# Foliations, projections and nucleon charge profiles in hypersphere soliton
model

Soon-Tae Hong<EMAIL_ADDRESS>Center for Quantum Spacetime and
Department of Physics, Sogang University, Seoul 04107, Korea

###### Abstract

In the hypersphere soliton model, we investigate foliations related with the
geometry on a hypersphere $S^{3}$ which is described by $(\mu,\theta,\phi)$.
To do this, we introduce a projection from a northern hemi-hypersphere
$S^{3}_{+}$ to an equatorial volume $E^{3}$ which contains the center of
$S^{3}$ and is described in terms of the spherical coordinates of the solid
sphere. Exploiting the projection and the so-called scanning algorithm we
study geometrical relations between spherical shell foliation leave on the
curved $S^{3}_{+}$ and that on the flat equatorial volume $E^{3}$. We then
elucidate the physical meaning of $\mu$ in $S^{3}$ of radius $\lambda$ by
showing that $\mu$ plays the role of an auxiliary angle to fix the radius
$\lambda\sin\mu$ of the $S^{2}$ spherical shell sharing the center of
$S^{3}(=S^{2}\times S^{1})$, at a given angle $\mu$. Next, we study charge
density profiles of nucleons which are formulated in terms of the angle $\mu$
in $S^{3}$. To be specific, we construct the nucleon fractional three
dimensional volumes and the corresponding fractional charges, which are
spherically symmetric and possesses the non-trivial distribution structures.
In particular we find that, in the neutron charge distribution, the positive
and negative charges are confined inside and outside the neutron core,
respectively. We next observe that the proton and neutron charges do not leak
out from the hypersphere soliton and the characteristic ratio of the
hypersphere volume to the corresponding solid sphere one is given by the
geometrical invariant. This invariant is also shown to originate from the
projection and scanning algorithm procedure over the angle range
$0\leq\mu\leq\pi$ of $S^{3}$.

foliations in hypersphere; nucleon charge density profiles; fractional neutron
distributions; projection; scanning

##  I Introduction

Exploiting the Dirac quantization in the first class formalism di ; hong15 ,
the hadron phenomenology theorists have tried to quantize constrained systems.
In particular, the standard Skyrmion skyrme61 ; anw83 ; hong98plb ; hong15
has been studied as one of the constrained Hamiltonian systems. In this model,
the three dimensional physical space is assumed to be topologically
compactified to $S^{3}$ and the spatial infinity is located on the north-pole
of $S^{3}$ anw83 . Next, the hypersphere soliton model (HSM) manton1 ;
hong98plb ; hong21  has been proposed to find a topological lower bound on the
soliton energy and a set of equations of motion via the canonical quantization
in the second class formalism manton1 . Moreover, making use of the canonical
quantization in the HSM, we have evaluated the baryon physical quantities such
as baryon masses, charge radii and magnetic moments, to propose that a
realistic hadron physics can be described by this phenomenological soliton on
the hypersphere $S^{3}$ hong98plb .

On the other hand, a pulsating charged conducting surface has been proposed as
the electron model by Dirac di62 . Following his idea we have investigated
pulsating baryons in the first class Hamiltonian quantization formalism in the
HSM, to predict the intrinsic frequencies of the baryons hong21 . In
particular, we have predicted the intrinsic frequencies $\omega_{N}=0.87\times
10^{23}~{}{\rm sec}^{-1}$ and $\omega_{\Delta}=1.74\times 10^{23}~{}{\rm
sec}^{-1}$ of the nucleon and delta baryon, respectively, to find the relation
$\omega_{\Delta}=2\omega_{N}$. Note that in the stringy photon model hong22
we have evaluated the intrinsic frequency for the photon
$\omega_{\gamma}=9.00\times 10^{23}~{}{\rm sec}^{-1}$ which is comparable to
the above intrinsic frequencies for the nucleon and delta baryon. Next we have
evaluated the nucleon and delta baryon masses, charge radii, magnetic moments
and axial coupling constant, most predicted values of which are in good
agreement with the corresponding experimental data hong21 .

Even though we have seen the phenomenological successes in predicting the
baryon static properties hong98plb ; hong21  which are obtained after
integrating out the angle $\mu$ of $S^{3}$ defined by the hypersphere
coordinates $(\mu,\theta,\phi)$ in the HSM, it remains unclear to explain the
exact physical meaning and role of $\mu$ associated with $S^{3}$ geometry.
Next, note that some geometrical aspects of the static quantities of the
baryons have been well explained hong21  on the $S^{3}$ hypersphere in terms
of the foliations foli ; hong21 . However the foliations related with the
physical quantities such as the nucleon charge density profiles have not been
studied in terms of $\mu$ of the three dimensional hypersphere $S^{3}$.

In this paper, in order to elucidate the fractional charge distributions in
the nucleon charge density profiles in $S^{3}$, in the HSM we will investigate
the foliations related with the geometry on the three dimensional hypersphere.
To do this, we will consider the projection from a northern hemi-hypersphere
$S^{3}_{+}$ to a three dimensional equatorial volume $E^{3}$ which contains
the center of $S^{3}$ and is described in terms of $(r,\theta,\phi)$. From now
on, we will name the mathematical algorithm to fill a compact manifold by
using the foliation leaves on the manifold, scanning. Exploiting the
projection and scanning algorithm, we will then investigate the connections
between spherical shell foliation leave on the curved $S^{3}_{+}$ and
spherical shell foliation leave on the equatorial volume $E^{3}$. Next, we
will formulate the charge density profiles of nucleons which are given in
terms of the angle $\mu$ in $S^{3}$. Making use of the geometrical
descriptions on the three dimensional volume of the hypersphere soliton, we
will find the nucleon charge distributions and the corresponding volumes which
are spherically symmetric.

In Sec. 2, we will study the hypersphere geometry, foliations, and projection
and scanning algorithm in the HSM. In Sec. 3, we will construct the three
dimensional volumes associated with the spherically symmetric nucleon charge
density profiles in $S^{3}$. Sec. 4 includes conclusions.

##  II Foliations, and projection and scanning algorithm in $S^{3}$ geometry

In this section, we will investigate the hypersphere $S^{3}$ geometry
associated with the foliation foli , and the projection and scanning
algorithm. To do this, we introduce the hypersphere manifold
$S^{3}(=S^{2}\times S^{1})$. Note $S^{3}$ itself is a three dimensional
manifold. The three metric on $S^{3}$ described in terms of the hypersphere
coordinates $(\mu,\theta,\phi)$ is given by

$ds^{2}(S^{3})=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu~{}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}),$
(2.1)

where the ranges of $(\mu,\theta,\phi)$ are given by $0\leq\mu\leq\pi$,
$0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$, and $\lambda$
($0\leq\lambda<\infty$) is a radius of $S^{3}$. Note that from the three
metric on $S^{3}$ in (2.1) one can read off the radius of $S^{2}$ in
$S^{3}(=S^{2}\times S^{1})$ which is given by $\lambda\sin\mu$ for a given
angle $\mu$. This geometrical property of the radius $\lambda\sin\mu$ will be
exploited later in the projection and scanning algorithm in the hypersphere
geometry.

On the $S^{3}$ hypersphere, the line element is defined as

$d\vec{l}=\hat{e}_{\mu}\lambda~{}d\mu+\hat{e}_{\theta}\lambda\sin\mu~{}d\theta+\hat{e}_{\phi}\lambda\sin\mu\sin\theta~{}d\phi,$
(2.2)

where $(\hat{e}_{\mu},\hat{e}_{\theta},\hat{e}_{\phi})$ are the unit vectors
along the three directions. Exploiting the three arc length elements $\lambda
d\mu$, $\lambda\sin\mu d\theta$ and $\lambda\sin\mu\sin\theta d\phi$ which can
be read off from (2.2), we find the two dimensional area elements given by

$d\vec{a}_{\mu}=\hat{e}_{\mu}\lambda^{2}\sin^{2}\mu\sin\theta~{}d\theta~{}d\phi,~{}~{}~{}d\vec{a}_{\theta}=\hat{e}_{\theta}\lambda^{2}\sin\mu\sin\theta~{}d\mu~{}d\phi,~{}~{}~{}d\vec{a}_{\phi}=\hat{e}_{\phi}\lambda^{2}\sin\mu~{}d\mu~{}d\theta,$
(2.3)

and the three dimensional volume element defined as

$dV=\lambda^{3}\sin^{2}\mu\sin\theta~{}d\mu~{}d\theta~{}d\phi.$ (2.4)

Note that in Ref. hong21  we have treated $dV$ as the thin bubble-like three
dimensional hypersurface area element (or in fact volume element) in
evaluating the physical quantities such as the magnetic isovector mean square
charge radius $\langle r^{2}\rangle^{1/2}_{M,I=1}$.

Making use of $da_{\mu}$ in (2.3), for a given angle $\mu$ we find the two
dimensional area perpendicular to $\hat{e}_{\mu}$ of the form

$A_{2}(S^{3})=4\pi\lambda^{2}\sin^{2}\mu.$ (2.5)

Note that the surface area $A_{2}(S^{3})$ is a function of the angle $\mu$ and
at $\mu=\frac{\pi}{2}$ it has a maximum value
$A_{2}^{max}(S^{3})=4\pi\lambda^{2}$ which is equivalent to the surface area
$A_{2}(shell)$ of the spherical shell of radius $\lambda$, as expected. Next
exploiting $dV$ in (2.4), for the range $0\leq\mu\leq\mu_{0}$ we have the
three dimensional volume

$V(S^{3};0\leq\mu\leq\mu_{0})=2\pi\lambda^{3}\left(\mu_{0}-\frac{1}{2}\sin
2\mu_{0}\right).$ (2.6)

In particular, inserting $\mu_{0}=\pi$ into (2.6), we obtain the three
dimensional volume of the hypersphere for the range $0\leq\mu\leq\pi$

$V(S^{3})=2\pi^{2}\lambda^{3}.$ (2.7)

Now we have a couple of comments on the three dimensional volume of the
hypersphere $V(S^{3})$ in (2.7). First, we can also obtain
$V(S^{3})=2\pi^{2}\lambda^{3}$ in (2.7) by using the two dimensional area
$A_{2}(S^{3})$ in (2.5) and the line element $\lambda d\mu$

$V(S^{3})=\int_{0}^{\pi}A_{2}(S^{3})\lambda d\mu.$ (2.8)

Note that $A_{2}(S^{3})$ ($0\leq\mu\leq\pi$) consists of the foliation leaves
sharing the center of the hypersphere $S^{3}(=S^{2}\times S^{1})$.

Second, the three dimensional volume $V(S^{3})$ of the hypersphere of radius
$\lambda$ is much larger than the three dimensional volume
$V(R^{3})=\frac{4\pi}{3}\lambda^{3}$ of the solid sphere of radius $\lambda$
to yield the characteristic ratio $\chi$ of the form

$\chi=\frac{V(S^{3})}{V(R^{3})}=\frac{3\pi}{2},$ (2.9)

which is greater than unity. Note that the hypersphere of radius $\lambda$ is
delineated by the compact manifold which corresponds to the three dimensional
volume $V(S^{3})$ in (2.7) and the ratio $\chi$ in (2.9).

Figure 1: The schematic projection and scanning algorithm diagrams for (a) the
two dimensional spherical shell $S^{2}$ geometry and (b) the three dimensional
hypersphere $S^{3}$ one depicted in terms of $S^{2}\times S^{1}$, where
$S^{2}$ is simplified by $S^{1}$ for convenience. In (b), the perpendicular
unit vectors along the three directions are described by
$(\hat{e}_{\mu},\hat{e}_{\theta},\hat{e}_{\phi})$, where the unit vectors
$\hat{e}_{\theta}$ and $\hat{e}_{\phi}$ are perpendicular to each other even
though they cannot be explicitly depicted in this schematic diagram. Note that
the unit vector $\hat{e}_{r}$ is perpendicular to the spherical shell $S^{2}$
which is described by $(\hat{e}_{\theta},\hat{e}_{\phi})$ and shares the
center $o$ of $S^{3}$. For more details of the spherical shell $S^{2}$
geometry in (b), see the text in Section 2.

Next, keeping the ratio $\chi$ in mind, we investigate the foliations related
with the geometry inside the hypersphere. To do this, we first consider a
simple case of the spherical shell $S^{2}$ of radius $\rho$ which is described
in terms of $(\theta,\phi)$, as shown in Fig. 1(a). Here the perpendicular
unit vectors along the three directions are denoted by
$(\hat{e}_{r},\hat{e}_{\theta},\hat{e}_{\phi})$. Note that $\hat{e}_{\phi}$ is
into the page at the observation point $p$. On the curved northern hemisphere
$S^{2}_{+}$, we have foliation leaves of circle of radius $\rho\sin\theta$
$(0\leq\theta\leq\frac{\pi}{2})$. Note that these foliation leaves fill
$S^{2}_{+}$ after scanning completely over the range
$0\leq\theta\leq\frac{\pi}{2}$. Similarly the southern hemisphere $S^{2}_{-}$
is filled with the foliation leaves of circle of radius $\rho\sin\theta$
$(\frac{\pi}{2}\leq\theta\leq\pi)$. Completing the scanning over the range
$0\leq\theta\leq\pi$ one finds the two dimensional surface area
$A_{2}(S^{2})=4\pi\rho^{2}$. We next consider a projection from the northern
hemisphere $S^{2}_{+}$ to the two dimensional equatorial plane $E^{2}$ as
shown in Fig. 1(a). To be specific, the circle of radius $\rho\sin\theta$ on
the curved manifold $S^{2}_{+}$ is projected on the circle of radius
$\rho\sin\theta$ on the equatorial plane $E^{2}$. In other words, the radius
$\rho\sin\theta$ on $S^{2}_{+}$ is projected on the radial distance $oq$ on
$E^{2}$. As we exploit the foliation leaves of circle of radius
$\rho\sin\theta$ to scan $S^{2}_{+}$ over the range
$0\leq\theta\leq\frac{\pi}{2}$, we can also scan the flat equatorial plane
$E^{2}$ with the same foliation leaves of circle of radius $\rho\sin\theta$ to
produce the surface area $A_{2}(E^{2})=\pi\rho^{2}$ which is less than
$A_{2}(S^{2}_{+})=2\pi\rho^{2}$. Note that
$A_{2}(S^{2})=A_{2}(S^{2}_{+})+A_{2}(S^{2}_{-})=4\pi\rho^{2}$.

Now we apply the same mathematical logic used in the projection and scanning
algorithm for the above toy model $S^{2}$, to the three dimensional volume of
the hypersphere $S^{3}$ of radius $\lambda$ which is described in terms of
$(\mu,\theta,\phi)$. To do this, we exploit the hypersphere $S^{3}$ geometry
schematically depicted in terms of $S^{2}\times S^{1}$ as shown in Fig. 1(b),
where $S^{2}$ is simplified by $S^{1}$ for convenience. Here the perpendicular
unit vectors along the three directions are denoted by
$(\hat{e}_{\mu},\hat{e}_{\theta},\hat{e}_{\phi})$. Moreover, the unit vectors
$\hat{e}_{\theta}$ and $\hat{e}_{\phi}$ are perpendicular to each other even
though they cannot be explicitly depicted in the schematic diagram in Fig.
1(b). Note also that the unit vector $\hat{e}_{r}$ is perpendicular to the
spherical shell $S^{2}$.

Next, we consider a projection from the northern hemi-hypersphere $S^{3}_{+}$
to the equatorial plane $E^{3}$, which contains the center of $S^{3}$ and is
described in terms of $(r,\theta,\phi)$, as shown in Fig. 1(b). The spherical
shell of radius $\lambda\sin\mu$ at a given angle $\mu$ on the curved manifold
$S^{3}_{+}$ is projected on the spherical shell of radius $\lambda\sin\mu$ on
the flat equatorial plane $E^{3}$. In other words, the radius $\lambda\sin\mu$
on $S^{3}_{+}$ is projected on the radial distance $oq$ on $E^{3}$ in Fig.
1(b). Note that the spherical shell foliation leave on the curved $S^{3}_{+}$
and the spherical shell foliation leave on the equatorial volume $E^{3}$ share
their corresponding centers at the same point $o$. Note also that, as
discussed above, $\hat{e}_{r}$ in Fig. 1(b) is perpendicular to the $S^{2}$
spherical shell used as the foliation leave on $S^{3}$, and thus the
corresponding center of $S^{2}$ is located at the origin $o$ of the
hypersphere $S^{3}$. This aspect is consistent with the statements that the
foliation leaves associated with the integrand $A_{2}(S^{3})$ in $V(S^{3})$ in
(2.8) share the center of $S^{3}$.

As we make use of the foliation leaves of spherical shell of radius
$\lambda\sin\mu$ to scan the curved manifold $S^{3}_{+}$ over the range
$0\leq\mu\leq\frac{\pi}{2}$, we can also scan the flat equatorial volume
$E^{3}$ with the foliation leaves of spherical shell of radius
$\lambda\sin\mu$ to yield the volume $V(R^{3})=\frac{4\pi}{3}\lambda^{3}$
which is less than $V(S^{3}_{+})=\pi^{2}\lambda^{3}$. Note that the foliation
leave of spherical shell of radius $\lambda\sin\mu$ on $S^{3}_{+}$ is distinct
from that on the equatorial volume $E^{3}$, since the former is attached on
the curved manifold $S^{3}_{+}$ while the latter is attached on the flat
$E^{3}$. On $S^{3}_{+}$, we have foliation leaves of two dimensional surface
of radius $\lambda\sin\mu$ $(0\leq\mu\leq\frac{\pi}{2})$ to yield its
corresponding area $A_{2}(S^{3})=4\pi\lambda^{2}\sin^{2}\mu$ given by (2.5).
Here these foliation leaves cover $S^{3}_{+}$ after scanning completely over
the range $0\leq\mu\leq\frac{\pi}{2}$. In the same way, the southern hemi-
hypersphere $S^{3}_{-}$ is covered with the foliation leaves of spherical
shell of radius $\lambda\sin\mu$ $(\frac{\pi}{2}\leq\mu\leq\pi)$.

Exploiting (2.1) at a fixed angle $\mu$ in $S^{3}$, in fact we have the two
metric
$ds^{2}=\lambda^{2}\sin^{2}\mu~{}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2})$ so
that we can obtain a concentric $S^{2}$ spherical shell (or foliation leave)
of radius $\lambda\sin\mu$, sharing the center $o$ of the hypersphere in Fig.
1(b).111In the $S^{2}$ toy model, the $S^{1}$ circles (or foliation leaves) of
radius $\rho\sin\theta$ do not share the center $o$ of the sherical shell
$S^{2}$ as in Fig. 1(a). Here the radius of the foliation leave $S^{2}$ at the
fixed angle $\mu$ is independent of the angles $(\theta,\phi)$ to produce the
radial distance $\lambda\sin\mu$. These discussions on the two metric at a
fixed angle $\mu$ in $S^{3}$ are consistent with the projection and scanning
algorithm for the $S^{3}$ geometry which is schematically depicted in Fig.
1(b). As a result, $\mu$ can be interpreted as an auxiliary angle to assign to
a given angle $\mu$ of $S^{1}$, the radius $\lambda\sin\mu$ of the $S^{2}$
spherical shell sharing the center of $S^{3}(=S^{2}\times S^{1})$. Moreover,
as shown in (2.2), $\hat{e}_{\mu}$ of the line element $\lambda d\mu$ in
$S^{1}$ is perpendicular to ($\hat{e}_{\theta},\hat{e}_{\phi}$) of the two
dimensional area $A_{2}(S^{3})$ of radius $\lambda\sin\mu$.

Completing the scanning over the range $0\leq\mu\leq\frac{\pi}{2}$
corresponding to $S^{3}_{+}$, we find the three dimensional volume
$V(S_{+}^{3})=\pi^{2}\lambda^{3}$. Similarly, we scan over the range
$\frac{\pi}{2}\leq\mu\leq\pi$ related with $S^{3}_{-}$ to yield the volume
$V(S_{-}^{3})=\pi^{2}\lambda^{3}$. Making use of the projection and scanning
algorithm, we thus scan twice over the range $0\leq\mu\leq\pi$ associated with
the total manifold $S^{3}$ to yield

$V(S^{3})=V(S^{3}_{+})+V(S^{3}_{-})=2\pi^{2}\lambda^{3},$ (2.10)

which produces the geometrical ratio $\chi$ in (2.9), and is consistent with
(2.7) obtained by exploiting the pure geometrical approach without resorting
to the projection and scanning algorithm procedures. Note that this
consistency of the volumes in (2.7) and (2.10) confirms the above statements
on the role of $\mu$ as an auxiliary angle. Next we also observe that there
exists a difference between the scanning in $S^{3}$ and that in $E^{3}$, since
in $S^{3}$ we scan twice over the $S^{3}_{+}$ and $S^{3}_{-}$ semi-
hyperspheres differently from the case in the $E^{3}$ manifold over which we
scan only once, to yield the ratio $\chi$ in (2.9).

##  III Nucleon charge distribution profiles in HSM

In this section, we will investigate the charge profiles of nucleons. To do
this we first study the magnetic isovector mean square charge radius on the
hypersphere $S^{3}$ possessing the fixed value of radius $\lambda_{B}$ in the
HSM hong21

$\langle r^{2}\rangle_{M,I=1}=\frac{2}{3e^{2}{\cal
I}}\int_{S^{3}}dV_{B}\sin^{2}\mu\sin^{2}f\left(1+\left(\frac{df}{d\mu}\right)^{2}+\frac{\sin^{2}f}{\sin^{2}\mu}\right),$
(3.1)

where the subscript $M$ denotes the magnetic radius and $e$ and ${\cal I}$ are
the dimensionless Skyrme parameter and moment of inertia, respectively. Here
the three dimensional volume element $dV_{B}$ defined on the hypersphere is
given by
$dV_{B}=\lambda_{B}^{3}\sin^{2}\mu\sin\theta~{}d\mu~{}d\theta~{}d\phi$ which
is obtainable with the specific value $\lambda=\lambda_{B}$ in (2.4).
Integrating (3.1) over the hypersphere $S^{3}$ in the HSM, we obtain hong98plb
; hong21

$\langle r^{2}\rangle^{1/2}_{M,I=1}=\sqrt{\frac{5}{6}}\frac{1}{ef_{\pi}}.$
(3.2)

Inserting the experimental value $\langle
r^{2}\rangle^{1/2}_{M,I=1}=0.80~{}{\rm fm}$ liu87 ; hong98plb ; hong21  into
(3.2), we find $ef_{\pi}=(0.876~{}{\rm fm})^{-1}$ which corresponds to
$\lambda_{B}=0.876~{}{\rm fm}$. In other words, the hypersphere soliton is
defined in terms of the fixed radius $\lambda_{B}=0.876~{}{\rm fm}$ which is
comparable to the charge radius $\langle
r^{2}\rangle^{1/2}_{M,I=1}=0.80~{}{\rm fm}$. Note that the calculated physical
quantity $\langle r^{2}\rangle^{1/2}_{M,I=1}$ in (3.2), which is related with
the charge effects of the hypersphere soliton, can be phenomenologically
compared with the corresponding experimental value, similar to the other
physical quantities such as the nucleon mass $M_{N}$ and delta baryon one
$M_{\Delta}$. Even though we evaluate $\langle r^{2}\rangle^{1/2}_{M,I=1}$ in
(3.2) which is obtained after integrating out the angle $\mu$ in (3.1) and
thus is an invariant independent of $\mu$, it remains unclear to explain the
exact physical meaning and role of $\mu$ associated with $S^{3}$ geometry,
which will be treated in (3.6) and the ensuing discussions on the charge
density profiles $\rho_{p}(\mu)$ and $\rho_{n}(\mu)$ having $\mu$ dependences.

Now the hypersphere of radius $\lambda_{B}$ has the manifold volume
corresponding to the three dimensional volume $V(S^{3};\lambda=\lambda_{B})$
which is given with the specific value $\lambda=\lambda_{B}$ in (2.7).
Explicitly, exploiting $\lambda_{B}=0.876~{}{\rm fm}$ in the HSM, we find the
compact manifold volume $V(S^{3};\lambda=\lambda_{B})$ possessing the
characteristic ratio $\chi=\frac{3\pi}{2}$ in (2.9)

$V(S^{3};\lambda=\lambda_{B})=2\pi^{2}\lambda_{B}^{3}=13.269~{}{\rm fm}^{3},$
(3.3)

which is greater than the corresponding solid sphere volume
$V(R^{3};\lambda=\lambda_{B})=2.816~{}{\rm fm}^{3}$. Note that the universe
${\cal U}$ in the HSM is given by a sum of two subspaces, namely ${\cal
U}=V(S^{3};\lambda=\lambda_{B})\oplus\bar{V}(S^{3};\lambda=\lambda_{B})$ where
$V(S^{3};\lambda=\lambda_{B})$ and $\bar{V}(S^{3};\lambda=\lambda_{B})$ are
the volume subspaces inside and outside the hypersphere soliton, respectively.

Figure 2: The charge density profiles (a) $\rho_{p}(\mu)$ and (b)
$\rho_{n}(\mu)$, for proton and neutron respectively, are plotted versus
$\mu$. Here $\rho_{p}(\mu)$ and $\rho_{n}(\mu)$ are associated with the
charges inside the nucleons.

Next on the hypersphere we obtain the proton and neutron charges as follows

$Q_{p,n}=\frac{e^{3}f_{\pi}^{3}}{4\pi^{2}}\int_{S^{3}}dV_{B}\left(1\pm\frac{4}{3}\sin^{2}\mu\right).$
(3.4)

Note that, after performing the integration of the nucleon charges $Q_{p,n}$
over the hypersphere $S^{3}$, we find the desired results $Q_{p}=1$ and
$Q_{n}=0$. As a result, the integrands in (3.1) and (3.4) do not possess any
dependences on the coordinates $(\theta,\phi)$, and thus they are spherically
symmetric physical densities on the hypersphere. In other words, the
corresponding static densities of the physical quantities defined on the
hypersphere possess the characteristics of spherical symmetries. In
particular, we notice that the proton and neutron charge densities
corresponding to the integrand of the integral in (3.4) are shown to have
spherical symmetries in the HSM. Integrating the integrand of $Q_{p,n}$ in
(3.4) over the solid angle associated with $(\theta,\phi)$, we arrive at

$Q_{p,n}=\int_{0}^{\pi}d\mu\rho_{p,n}(\mu),$ (3.5)

where the charge density profiles $\rho_{p}(\mu)$ and $\rho_{n}(\mu)$ are
given by

$\displaystyle\rho_{p}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sin^{2}\mu\left(1+\frac{4}{3}\sin^{2}\mu\right),$
$\displaystyle\rho_{n}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sin^{2}\mu\left(1-\frac{4}{3}\sin^{2}\mu\right).$
(3.6)

Note that $\rho_{p}(\mu)$ and $\rho_{n}(\mu)$ are spherically symmetric. The
proton and neutron charge densities in (3.6) are then effectively depicted in
terms of the angle $\mu$ only as shown in Fig. 2(a) and Fig. 2(b),
respectively.

Figure 3: The schematic charge distributions for (a) the proton and (b) the
neutron in the hypersphere of radius $\lambda_{B}$. In (a) the positive charge
$Q_{p}=1$ is filled in the proton. In (b) the positive charge
$Q_{n}^{I}+Q_{n}^{III}=\frac{\sqrt{3}}{8\pi}$ is confined inside the core of
the neutron, while the negative charge $Q_{n}^{II}=-\frac{\sqrt{3}}{8\pi}$ is
distributed outside the neutron core. Here these nucleon charges do not leak
out from the hypersphere soliton.

Now it seems appropriate to comment on the nucleon charge distributions
calculated on $S^{3}$ of radius $\lambda_{B}$. First, the arc length element
$\lambda_{B}d\mu$ in the spherically symmetric hemi-hypersphere $S^{3}_{+}$ of
radius $\lambda_{B}$ takes a part of the line element $dr$ in the spherically
symmetric solid sphere associated with $E^{3}$. The $\mu$ dependences in
$\rho_{p}(\mu)$ and $\rho_{n}(\mu)$ are thus closely related with the radial
dependences in the charge densities inside the nucleons which are located on
the equatorial volume $E^{3}$. Even though on the $S^{3}$ soliton in the HSM
we have the three dimensional volume on $S^{3}$ instead of the three
dimensional solid sphere volume, following the above arguments on the role of
the arc length element $\lambda_{B}d\mu$ in $S^{3}_{+}$ and the line element
$dr$ in $E^{3}$, we can treat
$dV_{B}=\lambda_{B}^{3}\sin^{2}\mu\sin\theta~{}d\mu~{}d\theta~{}d\phi$ as a
volume element inside the hypersphere describing the nucleon.

Second, we define $\lambda_{B}$ as the radial distance from the center of
$S^{3}$ to the boundary of the hypersphere $S^{3}$ hong21 . Note that the
calculated charge radius $\langle r^{2}\rangle^{1/2}_{M,I=1}$ in (3.2) can be
defined as the fixed radial distance to the point on a manifold which does not
need to be located only on the compact manifold $S^{3}$ of radius
$\lambda_{B}$. To be specific, the magnetic isovector charge radius $\langle
r^{2}\rangle_{M,I=1}^{1/2}$ denotes the radial distance which is a geometrical
invariant hong21 .

Third, we can also define the radial distance inside the hypersphere, and this
distance is related with the charge density inside the nucleon. Next the
physical quantities $Q_{p}$ and $Q_{n}$ can be compared with the corresponding
experimental values $Q_{p}^{\rm exp}=1$ and $Q_{n}^{\rm exp}=0$, similar to
the other physical quantities such as the nucleon and delta baryon masses
$M_{N}$ and $M_{\Delta}$.

Next, exploiting the neutron charge density in (3.6), we find two root values
of $\rho_{n}(\mu)$=0: $\mu=\frac{\pi}{3}$ and $\mu=\frac{2\pi}{3}$. We now
have three regions $I$, $II$ and $III$ in the neutron charge density as shown
in Fig. 2(b) and for each region, we obtain the charge fractions as follows

$Q_{n}^{I}=Q_{n}^{III}=\frac{\sqrt{3}}{16\pi},~{}~{}~{}Q_{n}^{II}=-\frac{\sqrt{3}}{8\pi},$
(3.7)

which are in good agreement with the result $Q_{n}=0$. Note that the charge
density profiles of proton and neutron in (3.6) have no dependence on the
coordinates $\theta$ and $\phi$ so that they can be spherically symmetric.
Moreover all the charges fill the three dimensional volume
$V(S^{3};\lambda=\lambda_{B})=2\pi^{2}\lambda_{B}^{3}$ of the hypersphere of
radius $\lambda_{B}$. Now we depict the diagrams for the proton and neutron
charge density distributions inside the hypersphere in Fig. 3. The proton
charge is confined inside the hypersphere of radius $\lambda_{B}$ with
positive charge as shown in Fig. 3(a). In contrast, the positive charge
$Q_{n}^{I}+Q_{n}^{III}$ of the neutron is located inside the core of the
hypersphere of radius $\frac{\sqrt{3}}{2}\lambda_{B}$ corresponding to the
boundaries at $\mu=\frac{\pi}{3}$ and $\mu=\frac{2\pi}{3}$, while the neutron
negative charge $Q_{n}^{II}$ is distributed outside the core of the
hypersphere, as shown in Fig. 3(b). Note that the proton and neutron charges
do not leak out from the soliton, and the radial distances $\lambda_{B}$ and
$\frac{\sqrt{3}}{2}\lambda_{B}$ in Fig. 3(b) are geometrical invariants.

Next exploiting the geometrical descriptions on the hypersphere soliton, we
evaluate the three dimensional volume associated with the neutron charge
distributions possessing the three pieces in (3.7) which are spherically
symmetric. To be specific, making use of (2.6), we arrive at the three
dimensional volume for the fractional neutron distributions

$\displaystyle V_{n}^{I}(S^{3};\lambda=\lambda_{B})$ $\displaystyle=$
$\displaystyle
V_{n}^{III}(S^{3};\lambda=\lambda_{B})=2\pi\lambda_{B}^{3}\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right),$
$\displaystyle V_{n}^{II}(S^{3};\lambda=\lambda_{B})$ $\displaystyle=$
$\displaystyle
2\pi\lambda_{B}^{3}\left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right),$ (3.8)

consistent with the total three dimensional volume
$V(S^{3};\lambda=\lambda_{B})=2\pi^{2}\lambda_{B}^{3}$ in (3.3), which is
applicable to the neutron and the proton. Following the projection algorithm
discussed in the previous section, we notice that
$V_{n}^{I}(S^{3};\lambda=\lambda_{B})+V_{n}^{III}(S^{3};\lambda=\lambda_{B})$
is located in the inner portion of the neutron, while
$V_{n}^{II}(S^{3};\lambda=\lambda_{B})$ is situated in the outer portion of
the neutron. Similar arguments can be applied to the neutron charge fractions
in (3.7) as shown in Fig. 3(a) and Fig. 3(b).

Now it seems appropriate to comment on the numerical analysis of the neutron
in the HSM. The predicted three dimensional volumes for the fractional neutron
distributions in (3.8) and the corresponding total volume in (3.3) are listed
in Table 1. Note that in the neutron hypersphere soliton the mass and charge
fill the enlarged $S^{3}$ volumes comparing to the corresponding ordinary
$R^{3}$ ones, to yield

$\frac{V_{n}(S^{3};\lambda=\lambda_{B})}{V_{n}(R^{3};\lambda=\lambda_{B})}=\left\\{\begin{array}[]{ll}4\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right)=2.457,~{}~{}~{}~{}{\rm
for~{}volume~{}inside~{}the~{}core,}\\\
6\left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right)=11.479,~{}~{}~{}{\rm
for~{}volume~{}outside~{}the~{}core,}\end{array}\right.$ (3.9)

implying that, in the neutron in the HSM, the volume ratio outside the core is
much more enhanced than that inside the core. Note also that the volume inside
the core in $S^{3}$ hypersphere originates from two regions
$V_{n}^{I}(S^{3};\lambda=\lambda_{B})$ and
$V_{n}^{III}(S^{3};\lambda=\lambda_{B})$ shown in Figure 3 depicted in terms
of the angle $\mu$, while that inside the core in $R^{3}$ comes from one
region $V_{n}^{inside}(R^{3};\lambda=\lambda_{B})$ associated with the
ordinary solid sphere of radius $\frac{\sqrt{3}}{2}\lambda_{B}$. Moreover, the
ratio of total volumes listed in Table 1 reproduce the characteristic ratio
$\chi=\frac{3\pi}{2}=4.712$ in (2.9).

Table 1: The numerical predictions of the three dimensional volumes for the fractional neutron distributions. Here the volume inside (outside) the neutron core is positively (negatively) charged as in Fig. 3(b). Volume type |  $S^{3}$ |  $R^{3}$
---|---|---
Volume inside the core |  $5.188~{}{\rm fm}^{3}$ |  $2.112~{}{\rm fm}^{3}$
Volume outside the core |  $8.081~{}{\rm fm}^{3}$ |  $0.704~{}{\rm fm}^{3}$
Total volume |  $13.269~{}{\rm fm}^{3}$ |  $2.816~{}{\rm fm}^{3}$

##  IV Conclusions

In summary, in the HSM we have investigated the foliations associated with the
geometry on the three dimensional hypersphere. To do this, we first have
studied a simple toy model of the spherical shell $S^{2}$ of radius $\rho$
which is described in terms of $(\theta,\phi)$. On the curved northern
hemisphere $S^{2}_{+}$, we have introduced foliation leaves of circle of
radius $\rho\sin\theta$ $(0\leq\theta\leq\frac{\pi}{2})$. Note that these
foliation leaves fill $S^{2}_{+}$ after scanning completely over the range
$0\leq\theta\leq\frac{\pi}{2}$, while the southern hemisphere $S^{2}_{-}$ is
filled with the foliation leaves of circle of radius $\rho\sin\theta$
$(\frac{\pi}{2}\leq\theta\leq\pi)$. Next, we have introduced a projection from
$S^{2}_{+}$ to the flat two dimensional equatorial plane $E^{2}$ of $S^{2}$.
Making use of the foliation leaves of circle of radius $\rho\sin\theta$ to
scan $S^{2}_{+}$ over the range $0\leq\theta\leq\frac{\pi}{2}$, we can obtain
the surface area $A_{2}(S^{2}_{+})=2\pi\rho^{2}$ which is greater than
$A_{2}(E^{2})=\pi\rho^{2}$. Applying the same mathematical logic for the
$S^{2}$ toy model to the three dimensional hypersphere $S^{3}$ of radius
$\lambda$ which is delineated by $(\mu,\theta,\phi)$, we have introduced the
projection from the northern hemi-hypersphere $S^{3}_{+}$ to the three
dimensional equatorial volume $E^{3}$ which possesses the center of $S^{3}$
and is described in terms of the spherical coordinates $(r,\theta,\phi)$.
Using the projection, we have studied similarities between spherical shell
foliation leave on the curved manifold $S^{3}_{+}$ and that on $E^{3}$. Here
the spherical shell foliation leave on the curved manifold $S^{3}_{+}$ and the
spherical shell foliation leave on $E^{3}$ share their corresponding centers,
differently from the $S^{2}$ toy model case. Making use of the foliation
leaves of spherical shell of radius $\lambda\sin\mu$ to scan the curved
manifold $S^{3}_{+}$ over the range $0\leq\mu\leq\frac{\pi}{2}$, we have found
the volume $V(S^{3}_{+})=\pi^{2}\lambda^{3}$. Similarly, we have exploited the
southern hemi-hypersphere $S^{3}_{-}$ to yield the volume
$V(S_{-}^{3})=\pi^{2}\lambda^{3}$ to obtain
$V(S^{3})=V(S^{3}_{+})+V(S^{3}_{-})=2\pi^{2}\lambda^{3}$ which is consistent
with the result obtained by using the pure geometrical approach.

Next, we have constructed the charge density profiles of nucleons in terms of
the angle $\mu$ in $S^{3}$. Exploiting the geometry on the three dimensional
volume of the hypersphere soliton, we have formulated the three dimensional
volume related with the spherically symmetric neutron distributions. We next
have found that the proton and neutron charge densities are spherically
symmetric with the non-trivial distribution structures without leaking out
from the soliton. We also have shown that in the neutron charge distribution,
the positive (negative) charge is confined inside (outside) the core of the
neutron. Next, we have constructed the three dimensional fractional volumes
corresponding to the fractional neutron charge distributions, to explicitly
yield the corresponding numerical predictions. Moreover, we have obtained the
charge fractions in the neutron charge distributions. We thus have found that
the characteristic ratio of the hypersphere volume
$V(S^{3};\lambda=\lambda_{B})$ to the corresponding solid sphere one
$V(R^{3};\lambda=\lambda_{B})$ is given by the geometrical invariant. Next we
have clarified the physical meaning of $\mu$ in $S^{3}$ of radius $\lambda$.
Note that, even though the HSM possesses complicated geometrical structures as
shown in this paper, the model shows the successful numerical predictions of
the static properties of baryons hong98plb ; hong21 .

###### Acknowledgements.

The author was supported by Basic Science Research Program through the
National Research Foundation of Korea funded by the Ministry of Education,
NRF-2019R1I1A1A01058449.

## References

  * (1) P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University Press, New York, 1964).
  * (2) S.T. Hong, BRST Symmetry and de Rham Cohomology (Springer, Heidelberg, 2015), and references therein.
  * (3) G.S. Adkins, C.R. Nappi, E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228, 552 (1983).
  * (4) T.H.R. Skyrme, A nonlinear field theory, Proc. R. Soc. Lond. A 260, 127 (1961).
  * (5) S.T. Hong, The static properties of Skyrmions on a hypersphere, Phys. Lett. B 417, 211 (1998) [Corrigendum-ibid. 803, 135179 (2020)].
  * (6) N.S. Manton, P.J. Ruback, Skyrmions in flat space and curved space, Phys. Lett. B 181, 137 (1986).
  * (7) S.T. Hong, Dirac quantization and baryon intrinsic frequencies in hypersphere soliton model, Nucl. Phys. B 973, 115611 (2021), arXiv:2105.11456.
  * (8) P.A.M. Dirac, An extensible model of the electron, Proc. Roy. Soc. Lond. A 268, 57 (1962).
  * (9) S.T. Hong, Photon intrinsic frequency and size in stringy photon model, Nucl. Phys. B 976, 115720 (2022), arXiv:2111.11852.
  * (10) A. Candel, L. Conlon, Foliations I (American Mathematical Society, Providence, 2000).
  * (11) G.S. Adkins, Chiral Solitons (World Scientific, Singapore, 1987), Editor: K.F. Liu.


RHDE models in FRW Universe with two IR cut-offs with redshift parametrization
Archana Dixit1, Vinod Kumar Bhardwaj 2 Anirudh Pradhan3

1,2,3Department of Mathematics, Institute of Applied Sciences and Humanities,
GLA University, Mathura,

Uttar Pradesh-281 406, India

1E-mail<EMAIL_ADDRESS>
2E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
3E-mail<EMAIL_ADDRESS>
###### Abstract

In this manuscript, we have researched the cosmic expansion phenomenon in flat
FRW Universe through the interaction of the recently proposed R$\grave{e}$nyi
holographic dark energy (RHDE). For this reason, we assumed Hubble (H) and
Granda–Oliveros (GO) horizons as IR cut-off in the framework of $f(R,T)$
gravity. With this choice for IR cut-off, we can obtain some important
cosmological quantities such as the equation of state $\omega_{{}_{T}}$,
energy density $\rho_{{}_{T}}$, density parameter $\Omega_{{}_{T}}$, and
pressure $p_{{}_{T}}$, which are the function of the redshift $z$. It is
observed that in both IR cut-offs the EoS parameter displays quintom-like
behaviour for three different values of $\delta$. Here we plot these
parameters versus redshift $z$ and discuss the consistency of the recent
findings. Next, we explore the
$\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$ plane and the stability
analysis of the dark energy model by a perturbation method. Our findings
demonstrate that the Universe is an accelerating model of rapid growth that is
explained by quintom like behaviour. Hence the feasibility of the RHDE model
with Hubble and GO cut-off is supported by our model. The results indicate
that the IR cut-offs play a significant role in the understanding of the
dynamics of the universe.

Keywords :FRW universe, RHDE, Hubble horizon, GO- horizon.
PACS: 98.80.-k, 98.80.Jk, 04.20.Jb

##  1 Introduction

Observational information received by (SNIa) [1]-[2], large scale structures
(LSS) [3]-[5] and cosmic microwave background (CMB), anisotropies [6]-[7]
confirmed the present accelerated expansion of the universe. The dark energy
is assumed as a responsible candidate for this present scenario of accelerated
Universe [8]-[10]. The character of DE is unknown, and mysterious. The easiest
choice for DE is cosmological constant with positive energy density and
negative pressure.

The cosmological constant faces several challenges, such as the issue of fine
tuning, and the problem of coincidence [8]. One reasonable way of relieving
the question of cosmic coincidence is to assume that dark matter (DM) and dark
energy (DE) interact. For the complex DE scenario, there are different
alternate theories suggested by observing the accelerating universe: (a) the
scalar-field models of DE including quintessence [11], tachyon [12], k-essence
[13], (b) the interacting DE models including chaplygin gas [14], polytropic
gas [15]-[16], phantom [17] and holographic dark energy (HDE) [18]-[19]. On
the other hand, the nature of DE can be investigated on the basis of certain
principles of quantum gravity, which would be the yield to the (HDE)
model[20]-[24].

Another approach to solving the dark energy problem is to connected the
certain aspects of the theory of quantum gravity, known as the holographic
principle [25]-[28]. Several authors are focusing on the numerous cosmological
implications of new and modified HDE models [29]-[33]. Another significant
factor of this model is the IR cut-off. A lot of HDE models are discussed in
the literature, depending on the IR cut-off.

Many authors are working on the different IR cut-offs like: Sharma and Dubey
[34] worked on the interacting R$\grave{e}$nyi holographic dark energy with
parametrization on the interaction term and studied the graphical behaviour of
cosmological parameters. Vipin et al. [35] have also observed the growth rate
of perturbations by using hierarchy. The anisotropic and spatially homogeneous
Bianchi type-$VI_{0}$ (RHDE) models of general relativity discussed in [36].
In this model authors assumed both (H) and (GO) horizons as IR cut-off and
obtained cosmological parameters. In the same context, Tayeb et al. [37]
analysed the anisotropic R$\grave{e}$nyi holographic dark energy models in
flat space-time. Qolibikloo and Ghodsib [38] investigate the Rényi entropy and
inequalities under the phase transition. The expansion and growth data were
combined by Akhlaghi [39] to investigate the ability of the three most popular
HDE models, namely the Ricci scale, future event horizon, and Granda–Oliveros
IR cut-offs.

In this sequence, Ghaffari [40] examined the cosmological models of HDE in a
DGP braneworld with GO cut-off. In recent year many entropy formalisms has
been used and explore the cosmological models. Some new HDE models are being
developed, such as the RHDE model, [41], Sharma–Mittal HDE (SMHDE) [42] and
Tsallis HDE (THDE). Among these models RHDE is more stable model which is
based on non-interactions between cosmic region [43]-[44]. Younas et al. [45]
have investigated entropies of R$\grave{e}$nyi, Tsallis and Sharma–Mittal in
flat FRW Universe within Chern–Simons modified gravity. HDE model was
conjectured as IR cut-off with Benkenstein entropy and Hubble horizon that
does not provide an appropriate explanation for the level of a flat FRW
universe [46]-[48]. Recently, a new HDE model was proposed and explored by
changing the standard HDE as $S_{\delta}=\gamma A^{\delta}_{r}$, where
$\gamma$ is an unspecified constant $A_{r}=4\pi L^{2}$ represents the area of
the horizon and $\delta$ is a non-additivity parameter, called Tsallis
holographic dark energy (THDE) [44, 49]. Another possibility for DE showed up
when Cohen et al. [50] applied some speculation on the mutual connection
between UV $(\Lambda)$ and IR (L) cut-offs and the entropy of framework,
expressing as $\rho_{{}_{\Lambda}}\varpropto{\frac{S}{L^{4}}}$. where
$\rho_{{}_{\Lambda}}$ is the vacuum energy density [50]. Bekenstein and
Hawking [51, 52] studied the thermodynamics of the black hole. It is indicated
that the Bekenstein–Hawking entropy bound $S_{{}_{B}H}\sim M_{p}^{2}L^{2}$,
scales as the region $A_{r}\backsim L^{2}$ rather than the volume $V\backsim
L^{3}$ and $M_{p}$ is the reduced Plank mass ($8\pi G=1/M_{p}^{2}=1$). On the
other hand, the Bekenstein entropy bound $S_{{}_{B}}$ is $EL$ for the case
where $E=\rho_{{}_{\Lambda}}L^{3}$ is the energy, and $L$ is the IR cut-off.
For $\rho_{{}_{\Lambda}}\leq M_{p}^{2}L^{-2}$ by using $S_{B}<S_{{}_{B}H}$.
Here $C^{2}_{1}$ is a numerical constant, given the HDE:

$\rho_{d}=3C^{2}_{1}M_{p}^{2}L^{-2}.$ (1)

Observational information, which is acquired by restricting the HDE model,
clearly demonstrate $C_{1}=0.818+_{-0.097}^{0.113}$ [53] and
$C_{1}=0.815+_{-0.039}^{0.179}$ [54] for flat and non-flat space time
separately. Several research on the HDE model and its features are reported in
[55]-[58]. The two latest entropies R$\grave{e}$nyi and Tsallis [59]-[62] are
commonly used to study various gravitational and cosmological phenomena
[63]-[71]. HDE density derivation is based on the relationship of entropy-area
$S_{BH}=A_{r}/4$, where $A_{r}$ is the area of black hole horizon.
Subsequently, by changing the entropy relation, one can locate new type of
HDE. It is surprising that the form of HDE and gravity model equations can be
generalized by using a generalized method of entropy-area relationship. So, we
have a generalized Friedmann equation to define evolution of the universe.
This idea motivates us to look through R$\grave{e}$nyi entropy into the
constructive accelerating phases of the Universe. The entropy in RHDE was
discussed in Refs.[65]-[67].

$S_{{}_{R}}=\frac{1}{\delta}ln(1+\delta S_{{}_{T}}).$ (2)

Here $\delta$ is a constant and $S_{{}_{T}}$ is Tsallis entropy. Bekenstein
and Tsallis is equal [65, 66, 67, 71] then Eq. (2) becomes:

$S_{{}_{R}}=\frac{1}{\delta}ln(1+\delta\frac{A_{r}}{4}),$ (3)

if $\delta$ tends to zero, the R$\grave{e}$nyi entropy reduces to $A_{r}/4$.
During the current work, we are considering HDE by applying the
R$\grave{e}$nyi entropy. In this direction a brief survey of the theories of
modified gravity as the new participation explored a cosmological
reconstruction. Harko et al. [72] developed new generalized theory known as
$F(R,T)$ gravity.

Several cosmologist [73]-[75] have studied $f(R,T)$ gravity in distinct
context. The main aim of our proposed model is to consider $f(R,T)$ model with
R$\grave{e}$nyi HDE by assuming two IR cut-offs. The work in this manuscript
is configured as: The field equations of $f(R,T)$ gravity is set out in
Sect.$2$. R$\grave{e}$nyi HDE models are considered in Sect.$3$.
R$\grave{e}$nyi HDE model with Hubble horizon cut-off are discussed in
Sect.$3.1$. R$\grave{e}$nyi HDE model with GO horizon cut-off is examined in
Sect.$3.2$. $\omega_{{}_{T}}-\omega_{{}_{T}}^{{}^{\prime}}$ plane is discussed
in Sect.$4$. The stability of the model is discussed in Sect.$5$ and the
outcomes are summarized with conclusions that are discussed in Sect.$6$.

##  2 Basic field equations of f(R,T) gravity

We assume that the behaviour for the modified gravity theories takes the form:

${\bf\mathbb{S}}=\frac{1}{16\pi}\int f(R,T)\sqrt{-g}d^{4}x+\int
L_{m}\sqrt{-g}d^{4}x.$ (4)

Here $f(R,T)$ is an arbitrary function of the trace ($T$) and Ricci scalar
($R$). Here $L_{m}$ is the matter Lagrangian density.

The stress-energy tensor of matter is defined as [76]

$T_{ij}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta g^{ij}}.$ (5)

The field equations of the $F(R,T)$ model are obtained as

$2F_{{}_{R}}(R,T)R_{ij}-F(R,T)g_{ij}+2(g_{ij}\square-\nabla_{i}\nabla_{j})f_{{}_{R}}(R,T)=16\pi
T_{ij}-2F_{{}_{T}}(R,T)T_{ij}-2F_{{}_{T}}(R,T)\ominus_{ij}.$ (6)

However, the stress-energy tensor for the current work is considered as

$T_{ij}=-pg_{ij}+(p+\rho)u_{i}u_{j}.$ (7)

Also, the matter Lagrangian can be assumed as $L_{m}=-p$. The conditions
$u_{i}u^{i}=1$ and $u^{i}\nabla_{j}u_{i}=0$ satisfy the four-velocity. In our
model, we assumed the particular case of $f(R,T)$ expressing by the function
$f(R,T)=2f(T)+R$, where f(T) is an arbitrary function of the stress-energy
tensor of matter. The gravitational field equations follows by Eq. (6) written
as

$2R_{ij}-Rg_{ij}=16\pi T_{ij}+4\dot{f}(T)T_{ij}+2[2p\dot{f}(T)+f(T)]g_{ij},$
(8)

where the dot indicates the derivative is related to the argument. For the
dust filled universe ($p=0$), the gravitational field equations are given by

$2R_{ij}-Rg_{ij}=16\pi T_{ij}+4\dot{f}(T)T_{ij}+2f(T)g_{ij}.$ (9)

These field equations are suggested in [77]. We can obtain a cosmological
model by considering the dust Universe, by select the function $f(T)=\xi T$,
$\xi\rightarrow$ constant.

The metric of a flat FRW Universe is considered as:

$ds^{2}=dt^{2}-a^{2}\left(dx^{2}+dy^{2}+dz^{2}\right).$ (10)

The gravitational field equations are written as,

$3\left(\frac{\dot{a}^{2}}{a^{2}}\right)=(8\pi+3\xi)\rho_{T},$ (11)
$2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}=\xi\rho_{T}.$ (12)

This $f(R,T)$ gravitational model is equal to efficient cosmological constant
$\Lambda_{eff}\varpropto H^{2}$, where $H=\frac{\dot{a}}{a}$ is the Hubble
function [77]. Field equations are reduced in terms of $H$,

$2\dot{H}+3\frac{8\pi+2\xi}{8\pi+3\xi}H^{2}=0.$ (13)

The general solution of the above Eq. (13) obtained as

$H(t)=\frac{2}{3t}\frac{(8\pi+3\xi)}{(8\pi+2\xi)}=\frac{2\beta}{3t},$ (14)

where ${\bf a(t)=t^{\frac{2}{3}\beta}}$ is the scale factor where
$\beta=\frac{(8\pi+3\xi)}{(8\pi+2\xi)}$.

Here we are using different expression for $H$, like future event Hubble
horizon as the infrared cut-off, Ref.[78] and GO infrared cut-off. By the
conservation equation we obtained as

$\frac{\partial\rho_{{}_{T}}}{\partial t}+3H(\rho_{{}_{T}}+p_{{}_{T}})=0.$
(15)

Here $\rho_{{}_{T}}$ is the R$\grave{e}$nyi holographic energy density,
solving Eq.(15) the EoS parameter $\omega_{{}_{T}}$ can be rewrite as:

$\omega_{T}=-1-\frac{1}{3H}\left(\frac{\dot{\rho}_{{}_{T}}}{\rho_{{}_{T}}}\right).$
(16)

##  3  R$\grave{e}$nyi HDE model

We have taken a system with $m$, states with $P_{i}$ probability distribution
and satisfies the condition $\sum_{i=1}^{m}P_{i}=1$. R$\grave{e}$nyi and
Tsallis entropy are well known parameter of generalized entropy [49].

$S_{{}_{R}}=\frac{1}{\delta}ln\sum_{i}^{m}P_{i}^{1-\delta}~{}~{}~{}~{}~{}S_{{}_{T}}=\frac{1}{\delta}ln\sum_{i=1}^{m}P_{i}^{1-\delta}-P_{i},$
(17)

where $\delta=1-U$ and $U$ is a real parameter. Now using the above equations
we get the relation,

$S_{{}_{R}}=\frac{1}{\delta}ln(1+\delta S_{T}).$ (18)

In Eq. (18) the Bekenstein entropy is $S_{T}=\frac{A_{r}}{4}$, and $A_{r}=4\pi
L^{2}$. This gives the R$\grave{e}$nyi entropy of the system as
$S_{R}=\frac{1}{\delta}ln(1+\delta\pi L^{2})$ [60].

In this section we take the assumption $\rho_{{}_{T}}dV\varpropto Tds$ [69]
then we can get the R$\grave{e}$nyi HDE density in IR cut-off written as

$\rho_{{}_{T}}=\frac{3C^{2}_{1}}{8\pi L^{2}}\frac{1}{\left(1+\pi\delta
L^{2}\right)}.$ (19)

In this case, $C^{2}_{1}$ is the numerical constant, $V$ is the volume, and
$T$ is the temperature of the system. We have used $T=\frac{H}{2\pi}$ and
$A=\frac{4\pi}{H^{2}}=4\pi(\frac{3V}{4\pi})^{2/3}$, relationships valid for
the flat FRW Universe. It is clear that we have
$\rho_{{}_{T}}=\frac{3C^{2}_{1}H^{2}}{8\pi}$ is a complete agreement with OHDE
[21, 46, 47] in the absence of $\delta$. In this section, we use the
R$\grave{e}$nyi entropy by considering the Bekenstein entropy as the Tsallis
entropy [63]-[69].

Finally, we are proposing a new holographic dark energy model, which is RHDE.
Next we have examined the well-known cosmological parameters of $H$ and GO IR
cut-off in Sect.3.1 $\&$ Sect.3.2.

###  3.1 Model-I: R$\grave{e}$nyi HDE model with hubble horizon cut-off with
redshift parameterization

Hubble horizon is the simplest choice with the Hubble length $L=H^{-1}$ [48].
According to the holographic principle the energy density of DE is
proportional to the square of the Hubble parameter i.e., $\rho_{T}\propto
H^{2}$ [39]. The principle states that the given choice can solve the fine
tuning problem.

Here the R$\grave{e}$nyi energy density is obtained as

$\rho_{{}_{T}}=\frac{3C^{2}_{1}H^{2}}{8\pi}\left(1+\frac{\pi\delta}{H^{2}}\right)^{-1}$
(20)

Utilizing Hubble horizon as a possibility for IR cut-off i.e., $L=\frac{1}{H}$
and $8\pi=1$. In this analysis we use the $a=\frac{a_{0}}{(1+z)}$ redshift
parameterization,

$\rho_{{}_{T}}=\frac{2\beta^{4}C^{2}_{1}}{12\pi\beta^{2}\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+27\pi^{2}\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}}.$
(21)

By using Eqs.(16) and (21), we get EoS parameter defined as,

$\omega_{{}_{T}}=\frac{-9\pi\beta\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+18\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}-4\beta^{3}+4\beta^{2}}{9\pi\beta\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+4\beta^{3}}.$
(22)

By using Eqs.(21) and (22) we get pressure as,

$p_{{}_{T}}=-\frac{2\beta^{3}c^{2}\left(9\pi\beta\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}-18\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+4\beta^{3}-4\beta^{2}\right)}{3\pi\left(4\beta^{2}\left(\frac{a_{0}}{z+1}\right){}^{\frac{3}{2\beta}}+9\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{\frac{9}{2\beta}}\right){}^{2}}.$
(23)

Using Eqs.(14) and (21), we obtain density parameter as

$\Omega_{{}_{T}}=\frac{\beta^{2}C^{2}_{1}}{18\pi^{2}\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+8\pi\beta^{2}}.$
(24)

The behaviour of cosmological parameters such as energy density, Eos
parameter, pressure and density parameter are shown in this section, where the
density parameter $\Omega_{T}$ is defined as the ratio between the actual (or
observed) density $\rho_{T}$ and the critical density $\rho_{c}$ of the
Friedmann Universe. Here $\rho{c}=\frac{3H^{2}}{8\pi}$, the relation between
the actual density and the critical density determines the overall geometry of
the Universe; when they are equal, the geometry of the Universe is flat
(Euclidean) of the total RHDE in IR cut-off.

###  3.2 Model-II: R$\grave{e}$nyi HDE model with Granda–Oliveros (GO) horizon
cut-off with redshift parameterization

Granda and Oliveros first [79, 80] introduced the cut-off
$L=(kH^{2}+m\dot{H})^{-1/2}$ and attempt to resolve the well known
cosmological problems like cosmic coincidence and stability. In this sequence,
the acceleration combined the Hubble parameter together with its time
derivative. On substituting GO cut-off in Eq. (14) and taking $(8\pi=1)$, we
get
$L=\frac{1}{\frac{4}{9}\beta^{2}k\left(\frac{a_{0}}{z+1}\right){}^{-3/\beta}-\frac{2}{3}\beta
m\left(\frac{a_{0}}{z+1}\right){}^{-3/\beta}}.$

By using Eq. (20), we get energy density as,

$\rho_{{}_{T}}=\frac{2\beta^{4}c^{2}\left(\frac{a_{0}}{z+1}\right){}^{-6/\beta}(3m-2\beta
k)^{4}}{27\pi\left(81\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}+16\beta^{4}k^{2}-48\beta^{3}km+36\beta^{2}m^{2}\right)}.$
(25)

By using Eqs. (16) and (25), we get EoS parameter defined as

$\omega_{{}_{T}}=\frac{-81\pi(\beta-4)\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}-16(\beta-2)\beta^{4}k^{2}+48(\beta-2)\beta^{3}km-36(\beta-2)\beta^{2}m^{2}}{81\pi\beta\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}+16\beta^{5}k^{2}-48\beta^{4}km+36\beta^{3}m^{2}}.$
(26)

By using Eqs.(25) and (26) we obtained pressure as

$p_{{}_{T}}=\frac{2\beta^{3}c^{2}\left(\frac{a_{0}}{z+1}\right){}^{-6/\beta}(3m-2\beta
k)^{4}\left(-81\pi(\beta-4)\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}-4(\beta-2)\beta^{2}(3m-2\beta
k)^{2}\right)}{27\pi\left(81\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}+4\beta^{2}(3m-2\beta
k)^{2}\right){}^{2}}.$ (27)

Using Eqs.(14) and (25) we get density parameter as

$\Omega_{{}_{T}}=\frac{\beta^{2}c^{2}\left(\frac{a_{0}}{z+1}\right){}^{-3/\beta}(3m-2\beta
k)^{4}}{18\pi\left(81\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}+16\beta^{4}k^{2}-48\beta^{3}km+36\beta^{2}m^{2}\right)}.$
(28)

Here we also calculate the well known cosmological parameters like energy
density, Eos Parameter, pressure and density parameter $\Omega_{{}_{T}}$ of
the Friedmann Universe. We noticed that the geometry of the universe is flat
(Euclidean) of the total RHDE in GO cut-off.

(a) (b)

Figure 1: a. Plot of $\rho_{{}_{T}}$ versus $z$ with (H hoz.) cut-off b. Plot
of $\rho_{{}_{T}}$ versus $z$ with (GO hoz.) cut-off

Figure $1a,b$ show the energy density of RHDE with Hubble horizon and GO
horizon cut-off versus redshift $z$ respectively. All the trajectories of
$\rho_{{}_{T}}$ indicate the positive behaviour throughout the evolution of
the universe for various estimations of $\delta$. Likewise it can be seen that
$\rho_{{}_{T}}$ is a positive decrease function and decreases, more sharply as
$\delta$ increases. We observed RHDE is the decreasing function of redshift
$z$ in both IR cut-offs and we also found that in high redshift
$\rho_{{}_{T}}$ tends to zero.

(a) (b)

Figure 2: a. Plot of pressure $p_{{}_{T}}$ versus $z$ (H hoz.) cut-off b. Plot
of pressure $p_{{}_{T}}$ versus $z$ (GO hoz.) cut-off

Figure $2a,b$ demonstrates the behaviour of pressure of RHDE with Hubble
horizon and GO horizon cut-off versus redshift $z$ respectively. Recent
cosmological perceptions firmly show that our universe is dominated by the
component with negative pressure called dark energy.

It is ascertained that the isotropic pressure is negative throughout the
evolution. From the figures we noticed that the pressure is decreasing
function of the redshift in both IR cut-offs.

(a) (b)

Figure 3: a. Plot of EoS parameter $\omega_{{}_{T}}$ versus $z$ (H hoz.) cut-off b. Plot of EoS parameter $\omega_{{}_{T}}$ versus $z$ (GO hoz.) cut-off Table 1: The behaviour of EoS parameter $\omega_{{}_{T}}$ with redshift parameterization for different values of $\beta$ in Hubble Horizon cut-off | $\delta=10$ |  | $\delta=100$ |  | $\delta=500$
---|---|---|---|---|---
$\beta<0$ | Phantom | $\beta<0$ | Phantom | $\beta<0$ | Phantom
$0<\beta\leq 1.88$ | Quintessence | $0<\beta\leq 1.98$ | Quintessence | $0<\beta\leq 1.99$ | Quintessence
$1.88<\beta\leq 2$ | Quintessence$\to$ Phantom | $1.98<\beta\leq 2$ | Quintessence$\to$ Phantom | $1.99<\beta\leq 10$ | Quintessence$\to$ Phantom
$2<\beta\leq 10$ | Phantom | $2<\beta\leq 10$ | Phantom | $2<\beta\leq 10$ | Phantom
Table 2: The behaviour of EoS parameter $\omega_{{}_{T}}$ with redshift parameterization for different values of $\beta$ in Granda–Oliver (GO) cut-off | $\delta=10$ |  | $\delta=100$ |  | $\delta=500$
---|---|---|---|---|---
$\beta<0$ | Phantom | $\beta<0$ | Phantom | $\beta<0$ | Phantom
$0<\beta\leq 3$ | Quintessence | $0<\beta\leq 3.7$ | Quintessence | $0<\beta\leq 3.9$ | Quintessence
$3<\beta\leq 3.99$ | quintessence$\to$ Phantom | $3.7<\beta\leq 3.99$ | Quintessence$\to$ Phantom | $3.90<\beta\leq 4$ | Quintessence$\to$ Phantom
$4<\beta\leq 10$ | Phantom | $3.99<\beta\leq 10$ | Phantom | $4<\beta\leq 10$ | Phantom

Figure $3a$ implies that the EoS parameter $\omega_{{}_{T}}$ is broadly used
to categorize the different phases of the expanding universe. The EoS
parameter of RHDE with $H$ hoz. cut-off is given in Eq. (22). The mathematical
analysis of Eos parameter regarding quintessence / phantom for different
values of $\beta$ in [–10, 10] has been presented in Table $1$ and $2$
respectively. Here we notice that the EoS parameter initially lies in low
phantom region or phantom era $\omega_{{}_{T}}<-1$ at high redshift $z$ and
crossing the phantom divide line(PDL) $\omega_{{}_{T}}=-1$. It enters in
quintessence region and again lies in phantom region for different values of
$\delta$ are shown in Table $1$ and $2$. Similarly in Figure $3b$ additionally
demonstrations the analysis of the EoS parameter for the (RHDE-GO cut-off)
which is shown in Eq. (26). In this figure we also observed that EoS parameter
$\omega_{{}_{T}}$ moves towards the high phantom region to quintessence region
and after that again lies in phantom region for different values of $\delta$.
The trajectories of EoS parameter show the transition from phantom region to
quintessence region by evolving the vacuum era of the universe in both IR cut-
offs. This is called quintom-like nature of the universe.

(a) (b)

Figure 4: a. Plot of density parameter $\Omega_{{}_{T}}$ versus $z$ (H hoz.)
cut-off b. Plot of density parameter $\Omega_{{}_{T}}$ versus $z$ (GO hoz.)
cut-off.

Fig. $4$ depicts the behaviour of density parameter $\Omega_{{}_{T}}$ versus
redshift $z$ for various estimations of $\delta$. In this figure the density
parameter of the RHDE is seen to be lower region for the high redshift. We
also noticed that the density parameter is positive throughout the evolution
in both IR cut-offs.

##  4 Dynamics of $\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$ plane

Caldwell and Linder [81, 82], recommend the
$\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$ plane to describe the
dynamical property of DE model. Here $\omega_{{}_{T}}$ is the EoS parameter
and $\omega_{{}_{T}}^{{}^{\prime}}$ is its evolutionary structure. Where dot
means the derivative concerning $lna$. Here the
($\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$) plane was segregated in
two regions, the thawing region $\omega_{{}_{T}}<0,$
$\omega_{{}_{T}}^{{}^{\prime}}>0$ is the region where the EoS parameter
evolves $\omega_{{}_{T}}<-1$ increases with time and its evolution parameter
shows positive behaviour, and the freezing region $\omega_{{}_{T}}<0,$
$\omega_{{}_{T}}^{{}^{\prime}}<0$ in this region evolution parameter is
negative.

For R$\grave{e}$nyi HDE with (H hoz.) IR-cut-off with redshift

We can obtain the $\omega_{{}_{T}}^{{}^{\prime}}$ by differentiating the EoS
parameter which is in Eqs. (22) and (26) with respect to $lna$ then we get

$\omega^{\prime}_{{}_{T}}=\frac{d\omega_{{}_{T}}}{d\ln
a}=\frac{108\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}}{\left(9\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{3/\beta}+4\beta^{2}\right){}^{2}}.$
(29)

For R$\grave{e}$nyi HDE with (GO hoz) IR cut-off with redshift

$\omega^{\prime}_{{}_{T}}=\frac{d\omega_{{}_{T}}}{d\ln
a}=\frac{3888\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}(3m-2\beta
k)^{2}}{\left(81\pi\delta\left(\frac{a_{0}}{z+1}\right){}^{6/\beta}+16\beta^{4}k^{2}-48\beta^{3}km+36\beta^{2}m^{2}\right){}^{2}}.$
(30)

(a) (b)

Figure 5: a. Plot of $\omega_{{}_{T}}-\omega_{{}_{T}}^{{}^{\prime}}$ versus
$z$ (H hoz) cut-off b. Plot of $\omega_{{}_{T}}-\omega_{{}_{T}}^{{}^{\prime}}$
versus $z$ (GO hoz.) cut-off

From figure $5a,b$ we discuss the behaviour of
($\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$) plane corresponding to
Eqs. (29) and (30) for three different values of $\delta$. These figures shows
that ($\omega_{{}_{T}}$-$\omega_{{}_{T}}^{{}^{\prime}}$) trajectories
represents thawing region and are overlapped together during the whole
evaluation.

##  5 Stability of the model

One may check the stability of the derived solution with respect to the
perturbation of the space-time [83]-[84]. To this purpose, we considered the
perturbations of volume scalar, directional Hubble factors and mean Hubble
factor as

$V\rightarrow V_{B}+V_{B}\sum_{i=1}^{3}\delta b_{i},~{}~{}~{}H_{i}\rightarrow
H_{B_{i}}+\delta\dot{b}_{i},~{}~{}~{}H\rightarrow
H_{B}+\frac{1}{3}\sum_{i=1}^{3}\delta\dot{b}_{i},~{}~{}~{}\sum_{i=1}^{3}H_{i}^{2}\rightarrow\sum_{i=1}^{3}H_{{B}_{i}}^{2}+2\sum_{i=1}^{3}H_{{B}_{i}}.\delta
b_{i}.$ (31)

For metric perturbation $\delta b_{i}$ to be linear the following equations
must be satisfied

$\sum_{i}{\ddot{\delta b_{i}}}+2\sum_{i}\theta_{B_{i}}\dot{\delta b_{i}}=0,$
(32) $\ddot{\delta b_{i}}+\frac{\dot{V_{B}}}{V_{B}}\dot{\delta
b_{i}}+\sum_{j}\dot{\delta b_{j}}\theta_{B_{i}}=0,$ (33) $\sum\dot{\delta
b_{i}}=0.$ (34)

From the simplification of above equations

$\ddot{\delta b_{i}}+\frac{\dot{V_{B}}}{V_{B}}\dot{\delta b_{i}}=0,$ (35)

where the background volume scalar $V_{B}$ leads to

$V_{B}=t^{2\beta}.$ (36)

From Eqs.(35) and (36), the metric perturbation becomes

$\delta b_{i}=c_{2}+c_{1}\frac{t^{1-2\beta}}{(1-2\beta)},$ (37)

where $c_{1}$ and $c_{2}$ are constants of integration.

Thus, the actual fluctuation for each expansion factor $\delta
a_{i}=a_{B_{i}}\delta b_{i}$ are expressed as

$\delta a_{i}=c_{2}t^{2\beta}+c_{1}\frac{t}{(1-2\beta)}.$ (38)

Figure 6:  Plot of stability analysis with redshift $z$

From Eq.(38) and Fig. $6$, we notice that the positive estimation of $\beta$,
the $\delta_{ai}$ approaches to zero for large $z\to\infty$ i.e.
$\delta_{ai}\to 0$. Thus, the background solution is stable against the metric
perturbation. This figure depicts the behaviour of actual fluctuations with
respect to redshift. In past, the actual fluctuation is null but it increases
minimally on later time.

##  6 Conclusion

In this work, the R$\grave{e}$nyi holographic type DE model with two IR cut-
offs: Hubble horizon $L=\frac{1}{H}$ and Granda & Oliveros horizon
$L=(kH^{2}+m\dot{H})^{-1/2}$ cut-off have been discussed in $f(R,T)$ gravity
by considering the parameters $k=0.8$ and $m=0.89$. For this purpose , we have
analysed the cosmological features like: energy density, pressure, EoS
parameter, density parameter, ($\omega_{T}$-$\omega_{T}^{{}^{\prime}}$) plane
and stability of the models are discussed. We have summarized our results as
follows:

  * •

The energy density of RHDE $\rho_{{}_{T}}$ is positive decreasing function for
the three values of $\delta$ with redshift $z$. We found this behaviour is
similar in both IR cut-offs as seen in Fig.$1a,b$ respectively.

  * •

Both models (I, II) indicate that the pressure is negative in all estimations
throughout the evaluation seen in Fig.$2a,b$.

  * •

The trajectories of EoS parameter show the transition from phantom region to
quintessence region by evolving the vacuum era of the universe in both IR cut-
offs as clear from Fig.$3a,b$.

  * •

We also notice that the density parameter $\Omega_{{}_{T}}$ is positive
throughout the evolution in both IR cut-offs as seen in Fig.$4a,b$.

  * •

The ($\omega_{{}_{T}}$-$\omega_{{}_{T}}{{}^{\prime}}$) trajectories indicate
the thawing region for the R$\grave{e}$nyi HDE model with Hubble horizon cut-
off and GO cut-off as seen in Fig.$5a,b$.

  * •

We have conducted that the perturbation analysis to examine the stability of
the considered dark energy model and found that RHDE shows more stability
during the comic evolution seen in (Fig.$6$). Subsequently our created model
is stable.

  * •

Here, we also include a comparative study of our work with recent works on
this subject[34, 35] and found that our findings are in good agreement with
their work but their models are unstable whereas our both models are stable.

The feasibility of the R$\grave{e}$nyi holographic dark energy model with
Hubble and GO cut-off is supported by our research and combined observational
data is consistent with the future horizon.

## Acknowledgments

The authors are heartily grateful to the anonymous referee for his
constructive comments which improved the paper in the present form.

## References

  * [1] A.G. Riess et al., Astron. J. 116, 1009 (1998).
  * [2] P.A. Bernardis et al., Nature 404, 955 2000.
  * [3] M. Colless et al., Mon. Not. R. Astron. Soc. 328, 1039 (2001).
  * [4] S. Cole et al., Mon. Not. R. Astron. Soc. 362, 505 (2005).
  * [5] V. Springel, C.S. Frenk, Nature 440, 1137 (2006).
  * [6] S. Ade Hanany et al., Astrophys. J. Lett. 545, L5 (2000).
  * [7] D.N. Spergel et al. Astrophys. J. Suppl. 148, 175 (2003).
  * [8] M. Roos, Wiley Chichester (2003).
  * [9] S. Nojiri, S.D. Odintsov, Phys. Lett. B 639, 144 (2006).
  * [10] K. Bamba, S. Capozziello, S. Nojiri, S.D. Odintsov, Astrophys. Space Sci. 342, 155 (2012).
  * [11] M. Malekjani, T. Naderi, F. Pace, MNRAS 453, 4148 (2015).
  * [12] M.R. Setare, Phys. Lett. B 644, 99 (2007).
  * [13] T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D 62, 023511 (2000).
  * [14] V. Pasquier, U. Moschella, A.Y. Kamenshchick, U. Moschella, V. Pasquier, Phys. Lett. B 511, 265 (2001).
  * [15] K. Kleidis, N.K. Spyrou, Astron. Astrographys. 576, A23 (2015).
  * [16] K. Kleidis, N.K. Spyrou, Entropy 18, 94 (2016).
  * [17] S. Nojiri, S.D. Odintsov, Phys. Lett. B 562, 147 (2003).
  * [18] S. Weinberg, Rev. Mod. Phys. 61, 1 (1998).
  * [19] E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
  * [20] M.R. Setare, Chin. Phys. Lett. 26, 029501 (2009).
  * [21] S.D.H Hsu, Phys. Lett. B 594, 13 (2004).
  * [22] M.R. Setare, E.N. Saridakis, Phys. Lett. B 671, 331 (2009).
  * [23] M. Jamil, E.N. Saridakis, M.R. Setare, Phys. Lett. 679, 172 (2009).
  * [24] J. Lu, E.N. Saridakis, M.R. Setare, L. Xu, J. Cosmol. Astropart. Phys. 031, 26 (2010).
  * [25] J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * [26] R. Bousso, JHEP 004, 9907 (1999).
  * [27] A. Cohen, D. Kaplan, A. Nelson, Phys. Rev. Lett. 82, 4971 (1999).
  * [28] L. Susskind, J. Math. Phys. (N.Y.) 36 , 6377 (1994).
  * [29] D.R.K. Reddy et al., Astrophys. Space Sci. 361, 356 (2016).
  * [30] Y. Aditya, D.R.K. Reddy, Eur. Phys. J. C 78, 619 (2018).
  * [31] V.U.M. Rao et al., Results Phys. 10 , 469 (2018).
  * [32] M.V. Santhi, et al., Can. J. Phys. 95, 381 (2017).
  * [33] K.D. Naidu et al., Eur. Phys. J. Plus 133, 303 (2018).
  * [34] U.K. Sharma, V.C. Dubey, arXiv:2001.02368, (2020).
  * [35] V.C. Dubey, A.K. Mishra, U.K. Sharma, arXiv:2003.07883, (2020).
  * [36] U.Y.D. Prasanthi, Y. Aditya, Results Phys. 17, 103101 (2020).
  * [37] T. Golanbari, K. Saaidi, P. Karimi, arXiv:2002.04097[astro-ph.CO] (2020).
  * [38] S. Qolibiklooa, A. Ghodsib, Eur. Phys. J. C 79, 406 (2019).
  * [39] I.A. Akhlaghi et al., MNRAS 477, 3659 (2018).
  * [40] S. Ghaffari, New Astron. 67, 76 (2019).
  * [41] H. Moradpour, et al., Eur. Phys. J. C 78, 829 (2018).
  * [42] A.S. Jahromi, et al., Phys. Lett. B 21, 780 (2018).
  * [43] M. Tavayef, A. Sheykhi, K. Bamba, H. Moradpour, Phys. Lett. B 781 195 (2018).
  * [44] C. Tsallis, L.J.L. Cirto, Eur. Phys. J. C 73, 2487 (2013).
  * [45] M. Younas et al., Adv. High Energy Phys. 2019, 1287932 (2019).
  * [46] P. Horava, D. Minic, Phys. Rev. Lett. 85, 1610 (2000).
  * [47] S. Thomas, Phys. Rev. Lett. 89, 081301 (2002).
  * [48] L.N. Granda, A. Oliveros, Phy. Lett.B 671, 199 (2009)
  * [49] A. Jawad, K. Bamba, M. Younas, S. Qummer, S. Rani, Symmetry 10, 635 (2018).
  * [50] A.G. Cohen, D.B. Kaplan, A.E. Nelson, Phys. Rev. Lett. 82, 4971 (1999).
  * [51] J.D. Bekenstein, Phys. Rev. D7 , 2333 (1973).
  * [52] S.W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * [53] M. Li, X. D. Li, S. Wang, X. Zhang, J. Cosmol. Astropart. Phys. 2009, 036 (2009).
  * [54] M. Li, X. D. Li,S. Wang, Y. Wang, X. Zhang, J. Cosmol. Astropart. Phys. 2009, 014 (2009)
  * [55] B. Guberina, R. Horvat, H. Nikolic, J. Cosmol. Astropart. Phys. 2007, 012 (2007).
  * [56] S. Wang, Y. Wang, M. Li, Phys. Rep. 1, 696 (2017).
  * [57] B. Wang, E. Abdalla, F. Atrio-Barandela, D. Pavon, Rep. Prog. Phys. 79, 096901 (2016)
  * [58] K. karami, A. Abdolmaleki, N. Sahraei, S. Ghaffari, JHEP 150, 1108 (2011).
  * [59] C. Tsallis, Entropy 13, 1765 (2011).
  * [60] A. R$\grave{e}$nyi, Probability Theory (North-Holland, Amsterdam, 1970).
  * [61] C. Tsallis, J. Stat. Phys. 52, 479 (1988)
  * [62] S. Nojiri, S.D. Odintsov and E.N. Saridakis, Eur. Phys. J. C 79, 242 (2019)
  * [63] A. Majhi, Phys. Lett. B 32, 772 (2017).
  * [64] A.S. Jahromi et al., Phys. Lett. B 780, 21 (2018).
  * [65] N. Komatsu, Eur. Phys. J. C 77, 229 (2017).
  * [66] H. Moradpour, A. Bonilla, E.M.C. Abreu, J.A. Neto, Phys. Rev. D 96, 123504 (2017).
  * [67] H. Moradpour, A. Sheykhi, C. Corda, I.G. Salako, Phys. Lett. B 783, 82 (2018).
  * [68] H. Moradpour, Int. J. Theor. Phys. 55, 4176 (2016).
  * [69] H. Moradpour, et. al. Eur. Phys. J. C 78, 829 (2018).
  * [70] E.M. Barboza , R.C. Nunes, E.M.C. Abreu, J.A. Neto, Phys. A Stat. Mech. Appl. 436, 301 (2015).
  * [71] V.G. Czinner, H. Iguchi, Phys. Lett. B 752, 306 (2016)
  * [72] T. Harko, Phys. Rev. D 81, 044021 (2010).
  * [73] S. Ram S.K Singh, M.K. Verma, Phys. Astron. Int. J. 4, 330 (2018).
  * [74] D.D. Pawar, R.V. Mapari, P.K. Agarwal, J. Astrophys. Astron. 40 13 (2019).
  * [75] P.K. Sahoo, P. Sahoo, B.K. Bishi, Int. J. Geom. Method Mod. Phys. 7, 17 (2018).
  * [76] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, Oxford (1998).
  * [77] N. J. Poplawski, arXiv:gr-qc/0608031.
  * [78] V. Faraoni, Cosmology in Scalar-Tensor Theory, Kluwer Academis Publishers (2004).
  * [79] S. Mizuno, S.J. Lee, E.J. Copeland, Phys. Rev. D 70, 043525 (2004), astro-ph/ 0405490.
  * [80] E.J. Copeland, M.R. Garousi, M. Sami, S. Tsujikawa, Phys. Rev. D 71, (2005) 043003.
  * [81] R.R. Caldwell, E.V. Linder, Phys. Rev. Lett. 2005, 95, 141301–141304.
  * [82] S. Bhattacharjee, arxiv:2006.04339v1[gr-qc] (2020).
  * [83] L.K. Sharma, B.K. Singh, A.K. Yadav, Int. J. Geom. Method Mod. Phys., 1, 2050111 (2020)
  * [84] C.M. Chen, W.F. Kao, Phys. Rev. D 64 (2001) 124019.