s) appear.

###  3.1 Stellar Sequences

We first show results for stellar sequences with a constant central density
range in order to demonstrate the expected increase in mass due to
differential rotation. Because the rotation parameter repeatedly appears as
$\hat{A}^{-1}$ in the equations described in Section 2, we follow the lead of
previous work which parameterized sequences by values of $\hat{A}^{-1}$ = 0.3,
0.5, 0.7, and 1.0 (Morrison ., 2004; Cook ., 1992; Galeazzi ., 2012). As
mentioned previously, uniform rotation is obtained in the limit
$\hat{A}^{-1}\rightarrow 0$, and an upper bound of the scaled rotation
parameter is $\hat{A}^{-1}=1.0$. As differential rotation becomes more extreme
in the star ($\hat{A}^{-1}\rightarrow 1.0$), the maximum mass and equatorial
radius of a stellar sequence is expected to increase.

This is indeed the case for the EOS containing hyperonic matter (npY) and the
three EOS which vary $\Delta$ potential (VΔ) depths. Figures 2 and 3 show
mass-radius (equatorial) relations for four degrees of differential rotation,
parameterized by $\hat{A}^{-1}$, compared to the non-rotating curves
calculated with the Tolman-Oppenheimer-Volkoff (TOV) equation; Figure 2 uses
the hyperonic EOS model denoted by npY, and Figure 3 uses the EOS model with
largest potential depth value of VΔ = 4/3 V${}_{\text{N}}$ Li . (2018).

They highlight that the inclusion of $\Delta$ in the EOS composition will
soften the EOS at lower densities and stiffens it at higher densities. This
impacts the corresponding mass-radius relations for non-rotating stars by
decreasing the radius but increasing the maximum mass. When comparing the
mass-radius curves under differential rotation resulting from the hyperonic
EOS in Figure 2 to those resulting from the largest value of VΔ in Figure 3,
the same trend is observed. For the highest degree of differential rotation
($\hat{A}^{-1}=1.0$), the canonical 1.4 M⊙ star containing $\Delta$’s has a
radius of 15.46 km for the EOS with VΔ = 4/3 V${}_{\text{N}}$, compared to a
radius of 17.65, reflecting the same decrease in radius observed with the non-
rotating curves in Li . (2018).

The maximum masses of the hyperonic EOS and the three EOS models containing
both hyperons and $\Delta$’s are given in Table 1 for non-rotating (TOV) and
differentially rotating stars. For values of parameter $\hat{A}^{-1}$
corresponding to different degrees of differential rotation are used. Both
Table 1 and Figures 2 and 3 reflect calculated values for stellar models at a
fixed value of the ratio between the polar and equatorial radius,
r${}_{\text{ratio}}=0.7$. For all employed EOS models, we observe a similar
16-17% increase in the maximum mass of a maximally differentially rotating
star when compared to the maximum mass of a static (TOV) star. This shows that
the difference in particle compositions that were accounted for has no
significant impact on how much the maximum mass will increase as the rotation
parameter $\hat{A}^{-1}\rightarrow 1.0$.

Figure 2: Mass-radius relations for the hyperonic EOS npY. The dashed black line shows the non-rotating curve resulting from the TOV equation, while the solid-colored lines are curves under different degrees of differential rotation. Differential rotation is parameterized by values of the rotation parameter, $\hat{A}^{-1}$ (see text for more details).  Figure 3: Mass-radius relations for the mixed hyperon-$\Delta$ EOS with VΔ = 4/3 V${}_{\text{N}}$. The dashed black line shows the non-rotating curve resulting from the TOV equation, while the solid-colored lines are curves under different degrees of differential rotation. Differential rotation is parameterized by values of the rotation parameter, $\hat{A}^{-1}$ (see text for more details). Table 1: Maximum gravitational mass (in units of M⊙) of four RHF EOS models which account for hyperons (npY) or hyperons and $\Delta$ particles (given as values of the VΔ potential): EOS Model | TOV |  $\hat{A}^{-1}$ = 0.3 |  $\hat{A}^{-1}$ = 0.5 |  $\hat{A}^{-1}$ = 0.7 |  $\hat{A}^{-1}$ = 1.0
---|---|---|---|---|---
npY | 2.011 | 2.183 | 2.253 | 2.319 | 2.356
2/3 V${}_{\text{N}}$ | 2.034 | 2.200 | 2.270 | 2.334 | 2.366
1.0 V${}_{\text{N}}$ | 2.054 | 2.216 | 2.287 | 2.355 | 2.386
4/3 V${}_{\text{N}}$ | 2.103 | 2.262 | 2.339 | 2.418 | 2.461

###  3.2 Particle Composition

In this section, density distributions of massive, differentially rotating
stars are used to demonstrate where in the star different particle species
appear.

Individual stellar models are constructed with a series of set parameters: the
central density, the ratio between the polar and equatorial radius
r${}_{\text{ratio}}$, and the rotation parameter $\hat{A}^{-1}$ which
specifies the degree of differential rotation. The parameter
r${}_{\text{ratio}}$ reflects the structural deformation observed with rapid
rotation, which has been extensively showed through numerical simulation
(Hartle Thorne, 1968; Huber ., 1998; Weber, 1999) to result in a lengthening
of the radius at the equator and shortening at the pole. In the case of
differential rotation, more extreme structural deformations can occur as
r${}_{\text{ratio}}\rightarrow 0$, in some cases leading to the formation of
toroidally shaped objects (Lyford ., 2003; Morrison ., 2004). In this section,
we fix r${}_{\text{ratio}}$ at a value of 0.6, slightly lower than that used
in Section 3.1, resulting in massive, rapidly rotating, ellipsoid-shaped
stars.

As in Section 3.1, we construct density distribution maps for the hyperonic
EOS model and the VΔ = 4/3 V${}_{\text{N}}$ EOS model additionally containing
$\Delta$’s. Particle fractions for both EOS models can be found in Figure 4.
For the hyperonic EOS model, the three hyperon species accounted for are the
$\Lambda$, the $\Xi^{-}$, and the $\Sigma^{-}$ hyperons, which appear in that
order. Figure 5 shows the density map of a 2.48 M⊙ star constructed from the
npY EOS model, with the following set parameters: central density
$\epsilon_{c}$ = 900 MeV, r${}_{\text{ratio}}$ = 0.6, and rotation parameter
$\hat{A}^{-1}$ = 0.7. The $\Lambda$ hyperon first appears at an equatorial
radius, r${}_{\text{e}}$, of $\sim$5 km and a density of 329.4 MeV/fm3 (shown
in cyan); the $\Lambda$ hyperon would be present from this depth until the
center of the star. The $\Sigma^{-}$ and $\Xi^{-}$ both appear at
r${}_{\text{e}}\approx$ 3 km and densities of 479.1 and 510.9 MeV/fm3,
respectively (shown in green). The $\Sigma^{-}$ only appear briefly due to
their repulsive potential at nuclear saturation density (Li ., 2018), but the
$\Xi^{-}$ would be populated throughout the star up to its center.

The density distribution map for a 2.55 M⊙ star constructed with the mixed
hyperon and $\Delta$ EOS with a potential depth of VΔ = 4/3 V${}_{\text{N}}$
is shown in Figure 6. The same values for the central density,
r${}_{\text{ratio}}$, and rotation parameter $\hat{A}^{-1}$ were used as in
Figure 5. As shown in both Figure 4 and Figure 2 of Li . (2018), a larger
value of the potential VΔ results in a lowered onset threshold for $\Delta$’s,
and their threshold density is much lower than the one for the first hyperon.
For the largest potential depth VΔ = 4/3 V${}_{\text{N}}$, the entire spin-3/2
zero-strangeness quartet appears in the star. In this case, only the $\Lambda$
and $\Xi^{-}$ hyperons appear. The $\Sigma^{-}$ hyperon is effectively
replaced by the same-charge $\Delta^{-}$ isobar, as it is energetically more
favorable due to its large negative potential. In Figure 6, the $\Delta^{-}$
appears at a density of 168.66 MeV/fm3, corresponding to an r${}_{\text{e}}$
of $\sim$8 km (shown in royal blue). The next species to appear, $\Delta^{0}$,
appears at a density of 344.1 MeV/fm3 corresponding to an r${}_{\text{e}}$ of
$\sim$8 km (shown in cyan). The $\Lambda$ and $\Delta^{+}$ become present at
densities of 465.9 and 486.3 MeV/fm3 respectively, shown in green at a radius
of $\sim$ 5 km. The particle species to appear is the $\Delta^{++}$, at a
density of 763.6 MeV/fm3 and radius of $<$ 3 km shown in red. All particles
would be present from the density they appear through to the center of the
star.

It is important to note here that neutrons, protons, electrons, and muons are
additionally present within the stellar models discussed above but not shown
explicitly in Figures 5 and 6.

Figure 4: Relative particle fractions Yi for the hypernuclear EOS npY (top
panel) and the hyperon-$\Delta$ admixed EOS with a potential depth of 4/3
V${}_{\text{N}}$ (bottom panel).  Figure 5: Energy density map of a 2.48 M⊙
differentially rotating star constructed with hyperonic EOS npY. The star has
an r${}_{\text{ratio}}$ = 0.6 and rotation parameter $\hat{A}^{-1}$ = 0.7.
Figure 6: Energy density map of a 2.55 M⊙ differentially rotating star
constructed with hyperon-$\Delta$ admixed EOS with a potential depth of 4/3
V${}_{\text{N}}$. The star has an r${}_{\text{ratio}}$ = 0.6 and rotation
parameter $\hat{A}^{-1}$ = 0.7.

##  4 Discussion

In this work, we extended the previous work of Li . (2018) and Li . (2023)
that studied static and uniformly rotating stellar configurations of hyperonic
and $\Delta$-admixed stars to stellar models that support differential
rotation. Specifically, we examined how the inclusion of $\Delta$ isobars in
the EOS of dense matter impacts the corresponding stellar properties like mass
and equatorial radius for non-vanishing differential rotations. In Section
3.1, we demonstrated that the inclusion of $\Delta$’s in hypernuclear EOS
results in the reduction of the equatorial radius of the differentially
rotating star and an increase in its maximum mass. These trends are consistent
with analogous findings for uniformly rotating and static compact stars. An
interesting finding is that when comparing the maximum masses from the highest
degree of differential rotation (where the rotation parameter
$\hat{A}^{-1}=1.0$) to the lowest ($\hat{A}^{-1}=0.3$), the magnitude of the
increase in mass remained constant for all four EOS models considered. In
particular, the inclusion of $\Delta$’s in the hypernuclear EOS does not
change the total amount of mass increase once the differential rotation is
allowed. In Section 3.2, density distribution maps of massive, differentially
rotating stars were shown to demonstrate at what radial depth different
particle species appear. The competition between the nucleation of
$\Sigma^{-}$ hyperon and $\Delta^{-}$ isobar was demonstrated, whose outcome
depends on the value of the $\Delta^{-}$ potential in nuclear matter.

##  5 Acknowledgements

DF and FW are supported by the National Science Foundation (USA) under Grant
No. PHY-2012152. J. L. acknowledges the support of the National Natural
Science Foundation of China (Grant No. 12105232), the Fundamental Research
Funds for the Central Universities (Grant No. SWU-020021), and by the Venture
& Innovation Support Program for Chongqing Overseas Returnees (Grant No.
CX2021007). A. S. is supported by the Deutsche Forschungsgemeinschaft Grant
No. SE 1836/5-2 and the Polish NCN Grant No. 2020/37/B/ST9/01937 at Wroclaw
University.

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