# Saez-Ballester gravity in Kantowski-Sachs Universe: a new reconstruction
paradigm for Barrow Holographic Dark Energy

G. G. Luciano<EMAIL_ADDRESS>Applied Physics Section of
Environmental Science Department, Escola Politècnica Superior, Universitat de
Lleida, Av. Jaume II, 69, 25001 Lleida, Spain

###### Abstract

We reconstruct Barrow Holographic Dark Energy (BHDE) within the framework of
Saez-Ballester Scalar Tensor Theory. As a specific background, we consider a
homogeneous and anisotropic Kantowski-Sachs Universe filled up with BHDE and
dark matter. By assuming the Hubble radius as an IR cutoff, we investigate
both the cases of non-interacting and interacting dark energy scenarios. We
analyze the evolutionary behavior of various model parameters, such as
skewness parameter, Equation-of-State parameter, deceleration parameter, jerk
parameter and squared sound speed. We also draw the trajectories of
$\omega_{D}-\omega^{\prime}_{D}$ phase plane and examine statefinder
diagnosis. Observational consistency is discussed by inferring the current
value of Hubble’s parameter through the best fit curve of data points from
Differential Age (DA) and Baryon Acoustic Oscillations (BAO) and commenting on
cosmological perturbations and growth rate of matter fluctuations in BHDE. We
show that our model satisfactorily retraces the history of the Universe, thus
providing a potential candidate explanation for dark energy. Comparison with
other reconstructions of BHDE is finally analyzed.

##  I Introduction

Precision Cosmology Primack:2006it  measurements have definitively shown that
our Universe is experiencing an accelerated phase expansion Supern ; Supernbis
; Supernter ; Supernquar ; Supernquin ; ER ; Vag1 ; Vag2 . However, the fuel
of this mechanism is not yet known, leaving room for disparate explanations.
Tentative descriptions can be basically grouped into two classes: on one side,
Extended Gravity Theories CapozDela  aim at solving the puzzle by modifying
the geometric part of Einstein-Hilbert action in General Relativity. On the
other side, one can introduce new degrees of freedom (DoF) in the matter
sector, giving rise to dynamical Dark Energy models. In this context, a
largely followed approach is the so-called Holographic Dark Energy (HDE) model
Cohen:1998zx ; Horava:2000tb ; Thomas:2002pq ; Li:2004rb ; Hsu:2004ri ;
Huang:2004ai ; Nojiri:2005pu ; Wang:2005ph ; Setare:2006sv ; Guberina:2006qh ;
Granda ; Sheykhi:2011cn ; Bamba:2012cp ; Ghaffari:2014pxa ; Wang:2016och ;
Odi1 ; Moradpour:2020dfm ; Zhang ; Li ; Zhang2 ; Lu ; Nojiri:2019kkp , which
is based on the use of the holographic principle at cosmological scales.

In the lines of gravity-thermodynamic conjecture, HDE describes our Universe
as a hologram, the DoF of which are encoded by Bekenstein-Hawking entropy.
Nevertheless, it fails to retrace the evolution of the cosmos properly
Horava:2000tb ; Thomas:2002pq ; Li:2004rb , thus motivating suitable
amendments to be implemented. Along this line, a promising framework is
offered by HDE with deformed horizon entropies Odi ; Odi2 , such as Tsallis
Tsallis1 ; Tsallis2 ; Tsallis3 ; Tsallis4 , Kaniadakis Kana1 ; Kana2 ; Kana3
and Barrow Bar1 ; Bar2 ; Bar3 ; Bar4 ; Bar5 ; Bar6 ; Bar7 ; Lucianoarx ;
BarUlt  entropies, which arise from the effort to introduce non-extensive,
relativistic and quantum gravity corrections in the classical Boltzmann-Gibbs
statistics, respectively. While predicting a richer phenomenology comparing to
the standard Cosmology, generalized HDE models suffer from the absence of an
underlying Lagrangian. This somehow questions their relevance in improving our
knowledge of Universe at fundamental level.

Preliminary attempts to overcome the above issue have been made by considering
reconstructing scenarios, where effective Lagrangian models are built by
comparing extended HDE and modified gravity. So far, this recipe has
extensively been used for Tsallis HDE, with a number of applications in $f(R)$
EPL , $f(R,T)$ Chinese , $f(G,T)$ fgtgrav , teleparallel Wahe , Brans-Dicke
GhaffariBD , logarithmic Brans-Dicke LogBD  and tachyon LiuT  models, among
others. By contrast, comparably less attention has been devoted to Barrow HDE
Sarkar ; LucianoPRD ; TeleBar . However, it is such a framework that can
potentially open new perspectives in modern theoretical Cosmology, especially
in light of the quantum gravitational nature of the underlying Barrow’s
conjecture B2020 . And, in fact, in the absence of any fully quantum theory of
Universe, the best we can do toward formulating the quantum effective action
of cosmological model at this stage is to frame common inputs and empirical
predictions of existing quantum gravity-oriented cosmological models in
suitable modifications of General Relativity.

Among the several modifications of Einstein’s theory SB , Saez-Ballester
Theory (SBT) has recently proven to be versatile enough to both address the
dark energy problem and accommodate reconstructing scenarios SBTBianchi1 ;
Rasouli1 ; SBTBianchi2 ; Rasouli2 ; Santhi . SBT is a member of the class of
Scalar Tensor Theory of gravity. In this theory the metric potentials are
coupled to a scalar field, which notoriously plays a key role in gravitation
and cosmology (see Kim ; Guth ; Linde ). In a broader context, SBT has been
discussed in Bianchi Cosmology in SBTBianchi1 ; SBTBianchi2 , reproducing the
transition from decelerating Universe to accelerating phase. On the other
hand, in Santhi  it has been considered as a background to investigate Tsallis
HDE. The ensuing model exhibits qualitative consistency with observations,
though it is classically unstable. Currently, SBT and, in general, Extended
Gravity are held to align with Precision Cosmology data.

Starting from the above premises, in this work we propose a reconstruction of
BHDE in SBT. We frame our analysis in Kantowski-Sachs (KS) geometry KSM ,
which describes a homogeneous but anisotropic Universe, the spatial section of
which has the topology of $\mathbb{R}\times S^{2}$ (see also Req1  for a
recent study of the evolution of KS Universe towards de Sitter at late time).
The reason why we consider such a type of Universe is that theoretical
investigations and new probes such as Cosmic Background Explorer (COBE),
Wilkinson Microwave Anisotropy Probe (WMAP) and Planck have recorded the
presence of anisotropy in our Universe Ani0 ; Ani1 , thus requiring a
generalization of the canonical Friedmann-Lemaitre-Robertson-Walker model.
This is also confirmed by recent space-based X-ray observations of hundreds of
galaxy clusters Ani . Motivated by these arguments, we explore the history of
a KS Universe filled up with anisotropic BHDE and dark matter in SBT. We
construct both non-interacting and interacting models by assuming the Hubble
radius as an IR cutoff and solving the field equations for a particular
relationship between the metric potentials. We focus on the evaluation of
skewness parameter, Equation-of-State parameter, deceleration parameter, jerk
parameter and squared sound speed. Also, we draw the trajectories of
$\omega_{D}-\omega^{\prime}_{D}$ phase plane and discuss the statefinder
diagnosis. We discuss observational consistency by deriving the current value
of Hubble’s parameter through the best fit curve of 57 data points measured
from Differential Age (DA) and Baryon Acoustic Oscillations (BAO) and
commenting on cosmological perturbations and structure formation. We show that
our model explains the current expansion satisfactorily, thus providing a
potential candidate for dark energy. Comparison with observations enables us
to constrain the values of free parameters in SBT.

The remainder of the work is organized as follows: in the next Section we
review the basics of BHDE and SBT in Kantowski-Sachs Universe. To this aim, we
follow Santhi . Section III is devoted to analyze the cosmic history of
reconstructed BHDE, while in Sec. IV we discuss Hubble’s parameter evolution
and cosmological perturbations. Conclusions and outlook are finally summarized
in Sec. V. Throughout the whole manuscript, we use natural units.

##  II Saez-Ballester Theory and Barrow Holographic Dark Energy: An
Introduction

In this Section we set the notation and provide the basic ingredients for the
core analysis of this work. We first define the geometry of a KS Universe in
SBT, deriving the corresponding field equations. Then, we focus on BHDE
framework and its main advantages over the standard HDE scenario. In passing,
we mention that a similar study has recently been performed in Bianchi_I  in
Bianchi-I anisotropic Universe with BHDE.

###  II.1 Saez-Ballester Theory of Gravity

In SB Scalar Tensor Theory of gravity the Lagrangian is written in the form SB

$\mathcal{L}_{SB}\,=\,R-w\phi^{n}\phi,_{\gamma}\phi^{,\gamma}\,,$ (1)

where $R$ is the scalar curvature, $\phi$ a dimensionless scalar field, $w$
and $n$ arbitrary dimensionless constants and
$\phi^{,\gamma}\equiv\phi,_{\alpha}g^{\alpha\gamma}$ (as usual we denote
partial derivatives by a comma, while covariant derivates by a semicolon).

From the above Lagrangian, one can build the action

$I_{SB}\,=\,\int_{\sigma}(\mathcal{L}_{SB}+\mathcal{L}_{m})(-g)^{1/2}\,dX^{1}dX^{2}dX^{3}dX^{4}\,,$
(2)

up to an overall factor multiplying the matter Lagrangian $\mathcal{L}_{m}$.
Here $g$ is the determinant of the metric, $X^{i}$ the coordinates and
$\Sigma$ an arbitrary domain of integration.

Now, for arbitrary independent variations of the metric and the scalar field
vanishing at the boundary surface of $\Sigma$, the variational principle
$\delta I_{SB}=0$ leads to the field equations

$\displaystyle
G_{\alpha\beta}-w\phi^{n}\left(\phi,_{\alpha}\phi,_{\beta}-\frac{1}{2}g_{\alpha\beta}\phi,_{\gamma}\phi^{,\gamma}\right)$
$\displaystyle=$
$\displaystyle-(T_{\alpha\beta}\,+\,\widetilde{T}_{\alpha\beta})\,,$ (3)
$\displaystyle
2\phi^{n}\phi^{,\gamma}_{;\gamma}\,+\,n\phi^{n-1}\phi,_{\gamma}\phi^{,\gamma}$
$\displaystyle=$ $\displaystyle 0\,,$ (4)

where $G_{\alpha\beta}$ is the Einstein tensor and $T_{\alpha\beta}$ the
energy-momentum tensor (EMT) defined form $\mathcal{L}_{m}$ in the usual way.
For later convenience, here we have separated out the contribution
$\widetilde{T}_{\alpha\beta}$ due to dark energy.

From relations (3) and (4), it is easy to prove that the following
conservation equation holds

$(T^{\alpha\beta}+\widetilde{T}^{\alpha\beta})_{;\alpha}\,=\,0\,.$ (5)

For dark matter of density $\rho_{m}$ and anisotropic DE of density
$\rho_{D}$, the EMTs read Santhi

$\displaystyle T_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathrm{diag}\left[1,0,0,0\right]\rho_{m}\,,$ (6)
$\displaystyle\widetilde{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathrm{diag}\left[\rho_{D},-p^{x}_{D},-p^{y}_{D},-p^{z}_{D}\right]$
(7) $\displaystyle=$
$\displaystyle\mathrm{diag}\left[1,-\omega^{x}_{D},-\omega^{y}_{D},-\omega^{z}_{D}\right]\rho_{D}$
$\displaystyle=$
$\displaystyle\mathrm{diag}\left[1,-\omega_{D},-(\omega_{D}+\alpha),-(\omega_{D}+\alpha)\right]\rho_{D}\,,$

respectively, where $p_{D}$ is the dark energy pressure and
$\omega_{D}=p_{D}/\rho_{D}$ the related Equation of State (EoS) parameter. The
deviation from isotropy is parametrized by setting $\omega^{x}_{D}=\omega_{D}$
and introducing the deviations along $y$ and $z$ axes by the (time-dependent)
skewness parameter $\alpha$. Clearly, the standard isotropic framework is
recovered in the $\alpha\rightarrow 0$ limit.

Let us now consider a homogeneous and anisotropic KS Universe of metric

$ds^{2}\,=\,dt^{2}-A^{2}(t)\,dr^{2}-B^{2}(t)\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,,$
(8)

where $A$ and $B$ are the (time-dependent) metric potentials. In this
framework, the field equations (3) become

$\displaystyle
2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B}+\frac{1}{B^{2}}-\frac{w}{2}\phi^{n}\dot{\phi}^{2}$
$\displaystyle=$ $\displaystyle-\omega_{D}\rho_{D}\,,$ (9)
$\displaystyle\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}-\frac{w}{2}\phi^{n}\dot{\phi}^{2}$
$\displaystyle=$ $\displaystyle-(\omega_{D}+\alpha)\rho_{D}\,,$ (10)
$\displaystyle
2\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}^{2}}{B^{2}}+\frac{1}{B^{2}}+\frac{w}{2}\phi^{n}\dot{\phi}^{2}$
$\displaystyle=$ $\displaystyle\rho_{m}+\rho_{D}\,,$ (11)
$\displaystyle\ddot{\phi}+\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\dot{\phi}+\frac{n}{2}\frac{\dot{\phi}^{2}}{\phi}$
$\displaystyle=$ $\displaystyle 0\,,$ (12)

where the dot denotes ordinary derivative with respect to the cosmic time $t$.

Similarly, the continuity equation (5) can be cast as

$\displaystyle\dot{\rho}_{m}+\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\rho_{m}+\dot{\rho}_{D}+\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)\left(1+\omega_{D}\right)\rho_{D}$
$\displaystyle+2\frac{\dot{B}}{B}\alpha\rho_{D}=0\,.$ (13)

In order to solve the system of four equations (9)-(12) in the seven unknowns
$A,B,\rho_{m},\rho_{D},\omega_{D}$, $\alpha$ and $\phi$, we need to impose
some extra conditions. Following Collins , we require the metric potentials to
be related by

$A\,=\,B^{k}\,,$ (14)

with $k\neq 1$ being a positive constant. Furthermore, we set Skewness1 ;
Skewness2

$\alpha\,=\,\frac{\alpha_{0}\left(k-1\right)\dot{B}B-1}{B^{2}\rho_{D}}\,,$
(15)

where $\alpha_{0}$ is an arbitrary constant.

In so doing, the metric potentials take the form

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left[\frac{\alpha_{1}\left(k+2\right)}{\alpha_{0}}e^{\alpha_{0}t}+\alpha_{2}(k+2)\right]^{\frac{k}{k+2}}\,,$
(16) $\displaystyle B$ $\displaystyle=$
$\displaystyle\left[\frac{\alpha_{1}\left(k+2\right)}{\alpha_{0}}e^{\alpha_{0}t}+\alpha_{2}(k+2)\right]^{\frac{1}{k+2}}\,,$
(17)

where $\alpha_{1}$ and $\alpha_{2}$ are integration constants. Hence, Eq. (8)
with the substitution of Eqs. (16) and (17) describes the geometry of KS
Universe in SBT.

Finally, we observe that the Hubble parameter for our model is given by Santhi

$H\,=\,\left(\frac{\dot{A}}{A}+2\,\frac{\dot{B}}{B}\right)\,$ (18)

where we have absorbed an overall $1/3$ in the redefinition of $A$ and $B$.

###  II.2 Barrow Holographic Dark Energy

HDE in its most common formulation avails of the Hubble horizon as an IR
cutoff and Bekenstein-Hawking area law for the horizon DoF of Universe.
However, its failures to reproduce the whole cosmic evolution have motivated
tentative changes over the years. Some proposals have been put forward by
considering either different IR cutoffs Wang:2016och ; Wang:2016lxa  or
modified horizon entropies Tsallis1 ; Tsallis2 ; Tsallis3 ; Tsallis4 ; Kana1 ;
Kana2 ; Kana3 ; Bar1 ; Bar2 ; Bar3 ; Bar4 ; Bar5 ; Bar6 ; Bar7 . Among the
latter models, HDE based on Barrow entropy B2020  has been attracting great
attention in the last years Bar1 ; Bar2 ; Bar3 ; Bar4 ; Bar5 ; Bar6 ; Bar7 .

Inspired by the Covid-19 virus structure, Barrow has proposed that quantum
gravity effects might affect black hole horizon structure, introducing
intricate, fractal features B2020 . In turn, this would generalize black hole
entropy formula to

$S\sim{A}^{1+\Delta/2}\,,$ (19)

where $0\leq\Delta\leq 1$, with $\Delta=1$ ($\Delta=0$) corresponding to the
maximal (vanishing) deviation from the standard entropy-area law. Notice that
observational constraints on $\Delta$ have been derived in Bar6 ;
Anagnostopoulos:2020ctz ; Barrow:2020kug ; LucianoInf .

Based on Eq. (19) and exploiting the deep connection between gravity and
thermodynamics, Barrow’s paradigm has recently been extended to Cosmology.
Specifically, as argued in Bar1  the definition of HDE density in Barrow’s
picture appears as

$\rho_{D}\,=\,CL^{\Delta-2}\,,$ (20)

where $C$ is an unknown parameter with dimensions $[L]^{-2-\Delta}$. By
setting the Hubble horizon as IR cutoff, we then get

$\rho_{D}\,=\,CH^{2-\Delta}\,.$ (21)

where $H$ is given by Eq. (18). From this relation, we easily get

$\dot{\rho}_{D}\,=\,C\left(2-\Delta\right)H^{1-\Delta}\dot{H}\,.$ (22)

Equations (16), (17) and (21) provide the necessary tools for our next
reconstruction of BHDE in SBT.

##  III BHDE reconstruction in Saez-Ballester Theory

Let us now describe the evolution of a KS Universe with anisotropic BHDE and
dark matter. We analyze separately the cases where: _i_) there is no energy
exchange between the cosmos sectors and _ii_) a suitable interaction is
assumed to exist.

###  III.1 Non-interacting model

As a first step, we observe that in this model the energy conservation
equations for dark matter and BHDE can be decoupled to give

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}\,=\,0\,,$ (23)
$\displaystyle\dot{\rho}_{D}+3H(1+\omega_{D})\rho_{D}+2\alpha\rho_{D}\frac{\dot{B}}{B}\,=\,0\,.$
(24)

respectively. Also, from Eq. (12) we obtain

$\displaystyle\phi^{\frac{n+2}{2}}$ $\displaystyle=$
$\displaystyle\left(\frac{n+2}{2}\right)\phi_{0}$ (25)
$\displaystyle\times\,\int\left[\frac{\alpha_{1}\left(k+2\right)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-3}dt\,+\,\phi_{1}\,,$

where $\phi_{0}$ and $\phi_{1}$ are integration constants.

By plugging Eqs. (17), (21) and (25) into (11), we are led to

$\displaystyle\rho_{m}$ $\displaystyle=$
$\displaystyle\frac{\left(2k+1\right)\alpha_{0}^{2}\,\alpha_{1}^{2}\,e^{2\alpha_{0}t}}{\left(k+2\right)^{2}\left(\alpha_{1}e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)^{2}}$
(26)
$\displaystyle+\,\left[\frac{\left(k+2\right)\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)}{\alpha_{0}}\right]^{-\frac{2}{k+2}}$
$\displaystyle-\,C\left(\frac{\alpha_{0}\alpha_{1}}{\alpha_{1}+\alpha_{0}\alpha_{2}\,e^{-\alpha_{0}t}}\right)^{2-\Delta}$
$\displaystyle+\,\frac{w\,\alpha_{0}^{6}\,\phi_{0}^{2}}{2\left(k+2\right)^{6}\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)^{6}}\,.$

In order to understand the role of skewness and Barrow parameters in the
evolution of the Universe, we now analyze the dynamics of model parameters for
various values of $\alpha_{0}$ and $\Delta$. The evolution of $\rho_{D}$
versus the redshift $z$ is plotted in Fig. 1. One can see that BHDE increases
as $z$ decreases and comes to dominate the energy budget of the Universe in
the far future. Also, $\rho_{D}$ increases with increasing $\alpha_{0}$ (see
the upper panel of Fig. 1), while it is only slightly affected by variation of
$\Delta$ (see the lower panel of Fig. 1).

Figure 1: Evolution of $\rho_{D}$ versus $z$ for different values of
$\alpha_{0}$ (upper panel) and $\Delta$ (lower panel) in non-interacting
model. We have set $C=0.1$, $k=1.1$, $\alpha_{1}=0.9$, $\alpha_{2}=0.52$,
$w=10$, $\phi_{0}=5$ and $\Delta=0.5$ in the upper panel, while
$\alpha_{0}=1.1$ in the lower one (online colors).

In Fig. 2 we depict the evolution of the skewness parameter (15). We observe
that it is negative and approaches constant values in the far future for
selected values of $\alpha_{0}$ (upper panel) and $\Delta$ (lower panel). The
same asymptotic behavior is exhibited in Raju  for the case of a Kantowski-
Sachs cosmological model with anisotropic dark energy fluid and massive scalar
field.

Figure 2: Evolution of skewness parameter versus $z$ for different values of
$\alpha_{0}$ (upper panel) and $\Delta$ (lower panel) in non-interacting
model. For all model parameters, we have set the same values as in Fig. 1
(online colors).

Now, from Eqs. (17), (21) and (24) we can derive the expression of the EoS
parameter of BHDE as

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-1\,-\frac{2\,e^{3\alpha_{0}t}\,\alpha_{0}^{5}\,\alpha_{1}^{\Delta-2}\left(\frac{\alpha_{0}}{\alpha_{1}+e^{-\alpha_{0}t}\,\alpha_{0}\alpha_{2}}\right)^{-\Delta}}{3C\left(k+2\right)^{2}\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)^{5}}$
(27)
$\displaystyle\times\,\left\\{\left(k-1\right)e^{\alpha_{0}t}\,\alpha_{0}\alpha_{1}-\left[\frac{\left(k+2\right)\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)}{\alpha_{0}}\right]^{\frac{k}{2+k}}\right\\}$
$\displaystyle+\,C\,e^{\alpha_{0}t}\,\alpha_{0}\,\alpha_{1}\,\alpha_{2}\left(\frac{\alpha_{0}\alpha_{1}}{\alpha_{1}+e^{-\alpha_{0}t}\alpha_{0}\alpha_{2}}\right)^{1-\Delta}\left(\Delta-2\right).$

This is plotted in Fig. 3: from the upper panel we see that BHDE evolves from
quintessence ($-1<\omega_{D}<-1/3$) at late time to approximately cosmological
constant ($\omega_{D}=-1$) at present and phantom ($\omega_{D}<-1$) in the far
future. In this regard, it is worth noting that largely negative values of
$\omega_{D}$ indicate that the Universe might either end up with a big-rip or
remain in the same current accelerating status.

By comparison with results of Santhi , we infer that the obtained behavior is
peculiar to BHDE model in KS Universe. In fact, Tsallis HDE always lies in a
quintessence-like regime and approaches cosmological constant in the far
future. On the other hand, the same evolution is exhibited in the context of
BHDE in Brans-Dicke Cosmology with a linear interaction Bar7  and Bianchi-type
I BHDE in teleparallel gravity TeleBar . Furthermore, quantitative analysis
gives us $\omega_{D_{0}}\in[-1.07,-0.99]$ for the current value of the EoS
parameter and the considered values of $\alpha_{0}$. This is in good agreement
with the recent constraint $\omega_{0}\in[-1.38,-0.89]$ obtained from
Planck+WP+BAO measurements Planck . A qualitatively similar transition from
quintessence to cosmological constant and phantom is displayed in the lower
panel of Fig. 3 for fixed $\alpha_{0}$ and varying $\Delta$. In this case we
find $\omega_{D_{0}}\in[-0.99,-0.95]$, which is still consistent with
observations Planck .

Figure 3: Evolution of EoS parameter versus $z$ for different values of
$\alpha_{0}$ (upper panel) and $\Delta$ (lower panel) in non-interacting mode.
For all model parameters, we have set the same values as in Fig. 1 (online
colors).

Let us now investigate trajectories of $\omega_{D}-\omega_{D}^{\prime}$ plane.
Here the overhead prime denotes derivative respect to the logarithm of the
scale factor $a$, which is as usual related to the Hubble parameter by
$H=\dot{a}/a$. $\omega_{D}-\omega_{D}^{\prime}$ plane has been introduced by
Caldwell and Linder CL  and represents a useful tool to distinguish among dark
energy models. Firstly, it has been applied to quintessence model, which gives
two different regions in this plane, i.e. the _thawing_
($\omega_{D}<0,\omega_{D}^{\prime}>0$) and _freezing_
($\omega_{D}<0,\omega_{D}^{\prime}<0$) domains. Subsequently, it has been
generalized and extended to other dynamical dark energy models, see for
instance Sche ; Chiba ; Guo ; Sharif . Cosmic trajectories of
$\omega_{D}-\omega_{D}^{\prime}$ plane for the present framework are plotted
in Fig. 4. We observe that our model predicts freezing region for BHDE, which
is consistent with the current behavior of the Universe, since freezing regime
is associated to a more accelerating era of cosmic expansion respect to
thawing domain. The same result is exhibited in TeleBar  for the case of BHDE
in teleparallel gravity TeleBar , while the opposite scenario occurs in
Tsallis HDE in KS Universe Santhi .

Figure 4: Evolution of $\omega_{D}-\omega^{\prime}_{D}$ trajectories for
different values of $\alpha_{0}$ (upper panel) and $\Delta$ (lower panel) in
non-interacting model. For all model parameters, we have set the same values
as in Fig. 1 (online colors).

Another quantity that should be taken into account to establish whether a dark
energy model is phenomenologically consistent is the deceleration parameter

$q\,=\,-\frac{\ddot{a}}{aH^{2}}\,=\,-1-\frac{\dot{H}}{H^{2}}\,.$ (28)

From this equation, we infer that positive values of $q$ correspond to a
decelerated expansion of the Universe ($\ddot{a}<0$), while negative values
characterize the accelerated regime ($\ddot{a}>0$). The behavior of $q$ is
displayed in Fig. 5, showing that our model correctly reproduces the current
accelerating phase of the cosmos. We emphasize that this is an advantage of
BHDE scenario over standard HDE, which by contrast fails to explain the
present accelerated expansion. Quantitatively speaking, for the selected
values of $\alpha_{0}$ we find $q_{0}\in[-1.35,-1.23]$ for the current
deceleration parameter. Although it slightly deviates from the standard
$\Lambda$CDM model value $q_{0}=-0.55$ Planck , it overlaps with the
estimation $q_{0}\in[-1.37,-0.79]$ recently obtained in Camerana  via local
supernovae measurements.

In Fig. 6 we present the evolution of the jerk parameter, which is a
dimensionless third derivative of the scale factor respect to the cosmic time,
i.e. Blandford:2004ah

$\displaystyle j$ $\displaystyle=$
$\displaystyle\frac{1}{aH^{3}}\frac{d^{3}a}{dt^{3}}\,=\,q\left(2q+1\right)+\left(1+z\right)\frac{dq}{dz}$
(29)
$\displaystyle=\,1\,+\frac{9e^{-\frac{3\alpha_{0}}{1+(1+z)^{2}}}\alpha_{0}\alpha_{2}\left\\{\left[2+z\left(2+z\right)\right]^{2}-2\left(1+z\right)^{2}\alpha_{0}\right\\}}{\left[2+z\left(2+z\right)\right]^{2}\alpha_{1}}$
$\displaystyle+\,\frac{18e^{-\frac{6\alpha_{0}}{1+\left(1+z\right)^{2}}}\alpha_{0}^{2}\,\alpha_{2}^{2}}{\alpha_{1}^{2}}\,.$

Figure 5: Evolution of deceleration parameter for different values of
$\alpha_{0}$ in non-interacting model. For all model parameters, we have set
the same values as in Fig. 1 (online colors).

We point out that the this parameter allows us to quantify deviations from
$\Lambda$CDM model, which is indeed characterized by $j=1$. From Fig. 6 we can
see that our model departs from $\Lambda$CDM at early times, while consistency
is achieved in the far future. Also, we have $j_{0}\in[1.34,1.83]$ for the
selected values of $\alpha_{0}$.

Figure 6: Evolution of jerk parameter for different values of $\alpha_{0}$ in
non-interacting model. For all model parameters, we have set the same values
as in Fig. 1 (online colors).

Figure 7: Evolution of squared speed of sound for different values of
$\alpha_{0}$ (upper panel) and $\Delta$ (lower panel) in non-interacting
model. For all model parameters, we have set the same values as in Fig. 1
(online colors). Figure 8: Evolution of $(r,s)$ plane trajectories for
different values of $\alpha_{0}$. For all model parameters, we have set the
same values as in Fig. 1 (online colors).

In order to study the classical stability of our model against small
perturbations, let us now evaluate the squared speed of sound

$v_{s}^{2}\,=\,\frac{\dot{p}_{D}}{\dot{\rho}_{D}}\,.$ (30)

Notice that, in order for a given dark energy model to be stable, we must have
$v_{s}^{2}>0$. Indeed, for a density perturbation, positive values of
$v_{s}^{2}$ correspond to a regular propagation mode. On the other hand, for
$v_{s}^{2}<0$ one has that the perturbation equation becomes an irregular wave
equation, giving rise to an escalating mode Pert1 . In this setting, the
pressure turns out to decrease when the density perturbation increases, thus
favoring the development of an instability.

For the present BHDE model, we obtain the non-trivial expression

$\displaystyle v_{s}^{2}$ $\displaystyle=$
$\displaystyle-1+C\alpha_{0}\,\alpha_{1}\,\alpha_{2}\left(\Delta-2\right)e^{\alpha_{0}t}H^{1-\Delta}\,+\,\frac{2\,\alpha_{1}^{2\Delta-5}}{3C}\bigg{\\{}1-\alpha_{0}\alpha_{1}\left(k-1\right)e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}\bigg{\\}}$
(31)
$\displaystyle\times\,\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{4+k}{2+k}}\,H^{3-\Delta}\,+\,C\alpha_{0}\,\alpha_{1}\,\alpha_{2}\,\left(\Delta-2\right)e^{\alpha_{0}t}H^{-\left(1+\Delta\right)}\left[\left(1-\Delta\right)\dot{H}+\alpha_{0}\,H\right]\frac{H^{1-\Delta}}{(2-\Delta)\dot{H}}$
$\displaystyle+\,\frac{2\,\alpha_{1}^{2\Delta-5}H^{4-\Delta}}{3C\left(2-\Delta\right)\dot{H}}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{4+k}{2+k}}\Bigg{\\{}\bigg{\\{}1-\alpha_{0}\,\alpha_{1}\left(k-1\right)e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}\bigg{\\}}$
$\displaystyle\times\,\bigg{\\{}\left(3-\Delta\right)\frac{\dot{H}}{H}-\left(\frac{4+k}{2+k}\right)\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-1}\bigg{\\}}+\alpha_{0}^{2}\,\alpha_{1}\left(1-k\right)e^{\alpha_{0}t}\bigg{\\{}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}$
$\displaystyle-\,\frac{\alpha_{1}\,k}{\alpha_{0}}\,e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k+1}{2+k}}\bigg{\\}}\Bigg{\\}}\,.$

This is plotted in Fig. 7 for different values of $\alpha_{0}$ (upper panel)
and $\Delta$ (lower panel). From the upper panel, we see that the model is
classically stable throughout the whole evolution for higher skewness (red
curve), while it exhibits the opposite behavior as skewness decreases (green
and blue curves), no matter the value of Barrow parameter (see lower panel).
By contrast, SBT-based reconstruction of Tsallis HDE, as well as non-
interacting BHDE in Brans-Dicke Cosmology are always unstable Santhi . This is
a further advantage of our reconstruction.

Before moving onto the study of the interacting model, we focus on the
statefinder diagnosis of BHDE. The statefinder parameters $r$ and $s$ were
first introduced in rs  to differentiate among the plethora of dark energy
models. In the definition of these parameters, derivatives of the scale factor
exceed the second order. In particular, we have

$\displaystyle r$ $\displaystyle\equiv$
$\displaystyle\frac{\dddot{a}}{aH^{3}}\,,$ (32) $\displaystyle s$
$\displaystyle\equiv$ $\displaystyle\frac{r-1}{3(q-\frac{1}{2})}\,.$ (33)

We remind that the evolutionary trajectories of dark energy models in the
$(r,s)$ plane can be classified as quintessence if $r<1$ and $s>0$ or
Chaplygin gas if $r>1$ and $s<0$ class . For the present model we obtain
Santhi

$\displaystyle r$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{0}^{2}\alpha_{2}\,e^{\alpha_{0}t}}{\alpha_{1}^{2}}\left[\frac{\alpha_{1}}{\alpha_{0}}+\alpha_{2}^{2}\left(1+2\,e^{-\alpha_{0}t}\right)\right]$
(34)
$\displaystyle+\,\frac{3\alpha_{0}\alpha_{2}\,e^{-\alpha_{0}t}}{\alpha_{1}}-6\,,$
$\displaystyle s$ $\displaystyle=$
$\displaystyle-\frac{2\alpha_{0}^{2}\alpha_{2}\,e^{-\alpha_{0}t}}{3\alpha_{1}\left(2\alpha_{0}\alpha_{2}\,e^{-\alpha_{0}t}-3\alpha_{1}\right)}\left[\frac{\alpha_{1}}{\alpha_{0}}+\alpha_{2}^{2}\left(1+2\,e^{-\alpha_{0}t}\right)\right]$
(35)
$\displaystyle+\frac{2\alpha_{0}\alpha_{1}\alpha_{2}\,e^{-\alpha_{0}t}\,-\,4\alpha_{1}}{\left(2\alpha_{0}\alpha_{2}\,e^{-\alpha_{0}t}-3\alpha_{1}\right)},$

The trajectories of $(r,s)$ plane are plotted in Fig. 8, indicating that BHDE
in this framework gives a correspondence with quintessence model.

###  III.2 Interacting model

Let us now examine how the above framework gets modified when considering a
more realistic scenario with interacting dark matter and BHDE. In this case
the continuity equation (13) can be split into the two relations

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}\,=\,Q\,,$ (36)
$\displaystyle\dot{\rho}_{D}+3H(1+\omega_{D})\rho_{D}+2\alpha\rho_{D}\frac{\dot{B}}{B}\,=\,-Q\,,$
(37)

with $Q$ being the interaction term.

While there is no natural guidance from fundamental physics on the form of
$Q$, phenomenological arguments have led to explore many possible scenarios
over the years Q1 ; Q2 ; Q3 ; Q4 ; ComT0 ; ComT1 ; ComT2 . Following ComT0 ;
ComT1 ; ComT2 , here we assume111 Notice that $\rho$ in Eq. (38) might be in
principle $\rho_{m}$, $\rho_{D}$ or even $\rho_{tot}$. For consistency with
Santhi , we set $\rho=\rho_{D}$. However, the same considerations can be
carried out by resorting to the more general interaction
$Q=3H\left(b_{1}^{2}\rho_{m}+b_{2}^{2}\rho_{D}\right)$, with $b_{1,2}$ being
dimensionless constant much less than unity and positive, so as to be
consistent with the Le Chatelier-Braun principle. While being characterized by
one more free parameter, we expect that the new framework could yield similar
results. A more detailed analysis of this point is left for future
investigation.

$Q=3\beta Hq\rho_{D}\,,$ (38)

where $\beta$ is a dimensionless constant, which should take negative values
according to observational measurements ComT1 .

Figure 9: Evolution of EoS parameter for different values of $\alpha_{0}$
(upper panel), $\Delta$ (middle panel) and $\beta$ (lower panel) in
interacting model. For all model parameters, we have set the same values as in
Fig. 1. For the upper and middle panels, we have considered $\beta=-0.1$
(online colors).

Within this framework, the EoS parameter for dark energy becomes

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-1\,-\frac{2\,e^{3\alpha_{0}t}\,\alpha_{0}^{5}\,\alpha_{1}^{\Delta-2}\left(\frac{\alpha_{0}}{\alpha_{1}+e^{-\alpha_{0}t}\,\alpha_{0}\alpha_{2}}\right)^{-\Delta}}{3C\left(k+2\right)^{2}\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)^{5}}$
$\displaystyle\times\,\left\\{\left(k-1\right)e^{\alpha_{0}t}\,\alpha_{0}\alpha_{1}-\left[\frac{\left(k+2\right)\left(\alpha_{1}\,e^{\alpha_{0}t}+\alpha_{0}\alpha_{2}\right)}{\alpha_{0}}\right]^{\frac{k}{2+k}}\right\\}$
$\displaystyle+\,C\,e^{\alpha_{0}t}\,\alpha_{0}\,\alpha_{1}\,\alpha_{2}\left(\frac{\alpha_{0}\alpha_{1}}{\alpha_{1}+e^{-\alpha_{0}t}\alpha_{0}\alpha_{2}}\right)^{1-\Delta}\left(\Delta-2\right)\,-\,\beta
q\,.$

Figure 10:  Evolution of $\omega_{D}-\omega^{\prime}_{D}$ trajectories for
different values of $\alpha_{0}$ (upper panel), $\Delta$ (middle panel) and
$\beta$ (lower panel) panel in interacting model. For all model parameters, we
have set the same values as in Fig. 1. For the upper and middle panels, we
have considered $\beta=-0.1$ (online colors).

For interaction small enough, the predicted evolution is qualitatively similar
to the previous model, with the sequence of quintessence-, cosmological
constant- and phantom-like behaviors (see upper and middle panels of Fig. 9).
Estimation of the present value of $\omega_{D}$ now gives
$\omega_{D_{0}}\in[-1.06,-0.92]$ for fixed $\Delta=0.5,\beta=-0.1$ and varying
$\alpha_{0}$ (upper panel), and $\omega_{D_{0}}\in[-1.06,-1.02]$ for fixed
$\alpha_{0}=1.1,\beta=-0.1$ and varying $\Delta$ (middle panel). Both these
ranges are still in good agreement with observations (see the discussion below
Eq. (27)). However, by increasing the magnitude of $\beta$, we find that BHDE
always lies in the phantom regime (see blue and red curves in the lower
panel), yielding $\omega_{D_{0}}\in[-1.61,-1.04]$. Therefore, we infer that
large values of $\beta$ are phenomenologically disfavored, in line with the
result of ComT1 .

Figure 10 displays the trajectories of $\omega_{D}-\omega^{\prime}_{D}$ phase
plane. As for non-interacting model, they show that BHDE in SBT lies in the
freezing domain.

Figure 11: Evolution of squared speed of sound for different values of
$\alpha_{0}$ (upper panel), $\Delta$ (middle panel) and $\beta$ (lower panel)
in interacting model. For all model parameters, we have set the same values as
in Fig. 1. For the upper and middle panels, we have considered $\beta=-0.1$
(online colors).

Let us finally consider how the classical stability is affected by Eq. (38).
After some algebra, we find the following expression for the squared sound
speed

$\displaystyle v_{s}^{2}$ $\displaystyle=$
$\displaystyle-1+C\alpha_{0}\,\alpha_{1}\,\alpha_{2}\left(\Delta-2\right)e^{\alpha_{0}t}H^{1-\Delta}\,+\,\frac{2\,\alpha_{1}^{2\Delta-5}}{3C}\bigg{\\{}1-\alpha_{0}\alpha_{1}\left(k-1\right)e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}\bigg{\\}}$
(40)
$\displaystyle\times\,\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{4+k}{2+k}}\,H^{3-\Delta}\,+\,C\alpha_{0}\,\alpha_{1}\,\alpha_{2}\,\left(\Delta-2\right)e^{\alpha_{0}t}H^{-\left(1+\Delta\right)}\left[\left(1-\Delta\right)\dot{H}+\alpha_{0}\,H\right]\frac{H^{1-\Delta}}{(2-\Delta)\dot{H}}$
$\displaystyle+\,\frac{2\,\alpha_{1}^{2\Delta-5}H^{4-\Delta}}{3C\left(2-\Delta\right)\dot{H}}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{4+k}{2+k}}\Bigg{\\{}\bigg{\\{}1-\alpha_{0}\,\alpha_{1}\left(k-1\right)e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}\bigg{\\}}$
$\displaystyle\times\,\bigg{\\{}\left(3-\Delta\right)\frac{\dot{H}}{H}-\left(\frac{4+k}{2+k}\right)\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-1}\bigg{\\}}+\alpha_{0}^{2}\,\alpha_{1}\left(1-k\right)e^{\alpha_{0}t}\bigg{\\{}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k}{2+k}}$
$\displaystyle-\,\frac{\alpha_{1}\,k}{\alpha_{0}}\,e^{\alpha_{0}t}\left[\frac{\alpha_{1}(k+2)}{\alpha_{0}}\,e^{\alpha_{0}t}+\alpha_{2}\left(k+2\right)\right]^{-\frac{k+1}{2+k}}\bigg{\\}}\Bigg{\\}}\,-\,\frac{\beta\dot{q}}{H}\,.$

The behavior of $v_{s}^{2}$ is plotted in Fig. 11, indicating that increasing
interactions might work in favor of a classical stabilization of the model
against small perturbations (see the lower panel of Fig. 11, where it is shown
that the larger the magnitude of $\beta$, the less stable BHDE becomes). More
discussion on the above results can be found in the last Section, along with
further directions to explore.

##  IV Observational constraints

###  IV.1 Hubble’s parameter

In order to explore the experimental consistency of reconstructed BHDE and
constrain Barrow exponent $\Delta$, let us study the evolution of Hubble’s
parameter $H(z)$ from Eq. (28). We develop our analysis for the non-
interacting model and fixed parameters as in Sec. III, but similar
considerations can be extended to the case when Eq. (38) is taken into
account. Following TeleBar , we use data points obtained from 57 Hubble’s
parameter measurements in the range $0.07\leq z\leq 2.36$. These points have
been obtained through Differential Age (31 points), BAO and other methods (the
remaining 26 points) and are outlined in Tab. 1 (see also TeleBar  for more
details).

$z$ |  $H(z)\,$ |  $\sigma_{H}$ |  $z\,$ |  $H(z)$ |  $\sigma_{H}$
---|---|---|---|---|---
0.070 |  69.0 |  19.6 |  0.4783 | 80 | 99
0.90 | 69 | 12 | 0.480 | 97 | 62
0.120 | 68.6 | 26.2 | 0.593 | 104 | 13
0.170 | 83 | 8 | 0.6797 | 92 | 8
0.1791 | 75 | 4 | 0.7812 | 105 | 12
0.1993 | 75 | 5 | 0.8754 | 125 | 17
0.200 | 72.9 | 29.6 | 0.880 | 90 | 40
0.270 | 77 | 14 | 0.900 | 117 | 23
0.280 | 88.8 | 36.6 | 1.037 | 154 | 20
0.3519 | 83 | 14 | 1.300 | 168 | 17
0.3802 | 83.0 | 13.5 | 1.363 | 160.0 | 33.6
0.400 | 95 | 17 | 1.430 | 177 | 18
0.4004 | 77.0 | 10.2 | 1.530 | 140 | 14
0.4247 | 87.1 | 11.2 | 1.750 | 202 | 40
0.4497 | 92.8 | 12.9 | 1.965 | 186.5 | 50.4
0.470 | 89 | 34 |  |  |

$z$ |  $H(z)\,$ |  $\sigma_{H}$ |  $z\,$ |  $H(z)$ |  $\sigma_{H}$
---|---|---|---|---|---
0.24 |  79.69 |  2.99 |  0.52 |  94.35 |  2.64
0.30 |  81.70 | 6.22 | 0.56 | 93.34 | 2.30
0.31 | 78.18 | 4.74 | 0.57 | 87.6 | 7.8
0.34 | 83.80 | 3.66 | 0.57 | 96.8 | 3.4
0.35 | 82.7 | 9.1 | 0.59 | 98.48 | 3.18
0.36 | 79.94 | 3.38 | 0.60 | 87.9 | 6.1
0.38 | 81.5 | 1.9 | 0.61 | 97.3 | 2.1
0.40 | 82.04 | 2.03 | 0.64 | 98.82 | 2.98
0.43 | 86.45 | 3.97 | 0.73 | 97.3 | 7.0
0.44 | 82.6 | 7.8 | 2.30 | 224.0 | 8.6
0.44 | 84.81 | 1.83 | 2.33 | 224 | 8
0.48 | 87.90 | 2.03 | 2.34 | 222.0 | 8.5
0.51 | 90.4 | 1.9 | 2.36 | 226.0 | 9.3

Table 1: 57 experimental points of $H(z)$ ($H$ is expressed in
$\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\sigma_{H}$ represents the uncertainty
for each data point).

The best fit is found by employing the statistical $R^{2}$-test, which is
defined by

$R^{2}\,=\,1-\frac{\sum_{i=1}^{57}\left[(H_{i})_{ob}-(H_{i})_{th}\right]^{2}}{\sum_{i=1}^{57}\left[(H_{i})_{ob}-(H_{i})_{mean}\right]^{2}}\,,$
(41)

where $(H_{i})_{ob}$ and $(H_{i})_{th}$ are the observed and predicted values
of Hubble’s parameter, respectively. Minimizing the departure of $|R^{2}-1|$
from zero gives the constraint $\Delta\simeq 0.07$ at $95\%$ confidence level.
This estimate is in line with recent predictions in literature
Anagnostopoulos:2020ctz , though being less stringent than that obtained via
Big Bang Nucleosynthesis measurements Barrow:2020kug . The fit in Fig. 12 also
allows us to infer $H_{0}=(65.02\pm 4.3)\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$,
which is close to the recent observation from Planck Collaboration
$H_{0}=\left(67.27\pm 0.60\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$ Planck , but
deviates from $H_{0}=\left(74.03\pm 1.42\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$
derived from 2019 SH0ES collaboration SH0ES .

Figure 12: Best fit curve of Hubble’s parameter $H$ versus $z$. Dots represent
data in Tab. 1, while the curve is the theoretical fit (online colors).

###  IV.2 Cosmological perturbations in BHDE

We now preliminarily investigate cosmological perturbations and structure
formation in BHDE. For this analysis, we are inspired by Dago . In particular,
we work in the linear regime on sub-horizon scales, studying the growth rate
of matter fluctuations for clustering dark matter and a homogeneous dark
energy component. More discussion along this line can be found in
Sheykhi:2022gzb . For the case of weakly interacting dark components and
scalar fluctuations of the metric in the Newtonian gauge, the line element (8)
is modified by including a potential term $\phi$ as in Mukha . By introducing
the density contrasts $\delta_{i}\equiv{\delta\rho_{i}}/{\rho_{i}}$ and
divergences of the fluid velocities
$\theta_{i}\equiv\vec{\nabla}\cdot\vec{v}_{i}$ for dark energy and dark
matter, the evolution equations for the perturbations in the Fourier space
take the form given in Abramo . We notice that DE effects on the growth of
perturbations are only appreciable for $c^{2}_{eff}\equiv\delta
p_{D}/\delta{\rho}_{D}\ll 1$, since in this case dark energy and dark matter
cluster in a similar way (adiabaticity condition) Eric . On the other hand,
$c^{2}_{eff}\simeq 1$ implies no growth because fluctuations would be
suppressed by pressure. We can use evolution equations along with the relation
$\frac{d}{dt}=\dot{a}\frac{d}{da}=aH\frac{d}{da}$ to extract differential
equations for dark energy and matter perturbations in the form222Here we
neglect anisotropic effects because of some technicalities when solving
evolution differential equations. This aspect will be considered in more
detail in future work.

$\displaystyle\frac{d^{2}\delta_{m}}{da^{2}}+A_{m}\frac{d\delta_{m}}{da}+B_{m}\delta_{m}$
$\displaystyle=$ $\displaystyle S_{m}\,,$ (42)
$\displaystyle\frac{d^{2}\delta_{D}}{da^{2}}+A_{D}\frac{d\delta_{D}}{da}+B_{D}\delta_{D}$
$\displaystyle=$ $\displaystyle S_{D}\,,$ (43)

where

$\displaystyle A_{m}$ $\displaystyle=$
$\displaystyle\frac{3}{2a}\left(1-\omega_{D}\Omega_{D}\right)\,,$ (44)
$\displaystyle B_{m}$ $\displaystyle=$ $\displaystyle 0\,,$ (45)
$\displaystyle S_{m}$ $\displaystyle=$
$\displaystyle\frac{3}{2a^{2}}\left[\Omega_{m}\delta_{m}+\left(1+3c_{eff}^{2}\right)\Omega_{D}\delta_{D}\right],$
(46) $\displaystyle A_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{a}\left[\frac{3}{2}\left(1-\omega_{D}\Omega_{D}\right)-\frac{a}{1+\omega_{D}}\frac{d\omega_{D}}{da}-3\omega_{D}\right],$
(47) $\displaystyle B_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}\left[3\left(\frac{1}{2}-\frac{3}{2}\omega_{D}\Omega_{D}-\frac{a}{1+\omega_{D}}\frac{d\omega_{D}}{da}-3c^{2}_{eff}\right)\right.$
(48)
$\displaystyle\left.\times\,\left(c^{2}_{eff}-\omega_{D}\right)-3a\frac{d\omega_{D}}{da}+\frac{k}{a^{2}H^{2}}c^{2}_{eff}\right],$
$\displaystyle S_{D}$ $\displaystyle=$
$\displaystyle\frac{3}{2a^{2}}\left(1+\omega_{D}\right)\left[\Omega_{m}\delta_{m}+\left(1+3c_{eff}^{2}\right)\Omega_{D}\delta_{D}\right],$
(49)

where $\Omega_{m}=\rho_{m}/\rho_{c}$ and $\Omega_{D}=\rho_{D}/\rho_{c}$ are
the fractional energy density parameters of dark matter and BHDE,
respectively, and $\rho_{c}=3m_{p}^{2}H^{2}$ the critical energy density.
Hhere, $k$ identifies the (sub-horizon) scale in the Fourier space.

To solve the above equations, we need to impose initial conditions. Concerning
$\delta_{m}$, perturbed Einstein equations lead to Dago

$\displaystyle\delta^{(in)}_{m}$ $\displaystyle=$
$\displaystyle-2\phi_{(in)}\left(1+\frac{k^{2}}{3a^{2}_{(in)}H^{2}_{(in)}}\right),$
(50) $\displaystyle\left(\frac{d\delta_{m}}{da}\right)^{(in)}$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{k^{2}\,\phi_{(in)}}{a^{2}_{(in)}H^{2}_{(in)}}\,.$
(51)

Similarly, the adiabaticity condition gives for $\delta_{D}$ Kodama

$\displaystyle\delta_{D}^{(in)}$ $\displaystyle=$
$\displaystyle\left(1+\omega_{D}\right)\delta_{m}^{(in)}\,,$ (52)
$\displaystyle\left(\frac{d\delta_{D}}{da}\right)^{(in)}$ $\displaystyle=$
$\displaystyle\left(1+\omega_{D}\right)\left(\frac{d\delta_{m}}{da}\right)^{(in)}+\frac{d\omega_{D}}{da}\delta_{m}^{(in)}\,.$
(53)

For homogeneous BHDE (i.e. $\delta_{D}=0$), the following evolution for the
linear matter perturbation on sub-horizon scales is obtained

$\frac{d^{2}\delta_{m}}{da^{2}}+\left(\frac{3}{a}+\frac{1}{E}\frac{dE}{da}\right)\frac{d\delta_{m}}{da}-\frac{3\left(1-\Omega_{D}\right)}{2a^{2}}\delta_{m}\,=\,0\,,$
(54)

where $E(a)$ is defined by

$E(a)\equiv\frac{H(a)}{H_{0}}\,.$ (55)

Now, from Eq. (54) we can calculate the growth rate of matter density
perturbations as

$f(a)\,\equiv\,a\,\frac{d\delta_{m}(a)}{da}\,.$ (56)

We compare the above function with measurements of $f\sigma_{8}(a)\equiv
f(a)\sigma_{8}(a)$ from redshift-space distortion observations for $0<z<1.5$
Boss , where $\sigma_{8}(a)=\sigma_{8}\delta_{m}(a)/\delta_{m}(1)$ is the
linear-density field fluctuations in $8h^{-1}\,\mathrm{Mpc}$ radius and
$\sigma_{8}$ its current value. We use the Gold-2017 dataset of 18
measurements of $f\sigma_{8}$ Nesseris , which are rescaled with respect to a
given fiducial cosmological model. We consider $\Lambda$CDM as fiducial
framework and introduce

$r(z)\,=\,\frac{H(z)\hskip 0.56905ptd_{A}(z)}{H_{fid}(z)d_{A,fid}(z)}\,,$ (57)

where

$H_{fid}(z)\,=\,H_{0}\sqrt{\Omega_{m0}\left(1+z\right)^{3}+\left(1-\Omega_{m0}\right)}\,,$
(58)

is the Hubble expansion rate in $\Lambda$CDM model. Measurements can then be
corrected by means of the vector

$\textbf{Y}\,=\,r(z_{i})\hskip 0.56905ptf\hskip
0.56905pt\sigma_{8}^{obs}(z_{i})-f\hskip 0.56905pt\sigma_{8}^{th}(z_{i})\,.$
(59)

Thus, the likelihood function is given by
$\mathcal{L}\,\propto\,e^{-\frac{1}{2}\hskip 0.56905pt\textbf{X}_{GRF}}$,
where
$\textbf{X}_{GRF}\,\equiv\,\textbf{Y}^{T}\,\textbf{C}_{GRF}^{-1}\,\textbf{Y}$
and $\textbf{C}_{GRF}$ is the covariance matrix of data listed in Tab. 2.

$z$ |  $f\sigma_{8}\,$ |  $z$ |  $f\sigma_{8}\,$
---|---|---|---
0.02 |  0.428 $\pm$ 0.0465 |  0.37 |  0.4602 $\pm$ 0.0378
0.02 |  0.398 $\pm$ 0.065 |  0.32 |  0.384 $\pm$ 0.095
0.02 |  0.314 $\pm$ 0.048 |  0.59 |  0.488 $\pm$ 0.060
0.10 |  0.370 $\pm$ 0.130 |  0.44 |  0.413 $\pm$ 0.080
0.15 |  0.490 $\pm$ 0.145 |  0.60 |  0.390 $\pm$ 0.063
0.17 |  0.510 $\pm$ 0.060 |  0.73 |  0.437 $\pm$ 0.072
0.18 |  0.360 $\pm$ 0.090 |  0.60 |  0.550 $\pm$ 0.120
0.38 |  0.440 $\pm$ 0.060 |  0.86 |  0.400 $\pm$ 0.110
0.25 |  0.3512 $\pm$ 0.0583 |  1.40 |  0.482 $\pm$ 0.116
Table 2: Dataset of 18 $f\sigma_{8}(z)$ measurements from different surveys
Nesseris .

In Fig. 13 we show the growth rate of matter fluctuations for the BHDE (blue
curve) model compared to the $\Lambda$CDM scenario (green curve). We see that
the BHDE curve is nearly overlapped with the $\Lambda$CDM one at low
redshifts. On the other hand, the two plots depart as $z$ increases, with the
BHDE curve fitting high-redshift data better than $\Lambda$CDM. In particular,
perturbations are found to grow up faster compared to predictions from the
standard Cosmology. Such a discrepancy at high redshift can be understood by
stressing that Barrow entropy is an effort to account for quantum-gravity
corrections on the horizon surface. Clearly, these effects are expected to
have more impact in the early Universe, when gravity should be quantum. A
similar result has been recently exhibited in Sheykhi:2022gzb , where the
faster growth of perturbations has been attributed to the exquisitely fractal
structure of Universe horizon in Barrow framework, which can potentially work
in favor of the growth of fluctuations of energy density.

Figure 13: Growth of matter perturbations in BHDE (blue curve) and
$\Lambda$CDM (green curve). We have set $\Delta=0.07$ and $\sigma_{8}=0.895$,
$\delta^{(in)}_{m}=10^{-2}$, $(d\delta_{m}/da)^{(in)}=1$ as initial conditions
Dago , while all other relevant parameters have been fixed as in Fig. 1
(online colors).

A final comment is in order here: to show the viability of a model, one needs
complementary constraints by combing different observables. Indeed, single
observational test cannot give solid sound result in principle. In this
regard, we emphasize that cosmological models attempting to include quantum
gravity effects in the standard Cosmology - like BHDE - are quite tough to
test experimentally at present time, since quantum gravity effects (and their
implications on the history of the Universe) are expected to be prominent in
the very early Universe, where observational data are still lacking or not
accurate enough. And in fact, constraints on Barrow entropy from relatively
recent astrophysical/cosmological phenomena give very tiny deviations of
Barrow parameter from zero, which in turn signal very small departure (if any)
from Boltzmann-Gibbs entropy at that age (see, for instance, Bar6 ;
Anagnostopoulos:2020ctz ; Barrow:2020kug ; LucianoInf ). In this sense, and
also motivated by the analysis of Sheykhi:2022gzb , we have then realized that
a useful test bench for Barrow proposal could be the study of growth rate
factor data of matter fluctuations and structure formation, where even tiny
gravitational effects might appear amplified as a result of the amplification
of primordial density fluctuations. To further support the viability of this
study, we remark that a similar constraint based on the growth of
perturbations has been proposed in Dago  for the case of Tsallis Holographic
Dark Energy. Clearly, one can still insist on parallel tests of Barrow model
with different observables. We stress that investigation along this direction
is very active in recent literature Bar6 ; Anagnostopoulos:2020ctz ;
Barrow:2020kug ; LucianoInf ; Lucianoarx ; Bar7 .

##  V Conclusions and Outlook

|  Free parameters |  Non-Interacting Model |  Observational value
---|---|---|---
$\hskip 13.65732pt\omega_{D_{0}}$ |  (See Fig. 1) |  $\hskip 41.25648pt[-1.07,-0.99]$ |  $[-1.38,-0.89]$ Planck
$q_{0}$ |  // |  $\hskip 41.25648pt[-1.35,-1.23]$ |  $[-1.37,-0.79]$ Camerana
$j$ |  // |  $j>0$ (for all $z$), $j(z\rightarrow-1)\rightarrow 1$ |  $j=1$ ($\Lambda$CDM)
$v_{s}^{2}$ |  // |  $v_{s}^{2}<0$ (for $\alpha_{0}=1.1,1.3$) $v_{s}^{2}>0$ (for $\alpha_{0}=1.5$) |  -
$\omega_{GT}-\omega^{\prime}_{GT}$ |  // |  freezing |  -
$H_{0}$ |  // |  $(65.02\pm 4.3)\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ |  $\left(67.27\pm 0.60\right)\mathrm{km\,s^{-1}\,Mpc^{-1}}$ Planck
Table 3: Theoretical and observational values of EoS parameter, deceleration
parameter, jerk parameter and squared sound speed for the best combination of
arbitrary parameters of BHDE in SBT (for the jerk parameter we have considered
the $\Lambda$CDM prediction as reference value).

In this work we have proposed a reconstruction of Barrow Holographic Dark
Energy in Saez-Ballester theory of gravity. Motivated by recent observations
from COBE, WMAP and Planck Ani0 ; Ani1 ; Ani , we have considered a Kantowski-
Sachs Universe filled with dark matter and anisotropic BHDE as a background.
To the best of our knowledge, this is the first time that BHDE is
reconstructed in an anisotropic background like Kantowski-Sachs Universe.
Indeed, all previous studies are framed in the standard homogeneous and
isotropic FRW Universe. In this sense, all the results obtained in the present
analysis are novel, as they account for anisotropic effects in the evolution
of the Universe. By assuming the Hubble radius as an IR cutoff, we have
investigated both the cases of non-interacting and interacting dark energy
models, with special focus on the calculation of skewness parameter, Equation-
of-State parameter, deceleration parameter, jerk parameter and squared sound
speed. Among the main advantages over other descriptions of dark energy, we
have shown that our model correctly reproduces the current accelerating phase
of the cosmos, in contrast to standard HDE. Furthermore, the study of the
squared speed of sound has revealed that our reconstruction is classically
stable throughout the whole evolution of the Universe for higher skewness. By
contrast, Saez Ballester-based reconstruction of Tsallis HDE, as well as non-
interacting BHDE in Brans-Dicke Cosmology, are always unstable. Moreover, we
have drawn the trajectories of $\omega_{D}-\omega^{\prime}_{D}$ phase plane
and discussed statefinder diagnosis for non-interacting mdoel. In order to
constrain free parameters, we have estimated current values of EoS parameter,
jerk parameter and deceleration parameter, and compared them with recent
measurements from Planck+WP+BAO. We have finally discussed the evolution of
Hubble’s parameter and growth rate of matter fluctuations in BHDE. Results are
summarized in Tab. 3, showing that SBT-based reconstruction of non-interacting
BHDE is observationally consistent for $1.1<\alpha_{0}<1.5$, with large values
of $\alpha_{0}$ being compatible with classical stability too. On the other
hand, large (negative) interactions $\beta$ are phenomenologically disfavored,
although they may contribute to stabilize the model against small
perturbations.

Several aspects remain to be investigated:

  * -

as discussed in Sec. III, our investigation correctly explains the behavior of
various model parameters and the current accelerated expansion of the cosmos,
though it does not predict its early-time decelerating phase. A possible
explanation is that we are overlooking some ordinary matter DoF, which mostly
contributed to the energy content in the early Universe and caused its initial
deceleration. We reserve to improve our analysis in future investigation.

  * -

In line with the study of LucianoPRD ; Mamon:2020spa ; Chakraborty:2021uzp ,
we aim at analyzing the thermodynamic implications of our model in order to
establish whether it is thermally stable. This is essential to understand if
SBT-based reconstruction of BHDE could clarify the yet unknown nature of DE.
In this regard, we remark that thermal stability has been studied in
LucianoPRD  by considering the heat capacities and compressibilities of both
interacting and non-interacting BHDE. It has been shown that such a model does
not satisfy the Thermal Stability Condition. Moreover, in Lucianoarx  the
generalized second law (GSL) of thermodynamics has been analyzed by also
including radiation effects in the energy budget of the Universe. Since the
total entropy variation is not necessarily a non-negative function, a
violation of the GSL can potentially occur. We emphasize that these results
are in line with the achievement of Achiev  for the case of Tsallis
Holographic Dark Energy and, more general, with the outcome of Outc , where it
has been found that DE fluids with a time-dependent EoS parameter are in
conflict with the physical constraints imposed by thermodynamics. All the
above results have been obtained for the usual FRW background. It would be
interesting to explore whether and how they appear in the case of anisotropic
spacetime, such as Kantowski-Sachs Universe considered here. This aspect is
under active investigation and will be presented elsewhere.

  * -

In Alleviate  Abdalla et al. have argued that some HDE models can alleviate
the $H_{0}$ tension since they predict $\omega_{D}<-1$, which seems a
necessary condition to provide a solution based on late-time modifications LT
. This requirement is met in both the two models considered above, potentially
giving some glimpses toward the resolution of the $H_{0}$ tension. On other
hand, in CapozLB  it has been argued that the Hubble tension could be somehow
removed if the _look-back time_ is correctly referred to the redshift where
the measurement is performed. This in turn rests upon the usage of the correct
definition of $E(z)=H(z)/H_{0}$ in terms of the different contributions of
radiation, matter, cosmological constant and spatial curvature to the density
budget of the Universe. In this way, it has been shown that both values of the
Hubble constant $H_{0}$ reported by the SH0ES and Planck collaborations
Alleviate ; Khodadi  can be recovered. It is suggestive to explore whether a
similar dynamical resolution of the Hubble tension can be obtained in the
context of BHDE. Although we are here working with a fixed IR cutoff, we
envisage that some sort of correspondence between BHDE and look-back time
approach can be still established by allowing Barrow parameter $\Delta$ to be
varying on time. Preliminary studies along this direction have been presented
recently in DiGennaro . More work along this direction is inevitably needed.

  * -

BHDE is a generalization of standard Cosmology based on a deformation of the
horizon entropy of the Universe. Extended cosmological scenarios, however, can
also be obtained by modifying the geometric (i.e. gravitational) sector of
Einstein-Hilbert action or motivated by thermodynamic requirements over the
cosmological kinematics. In these directions, interesting approaches are
provided by the Extended Gravity Cosmography EGC , which is a model-
independent framework to tackle the dark energy/modified gravity problem, and
the thermodynamic parametrization of dark energy TPDE . Hence, a possible
outlook is to study BHDE in parallel with such generalized approaches and
possibly reinterpret Barrow’s conjecture in this alternative language.

  * -

In Sec. IV.2 we have discussed cosmological perturbations and structure
formation neglecting anisotropy of spacetime. Clearly, a more comprehensive
analysis requires including this feature too.

  * -

A further challenging perspective is to extend the present analysis to the
case of HDE based on Kaniadakis entropy Kana1  (Kaniadakis Holographic Dark
Energy, KHDE), which is a self-consistent relativistic generalization of
Boltzmann-Gibbs entropy parameterized by $-1<K<1$ GioKan ; LucRev , or HDE
built on other commonly used entropies in physics, such as Abe, Landsberg-
Vedral, Sharma-Mittal and Rény entropies. In this context, it would be
interesting to find any connection between BHDE and such other models, and
possibly constrain deformation parameters of these alternative modified
entropies.

  * -

A possible candidate for DE has been recently proposed in Capol1 ; Capol2 ;
Capol3  by looking at the properties of the vacuum condensate of flavor mixed
fields, in particular neutrinos Blas95 ; Luc23 . Although the origin of this
alternative explanation is rooted in particle physics, it would be interesting
to discuss these results in connection with our model of HDE. This may also
allows us to extend the paradigm of Barrow entropy to the framework of
particle physics.

  * -

Finally, since our model attempts to include quantum gravitational effects
into HDE, it is important to analyze our results in connection with
predictions of more fundamental candidate theories of quantum gravity, such as
String Theory, Loop Quantum Gravity and Asymptotically Safe Gravity, or
phenomenological approaches, such as Planck-scale deformations of the
Heisenberg principle KMM ; Scardigli ; LucGUP .

Work along these and other directions is still in progress and results will be
presented elsewhere.

###### Acknowledgements.

The author acknowledges S. Odintsov for comments on the original manuscript.
He is also grateful to the Spanish “Ministerio de Universidades” for the
awarded Maria Zambrano fellowship and funding received from the European Union
- NextGenerationEU. He finally expresses his gratitude for participation in
the COST Action CA18108 “Quantum Gravity Phenomenology in the Multimessenger
Approach” and LISA Cosmology Working group.

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