#  CREX and PREX-II motivated relativistic interactions and their implications
to the bulk properties of nuclear matter and neutron star

Mukul Kumar<EMAIL_ADDRESS>Department of Physics, Himachal
Pradesh University, Shimla-171005, India  Sunil Kumar  Department of Physics,
Himachal Pradesh University, Shimla-171005, India  Virender Thakur
<EMAIL_ADDRESS>Department of Physics, Himachal Pradesh University,
Shimla-171005, India  Raj Kumar<EMAIL_ADDRESS>Department of Physics,
Himachal Pradesh University, Shimla-171005, India  B.K. Agrawal
<EMAIL_ADDRESS>Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Kolkata 700064, India  Shashi K. Dhiman<EMAIL_ADDRESS>Department of
Physics, Himachal Pradesh University, Shimla-171005, India  School of Applied
Sciences, Himachal Pradesh Technical University, Hamirpur-177001, India

###### Abstract

We investigate the implications of parity-violating electron scattering
experiment on neutron skin thickness of 48Ca (CREX) and 208Pb (PREX-II) data
on the bulk properties of finite nuclei, nuclear matter and neutron stars. The
neutron skin thickness from the CREX and PREX-II data is employed to constrain
the parameters of relativistic mean field models which includes different non-
linear, self and cross-couplings among isoscalar-scalar $\sigma$, isoscalar-
vector $\omega$, isovector-scalar $\delta$ and isovector-vector $\rho$ meson
fields up to the quartic order. Three parametrizations of RMF model are
proposed by fitting CREX, PREX-II and both CREX as well as PREX-II data to
assess their implications. A covariance analysis is performed to assess the
theoretical uncertainties of model parameters and nuclear matter observables
along with correlations among them. The RMF model parametrization obtained
with the CREX data acquires much smaller value of symmetry energy (J=
28.97$\pm$ 0.99 MeV), its slope parameter (L= 30.61$\pm 6.74$ MeV) in
comparison to those obtained with PREX-II data. The neutron star properties
are studied by employing the equations of state (EoSs) composed of nucleons
and leptons in $\beta$ equilibrium.

Equation of State; Neutron star

††preprint: APS/123-QED

##  I INTRODUCTION

The nuclear equation of state (EoS) plays a vital role for understanding the
properties of strongly interacting many body systems like atomic nuclei and
neutron stars [1, 2, 3]. The nuclear symmetry energy and its density
dependence are key features of nuclear EoS. Constraining the density
dependence of symmetry energy represents a long-standing and unresolved
question in nuclear physics and astrophysics [3]. The density dependence of
the symmetry energy has implications in a variety of phenomena such as heavy-
ion collisions, core-collapse supernovas and neutron-star structures. Although
important, this quantity cannot be directly measured in the laboratory, it can
only be derived from theories and thus to constrain their values it is
necessary to identify and use relevant observables on finite nuclei. A neutron
star is a highly dense and asymmetric nuclear system that has a central
density of about 5-6 times the nuclear saturation density [4]. The study of
the neutron star proclaims that its internal structure is more complex as new
degrees of freedom like hyperons and quarks appear in the core. The properties
of the neutron star like mass, radius, and tidal deformability can be
explained by taking into account the interaction between nucleons and the
mesonic degree of freedom in the form of Lagrangian. This provides an EoS
which is the main input for the calculation of neutron star properties. The
several relativistic mean fields (RMF) models having effective lagrangian
density consisting of nonlinear $\sigma$, $\omega$, $\rho$ and $\delta$ terms
and cross terms have been analyzed for nucleonic matter and nucleonic along
with hyperonic matter and accosted with the constraints of nuclear matter
properties and astrophysical observations of compact star masses [5, 6, 7].
The nuclear theory studies [8, 9, 10] are mainly focusing on understanding the
dense matter in compact stars (CS). The constraints on EoS at high density are
imposed with accurate information of a neutron star’s maximum mass and radius
[11, 12, 1]. The precise measurement of masses of millisecond pulsars such as
PSR J1614-2230 [13], PSR J0348+0432 [14] show that the maximum mass of the
neutron star should be around 2 M$\odot$. The recent observations with LIGO
and Virgo of GW170817 event [15, 16] of Binary Neutron Stars merger and the
discovery of CS with masses around 2$M_{\odot}$ [13, 14, 17, 18, 19, 20] have
intensified the interest in these intriguing objects. The analysis of GW170817
has demonstrated the potential of gravitational wave (GW) observations to
yield new information relating to the limits on CS tidal deformability.
The recent precise parity-violating electron scattering experiments on 48Ca
(CREX) [21] and 208Pb (PREX-II)[22] provide new insights into the neutron skin
thickness of nuclei. These experiments are helpful in determining the nuclear
weak charge form factor by measuring the parity-violating asymmetry. The weak
charge form factor has a strong correlation with the density dependence of
symmetry energy and neutron skin thickness of nuclei and plays an important
role in probing the isovector channels of energy density functionals. The
parity-violating electron scattering experiments give precise and model-
independent data for the nuclear weak charge form factor that can be used to
constrain the energy density functionals [23]. The weak charge form factors of
48Ca and 208Pb reported by CREX and PREX-II experiments and their measured
parity-violating asymmetry has been analyzed using density functionals and
reached on a conclusion that it is difficult to describe parity-violating
asymmetry simultaneously in both nuclei [24, 25]. The Calcium Radius
Experiment (CREX) has recently given a model-independent extraction of neutron
skin thickness of ${}^{48}{Ca}$ as $\Delta r_{np}$ = 0.121 $\pm$ 0.026 fm [21]
which suggests softness of density dependence of symmetry energy. The Lead
Radius Experiment (PREX) has recently given a model-independent extraction of
neutron skin thickness of ${}^{208}{Pb}$ as $\Delta r_{np}$ = 0.283 $\pm$
0.071 fm [22] by combining the original PREX result with the new PREX-II. The
$\Delta r_{np}$ has been identified as an ideal probe on symmetry energy - a
key but poorly known quantity that describes the isospin dependence of EoS of
nuclear matter and plays a crucial role in various issues in nuclear physics
and astrophysics. The neutron skin thickness of the Lead nucleus exhibits a
strong positive linear correlation with the slope of symmetry energy (L) at
saturation density. The value of L around the saturation density strongly
affects the Mass-Radius relation and tidal deformability ($\Lambda$) of a
neutron star and provides a unique bridge between atomic nuclei and neutron
star. The large value of $\Delta r_{np}$ = 0.283 $\pm$ 0.071 fm suggests a
very stiff EoS and large value of L around saturation density and generally
gives rise to a large value of neutron star radius and the tidal deformability
[26]. The upper limit on $\Lambda_{1.4}$ $\leq$ 580 for GW170817 requires
softer EoS and hence softer symmetry energy coefficient [15]. The heaviest
neutron star 2.14${}_{-0.09}^{+0.10}M_{\odot}$ of PSRJ0740+6620 [27] also
limits the EoS for symmetric nuclear matter (SNM). The flow data from heavy
ion collisions suggests that the EoS for SNM should be relatively softer [28].
The motivation of the present work is to generate three new relativistic
interactions for the Lagrangian density of the RMF model to investigate the
effect of CREX and PREX-II data on neutron skin thickness for 48Ca and 208Pb
nuclei on the bulk nuclear properties and observed astrophysical constraints
on neutron stars. A covariance analysis can also be performed to assess the
theoretical uncertainties of model parameters and nuclear matter observables
along with correlations among them.
The paper is organized as follows, in section II, a brief outlines of the RMF
Lagrangian, equations of motion and EoS for neutron stars is provided. In
section III, the procedure for optimization of the model parameters and
covariance analysis is discussed. Numerical results and detailed discussions
features of model parametrizations, finite nuclei, bulk nuclear matter,
neutron star matter and correlations amongst nuclear matter observables and
model parameters are presented in section IV. Finally, we give a summary in
section V.

##  II THEORETICAL MODEL

The Lagrangian density for the RMF model used in the present work is based
upon different non-linear, self and inter-couplings among isoscalar-scalar
$\sigma$, isoscalar-vector $\omega_{\mu}$, isovector-scalar $\delta$ and
isovector-vector $\rho_{\mu}$ meson fields and nucleonic Dirac field $\Psi$
[5, 29, 6, 7], is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{B}\overline{\Psi}_{B}[i\gamma^{\mu}\partial_{\mu}-(M_{B}-g_{\sigma
B}\sigma-g_{\delta B}\delta\cdot\tau_{3})-(g_{\omega
B}\gamma^{\mu}\omega_{\mu}$ (1) $\displaystyle+$
$\displaystyle\frac{1}{2}g_{\mathbf{\rho}B}\gamma^{\mu}\tau_{B}.\mathbf{\rho}_{\mu})]\Psi_{B}+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})$ $\displaystyle-$
$\displaystyle\frac{\overline{\kappa}}{3!}g_{\sigma
N}^{3}\sigma^{3}-\frac{\overline{\lambda}}{4!}g_{\sigma
N}^{4}\sigma^{4}-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}$
$\displaystyle+$ $\displaystyle\frac{1}{4!}\zeta g_{\omega
N}^{4}(\omega_{\mu}\omega^{\mu})^{2}-\frac{1}{4}\mathbf{\rho}_{\mu\nu}\mathbf{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\mathbf{\rho}_{\mu}\mathbf{\rho}^{\mu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}(\partial_{\mu}\delta\partial^{\mu}\delta-
m_{\delta}^{2}\delta^{2})+\frac{1}{2}\Lambda_{\omega\rho}g_{\omega}^{2}g_{\rho}^{2}\omega_{\mu}\omega^{\mu}\rho_{\mu}\rho^{\mu}$
$\displaystyle-$
$\displaystyle\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\sum_{B}e\overline{\Psi}_{B}\gamma_{\mu}\frac{1+\tau_{3B}}{2}A_{\mu}\Psi_{B}$
$\displaystyle+$
$\displaystyle\sum_{\ell=e,\mu}{\overline{\Psi}_{\ell}}\left(i\gamma^{\mu}\partial_{\mu}-M_{\ell}\right)\Psi_{\ell}.$

The equation of motion for baryons, mesons, and photons can be derived from
the Lagrangian density defined in Eq.(1). The equation of motion for baryons
can be given as,

$\displaystyle\bigg{[}\gamma^{\mu}\left(i\partial_{\mu}-g_{\omega
B}\omega_{\mu}-\frac{1}{2}g_{\rho
B}\tau_{B}.\rho_{\mu}-e\frac{1+\tau_{3B}}{2}A_{\mu}\right)-$ (2)
$\displaystyle(M_{B}-g_{\sigma B}\sigma-g_{\delta
B}\delta\cdot\tau_{3})\bigg{]}\Psi_{B}=\epsilon_{B}\Psi_{B}.$

The Euler-Lagrange equations for the ground-state expectation values of the
mesons fields are

$\displaystyle\left(-\Delta+m_{\sigma}^{2}\right)\sigma$ $\displaystyle=$
$\displaystyle\sum_{B}g_{\sigma
B}\rho_{sB}-\frac{\overline{\kappa}}{2}g_{\sigma
N}^{3}\sigma^{2}-\frac{\overline{\lambda}}{6}g_{\sigma N}^{4}\sigma^{3}$
$\displaystyle\left(-\Delta+m_{\omega}^{2}\right)\omega$ $\displaystyle=$
$\displaystyle\sum_{B}g_{\omega B}\rho_{B}-\frac{\zeta}{6}g_{\omega
N}^{4}\omega^{3}$ (4) $\displaystyle-$
$\displaystyle\Lambda_{\omega\rho}g_{\omega N}^{2}g_{\rho
N}^{2}\omega\rho^{2},$ $\displaystyle\left(-\Delta+m_{\rho}^{2}\right)\rho$
$\displaystyle=$ $\displaystyle\sum_{B}g_{\rho
B}\tau_{3B}\rho_{B}-\frac{\xi}{6}g_{\rho N}^{4}\rho^{3}$ (5)
$\displaystyle-\Lambda_{\omega\rho}g_{\omega N}^{2}g_{\rho
N}^{2}\omega^{2}\rho,$
$\left(-\Delta+m_{\delta}^{2}\right)\delta=\sum_{B}g_{\delta B}\rho_{s3B}$ (6)
$-\Delta A_{0}=e\rho_{p}.$ (7)

where the baryon vector density $\rho_{B}$, scalar density $\rho_{sB}$ and
charge density $\rho_{p}$ are, respectively,

$\rho_{B}=\left<\overline{\Psi}_{B}\gamma^{0}\Psi_{B}\right>=\frac{\gamma
k_{B}^{3}}{6\pi^{2}},$ (8)

$\rho_{sB}=\left<\overline{\Psi}_{B}\Psi_{B}\right>=\frac{\gamma}{(2\pi)^{3}}\int_{0}^{k_{B}}d^{3}k\frac{M_{B}^{*}}{\sqrt{k^{2}+M_{B}^{*2}}},$
(9)
$\rho_{p}=\left<\overline{\Psi}_{B}\gamma^{0}\frac{1+\tau_{3B}}{2}\Psi_{B}\right>,$
(10)

with $\gamma$ the spin-isospin degeneracy. The Dirac effective mass for the
neutron and proton can be written as

$M^{*}_{p}=(M-g_{\sigma}\sigma-g_{\delta}\delta),$ (11)
$M^{*}_{n}=(M-g_{\sigma}\sigma+g_{\delta}\delta),$ (12)

Following the Euler-Lagrange formalism one can readily find the expressions
for energy density ${\cal E}$ and pressure $P$ as a function of density from
Eq. (1) [30].

The energy density of the uniform matter within the framework of the RMF model
is given by;

$\begin{split}{\cal
E}&=\sum_{j=B,\ell}\frac{1}{\pi^{2}}\int_{0}^{k_{j}}k^{2}\sqrt{k^{2}+M_{j}^{*2}}dk\\\
&+\sum_{B}g_{\omega B}\omega\rho_{B}+\sum_{B}g_{\rho
B}\tau_{3B}\rho_{B}\rho+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}\\\
&+\frac{\overline{\kappa}}{6}g_{\sigma
N}^{3}\sigma^{3}+\frac{\overline{\lambda}}{24}g_{\sigma
N}^{4}\sigma^{4}-\frac{\zeta}{24}g_{\omega N}^{4}\omega^{4}\\\
&-\frac{\xi}{24}g_{\rho
N}^{4}\rho^{4}-\frac{1}{2}m_{\omega}^{2}\omega^{2}-\frac{1}{2}m_{\rho}^{2}\rho^{2}\\\
&-\frac{1}{2}\Lambda_{\omega\rho}g_{\omega N}^{2}g_{\rho
N}^{2}\omega^{2}\rho^{2}+\frac{1}{2}m_{\delta}^{2}\delta^{2}\\\ \end{split}$
(13)

The pressure of the uniform matter is given by

$\begin{split}P&=\sum_{j=B,\ell}\frac{1}{3\pi^{2}}\int_{0}^{k_{j}}\frac{k^{4}dk}{\sqrt{k^{2}+M_{j}^{*2}}}-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}\\\
&-\frac{\overline{\kappa}}{6}g_{\sigma
N}^{3}\sigma^{3}-\frac{\overline{\lambda}}{24}g_{\sigma
N}^{4}\sigma^{4}+\frac{\zeta}{24}g_{\omega N}^{4}\omega^{4}\\\
&+\frac{\xi}{24}g_{\rho
N}^{4}\rho^{4}+\frac{1}{2}m_{\omega}^{2}\omega^{2}+\frac{1}{2}m_{\rho}^{2}\rho^{2}\\\
&+\frac{1}{2}\Lambda_{\omega\rho}g_{\omega N}^{2}g_{\rho
N}^{2}\omega^{2}\rho^{2}-\frac{1}{2}m_{\delta}^{2}\delta^{2}\\\ \end{split}$
(14)

Here, the sum is taken over nucleons and leptons.

##  III OPTIMIZATION AND COVARIANCE ANALYSIS

The optimization of the parameters (p) appearing in the Lagrangian (Eq. 1) has
been performed by using the simulated annealing method (SAM) [31, 32] by
following $\chi^{2}$ minimization procedure which is given as,

${\chi^{2}}(\textbf{p})=\frac{1}{N_{d}-N_{p}}\sum_{i=1}^{N_{d}}\left(\frac{M_{i}^{exp}-M_{i}^{th}}{\sigma_{i}}\right)^{2},$
(15)

where $N_{d}$ is the number of experimental data points and $N_{p}$ is the
number of fitted parameters. The $\sigma_{i}$ denotes adopted errors [33] and
$M_{i}^{exp}$ and $M_{i}^{th}$ are the experimental and the corresponding
theoretical values, respectively, for a given observable. The minimum value of
${{\chi}}^{2}_{0}$ corresponds to the optimal values $\bf{p}_{0}$ of the
parameters. After the optimization of the energy density functional, it is
important to explore the richness of the covariance analysis. It enables one
to calculate the statistical uncertainties/errors on model parameters or any
calculated physical observables. The covariance analysis also provides
additional information about the sensitivity of the parameters to the physical
observables, and interdependence among the parameters [33, 34, 35, 36]. Having
obtained the parameter set, the correlation coefficient between two quantities
Y and Z can be calculated by covariance analysis [37, 38, 36, 33, 35] as

$\textit{c}_{YZ}=\frac{\overline{\Delta{Y}\Delta{Z}}}{\sqrt{\overline{\Delta{Y^{2}}}\quad\overline{\Delta{Z^{2}}}}},$
(16)

where covariance between Y and Z is expressed as

$\overline{\Delta{Y}\Delta{Z}}=\sum_{\alpha\beta}\left(\frac{\partial{Y}}{\partial{p}_{\alpha}}\right)_{\textbf{p}_{0}}C_{\alpha\beta}^{-1}\left(\frac{\partial{Z}}{\partial{p}_{\beta}}\right)_{\textbf{p}_{0}}.$
(17)

Here, $C_{\alpha\beta}^{-1}$ is an element of inverted curvature matrix given
by

$\textit{C}_{\alpha\beta}=\frac{1}{2}\left(\frac{\partial^{2}\chi^{2}(\textbf{p})}{\partial{p}_{\alpha}\partial{p}_{\beta}}\right)_{\textbf{p}_{0}}.$
(18)

The standard deviation, $\overline{\Delta{Y}^{2}}$, in Y can be computed using
Eq. (17) by substituting Z = Y.

##  IV RESULTS and DISCUSSION

We obtain parameterizations for RMF models by employing CREX, PREX-II and
combined CREX -PREX II data by following the procedure discussed in Section
III. The model parameterizations obtained are then used to calculate the
properties of finite nuclei and infinite nuclear matter and neutron stars. We
also discuss the correlations among nuclear matter observables and model
parameters.

###  IV.1 Parametrizations of RMF Model

In the present study, three new relativistic interactions BSRV-CREX, BSRV-PREX
and BSRV-CPREX have been generated for the Lagrangian density given by Eq. (1)
to investigate the effect of CREX and PREX-II data on neutron skin thickness
for ${}^{48}Ca$ and ${}^{208}Pb$ nuclei on the properties of finite nuclei and
neutron star matter. The parameters of the BSRV-CREX, BSRV-PREX and BSRV-CPREX
models are obtained by fitting exactly the available experimental data of [39]
on binding energies ($BE$) and charge rms radii ($r_{ch}$) [40] of some
closed/open-shell nuclei 16,24O, 40,48Ca, 56,68,78Ni, 88Sr,90Zr,
100,116,132Sn, 144Sm and 208Pb. In addition, we also include in our fit the
value of neutron skin thickness for 48Ca and 208 Pb nuclei, which is a very
important physical observable to constrain the value of L that determines the
linear dependence of symmetry energy.

Table 1:  Newly generated parameter sets BSRV-CREX, BSRV-PREX and BSRV-CPREX
for the Lagrangian of RMF model as given in Eq.(1) along with theoretical
uncertainties/errors. The parameters $\overline{\kappa}$, is in fm-1. The mass
for nucleon, $\omega$, $\rho$ and $\delta$ mesons are taken as $M_{N}$ = 939
MeV, $m_{\omega}$= 782.5 MeV, $m_{\rho}$= 762.468 MeV and $m_{\delta}$= 980
MeV respectively. The values of $\overline{\kappa}$, $\overline{\lambda}$, and
${\Lambda_{\omega\rho}}$ are multiplied by $10^{2}$. Parameters for NL3 [41],
FSUGarnet [34], IOPB-1 [42], and Big Apple [43] are also shown for comparison.
${\bf{Parameters}}$ | BSRV-CREX | BSRV-PREX | BSRV-CPREX | NL3 | FSUGarnet | IOPB-1 | Big Apple
---|---|---|---|---|---|---|---
${\bf g_{\sigma}}$ | 10.71506$\pm$ 0.02086 | 10.40613 $\pm$0.08977 | 10.44537$\pm$0.02480 | 10.21743 | 10.50315 | 10.41851 | 9.67810
${\bf g_{\omega}}$ | 13.82692$\pm$0.03539 | 13.37605$\pm$0.13078 | 13.43408$\pm$0.03477 | 12.86762 | 13.69695 | 13.38412 | 12.33541
${\bf g_{\rho}}$ | 16.18406$\pm$1.20866 | 10.27951$\pm$2.23787 | 10.28003$\pm$2.15232 | 8.94880 | 13.87880 | 11.11560 | 14.14256
${\bf g_{\delta}}$ | 4.27816$\pm$0.75631 | 1.19517$\pm$4.29242 | 1.70339$\pm$3.97826 | – | – | – | –
${\bf\overline{\kappa}}$ | 1.41726$\pm$0.04886 | 1.64259$\pm$0.04079 | 1.66238$\pm$0.05821 | 1.95734 | 1.65229 | 1.85581 | 2.61776
${\bf\overline{\lambda}}$ | 0.46733$\pm$0.08773 | -0.08316$\pm$0.09341 | -0.20868$\pm$0.19224 | -1.59137 | -0.035330 | -0.075516 | -2.16586
${\bf{\zeta}}$ | 0.03441$\pm$0.00130 | 0.02611$\pm$0.00212 | 0.02429$\pm$0.00326 | 0.00000 | 0.23486 | 0.017442 | 0.000699
${\bf\Lambda_{\omega\rho}}$ | 5.18286$\pm$1.21735 | 2.90293$\pm$2.60556 | 2.25722$\pm$1.70126 | 0.00000 | 8.6000 | 4.80000 | 9.40000
${\bf m_{\sigma}}$ | 504.679$\pm$0.689 | 502.050$\pm$1.099 | 501.933$\pm$1.600 | 508.194 | 496.731 | 500.487 | 492.975

The BSRV-CREX parametrization has been obtained by incorporating the recently
measured neutron skin thickness $\Delta r_{np}$= $0.121\pm 0.026$ fm for 48Ca
using the parity-violating electron scattering experiment [21]. The parameters
of BSRV-PREX model have been searched by incorporating the recently measured
neutron skin thickness $\Delta r_{np}$= $0.283\pm 0.071$ fm for 208Pb from the
PREX-II data [21] in our fit. The BSRV-CPREX parametrizations has been
obtained by incorporating both the CREX and PREX-II data for neutron skin
thicknesses for 48Ca and 208Pb nuclei in the fit data. In addition, we have
also included the maximum mass of neutron star [44] in our fit data.
For the open shell nuclei, the pairing has been included using BCS formalism
with constant pairing gaps [45, 46] that are taken from the nucleon separation
energies of neighboring nuclei [39]. Neutron and proton pairing gaps are
calculated by using the fourth-order finite difference mass formula (five-
point difference) [47]. The neutron and proton pairing gaps
($\Delta_{n}$,$\Delta_{p}$) in MeV for the open shell nuclei are
${}^{68}Ni$(1.46,0.0), ${}^{88}Sr$(0.0,1.284), ${}^{90}Zr$(0.0,1.239),
${}^{116}Sn$(1.189,0.0) and ${}^{144}Sm$(0.0,1.0) The neutron pairing gap for
${}^{24}O$ practically vanishes since the first unoccupied orbit 1$d_{3/2}$ is
almost 4.5 MeV above the completely filled 2$s_{1/2}$ orbit [34, 48]. The
pairing correlation energies for a fix gap $\Delta$ is calculated by using the
pairing window of 2$\hbar\omega$, where $\hbar\omega$ = $45A^{-1/3}$ \-
$25A^{-2/3}$ MeV [29, 7]. We have obtained three different parametrizations by
calibrating the parameters to a suitable set of finite nuclei as discussed
earlier. Three different parameterizations obtained in the present work give
an equally good fit to the properties of finite nuclei which were used for the
optimization procedure. In Table 1, We display the model parameters for the
newly generated parameter sets BSRV-CREX, BSRV-PREX and BSRV-CPREX along with
theoretical uncertainties/errors. We also list the value of parameters for NL3
[41], FSUGarnet [34], IOPB-1 [42] and Big Apple [43] for comparison. It can be
seen from the table that comparatively a large value of isovector scalar meson
coupling parameter $g_{\delta}$ (4.27816), isovector vector meson coupling
$g_{\rho}$ (16.18406) and cross-coupling between $\omega$ and $\rho$ meson
quantified by the term $\Lambda_{\omega\rho}$ (5.18286) are obtained for BSRV-
CREX parameterization in which only CREX data is included in the fit. The
values of coupling parameters $g_{\delta}$, $g_{\rho}$ and
$\Lambda_{\omega\rho}$ is more or less the same for BSRV-PREX and BSRV-CPREX
models. In Figs. 1, 2 and 3 we show the colour-coded plots for the correlation
coefficients between the coupling parameters appearing in Lagrangian (Eq.1)
for BSRV-CREX, BSRV-PREX and BSRV-CPREX models. A strong correlation exists
between the pairs of coupling parameters $g_{\sigma}$ \- $g_{\omega}$ ,
$\Lambda_{\omega\rho}-g_{\delta}$ , and $\overline{\kappa}-\overline{\lambda}$
for BSRV-CREX model. The strong correlation is also observed for pair of
coupling parameters $g_{\sigma}$ \- $g_{\omega}$, $g_{\rho}$ with $g_{\delta}$
and $\Lambda_{\omega\rho}$. For BSRV-CPREX model, a strong correlation is
observed for $g_{\delta}$ with $g_{\rho}$, $\overline{\lambda}$ and $\zeta$.
The coupling parameter $\zeta$ is found to be well correlated with $g_{\rho}$
and $\overline{\lambda}$. A strong correlation between the model parameters
indicates a strong interdependence i.e. if one parameter is fixed at a certain
value then the other must attain the precise value as suggested by their
correlation. It can be seen from Fig. 3 that when the skin thickness for 48 Ca
and 208 Pb fitted together there is an overall reduction among the model
parameters correlations as compared to those obtained by fitting the
individual neutron skin.

Figure 1:  (color online) Correlation coefficients among the model parameters
of the Lagrangian given by Eq. (1) for BSRV-CREX parametrization.  Figure 2:
(color online) Same as Fig. 1, but for BSRV-PREX parametrization. Figure 3:
(color online) Same as Fig. 1, but for for BSRV-CPREX parametrization).

###  IV.2 Finite Nuclei and Infinite Nuclear Matter

In Table 2, different observables fitted in the present work, their
experimental values [39, 40], adopted errors $\sigma$ on them along with their
calculated values for different BSRV’s parametrizations and theoretical
uncertainties are displayed. The newly generated parameterizations BSRV-CREX,
BSRV-PREX and BSRV-CPREX give an equally good fit to the properties of finite
nuclei. The fitted values of finite nuclei are quite close to their
experimental values. The root mean square (rms) errors in total binding energy
for all the nuclei considered in our fit are found to be 1.62, 1.39 and 1.39
MeV, whereas the root mean square (rms) errors in total charge radii are
0.071, 0.080, and 0.081 fm for BSRV-CREX, BSRV-PREX and BSRV-CPREX parameter
sets respectively. We also depict the predicted values of neutron skin
thickness $\Delta r_{np}=R_{n}-R_{p}$ and charge rms radii ($r_{ch}$)for all
our parametrizations. It can be observed from the table that for BSRV-CREX
parametrization, the value of $\Delta r_{np}$ for 48Ca nucleus is 0.146 $\pm
0.019$ fm and is consistent with the recently reported value of $\Delta
r_{np}$ of 48Ca from CREX data [21] while the value of $\Delta r_{np}$ for
208Pb nucleus comes out to be 0.13$\pm 0.018$ fm that is also in good
agreement with the value reported for neuton skin thickness of $\Delta
r_{np}=(0.18\pm 0.07)$ fm for 208Pb obtained by dispersive optical model
analysis of the Washington University group [49]. The values of $\Delta
r_{np}$ obtained for ${}^{208}Pb$ using the BSRV-CREX model does not satisfy
the PREX-II measurement. The values of $\Delta r_{np}$ for 208Pb obtained for
BSRV-PREX and BSRV-CPREX parametrizations is in good agreement with the
recently reported value $\Delta r_{np}=(0.283\pm 0.071)$ fm for 208Pb for
PREX-II data, but the value of $\Delta r_{np}$ predicted for 48Ca
overestimates the CREX data [21]. In Fig 4 we show the neutron skin thickness
$\Delta r_{np}$ for nuclei considered in our data fit as a function of
neutron-proton asymmetry parameter $\delta=\frac{N-Z}{A}$. The predicted
result of $\Delta r_{np}$ for the nuclei used in our fit data (Table 2) for
BSRV-CREX, BSRV-PREX and BSRV-CPREX parametrizations are compared with the
corresponding available experimental values along with error bars taken from
Refs.[50, 22, 21, 49]. The shaded regions represent the linear dependence of
neutron skin thickness on asymmetry (neutron excess) $\delta$ of a nucleus
that can be fitted by [51, 50].

Table 2:  The calculated values of binding energy (BE) and charge radii ($r_{ch}$) along with theoretical uncertainties/errors for various BSRV parametrizations are presented. The predicted value of neutron skin thickness $\Delta r_{np}$ =$r_{n}$ -$r_{p}$ is also depicted for various models. The corresponding experimental values of BE and $r_{ch}$ [39, 40] and $\Delta r_{np}$ [50, 22, 21, 49] are also listed. The adopted errors on the observables ($\sigma$) used for the optimization of parameters and the asymmetry parameter $\delta=(N-Z)/A$ for the nuclei are also displayed. The value of BE are given in units of MeV and $r_{ch}$, $\Delta r_{np}$ are in fm.  ${\bf{Nucleus}}$ | ${\bf{Observables}}$ | Exp. | $\sigma$ | $\delta$ | BSRV-CREX | BSRV-PREX | BSRV-CPREX |  NL3 |  IOPB-I
---|---|---|---|---|---|---|---|---|---
${}^{16}O$ | BE | 127.62 | 4.0 | 0.0 | 129.82$\pm$0.32 | 128.71$\pm$0.55 | 128.57$\pm$0.60 | 127.08 | 128.05
| $r_{ch}$ | 2.699 | 0.04 |  | 2.690$\pm$0.024 | 2.706$\pm$0.031 | 2.709$\pm$0.029 | 2.727 | 2.719
| $\Delta r_{np}$ | – | - |  | -0.029$\pm$0.020 | -0.028$\pm$0.009 | -0.028$\pm$0.008 | -0.027 | -0.029
${}^{24}O$ | BE | 168.96 | 2.0 | 0.33 | 166.89$\pm$1.41 | 170.71$\pm$1.29 | 171.03$\pm$1.15 | 170.54 | 169.48
| $r_{ch}$ | – | – |  | 2.753$\pm$0.033 | 2.734$\pm$0.011 | 2.732$\pm$0.015 | 2.737 | 2.741
| $\Delta r_{np}$ | - | - |  | 0.559$\pm$0.023 | 0.511$\pm$0.010 | 0.632$\pm$0.021 | 0.635 | 0.613
${}^{40}Ca$ | BE | 342.04 | 3.0 | 0.0 | 344.53$\pm$0.47 | 343.75$\pm$0.56 | 343.51$\pm$0.81 | 341.32 | 342.68
| $r_{ch}$ | 3.478 | 0.04 |  | 3.445$\pm$0.035 | 3.455$\pm$0.029 | 3.457$\pm$0.014 | 3.469 | 3.464
| $\Delta r_{np}$ | $-0.08^{+0.05}_{-1.0}$ | - |  | -0.052$\pm$0.002 | -0.050$\pm$0.012 | -0.050$\pm$0.002 | -0.048 | -0.050
48Ca | BE | 415.97 | 1.0 | 0.167 | 416.07$\pm$0.46 | 415.46$\pm$0.56 | 415.52$\pm$0.60 | 414.52 | 414.57
| $r_{ch}$ | 3.477 | 0.04 |  | 3.475$\pm$0.014 | 3.467$\pm$0.010 | 3.467$\pm$0.020 | 3.471 | 3.471
| $\Delta r_{np}$ | 0.121$\pm$ 0.026 | 0.026 |  | 0.146$\pm$0.019 | 0.212$\pm$0.022 | 0.215$\pm$0.018 | 0.226 | 0.199
${}^{56}Ni$ | BE | 484.01 | 5.0 | 0.0 | 483.92$\pm$0.62 | 481.55$\pm$1.00 | 482.14$\pm$0.78 | 482.12 | 482.48
| $r_{ch}$ | 3.750 | 0.02 |  | 3.709$\pm$0.015 | 3.721$\pm$0.015 | 3.718$\pm$0.016 | 3.716 | 3.707
| $\Delta r_{np}$ | $-0.03^{+0.08}_{-0.11}$ | - |  | -0.038$\pm$0.001 | -0.037$\pm$0.004 | -0.036$\pm$0.009 | -0.034 | -0.037
${}^{68}Ni$ | BE | 590.41 | 2.0 | 0.176 | 592.43$\pm$0.50 | 592.43$\pm$0.53 | 592.35$\pm$0.49 | 591.21 | 591.66
| $r_{ch}$ | – | – |  | 3.883$\pm$0.015 | 3.866$\pm$0.027 | 3.864$\pm$0.020 | 3.863 | 3.869
| $\Delta r_{np}$ | - | - |  | 0.239$\pm$0.019 | 0.310$\pm$0.027 | 0.317$\pm$0.029 | 0.333 | 0.299
${}^{78}Ni$ | BE | 642.564 | 3.0 | 0.282 | 640.19$\pm$1.24 | 641.61$\pm$1.41 | 641.83$\pm$1.27 | 643.04 | 640.55
| $r_{ch}$ | – | – |  | 3.971$\pm$0.024 | 3.952$\pm$0.015 | 3.950$\pm$0.017 | 3.942 | 3.950
| $\Delta r_{np}$ | - | - |  | 0.403$\pm$0.033 | 0.526$\pm$0.123 | 0.535$\pm$0.034 | 0.553 | 0.506
${}^{88}Sr$ | BE | 768.42 | 2.0 | 0.136 | 768.51$\pm$0.51 | 767.69$\pm$0.55 | 767.62$\pm$0.56 | 767.31 | 766.65
| $r_{ch}$ |  | 0.02 |  | 4.237$\pm$0.013 | 4.227$\pm$0.012 | 4.226$\pm$0.014 | 4.225 | 4.229
| $\Delta r_{np}$ | - | - |  | 0.069$\pm$0.009 | 0.130$\pm$0.026 | 0.134$\pm$0.021 | 0.148 | 0.118
${}^{90}Zr$ | BE | 783.81 | 1.0 | 0.111 | 783.92$\pm$0.59 | 783.19$\pm$0.60 | 783.16$\pm$0.62 | 782.95 | 782.44
| $r_{ch}$ | 4.269 | 0.02 |  | 4.289$\pm$0.018 | 4.282$\pm$0.012 | 4.282$\pm$0.014 | 4.280 | 4.284
| $\Delta r_{np}$ | $0.09^{+0.02}_{-0.02}$ | - |  | 0.033$\pm$0.013 | 0.083$\pm$0.022 | 0.086$\pm$0.018 | 0.097 | 0.072
${}^{100}Sn$ | BE | 825.10 | 2.0 | 0.0 | 827.55$\pm$0.97 | 826.96$\pm$1.04 | 827.55$\pm$1.34 | 829.33 | 827.52
| $r_{ch}$ | – | – |  | 4.515$\pm$0.014 | 4.522$\pm$0.014 | 4.521$\pm$0.026 | 4.511 | 4.514
| $\Delta r_{np}$ | - | - |  | -0.129$\pm$0.002 | -0.126$\pm$0.018 | -0.125$\pm$0.002 | -0.117 | -0.126
${}^{116}Sn$ | BE | 988.67 | 2.0 | 0.138 | 987.42$\pm$0.65 | 987.15$\pm$0.72 | 986.85$\pm$0.69 | 986.89 | 986.24
| $r_{ch}$ | 4.627 | 0.02 |  | 4.631$\pm$0.017 | 4.617$\pm$0.015 | 4.617$\pm$0.014 | 4.610 | 4.620
| $\Delta r_{np}$ | $0.10^{+0.03}_{-0.03}$ | - |  | 0.098$\pm$0.013 | 0.163$\pm$0.024 | 0.169$\pm$0.028 | 0.183 | 0.149
${}^{132}Sn$ | BE | 1102.22 | 2.0 | 0.242 | 1102.55$\pm$0.94 | 1102.51$\pm$1.05 | 1102.37$\pm$0.96 | 1104.81 | 1101.93
| $r_{ch}$ | 4.709 | 0.02 |  | 4.742$\pm$0.013 | 4.723$\pm$0.019 | 4.722$\pm$0.014 | 4.710 | 4.721
| $\Delta r_{np}$ | - | - |  | 0.226$\pm$0.025 | 0.348$\pm$0.048 | 0.357$\pm$0.025 | 0.383 | 0.325
${}^{144}Sm$ | BE | 1195.77 | 2.0 | 0.139 | 1197.69$\pm$0.82 | 1197.21$\pm$0.91 | 1196.90$\pm$0.82 | 1198.14 | 1195.82
| $r_{ch}$ | – | – |  | 4.978$\pm$0.021 | 4.965$\pm$0.102 | 4.965$\pm$0.014 | 4.956 | 4.967
| $\Delta r_{np}$ | - | - |  | 0.047$\pm$0.013 | 0.116$\pm$0.023 | 0.121$\pm$0.031 | 0.137 | 0.103
208Pb | BE | 1636.34 | 1.0 | 0.212 | 1637.11$\pm$0.92 | 1636.57$\pm$0.94 | 1636.00$\pm$0.93 | 1639.43 | 1636.75
| $r_{ch}$ | 5.501 | 0.04 |  | 5.551$\pm$0.016 | 5.533$\pm$0.018 | 5.532$\pm$0.029 | 5.517 | 5.532
| $\Delta r_{np}$ | $0.283\pm 0.071$ | 0.071 |  | 0.130$\pm$0.018 | 0.243$\pm$0.048 | 0.252$\pm$0.028 | 0.279 | 0.219
|  | $(0.18\pm 0.07)$ | - |  |  |  |  |  |

$\Delta r_{np}=(-0.09\pm 0.02)+(1.45\pm 0.12)\delta~{}~{}~{}~{}~{}(fm)$ (19)

The values of $\Delta r_{np}$ obtained for BSRV-PREX and BSRV-CPREX parameter
sets for some of the nuclei deviate from shaded region as can be observed from
the Fig. 4. This may be attributed to the fact that for these paramerizations
the value of $\omega$ -$\rho$ meson coupling parameter $\Lambda_{\omega\rho}$
that constrains the density dependence of symmetry energy and coupling
parameters $g_{\delta}$ and $g_{\rho}$ is relatively smaller as compared to
BSRV-CREX parameter set. For BSRV-CREX parameter set, the values of $\Delta
r_{np}$ lie in the shaded region or very close to it except for ${}^{24}O$
nucleus for which asymmetry is 0.33.

Figure 4:  (color online) Variation of neutron skin thickness predicted for
the nuclei considered in the fit data as a function of asymmetry parameter
($\delta$). The shaded region is calculated by using Eq. (19).

For BSRV-CREX, BSRV-PREX and BSRV-CPREX models, the calculated values of
neutron skin thickness for 208Pb nucleus are 0.130$\pm 0.018$ fm, 0.243$\pm
0.048$ fm and 0.252$\pm 0.028$ fm and for 48Ca nucleus are 0.146$\pm$0.019 fm,
0.212$\pm 0.022$ fm and 0.215$\pm$0.018 fm respectively. The value of $\Delta
r_{np}$=0.146$\pm$0.019 fm for 48Ca predicted by BSRV-CREX is consistent with
the recently measured $\Delta r_{np}=0.121\pm 0.026$ fm from parity-violating
electron scattering experiment (CREX) [21]. The values of neutron skin
thickness $\Delta r_{np}$= 0.243$\pm 0.048$ fm and 0.252$\pm 0.028$ fm
predicted for 208Pb by BSRV-PREX and BSRV-CPREX are also in agreement with the
recently reported neutron skin thickness from updated Lead Radius
Experiment(PREX-II) [22].
In the present work, we have tried to resolve the serious conflict of CREX and
PREX-II measurements by searching the model parameters of BSRV-CPREX
parameterization by including both the $\Delta r_{np}$ of 48Ca from the CREX
and 208Pb from PREX-II in our fit data for the optimization of model
parameters. But the $\Delta r_{np}$ predicted by this model overestimates the
recently reported value by CREX and the serious conflict between CREX and
PREX-II measurements continues as also discussed in Ref. [52].
In Table 3, we present our results for the symmetric nuclear matter (SNM)
properties such as binding energy per nucleon (E/A), incompressibility (K) ,
symmetry energy coefficient (J), density dependence of symmetry energy (L) and
ratio of effective mass to the mass of nucleon at the saturation density
($\rho_{0}$) and curvature of symmetry energy ($K_{sym}$) along with
theoretical uncertainties. These properties play a vital role for constructing
the EoS for nuclear matter. E/A is more or less the same for all BSRV’s
parameterizations. For newly generated parameterizations BSRV-PREX and BSRV-
CPREX, the value of J and L are consistent with the constraints from
observational analysis J = 38.1 $\pm$ 4.7 MeV and L = 106 $\pm$ 37 MeV as
reported by Reed et. al.,[26] and for BSRV-CREX parameter set, the value of J
= 28.97$\pm$0.99 MeV and L = 30.61$\pm$ 6.74 MeV is in close proximity to that
reported in Ref.[52], it is also consistent with the constraints from
observational analysis J = 31.6 $\pm$ 2.66 MeV [53]. The neutron skin
thickness as reported by CREX collaboration suggests softness i.e. low value
of symmetry energy coefficient (J) and its corresponding density dependence
(L). The value of K lies in the range 222.29$\pm$13.08 - 227.45$\pm$6.95 MeV
which is also in good agreement with the value of K = 240 $\pm$ 20 MeV
determined from isoscalar giant monopole resonance (ISGMR) for ${}^{90}Zr$ and
208Pb nuclei [54, 55]. The curvature of symmetry energy $K_{sym}$ also
satisfies the empirical limit discussed in [56]. The ratio of effective mass
to the nucleon mass is found to be similar for all BSRV’s parameterizations as
shown in Table 3. The SNM properties calculated with NL3, FSUGarnet, IOPB-1
and Big Apple are also shown for comparison.
In Fig. (5 and 6), we plot the EoS i.e. pressure as a function of baryon
density scaled to saturation density ($\frac{\rho}{\rho_{0}}$) for SNM and
pure neutron matter (PNM) using BSRV-CREX, BSRV-PREX and BSRV-CPREX
parameterizations which are in good agreement and lie in the allowed region
with the EoS extracted from the analysis of particle flow in heavy ion
collision [28]. These results are also compared with the NL3, IOPB-1,
FSUGarnet and Big Apple parameterizations. It can be easily seen that the EoSs
for SNM and PNM obtained from NL3 and Big Apple parameterizations are very
stiff and are ruled out by constraints imposed by heavy ion collision data.
The stiffness of the EoSs for NL3 and Big Apple parameter sets may be due to
the fact that the coupling parameter $\zeta$ which is responsible for varying
the high-density behavior of EoS is zero for NL3 and very small for Big Apple
parameter sets. The EoSs calculated using BSRV’s parameter sets are relatively
much softer and lie in the allowed region of heavy ion collision data [28].
The EoS calculated from BSRV-CREX parametrization is the softest amongst all
EoSs and it might be due to the relatively somewhat higher value of parameter
$\zeta$ and $\Lambda_{\omega\rho}$ obtained for this parameter set during the
calibration procedure which is responsible for varying high-density behavior
of EoS. In Fig.7 we plot the density dependence of symmetry energy (L) as a
function of baryon density for BSRV’s paramerizations. The results for
NL3,IOPB-1, FSUGarnet and Big Apple parameter sets are also displayed for
comparison. It can be seen from the figure that in low or medium density
regime the behvior of BSRV-CREX is softest (Low value of L at a given baryon
density) amongst all parametrizations and changes to stiffest in high density
regime even though the $\Delta r_{np}$ for 208Pb nucleus is smallest for this
parameter set. This may be attributed to the large value of coupling parameter
$g_{\delta}$ (4.27816) obtained for BSRV-CREX parameter set. The coupling
parameter $g_{\delta}$ is responsible for changing the behivior of L from soft
in low - medium density regime to stiff in high density regime. The stiffness
of L for BSRV-PREX and BSRV-CPREX parameter sets may also be due to the
coupling parameter $g_{\delta}$ and large value of $\Delta r_{np}$ for 208Pb
nucleus. A large value of $\Delta r_{np}$ for 208Pb NL3 parameter and its
small value for Big Apple may be responsible for for stiffness and softness
behaviour of L respectively.

Table 3:  The bulk nuclear matter properties at saturation density along with theoretical uncertainties for BSRV-CREX, BSRV-PREX and BSRV-CPREX parametrizations are listed along with NL3, IOPB-1, FSUGarnet and Big Apple models. $\rho_{0}$, E/A, K, J, L, $K_{sym}$ and $M^{*}/M$ denotes the saturation density, binding energy per nucleon, incompressibility coefficient, symmetry energy, density dependence of symmetry energy, the curvature of symmetry energy and the ratio of effective nucleon mass to the nucleon mass, respectively. ${\bf{Parameters}}$ | BSRV-CREX | BSRV-PREX | BSRV-CPREX | NL3 | IOPB-1 | FSUGarnet | BigApple
---|---|---|---|---|---|---|---
${\bf{\rho_{0}~{}(fm^{-3})}}$ | 0.148$\pm$0.003 | 0.148$\pm$0.001 | 0.148$\pm$0.002 | 0.148 | 0.149 | 0.153 | 0.155
${\bf E/A~{}(MeV)}$ | -15.99$\pm$0.03 | -16.10$\pm$0.06 | -16.09$\pm$0.05 | -16.24 | -16.09 | -16.23 | -16.34
${\bf K~{}(MeV)}$ | 222.29$\pm$13.08 | 227.45$\pm$6.95 | 226.99$\pm$3.74 | 271.56 | 222.57 | 229.62 | 227.09
${\bf~{}M^{*}/M}$ | 0.600$\pm$0.007 | 0.606$\pm$0.004 | 0.602$\pm$0.006 | 0.595 | 0.593 | 0.578 | 0.608
${\bf J~{}(MeV)}$ | 28.97$\pm$0.99 | 34.41$\pm$2.71 | 34.99$\pm$2.15 | 37.40 | 33.30 | 30.98 | 31.41
${\bf L~{}(MeV)}$ | 30.61$\pm$6.74 | 77.08$\pm$28.87 | 82.32$\pm$22.93 | 118.56 | 63.85 | 50.92 | 40.33
${\bf~{}K_{sym}(MeV)}$ | 61.79$\pm$32.74 | -71.48$\pm$5.53 | -65.65$\pm$20.32 | 100.90 | -37.79 | 58.46 | 89.58
Figure 5:  (color online) Variation of Pressure as a function of baryon
density for symmetric nuclear matter (SNM) computed with BSRV’s
parameterizations along with NL3, IOPB-1, FSUGarnet and Big Apple models. The
shaded region represents the experimental data taken from the reference [28].
Figure 6:  (color online) Variation of Pressure as a function of baryon
density for pure neutron matter (PNM) computed with BSRV’s parameterizations
along with NL3, IOPB-1, FSUGarnet and Big Apple models. The shaded region
represents the experimental data taken from the reference [28].  Figure 7:
(color online) Variation of density dependence of symmetry energy (L) as a
function of baryon density for various parametrizations considered in the
present work.

###  IV.3 Neutron star properties

Figure 8:  (Color online) EoSs i.e. pressure as a function of energy density
for $\beta$-stable equilibrated nuclear matter for various
parametrizations.The shaded region represents the observational constraints
reported in Ref. [57].

In Fig (8) we display the EoS i.e. the variation of pressure with the energy
density for the neutron star in $\beta$ equilibrium for all BSRV’s
parameterizations. The results calculated with NL3, IOPB-1, FSUGarnet and Big
Apple parameter sets are also shown for comparison. The shaded region in Fig.
8 represents the observational constraints at $r_{ph}$=R with the uncertainty
of 2$\sigma$ [57]. Here $r_{ph}$ and R are the photospheric and neutron star
radius respectively. It can be observed from the figure that the EoS computed
with newly generated BSRV’s parameter sets are softer and lie in the lower
boundary of the shaded region at very high density ($\cal E$ $\approx$
700-1000 MeV fm-3). The softness of EoSs for BSRV’s parameter sets may be due
to the moderate value of $\omega$ meson coupling parameter $\zeta$ that
governs the high-density behavior of EoS. The EoSs calculated for NL3 and Big
Apple parameter sets are stiffer and may be attributed to either zero or very
small value of coupling parameter $\zeta$ for these parameter sets
respectively.

Figure 9:  (Color online) The plot of proton fractions as a function of baryon
density. The dotted horizontal line represents the threshold for the direct
Urca process i.e. $(\rho_{p}/\rho)$ = 0.11 [58].

In Fig. 9 we illustrate the proton fraction in neutron star matter. One
remarkable point is that the $g_{\delta}$ has an influence on proton fraction
at high densities. The large value of $g_{\delta}$ for BSRV-CREX suppresses
the proton fraction and then delays the direct Urca process in which neutrinos
can be emitted rapidly. It can also be observed that $\Delta r_{np}$ of 208Pb
nucleus also plays a significant role in the direct Urca process. The large
value of $\Delta r_{np}$ for 208Pb may be attributed to the early start of the
direct Urca process for BSRV-PREX, BSRV-CPREX and NL3 whereas its small value
for BSRV-CREX parameter set may be responsible for delayed direct Urca process
for BSRV-CREX parameterization.

Table 4: The properties of nonrotating neutron stars for the various EoSs computed with BSRV’s parameter sets are displayed along with theoretical uncertainties on them. M${}_{max}(M_{\odot})$ and Rmax denote the Maximum Gravitational mass and radius corresponding to the maximum mass of the nonrotating neutron stars respectively. The values for Rmax, R1.4, and $\Lambda_{1.4}$ denote radius and dimensionless tidal deformability corresponding to $M_{max}$ and 1.4M⊙. SN | EoS | Mmax(M⊙) |  Rmax |  R1.4 | ${\bf{~{}~{}\Lambda_{1.4}}}$ |
---|---|---|---|---|---|---
|  |  | (km) |  (km) |  |
1. | BSRV-CREX | 1.95$\pm$0.04 | 11.34$\pm$0.21 | 12.66$\pm$0.39 | 525.84$\pm$151.48 |
2 | BSRV-PREX | 2.01$\pm$0.04 | 11.67$\pm$0.26 | 13.27$\pm$0.55 | 638.51$\pm$142.03 |
3. | BSRV-CPREX | 2.04$\pm$0.04 | 11.79$\pm$0.29 | 13.41$\pm$0.71 | 682.57$\pm$219.73 |
4. | NL3 | 2.77 | 13.28 | 14.65 | 1274 |
5 | IOPB-I | 2.15 | 11.95 | 13.29 | 682 |
6 | FSUGarnet | 2.06 | 11.65 | 12.98 | 622.51 |
7 | Big Apple | 2.60 | 12.41 | 13.12 | 715.96 |
Figure 10:  (Color online) Variation of the gravitational mass of non-rotating
neutron stars as a function of radius. for BSRV’s parameterizations. The
results for NL3,IOPB-1, FSUGarnet and Big Apple parameter sets are also shown.

The mass and radius of a neutron star are obtained by solving the Tolman-
Oppenheimer-Volkoff (TOV) equations [59, 60] given as:

$\frac{dP(r)}{dr}=-\frac{\\{\epsilon(r)+P(r)\\}\\{4\pi
r^{3}P(r)+m(r)\\}}{r^{2}(1-2m(r)/r)}$ (20) $\frac{dm}{dr}=4\pi
r^{2}\epsilon(r),$ (21) $m(r)=4\pi\int_{0}^{r}drr^{2}\epsilon(r)$ (22)

where $P(r)$ is the pressure at radial distance $r$ and $m(r)$ is the mass of
neutron stars enclosed in the sphere of radius $r$. In Fig. 10 we plot the
results for the gravitational mass of a non-rotating neutron star and its
radius for BSRV-CREX, BSRV-PREX and BSRV-CPREX parameter sets. The results are
also displayed for NL3, IOPB-1, FSUGarnet and Big Apple parameterizations. It
is observed that the maximum gravitational mass of the non-rotating neutron
star for BSRV’s parameter sets lies in the range 1.95$\pm$0.04M⊙ \-
2.04$\pm$0.04 M⊙ which is in good agreement with the mass constraints from
GW170817 event, pulsars PSRJ1614-2230, PSRJ0348+0432, and PSRJ0740+6620 [13,
61, 44, 62, 63]. The EoS for $\beta$\- equalibrated matter calculated with
BSRV-CREX model is the softest amongst all parametrizations considered in this
work. The radius of canonical mass ($R_{1.4}$) including BPS crust [64] for
low density region is 12.66 $\pm$ 0.39 Km, 13.27$\pm$0.55 Km and 13.41$\pm
0.71$ Km for BSRV-CREX, BSRV-PREX and BSRV-CPREX parameterizations which
satisfies the radius constraints from NICER on $R_{1.4}$ and PREX-II reported
in Ref. [26]. The value of $R_{1.4}$ for BSRV’s paramerization is consistent
with the softness and stiffness behaviour of symmetry energy coefficient. The
value of $R_{1.4}$ for NL3 parameterization is 14.65 Km which seems to be
ruled out by the constraints for $R_{1.4}$ extracted from Ref. [65].
The tidal deformability rendered by the companion stars on each other in a
binary system can provide significant information on the EoS of neutron stars
[66, 67]. The tidal influences of its companion in the binary neutron star
(BNS) system will deform neutron stars in the binary system and, the resulting
change in the gravitational potential modifies the BNS orbital motion and its
corresponding gravitational wave (GW) signal. This effect on GW phasing can be
parameterized by the dimensionless tidal deformability parameter,
$\Lambda_{i}=\lambda_{i}/M_{i}^{5},$ i = 1, 2. For each neutron star, its
quadrupole moment ${\cal{Q}}_{j,k}$ must be related to the tidal field
${\cal{E}}_{j,k}$ caused by its companion as,
${\cal{Q}}_{j,k}=-\lambda{\cal{E}}_{j,k}$, where, $j$ and $k$ are spatial
tensor indices. The dimensionless tidal deformability parameter $\Lambda$ of a
static, spherically symmetric compact star depends on the neutron star
compactness parameter C and a dimensionless quadrupole Love number k2 as,
$\Lambda$=(2k${}_{2}/{3)C^{-5}}$. The $\Lambda$ critically parameterizes the
deformation of neutron stars under the given tidal field, therefore it should
depend on the EoS of nuclear dense matter. To measure the Love number k2 along
with the evaluation of the TOV equations we have to compute y2 = y(R) with
initial boundary condition y(0) = 2 from the first-order differential equation
[66, 68, 67, 69] simultaneously,

$\displaystyle y^{\prime}=\frac{1}{r}[-r^{2}Q-ye^{\lambda}\\{1+4\pi
Gr^{2}(P-{\cal{E}})\\}-y^{2}],$ (23)

where Q $\equiv$ 4$\pi$Geλ(5${\cal{E}}$+9P+$\frac{{\cal{E}}+P}{c_{s}^{2}})$
-6$\frac{e^{\lambda}}{r^{2}}$-$\nu^{\prime^{2}}$ and
e${}^{\lambda}\equiv(1-\frac{2Gm}{r})^{-1}$ and, $\nu^{\prime}\equiv$ 2G eλ
($\frac{m+4\pi Pr^{3}}{r^{2}}$). First, we get the solutions of Eq.(23) with
boundary condition, y2 = y(R), then the electric tidal Love number k2 is
calculated from the expression as,

$\displaystyle
k_{2}=\frac{8}{5}C^{5}(1-2C)^{2}[2C(y_{2}-1)-y_{2}+2]\\{2C(4(y_{2}+1)C^{4}$
$\displaystyle+(6y_{2}-4)C^{3}+(26-22y_{2})C^{2}+3(5y_{2}-8)C-3y_{2}+6)$
$\displaystyle-3(1-2C)^{2}(2C(y{{}_{2}}-1)-y_{2}+2)\log(\frac{1}{1-2C})\\}^{-1}.$

Figure 11:  (Color online) Variation of dimensionless tidal deformability
($\Lambda$) with respect to gravitational mass for all BSRV’s
parameterizations. The results for NL3, IOPB-1, FSUGarnet and Big Apple
parameter sets are also shown.

In Fig. (11), we display the results of dimensionless tidal deformability
$\Lambda$ as a function of gravitational mass for neutron stars for BSRV’s
parametrizations. For the sake of comparison, we also display the results
calculated with NL3, IOPB-1, FSUGarnet and Big Apple parameterizations. The
values of $\Lambda_{1.4}$ obtained for canonical mass are 525.84$\pm$151.48,
638.51$\pm$142.03 and 682.57$\pm$219.73 corresponding to BSRV-CREX, BSRV-PREX
and BSRV-CPREX respectively and satisfy the constraints as reported in Ref.
[26, 70, 71]. It may be noted that the value of tidal deformability of
1.4$M_{\odot}$ neutron star obtained for BSRV-PREX and BSRV-CPREX models have
overlap with the revised limit of $\Lambda_{1.4}\leq 580$ within 1$\sigma$
uncertainty [15]. The value (525.84$\pm$151.48) of $\Lambda_{1.4}$ obtained
for the BSRV-CREX parameter set is also consistent with the constraints
imposed in Ref. [15]. This might be attributed to the high value of coupling
$g_{\delta}$ for this parametrization that governs the softness and stiffness
of EoS in the low and the high-density regime.
In Table 4, we summarize the results for non-rotating neutron star properties
such as maximum gravitational mass (M), neutron star radius corresponding to
the maximum mass ($R_{max}$), radius $R_{1.4}$, radius $R_{2.0}$ and tidal
deformability ($\Lambda$) corresponding to canonical and maximum mass of
neutron star along with theoretical uncertainties.

Figure 12:  (Color online) The thresholds for the onset of direct Urca Process
in neutron stars. The threshold density is represented by the solid black
circles and the corresponding neutron star mass for such central density is
depicted by solid magenta squares.

In Fig. 12 we plot the threshold central density (solid black circles) and
corresponding neutron star mass (solid magenta squares) as a function of
neutron skin thickness for the onset of the direct Urca process for BSRV’s
parameter sets. Similar results for NL3, IOPB-1, FSUGarnet and Big Apple
models are also displayed. It can be observed that the onset of the Urca
process has a strong dependence on neutron skin thickness of 208Pb and hence
on symmetry energy coefficient. In particular, stiff symmetry energy as
suggested by PREX-II favors large proton fractions that may trigger the onset
of direct Urca process at lower central density [26]. It is evident from the
figure that the onset of the direct Urca process is delayed with the decrease
in the $\Delta r_{np}$ of the ${}^{208}Pb$. The threshold density for the
onset of the direct Urca process for the BSRV-CREX model is 0.71 $fm^{-3}$.
This high value is attributed to the small neutron skin thickness and soft L
for BSRV-CREX. The threshold density for the onset of the direct Urca process
decreases from 0.71 $fm^{-3}$ to 0.21 $fm^{-3}$ as the value of $\Delta
r_{np}$ for 208Pb increases from 0.13 fm (BSRV-CREX) to 0.28 fm (NL3). The
neutron star mass corresponding to threshold central density for the direct
Urca process decreases from 2.53 to 0.86 ${M_{\odot}}$ for the
parametrizations considered in the present work.

###  IV.4 Correlations amongst nuclear matter observables and model parameters

In this sub-section, we discuss the correlations between nuclear matter
observables and model parameters. In Fig. 13 to Fig. 15, we display the
correlations of bulk nuclear matter properties at saturation density and
neutron star observables with the model parameters for BSRV’s
parametrizations.

Figure 13:  (Color online) Correlation coefficients amongst neutron star
observables as well as the bulk properties of nuclear matter at the saturation
density and model parameters for BSRV-CREX parametrization. Figure 14:  (Color
online) Same as Fig. 13, but for BSRV-PREX paramerization. Figure 15:  (Color
online) Same as Fig. 13, but for BSRV-CPREX paramerization.

For the BSRV-CREX model, the isoscalar bulk nuclear matter properties like K,
M*/M show strong correlations with isoscalar parameters $\overline{\kappa}$.
It can also be observed from the Fig. 13 that the symmetry energy coefficients
(J), its density dependence (L) and curvature of symmetry energy $K_{sym}$ can
be well constrained by the coupling parameter $g_{\rho}$ as suggested by their
correlations. The neutron star observables like $M_{max}$, $R_{1.4}$ and
$\Lambda_{1.4}$ display strong correlations with couplings
$\Lambda_{\omega\rho}$ and $g_{\delta}$. A strong correlation between
$M_{max}$ and $\omega$-meson self-coupling parameter $\zeta$ is missing in the
case of BSRV-CREX parameterization. The parameter $g_{\delta}$ might be
responsible for the large maximum mass of neutron star as it makes the EOS
somewhat stiffer at high density and thus weakens the correlation between
$M_{max}$ and $\zeta$.
For the BSRV-PREX model, the isoscalar bulk nuclear matter properties like
E/A, K and M*/M show strong correlations with isoscalar parameters
$g_{\sigma}$ and $g_{\omega}$. As expected, a strong correlation of the
isovector properties like J, L and $\Delta r_{np}$ for 208Pb with isovector
parameters $g_{\rho}$ and $\Lambda_{\omega\rho}$ is observed. The neutron star
observables like $M_{max}$, $R_{1.4}$ and $\Lambda_{1.4}$ display good
correlations with coupling $g_{\delta}$ and weak correlations with $\zeta$.
This suggests that the neutron star observables results from a competition
between $g_{\delta}$ and $\zeta$. For the BSRV-CPREX model, J, L and $\Delta
r_{np}$ for 208Pb are very well constrained by couplings $g_{\rho}$ and
$\Lambda_{\omega\rho}$ as can be observed from their correlations. $K_{sym}$
shows strong dependence upon $\Lambda_{\omega\rho}$. The $\Delta r_{np}$ for
48Ca is very well constrained by $g_{\rho}$, $g_{\delta}$ and
$\Lambda_{\omega\rho}$. The $R_{1.4}$ and $\Lambda_{1.4}$ is found to have
strong correlation with $g_{\delta}$. These findings are quite in harmony with
the results reported in Refs. [36, 34].

##  V Summary

Three relativistic interactions BSRV-CREX, BSRV-PREX and BSRV-CPREX for the
relativistic mean field model have been generated keeping in view the recently
reported constraints on neutron skin thickness of 48Ca (CREX) and 208Pb nuclei
by PREX-II data. The precise measurements of neutron skin thickness give an
opportunity to modify or readjust the coupling constants of RMF models without
compromising the bulk properties of nuclear matter and neutron stars. The
Lagrangian density for the RMF model used in the present work is based on
different non-linear, self and inter-couplings among isoscalar-scalar
$\sigma$, isoscalar-vector $\omega_{\mu}$, isovector-scalar $\delta$ and
isovector-vector $\rho_{\mu}$ meson fields and nucleonic Dirac field $\Psi$.
The BSRV-CREX parametrization has been obtained by incorporating the recently
measured neutron skin thickness $\Delta r_{np}$= $0.121\pm 0.026$ fm for 48Ca
using the parity-violating electron scattering experiment [21]. The parameters
of the BSRV-PREX model have been searched by incorporating the recently
measured neutron skin thickness $\Delta r_{np}$= $0.283\pm 0.071$ fm for 208Pb
from the PREX-II [21] in our fit. The BSRV-CPREX parametrizations have been
obtained by including both the CREX and PREX-II data for neutron skin
thicknesses for 48Ca and 208Pb nuclei in the fitting data. The BSRV’s
parameter sets reproduce the ground state properties of the finite nuclei,
bulk nuclear matter and also satisfy the constraints on mass and radius along
with dimensionless deformability ($\Lambda$) of a neutron star from recent
astrophysical observations [57, 65, 15, 26]. All the BSRV’s parametrizations
give an equally good fit to the finite nuclear properties. The Bulk nuclear
matter properties obtained are well consistent with the current empirical data
[26, 55, 52]. The maximum gravitational mass and radius ($R_{1.4}$) of the
neutron star lie in the range between 1.95$\pm$0.04 - 2.04$\pm$0.04 M$\odot$
and 12.66$\pm$0.39 - 13.41$\pm$0.71 km for BSRV’s parameter sets respectively.
The value of $\Lambda_{1.4}$ which lie in the range 525.84$\pm$151.48 -
682.57$\pm 219.73$ for BSRV’s parameterization also satisfies the GW170817
event [71, 15] and constraints obtained using Bayesian analysis and PREX-II
reported in Refs. [70, 71, 26]. Covariance analysis to measure the accuracy of
the model parameters is also performed. This enabled us to estimate the
statistical uncertainties in the model parameters along with various
correlations amongst the nuclear matter observables and parameters.
CREX and PREX-II data on neutron skin thickness has opened an important new
perspective to constrain the density dependence of symmetry energy and energy
density functionals and having a considerable effect on the isovector channel.
The BSRV-CREX parameterization obtained by incorporating CREX data provides
the smaller value of symmetry energy, its slope at saturation density and
neutron skin thickness are significantly very small. This may be due to the
large value of coupling $g_{\delta}$ which seems to play an important role
during the calibration procedure of model parameters and is responsible for
soft and stiff behaviour of symmetry energy in medium and high density regime.
But the radius and tidal deformability of 1.4$M_{\odot}$ neutron star reveals
some tension with the revised limit of $\Lambda_{1.4}\leq 580$ [15]. The BSRV-
PREX parametrization obtained keeping in view the PREX-II data suggests stiff
symmetry energy and stiff EoS. For BSRV-PREX and BSRV-CPREX models, the tidal
deformability of 1.4$M_{\odot}$ neutron star has some overlap with the revised
limit of $\Lambda_{1.4}\leq$ 580 within 1$\sigma$ uncertainty reported in
Ref.[15]. BSRV-CREX gives soft density dependence of symmetry energy whereas
BSRV-PREX and BSRV-CPREX provide its stiff value. The analysis of the present
work shows that no consistent conclusion from the theoretical side could be
obtained when using recently measured CREX and PREX-II data. We are hoping
that novel experimental studies are necessary to resolve this conflict or
discrepancies.

###### Acknowledgements.

Author(s) are thankful to Himachal Pradesh University for providing the
computational facility. BKA acknowledges partial support from the SERB,
Department of science and technology, Govt. of India with grant numbers
SIR/2022/000566 and CRG/2021/000101 respectively. SK is highly thankful to
CSIR-UGC (Govt. of India) for providing financial assistance (NTA/
211610029883 dated 19/04/2022) under Junior/Senior Research Fellowship scheme.

## References

  * [1] James M Lattimer.  The nuclear equation of state and neutron star masses.  Annu. Rev. Nucl. Part. Sci., 62:485–515, 2012.
  * [2] M. Oertel, M. Hempel, T. Klähn, and S. Typel.  Equations of state for supernovae and compact stars.  Rev. Mod. Phys., 89:015007, Mar 2017.
  * [3] X Roca-Maza and N Paar.  Nuclear equation of state from ground and collective excited state properties of nuclei.  Progress in Particle and Nuclear Physics, 101:96–176, 2018.
  * [4] J. M. Lattimer and M. Prakash.  Science, 304:536, 2004.
  * [5] Shashi K Dhiman, Raj Kumar, and B. K. Agrawal.  Nonrotating and rotating neutron stars in the extended field theoretical model.  Phys. Rev. C, 76(4):045801, 2007.
  * [6] Virender Thakur, Raj Kumar, Pankaj Kumar, Vikesh Kumar, BK Agrawal, and Shashi K Dhiman.  Relativistic mean field model parametrizations in the light of gw170817, gw190814, and psr j 0740+ 6620.  Physical Review C, 106(2):025803, 2022.
  * [7] Virender Thakur, Raj Kumar, Pankaj Kumar, Vikesh Kumar, Mukul Kumar, C Mondal, BK Agrawal, and Shashi K Dhiman.  Effects of an isovector scalar meson on the equation of state of dense matter within a relativistic mean field model.  Physical Review C, 106(4):045806, 2022.
  * [8] Paweł Haensel, Aleksander Yu Potekhin, and Dmitry G Yakovlev.  Neutron stars 1: Equation of state and structure, volume 326.  Springer Science & Business Media, 2007.
  * [9] J. M. Lattimer.  Neutron stars.  Gen. Rel. Grav., 46:1713, 2014.
  * [10] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, and T. Takatsuka.  From hadrons to quarks in neutron stars: a review.  Rep. Prog. Phys., 81:056902, 2018.
  * [11] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk.  Constraints on neutron star radii based on chiral effective field theory interactions.  Phys. Rev. Lett., 105:161102, Oct 2010.
  * [12] K Hebeler, JM Lattimer, Christopher J Pethick, and A Schwenk.  Equation of state and neutron star properties constrained by nuclear physics and observation.  The Astrophysical Journal, 773(1):11, 2013.
  * [13] P. B. Demorest, Tim Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels.  A two-solar-mass neutron star measured using shapiro delay.  nature, 467(7319):1081, 2010.
  * [14] John Antoniadis, Paulo C. C. Freire, Norbert Wex, Thomas M Tauris, Ryan S Lynch, Marten H van Kerkwijk, Michael Kramer, Cees Bassa, Vik S Dhillon, Thomas Driebe, et al.  A massive pulsar in a compact relativistic binary.  Science, 340(6131):1233232, 2013.
  * [15] B. P. Abbott, R Abbott, T. D. Abbott, F Acernese, K Ackley, C Adams, T Adams, P Addesso, R. X. Adhikari, V. B. Adya, et al.  Gw170817: Measurements of neutron star radii and equation of state.  Phys. Rev. Lett., 121(16):161101, 2018.
  * [16] B. P. Abbott, R Abbott, T. D. Abbott, F Acernese, K Ackley, C Adams, T Adams, P Addesso, R. X. Adhikari, V. B. Adya, et al.  Properties of the binary neutron star merger gw170817.  Phys. Rev. X, 9(1):011001, 2019.
  * [17] Zaven Arzoumanian, Adam Brazier, Sarah Burke-Spolaor, Sydney Chamberlin, Shami Chatterjee, Brian Christy, James M Cordes, Neil J Cornish, Fronefield Crawford, H Thankful Cromartie, et al.  The nanograv 11-year data set: high-precision timing of 45 millisecond pulsars.  Astrophys. J. Suppl. S., 235(2):37, 2018.
  * [18] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, A. K. Harding, W. C. G. Ho, J. M. Lattimer, R. M. Ludlam, S. Mahmoodifar, S. M. Morsink, P. S. Ray, T. E. Strohmayer, K. S. Wood, T. Enoto, R. Foster, T. Okajima, G. Prigozhin, and Y. Soong.  PSR j0030$+$0451 mass and radius from NICER data and implications for the properties of neutron star matter.  The Astrophysical Journal, 887(1):L24, dec 2019.
  * [19] T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E. Strohmayer.  A nicer view of psr j0030+0451: Millisecond pulsar parameter estimation.  The Astrophysical Journal Letters, 887(1):L21, dec 2019.
  * [20] G. Raaijmakers, T. E. Riley, A. L. Watts, S. K. Greif, S. M. Morsink, K. Hebeler, A. Schwenk, T. Hinderer, S. Nissanke, S. Guillot, Z. Arzoumanian, S. Bogdanov, D. Chakrabarty, K. C. Gendreau, W. C. G. Ho, J. M. Lattimer, R. M. Ludlam, and M. T. Wolff.  A NICER view of PSR j0030$+$0451: Implications for the dense matter equation of state.  The Astrophysical Journal, 887:L22, 2019.
  * [21] D. Adhikari, H. Albataineh, D. Androic, K. A. Aniol, D. S. Armstrong, T. Averett, C. Ayerbe Gayoso, S. K. Barcus, V. Bellini, R. S. Beminiwattha, J. F. Benesch, H. Bhatt, D. Bhatta Pathak, D. Bhetuwal, B. Blaikie, J. Boyd, Q. Campagna, A. Camsonne, G. D. Cates, Y. Chen, C. Clarke, J. C. Cornejo, S. Covrig Dusa, M. M. Dalton, P. Datta, A. Deshpande, D. Dutta, C. Feldman, E. Fuchey, C. Gal, D. Gaskell, T. Gautam, M. Gericke, C. Ghosh, I. Halilovic, J.-O. Hansen, O. Hassan, F. Hauenstein, W. Henry, C. J. Horowitz, C. Jantzi, S. Jian, S. Johnston, D. C. Jones, S. Kakkar, S. Katugampola, C. Keppel, P. M. King, D. E. King, K. S. Kumar, T. Kutz, N. Lashley-Colthirst, G. Leverick, H. Liu, N. Liyanage, J. Mammei, R. Mammei, M. McCaughan, D. McNulty, D. Meekins, C. Metts, R. Michaels, M. Mihovilovic, M. M. Mondal, J. Napolitano, A. Narayan, D. Nikolaev, V. Owen, C. Palatchi, J. Pan, B. Pandey, S. Park, K. D. Paschke, M. Petrusky, M. L. Pitt, S. Premathilake, B. Quinn, R. Radloff, S. Rahman, M. N. H. Rashad, A. Rathnayake, B. T. Reed, P. E. Reimer, R. Richards, S. Riordan, Y. R. Roblin, S. Seeds, A. Shahinyan, P. Souder, M. Thiel, Y. Tian, G. M. Urciuoli, E. W. Wertz, B. Wojtsekhowski, B. Yale, T. Ye, A. Yoon, W. Xiong, A. Zec, W. Zhang, J. Zhang, and X. Zheng.  Precision determination of the neutral weak form factor of ${}^{48}\mathrm{Ca}$.  Phys. Rev. Lett., 129:042501, Jul 2022.
  * [22] D Adhikari, H Albataineh, D Androic, K Aniol, DS Armstrong, T Averett, C Ayerbe Gayoso, S Barcus, V Bellini, RS Beminiwattha, et al.  Accurate determination of the neutron skin thickness of pb 208 through parity-violation in electron scattering.  Physical review letters, 126(17):172502, 2021.
  * [23] P-G Reinhard, J Piekarewicz, W Nazarewicz, BK Agrawal, Nils Paar, and X Roca-Maza.  Information content of the weak-charge form factor.  Physical Review C, 88(3):034325, 2013.
  * [24] Paul-Gerhard Reinhard, Xavier Roca-Maza, and Witold Nazarewicz.  Information content of the parity-violating asymmetry in ${}^{208}\mathrm{Pb}$.  Phys. Rev. Lett., 127:232501, Nov 2021.
  * [25] Paul-Gerhard Reinhard, Xavier Roca-Maza, and Witold Nazarewicz.  Combined theoretical analysis of the parity-violating asymmetry for ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$.  arXiv:2206.03134v1, 2022.
  * [26] Brendan T Reed, Farrukh J Fattoyev, Charles J Horowitz, and Jorge Piekarewicz.  Implications of prex-2 on the equation of state of neutron-rich matter.  Physical Review Letters, 126(17):172503, 2021.
  * [27] H. T. Cromartie, E. Fonseca, S. M. Ransom, P. B. Demorest, Z. Arzoumanian, H. Blumer, P. R. Brook, M. E. DeCesar, T. Dolch, J. A. Ellis, R. D. Ferdman, E. C. Ferrara, N. Garver-Daniels, P. A. Gentile, M. L. Jones, M. T. Lam, D. R. Lorimer, R. S. Lynch, M. A. McLaughlin, C. Ng, D. J. Nice, T. T. Pennucci, R. Spiewak, I. H. Stairs, K. Stovall, J. K. Swiggum, and W. W. Zhu.  Relativistic shapiro delay measurements of an extremely massive millisecond pulsar.  Nature Astronomy, 4:72, 2020.
  * [28] P. Danielewicz.  Science, 298:1592, 2002.
  * [29] Raj Kumar, B. K. Agrawal, and Shashi K. Dhiman.  Effects of $\omega$ meson self-coupling on the properties of finite nuclei and neutron stars.  Phys. Rev. C, 74:034323, Sep 2006.
  * [30] N. K. Glendenning.  Compact stars: Nuclear physics, particle physics, and general relativity.  Springer-Verlag, New York, 2000.
  * [31] TJ Bürvenich, DG Madland, and P-G Reinhard.  Adjustment studies in self-consistent relativistic mean-field models.  Nuclear Physics A, 744:92–107, 2004.
  * [32] Scott Kirkpatrick.  Optimization by simulated annealing: Quantitative studies.  Journal of statistical physics, 34(5):975–986, 1984.
  * [33] J Dobaczewski, W Nazarewicz, and PG Reinhard.  Error estimates of theoretical models: a guide.  Journal of Physics G: Nuclear and Particle Physics, 41(7):074001, 2014.
  * [34] Wei-Chia Chen and Jorge Piekarewicz.  Searching for isovector signatures in the neutron-rich oxygen and calcium isotopes.  Physics Letters B, 748:284–288, 2015.
  * [35] C Mondal, BK Agrawal, and JN De.  Constraining the symmetry energy content of nuclear matter from nuclear masses: A covariance analysis.  Physical Review C, 92(2):024302, 2015.
  * [36] FJ Fattoyev and J Piekarewicz.  Accurate calibration of relativistic mean-field models: Correlating observables and providing meaningful theoretical uncertainties.  Physical Review C, 84(6):064302, 2011.
  * [37] Siegmund Brandt.  Statistical and computational methods in data analysis.  Springer, 1997.
  * [38] P.-G. Reinhard and W. Nazarewicz.  Information content of a new observable: The case of the nuclear neutron skin.  Phys. Rev. C, 81:051303, 2010.
  * [39] Meng Wang, WJ Huang, Filip G Kondev, Georges Audi, and Sarah Naimi.  The ame 2020 atomic mass evaluation (ii). tables, graphs and references.  Chinese Physics C, 45(3):030003, 2021.
  * [40] István Angeli and Krassimira Petrova Marinova.  Table of experimental nuclear ground state charge radii: An update.  Atomic Data and Nuclear Data Tables, 99(1):69–95, 2013.
  * [41] GA Lalazissis, J König, and P Ring.  New parametrization for the lagrangian density of relativistic mean field theory.  Physical Review C, 55(1):540, 1997.
  * [42] Bharat Kumar, S. K. Patra, and B. K. Agrawal.  New relativistic effective interaction for finite nuclei, infinite nuclear matter, and neutron stars.  Phys. Rev. C, 97:045806, Apr 2018.
  * [43] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, and Brendan Reed.  Gw190814: Impact of a 2.6 solar mass neutron star on the nucleonic equations of state.  Phys. Rev. C, 102:065805, Dec 2020.
  * [44] Luciano Rezzolla, Elias R. Most, and Lukas R. Weih.  Using gravitational-wave observations and quasi-universal relations to constrain the maximum mass of neutron stars.  The Astrophysical Journal Letters, 852(2):L25, jan 2018.
  * [45] P. Ring and P. Schuck.  The Nuclear Many-body Problem.  Texts and monographs in physics. Springer, 1980.
  * [46] S Karatzikos, AV Afanasjev, GA Lalazissis, and P Ring.  The fission barriers in actinides and superheavy nuclei in covariant density functional theory.  Physics Letters B, 689(2-3):72–81, 2010.
  * [47] T. Duguet, P. Bonche, P.-H. Heenen, and J. Meyer.  Pairing correlations. ii. microscopic analysis of odd-even mass staggering in nuclei.  Phys. Rev. C, 65:014311, Dec 2001.
  * [48] C. Mondal, B. K. Agrawal, J. N. De, and S. K. Samaddar.  Sensitivity of elements of the symmetry energy of nuclear matter to the properties of neutron-rich systems.  Phys. Rev. C, 93:044328, Apr 2016.
  * [49] CD Pruitt, RJ Charity, LG Sobotka, MC Atkinson, and WH Dickhoff.  Systematic matter and binding-energy distributions from a dispersive optical model analysis.  Physical Review Letters, 125(10):102501, 2020.
  * [50] J Jastrzebski, A Trzcinska, P Lubiski, B Kos, FJ Hartmann, T von Egidy, and S Wycech.  Neutron density distributions from antiprotonic atoms compared with hadron scattering data.  International Journal of Modern Physics E, 13(01):343–351, 2004\.
  * [51] A Trzcinska, J Jastrzebski, P Lubinski, FJ Hartmann, R Schmidt, T Von Egidy, and B Klos.  Neutron density distributions deduced from antiprotonic atoms.  Physical review letters, 87(8):082501, 2001.
  * [52] Esra Yüksel and Nils Paar.  Implications of parity-violating electron scattering experiments on 48ca (crex) and 208pb (prex-ii) for nuclear energy density functionals.  Physics Letters B, 836:137622, 2023.
  * [53] Bao-An Li and Xiao Han.  Constraining the neutron–proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density.  Physics Letters B, 727(1-3):276–281, 2013.
  * [54] G Colo, U Garg, and H Sagawa.  Symmetry energy from the nuclear collective motion: constraints from dipole, quadrupole, monopole and spin-dipole resonances.  The European Physical Journal A, 50(2):1–12, 2014.
  * [55] J Piekarewicz.  Symmetry energy constraints from giant resonances: A relativistic mean-field theory overview.  The European Physical Journal A, 50(2):1–18, 2014.
  * [56] Josef Zimmerman, Zack Carson, Kristen Schumacher, Andrew W Steiner, and Kent Yagi.  Measuring nuclear matter parameters with nicer and ligo/virgo.  arXiv preprint arXiv:2002.03210, 2020.
  * [57] AW Steiner, JM Lattimer, and EF Brown.  The equation of state from observed masses and radii of neutron stars astrophys.  J, 722:33–54, 2010.
  * [58] Tomoyuki Maruyama and Satoshi Chiba.  Equation of state of neutron-star matter and the isovector nucleon optical model potential.  Journal of Physics G: Nuclear and Particle Physics, 25(12):2361, 1999.
  * [59] J. R. Oppenheimer and G. M. Volkoff.  On massive neutron cores.  Phys. Rev., 55:374–381, Feb 1939.
  * [60] Richard C. Tolman.  Static solutions of einstein’s field equations for spheres of fluid.  Phys. Rev., 55:364–373, Feb 1939.
  * [61] Emmanuel Fonseca, Timothy T Pennucci, Justin A Ellis, Ingrid H Stairs, David J Nice, Scott M Ransom, Paul B Demorest, Zaven Arzoumanian, Kathryn Crowter, Timothy Dolch, et al.  The nanograv nine-year data set: Mass and geometric measurements of binary millisecond pulsars.  Astrophys. J., 832(2):167, 2016.
  * [62] Thomas E Riley, Anna L Watts, Paul S Ray, Slavko Bogdanov, Sebastien Guillot, Sharon M Morsink, Anna V Bilous, Zaven Arzoumanian, Devarshi Choudhury, Julia S Deneva, et al.  A nicer view of the massive pulsar psr j0740+ 6620 informed by radio timing and xmm-newton spectroscopy.  The Astrophysical Journal Letters, 918(2):L27, 2021.
  * [63] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, S. M. Morsink, P. S. Ray, M. T. Wolff, C. L. Baker, T. Cazeau, S. Manthripragada, C. B. Markwardt, T. Okajima, S. Pollard, I. Cognard, H. T. Cromartie, E. Fonseca, L. Guillemot, M. Kerr, A. Parthasarathy, T. T. Pennucci, S. Ransom, and I. Stairs.  The radius of psr j0740+6620 from nicer and xmm-newton data.  The Astrophysical Journal Letters, 918(2):L28, sep 2021.
  * [64] Gordon Baym, Christopher Pethick, and Peter Sutherland.  The ground state of matter at high densities: equation of state and stellar models.  Astrophys. J., 170:299, 1971.
  * [65] Eemeli Annala, Tyler Gorda, Aleksi Kurkela, and Aleksi Vuorinen.  Gravitational-wave constraints on the neutron-star-matter equation of state.  Physical review letters, 120(17):172703, 2018.
  * [66] Tanja Hinderer.  Tidal love numbers of neutron stars.  Astrophys. J., 677(2):1216, 2008.
  * [67] Tanja Hinderer, Benjamin D Lackey, Ryan N Lang, and Jocelyn S Read.  Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral.  Phys. Rev. D, 81(12):123016, 2010.
  * [68] Tanja Hinderer.  Erratum:“tidal love numbers of neutron stars”(2008, apj, 677, 1216).  Astrophys. J., 697(1):964, 2009.
  * [69] Thibault Damour and Alessandro Nagar.  Effective one body description of tidal effects in inspiralling compact binaries.  Phys. Rev. D., 81(8):084016, 2010.
  * [70] Yuxi Li, Houyuan Chen, Dehua Wen, and Jing Zhang.  Constraining the nuclear symmetry energy and properties of the neutron star from gw170817 by bayesian analysis.  The European Physical Journal A, 57(1):1–10, 2021.
  * [71] Benjamin P Abbott, Rich Abbott, T. D. Abbott, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, R. X. Adhikari, V. B. Adya, et al.  Gw170817: observation of gravitational waves from a binary neutron star inspiral.  Phys. Rev. Lett., 119(16):161101, 2017.


# Characterizing Residential Load Patterns by Household Demographic and
Socioeconomic Factors

Zhuo Wei  Department of Data Science and AIMonash
UniversityMelbourneVictoriaAustraliaVIC 3800<EMAIL_ADDRESS>and  Hao Wang
Department of Data Science and AIMonash
UniversityMelbourneVictoriaAustralia3800<EMAIL_ADDRESS>
(2021)

###### Abstract.

The wide adoption of smart meters makes residential load data available and
thus improves the understanding of the energy consumption behavior. Many
existing studies have focused on smart-meter data analysis, but the drivers of
energy consumption behaviors are not well understood. This paper aims to
characterize and estimate users’ load patterns based on their demographic and
socioeconomic information. We adopt the symbolic aggregate approximation (SAX)
method to process the load data and use the K-Means method to extract key load
patterns. We develop a deep neural network (DNN) to analyze the relationship
between users’ load patterns and their demographic and socioeconomic features.
Using real-world load data, we validate our framework and demonstrate the
connections between load patterns and household demographic and socioeconomic
features. We also take two regression models as benchmarks for comparisons.

Smart Meter, Symbolic Aggregate Approximation, Load Pattern, Clustering, Deep
Neural Network

††journalyear: 2021††copyright: acmlicensed††conference: The Twelfth ACM
International Conference on Future Energy Systems; June 28–July 2, 2021;
Virtual Event, Italy††booktitle: The Twelfth ACM International Conference on
Future Energy Systems (e-Energy ’21), June 28–July 2, 2021, Virtual Event,
Italy††price: 15.00††doi: 10.1145/3447555.3464867††isbn:
978-1-4503-8333-2/21/06††ccs: Theory of computation Machine learning
theory††ccs: Hardware Power and energy

##  1\. Introduction

Digitalization is reshaping the energy sector, and smart meters have been
widely installed in households (Yildiz et al., 2017). A large amount of data
collected from smart meters has great potential to enhance the energy
efficiency, improve energy services, and secure system operations (Guo et al.,
2014). Many existing studies, such as (Haben et al., 2015), analyzed
residential customers’ energy consumption behaviors using smart meter data.
Studies in (Jones et al., 2015) took a further step revealing major factors
that determine energy consumption behaviors, including socioeconomic factors
and dwelling factors. The exploration of linkages between load patterns and
demographic and socioeconomic factors can help provide more personalized
services. Different social groups have different load patterns (Jain et al.,
2017). If these differences can be identified, there can be a better
understanding of the impact of energy policies and programs on energy equity
and justice. However, the relationship between users’ energy consumption
behaviors and their demographic and socioeconomic background (e.g., age,
income, and education) is less well understood. Therefore, this paper aims to
develop a methodology to model consumers’ load patterns using smart meter data
and establish the link between load patterns and demographic and socioeconomic
factors.

The relationship between consumers’ load patterns and their demographic and
socioeconomic information has been analyzed in the literature. For example,
the authors in (Han et al., 2014) used a random forest model to assess the
impact of socioeconomic and environmental factors on residential energy
consumption. The study in (Pombeiro et al., 2012) pointed out that the
consumption behavior is more strongly linked to intrinsic factors than to
characteristics of the residential environment. In (McLoughlin et al., 2012),
an interesting observation was made based on demographic information, and the
results showed that younger users had their peak consumption later in the day
than elderly users. In (Tang et al., 2019), socioeconomic information, such as
income and education levels, was selected as features to train a prediction
model for the load distribution. The studies mentioned above usually input the
raw load data into clustering or classification models, and the measurement
noises would affect the results. Moreover, from the operator’s perspective,
the information regarding load shape (e.g., single peak or dual peak) and peak
time weights more than subtle load variations in the off-peak period. Hence, a
more effective method is needed to model consumption behaviors using smart
meter data.

This paper develops an effective methodology to address the above shortcomings
in handling smart meter data and improve the understanding of the relationship
between loading patterns and demographic and socioeconomic factors.
Specifically, we use clustering techniques to extract consumers’ load patterns
and build a machine learning model to study the relationship between load
patterns and demographic and socioeconomic factors.

  * •

We use symbolic aggregate approximation (SAX) to process the smart meter data
and eliminate the effect of noises on the clustering results.

  * •

We model load patterns using a set of load clusters, representing different
consumption behaviors.

  * •

We build a deep neural network (DNN) model to study the relationship between
load patterns and demographic and socioeconomic information, such as
consumers’ age, income level, education level, and premise area.

  * •

Our developed methodology is validated using real-world data of households.

Note that the data involved in this work are anonymized and do not involve
personal privacy issues. Also, our study focuses on the energy consumption
patterns at the group level, aiming to improve the understanding of different
groups’ behaviors for better energy equity.

##  2\. Data Description

This paper uses data from the Pecan Street database (Pecan Street. Pecan
street online database. Pecan Street: Austin, 2019) as a case study to
validate our design and study the relationship between load patterns and
demographic and socioeconomic factors. This database includes smart meter data
of over a thousand households at a resolution of one hour. This dataset also
provides these households’ demographic and socioeconomic information,
including the number of residents in each age group, annual household income,
education level, and the household audit data.

###  2.1. Load Data

We first pre-process the load data to remove invalid and duplicate values and
obtain valid load data of $312$ households from January 1st, 2015 to December
31st, 2017. Specifically, the load data are recorded as the electricity
consumption in kWh for each household in each hour, thus forming daily load
profiles over three years with a total of $341,328$ data entries.

###  2.2. Demographic and Socioeconomic Data

Another essential part of the data is consumers’ demographic and socioeconomic
information. According to (Pecan Street. Pecan street online database. Pecan
Street: Austin, 2019), consumers’ demographic and socioeconomic information
includes many aspects, such as income level, education level, age of
residents, household location, and year built. Since not all the demographic
and socioeconomic data serve meaningful features, simply including them will
increase models’ complexity and reduce accuracy. It is necessary to eliminate
redundant features and select the appropriate variables. According to the
results from the entropy-based feature selection in (Tang et al., 2019), the
most relevant demographic and socioeconomic characteristics to load patterns
are age, education, income, and household square footage. Therefore, we choose
the above characteristics as the demographic and socioeconomic information in
our study.

##  3\. Methods for Characterizing Load Patterns

To study the relationship between load patterns and demographic and
socioeconomic factors, we process the load data to model representative load
patterns and then build a machine learning model to analyze the relationship.
Our method consists of three parts: processing load data, clustering load
patterns, and developing a model to find the relationship between load
patterns and demographic and socioeconomic factors. These three parts are
presented in Section 3.1-3.3, respectively.

###  3.1. Representing Load Data with SAX

We further process the original load data for the following reasons. The
hourly time series load data collected by smart meters often contain
measurement noises. Moreover, the operator or the utility cares more about the
peak load and load type, such as dual peak or single peak and when the peaks
appear. The subtle differences between load profiles, especially in off-peak
hours, do not provide many insights but make load clustering less effective.
So dimension reduction is necessary to eliminate these effects and capture the
key information in the load profiles. In (Fan et al., 2015), SAX has been used
to convert the load profiles into symbol strings. This process eliminates the
influence of noises and subtle variations and highlights the trend change of
the load profile. We use the SAX method to process the load data in our work.

The SAX algorithm proposed in (Lin et al., 2003) extended the Piece-wise
Aggregate Approximation (PAA) and kept the low complexity of the original
method (Lin et al., 2003). It can convert time series data into different
segments with a few discrete values as an approximation. In this paper, we use
PAA to process the original time-series load data
$\boldsymbol{L}=\\{l_{1},l_{2},\ldots,l_{H}\\}$, which represents the daily
load profile of $H=24$ hours. We evenly divide $24$ hours into $S$ segments,
each of which has an interval length of $\tau$. PAA converts the original
time-series load $\boldsymbol{L}$ into an approximating representation
$\bar{\boldsymbol{L}}=\\{\bar{l}_{1},\bar{l}_{2},\ldots,\bar{l}_{S}\\}$, where
$\bar{l}_{s},~{}s=1,...,S$ is calculated as

(1) $\bar{l}_{s}=\frac{1}{\tau}\sum_{t=\tau(s-1)+1}^{\tau*S}{l_{t}}.$

After the PAA representation is complete, we use the Gaussian distribution to
find interval breakpoints, each of which represents an approximation. Each
segment is allocated to the approximation corresponding to its closest break-
point, thus dividing the time-series load profile into a few segments with
several discrete values (Lin et al., 2003). In our case study, we choose
$\tau=3$ and divide the $24$-hour load profiles into $S=8$ segments. The load
data in each segment can be represented by five discrete values (Ma et al.,
2018) as an approximation of consumers’ load profiles.

###  3.2. Modeling Load Patterns via Clustering

After processing the load data with SAX, it is necessary to extract several
representative load patterns from the processed load data. Specifically, we
use the clustering technique to model load patterns by dividing the SAX-
processed load profiles into several clusters, each of which represents a
representative load pattern.

The partitioning clustering methods (Wen et al., 2018) have been widely used
to cluster time-series data, and K-means is one of the most commonly used
partitioning clustering methods. The main advantage of K-means is that it is
easy to implement and scale-up (Coates and Ng, 2012), and thus we choose
K-means as the clustering method in our work. The number of clusters $K$ must
be chosen carefully to represent typical load patterns. If $K$ is too large,
clusters can be very similar to each other and do not exhibit meaningful
differences. If $K$ is too small, some typical load patterns may be missing.
Therefore, we consider both Sum of Squared Errors (SSE) and the silhouette
index (Manochandar et al., 2020) to select the most appropriate value for $K$.

A smaller SSE indicates better clustering performance. As the number of
clusters $K$ increases, the SSE will gradually become smaller. When $K$
reaches an appropriate number, the declining rate of SSE diminishes and
stabilizes even with an increasing $K$. In our case, the SSE starts to
stabilize after the cluster number $K$ reaches $5$, so any $K$ greater than
$5$ can be considered. To determine the best $K$, we further consider the
silhouette index, as it is also an important measure of clustering performance
(Manochandar et al., 2020). A larger silhouette index means that the load
profiles are closer to each other within a cluster but more distant from each
other across different clusters, indicating better clustering performance. In
our study, the silhouette index peaks at a cluster number of $K=7$. Therefore,
we choose $K=7$ clusters to model load patterns for achieving a low SSE and
the highest silhouette index.

To sum up, this paper divides the load profiles into $7$ clusters by the
K-means method, representing $7$ different electricity consumption behaviors.
We further study how the consumers’ load patterns are correlated with the
demographic and socioeconomic features in Section 3.3.

###  3.3. Machine Learning Model Based on DNN

We develop a machine learning model to explore the relationship between
demographic and socioeconomic characteristics and consumers’ load patterns.
Our problem can be regarded as a regression problem, in which the independent
variables are consumers’ demographic and socioeconomic features, and the
dependent variables are the probabilities of load patterns. We tried to solve
the problem with regression models, such as linear regression and polynomial
regression, but the fit was not satisfactory due to unknown nonlinearity in
the data. Therefore, we tried more efficient models. DNN is a powerful machine
learning method with many successful applications (Tang et al., 2019). It
consists of a collection of neurons organized in a sequence of multiple
layers, which can incorporate nonlinear activation functions. Therefore, this
paper uses a DNN model and takes a linear regression model and a polynomial
regression model as two benchmark models.

After the clusters have been identified in our study, we have a set of
representative load patterns. Each consumer’s load profiles can be modeled as
a probability distribution over the representative load patterns. We take
users’ demographic and socioeconomic features as the input of the DNN model to
find the relationship between them and the load patterns. A set of percentages
of load profiles belonging to each representative load pattern is the
estimated target, i.e., the DNN model’s output. The probability for each
cluster is always in the range [0,1], and the sum of all probabilities is
always $1$. Therefore, we introduce the SoftMax function to normalize the
output as a valid probability distribution.

In summary, the developed DNN based model takes consumers’ demographic and
socioeconomic features as the input and the probability distribution of load
patterns as the output, thereby correlating the demographic and socioeconomic
features with load patterns and revealing their relationship.

##  4\. Case Study

To validate our developed method in Section 3, we use real-world data
described in Section 2 as a case study. We implement a DNN model with $3$
hidden layers, each of which includes $100$ neurons. The number of residents
in each age group, annual household income, the education level of the
residents, and total square footage of the house are used as inputs. The
outputs are the probability distributions of load patterns. Considering the
foreseeable difference in behaviors on weekdays and weekends, we split the
load data into weekday and weekend load data. We use 80% of the data for
training and the remaining 20% for the test based on the weekday and weekend
load data, respectively.

###  4.1. Model Performance

We compare our developed DNN model with a linear regression model and a
polynomial regression model. The data used for training and test are the same
for all of the models. Due to the page limit, we only depict the comparison
results for weekday load. Figure 1 shows the load patterns described as $7$
clusters. Figure 2 shows a comparison of weekday load patterns for three
typical consumers with different demographic and socioeconomic features.

Figure 1. The $7$ representative load patterns on weekdays.

Figure 2. The comparison of three different users on weekdays.

In Figure 1, we see that the lower electricity consumption periods may suggest
that residents are resting or not at home, while the first peak load (as shown
in Clusters 1, 2, and 4) usually means the start of activities in the morning,
such as preparing breakfast or doing laundry before going to work. In some
clusters (such as Clusters 1, 2, 5, and 6), we see peak load in the evening,
suggesting that residents get back home after work or start preparing dinner
and have recreational activities. In Cluster 3, the peak occurs during midday,
which indicates that residents are at home and residents’ activities are
concentrated throughout the day. Cluster 7 has a late-night peak, but the
electricity consumption is very low during the day. A possible reason for this
load pattern is that residents are sleeping or not at home during the daytime
but awake at night.

The residents’ load patterns reflect their household activities, which are
correlated with their demographic and socioeconomic background. Figure 2 shows
the estimated results on three residents’ probability distributions of load
patterns based on their demographic and socioeconomic features using our DNN
model and two regression benchmarks, compared with the ground truth. We see
that the probability distributions of the load patterns for both Household 59
and Household 585 are concentrated on two specific clusters, but the
probability distribution of the load patterns for household 121 is less
concentrated, which may be related to the age distribution of the residents.
Both Household 59 and Household 585 have residents with ages over 65, but
Household 121 also has children under 12 years of age, which reveals that the
elderly usually have a more regular routine. Moreover, both Household 59 and
Household 585 have load patterns concentrated in Cluster 2 and Cluster 4 with
a peak during morning hours. This could be because elderly users are more
inclined to wake up early and have a hot breakfast. The probability
distribution of Household 121 is less concentrated than those of the other two
households. But Household 121 has a higher fraction of load in Cluster 2 and
Cluster 6, indicating a peak during the evening hours. This can be caused by
households with children spending the evening for some family entertainment
activities. These households having peak load in the evening are more likely
to be called for demand response, and their household activities will be
affected.

Furthermore, the DNN model has a better fit than the two benchmark models. Due
to the complex relationship between consumers’ demographic and socioeconomic
characteristics and the probability distribution of their load patterns, the
two benchmark models may under-fit, while our DNN model can provide more
accurate results. The errors of the three models are measured by the Mean
Squared Error (MSE), and a smaller MSE means better fitting. The DNN model
reduces MSE by 90% compared to the linear regression model and more than 70%
compared to the polynomial regression model. Note that different results will
be obtained using a different dataset. Our work provides a methodology to
model the load patterns and study the relationship between load patterns and
demographic and socioeconomic features. We have no intention of emphasizing
the improvement in the estimation accuracy.

##  5\. Discussions and Conclusion

We developed an analytical method to model the residential load patterns and
discover the relationship between the load patterns and the demographic and
socioeconomic factors. We use the SAX method to remove the impact of
measurement noises in load data, symbolically represent the data and
approximate the load profiles. We use the K-means clustering to model the
representative load patterns. Using demographic and socioeconomic factors as
inputs and load patterns as outputs, we develop a DNN model to study the
relationship between the load data and demographic and socioeconomic data. Our
model generates new insights into how consumers with different demographic
backgrounds use electricity.

In our future work, we plan to reduce the impact of measurement noises in the
load data to model load patterns. The data availability also restricts the
current results, and we plan to find more a large dataset for validation. We
also aim to analyze the impact of energy tariffs and policy on consumers and
improve energy equity.

## References

  * (1)
  * Coates and Ng (2012) Adam Coates and Andrew Y Ng. 2012.  Learning feature representations with k-means.  In _Neural networks: Tricks of the trade_. Springer, 561–580.
  * Fan et al. (2015) Cheng Fan, Fu Xiao, Henrik Madsen, and Dan Wang. 2015\.  Temporal knowledge discovery in big BAS data for building energy management.  _Energy and Buildings_ 109 (2015), 75–89.
  * Guo et al. (2014) Zhenyu Guo, Z Jane Wang, and Ali Kashani. 2014.  Home appliance load modeling from aggregated smart meter data.  _IEEE Transactions on power systems_ 30, 1 (2014), 254–262.
  * Haben et al. (2015) Stephen Haben, Colin Singleton, and Peter Grindrod. 2015\.  Analysis and clustering of residential customers energy behavioral demand using smart meter data.  _IEEE transactions on smart grid_ 7, 1 (2015), 136–144.
  * Han et al. (2014) Yufei Han, Xiaolan Sha, Etta Grover-Silva, and Pietro Michiardi. 2014.  On the impact of socio-economic factors on power load forecasting. In _2014 IEEE International Conference on Big Data_. IEEE, 742–747.
  * Jain et al. (2017) Rishee K Jain, Junjie Qin, and Ram Rajagopal. 2017\.  Data-driven planning of distributed energy resources amidst socio-technical complexities.  _Nature Energy_ 2, 8 (2017), 1–11.
  * Jones et al. (2015) Rory V Jones, Alba Fuertes, and Kevin J Lomas. 2015\.  The socio-economic, dwelling and appliance related factors affecting electricity consumption in domestic buildings.  _Renewable and Sustainable Energy Reviews_ 43 (2015), 901–917.
  * Lin et al. (2003) Jessica Lin, Eamonn Keogh, Stefano Lonardi, and Bill Chiu. 2003.  A symbolic representation of time series, with implications for streaming algorithms. In _Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery_. 2–11.
  * Ma et al. (2018) Zhenjun Ma, Rui Yan, Kehua Li, and Natasa Nord. 2018\.  Building energy performance assessment using volatility change based symbolic transformation and hierarchical clustering.  _Energy and Buildings_ 166 (2018), 284–295.
  * Manochandar et al. (2020) S. Manochandar, Murugesan Punniyamoorthy, and R.K. Jeyachitra. 2020\.  Development of new seed with modified validity measures for k- means clustering.  _Computers & Industrial Engineering_ 141 (3 2020), 106290\.
  * McLoughlin et al. (2012) Fintan McLoughlin, Aidan Duffy, and Michael Conlon. 2012\.  Characterising domestic electricity consumption patterns by dwelling and occupant socio-economic variables: An Irish case study.  _Energy and buildings_ 48 (2012), 240–248.
  * Pecan Street. Pecan street online database. Pecan Street: Austin (2019) USA Pecan Street. Pecan street online database. Pecan Street: Austin, TX. 2019.
  * Pombeiro et al. (2012) Henrique Pombeiro, André Pina, and Carlos Silva. 2012\.  Analyzing residential electricity consumption patterns based on consumer’s segmentation. In _Proceedings of the first international workshop on information technology for Energy Applications, Lisbon, Portugal_. 29–38.
  * Tang et al. (2019) Wen-Jun Tang, Xian-Long Lee, Hao Wang, and Hong-Tzer Yang. 2019\.  Leveraging socioeconomic information and deep learning for residential load pattern prediction. In _2019 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe)_. IEEE, 1–5.
  * Wen et al. (2018) Lulu Wen, Kaile Zhou, Shanlin Yang, and Lanlan Li. 2018\.  Compression of smart meter big data: A survey.  _Renewable and Sustainable Energy Reviews_ 91 (2018), 59–69.
  * Yildiz et al. (2017) Baran Yildiz, Jose I Bilbao, Jonathon Dore, and Alistair B Sproul. 2017.  Recent advances in the analysis of residential electricity consumption and applications of smart meter data.  _Applied Energy_ 208 (2017), 402–427.