# Halos of dark energy

P.P. Avelino  Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
<EMAIL_ADDRESS>Instituto de Astrofísica e Ciências do Espaço,
Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, PT4150-762
Porto, Portugal

###### Abstract

We investigate the properties of dark energy halos in models with a nonminimal
coupling in the dark sector. We show, using a quasistatic approximation, that
a coupling of the mass of dark matter particles to a standard quintessence
scalar field $\phi$ generally leads to the formation of dark energy
concentrations in and around compact dark matter objects. These are associated
with regions where scalar field gradients are large and the dark energy
equation of state parameter is close to $-1/3$. We find that the energy and
radius of a dark energy halo are approximately given by $E_{\rm
halo}\sim\boldsymbol{\beta}^{2}\varphi\,m$ and $r_{\rm
halo}\sim\sqrt{\boldsymbol{\beta}\,\varphi({R}/{H})}$, where
$\varphi=Gm/(Rc^{2})$, $m$ and $R$ are, respectively, the mass and radius of
the associated dark matter object, $\boldsymbol{\beta}=-(8\pi G)^{-1/2}d\ln
m/d\phi$ is the nonminimal coupling strength parameter, $H$ is the Hubble
parameter, $G$ is the gravitational constant, and $c$ is the speed of light in
vacuum. We further show that current observational limits on
$\boldsymbol{\beta}$ over a wide redshift range lead to stringent constraints
on $E_{\rm halo}/m$ and, therefore, on the impact of dark energy halos on the
value of the dark energy equation of state parameter. We also briefly comment
on potential backreaction effects that may be associated with the breakdown of
the quasistatic approximation and determine the regions of parameter space
where such a breakdown might be expected to occur.

##  I Introduction

In general relativity an exotic dark energy (DE) fluid Frieman et al. (2008);
Caldwell and Kamionkowski (2009); Li et al. (2011) dominating the energy
density of the Universe is required in order to explain the recent
acceleration of the expansion of the Universe Alam et al. (2017); Scolnic et
al. (2018); Aghanim et al. (2020). Various observations also suggest that
matter in the Universe is primarily nonbaryonic and dark Cooke et al. (2018);
Aghanim et al. (2020). Despite their importance for the dynamics of the
Universe, the physical nature of nonbaryonic dark matter (DM) and DE remains
largely unknown. In particular, it is not known whether or not DM and DE are
nonminimally coupled Wetterich (1995); Amendola (2000); Zimdahl and Pavon
(2001); Farrar and Peebles (2004), or even if they could be associated with a
single DE fluid Avelino et al. (2003, 2008). It is also possible that general
relativity may not provide an accurate description of gravity on cosmological
scales, and that DE and/or DM could be manifestations of modified gravity
Nojiri and Odintsov (2011); Clifton et al. (2012); Avelino et al. (2016).

The coupling of the mass of DM particles to a DE scalar field Amendola (2000);
Zimdahl and Pavon (2001); Farrar and Peebles (2004) has been shown to give
rise to DE mediated fifth forces between DM particles as well as velocity
dependent forces. These forces can affect the linear growth of cosmological
perturbations Pettorino and Baccigalupi (2008); Caldera-Cabral et al. (2009);
Baldi (2011), and may also play a crucial role on nonlinear scales with a
potential impact on the dynamics of galaxies and clusters of galaxies Manera
and Mota (2006); Mainini and Bonometto (2006); Abdalla et al. (2009, 2010);
Avelino and Barreira (2012); Baldi and Salucci (2012); Cui et al. (2012).
Also, a nonminimal coupling between DM and DE has been claimed to alleviate
some cosmic tensions Ade et al. (2016); Di Valentino et al. (2017); Barros et
al. (2019); Yang et al. (2018); Pan et al. (2019); Di Valentino et al. (2020),
including the apparent discrepancy between local and high redshift constraints
on the value of the Hubble parameter, making this a promising avenue of
research.

Although many studies do not explicitly consider them, local variations in the
DE have been shown to play an important role in the context of growing
neutrino models, with a potentially significant backreaction on the background
evolution of the Universe Ayaita et al. (2012, 2013); Casas et al. (2016). It
has also been shown that the dynamics of DM particles nonminimally coupled to
a DE field may be affected by backreaction effects with a possible impact on
structure formation and on the large scale dynamics of the Universe Avelino
(2015).

This paper aims to characterize the properties of DE halos resulting from a
nonminimal coupling between DM and a DE scalar field. We shall work in the
context of the quasistatic approximation, which essentially consists in
neglecting terms involving time derivatives in the perturbed field equations.
This approximation has been previously used to investigate local variations of
the fine-structure constant inside virialized objects in the context of a DE
model with a nonminimal coupling to the electromagnetic field Avelino et al.
(2006). It has also been frequently used in the context of other DE and
modified gravity scenarios Noller et al. (2014); Bose et al. (2015); Sawicki
and Bellini (2015); Pace et al. (2021). The quasistatic approximation is
particularly useful when considering small subhorizon scales, since the
dynamics of perturbation modes with a wave number significantly smaller than
the Hubble radius is in general expected to be dominated by the terms
containing spatial derivatives in the equations of motion. Under these
conditions, the quasistatic approximation may significantly simplify the
perturbation equations, and allow for an analytical treatment.

The outline of this paper is as follows. In Sec. II we describe a generic
family of quintessence models with a nonminimal coupling to the DM and derive
the corresponding equations of motion. In Sec. III we use the quasistatic
approximation to compute the scalar field perturbations around compact DM
objects, discussing the conditions required for its applicability. In Sec. IV
we estimate the energy and radius of a DE halo surrounding a compact DM
object, as well as the corresponding DE equation of state parameter. We also
provide a lower bound to the contribution of DE perturbations inside compact
DM objects. Finally, we discuss the implications of our results and conclude
in Sec. V.

Throughout this paper we use units where the speed of light in vacuum is equal
$c=1$. We also adopt the metric signature $(-,+,+,+)$. The Einstein summation
convention will be used when a greek index appears twice in a single term,
once in an upper (superscript) and once in a lower (subscript) position.

##  II Nonminimally interacting dark sector

In this paper we consider a class of models for the dark sector with a
nonminimal coupling between DM and a standard quintessence scalar field (which
plays a DE role). These models are described by the action

$S=\int d^{2}x\,\sqrt{-g}\,{\mathcal{L}}\,,$ (1)

where the Lagrangian $\mathcal{L}$ is given by

${\mathcal{L}}={\mathcal{L}}_{\rm DE}+{\mathcal{L}}_{\rm DM}\,,$ (2)

with

$\displaystyle{\mathcal{L}}_{\rm DE}$ $\displaystyle=$ $\displaystyle
X-V(\phi)\,,$ (3) $\displaystyle{\mathcal{L}}_{\rm DM}$ $\displaystyle=$
$\displaystyle f(\phi){\mathcal{L}}_{\rm DM*}\,.$ (4)

Here $X=-\nabla^{\mu}\phi\nabla_{\mu}\phi/2$ is a standard kinetic term,
$V(\phi)$ is the scalar field potential,

${\mathcal{L}}_{\rm DM}=f(\phi){\mathcal{L}}_{\rm DM*}$ (5)

is the nonminimally coupled DM Lagrangian (${\mathcal{L}}_{\rm DM*}$ denoting
the minimally coupled DM Lagrangian). The components of the DM and DE energy-
momentum tensors are given by

$\displaystyle T^{\mu\nu}_{\rm DM}$ $\displaystyle=$
$\displaystyle\frac{2}{{\sqrt{-g}}}\frac{\delta({\sqrt{-g}}{\mathcal{L}}_{\rm
DM})}{\delta g_{\mu\nu}}=f(\phi)T^{\mu\nu}_{\rm DM*}\,,$ (6) $\displaystyle
T^{\mu\nu}_{\rm DE}$ $\displaystyle=$
$\displaystyle\frac{2}{{\sqrt{-g}}}\frac{\delta({\sqrt{-g}}{\mathcal{L}}_{\rm
DE})}{\delta g_{\mu\nu}}$ (7) $\displaystyle=$
$\displaystyle\nabla^{\mu}\phi\nabla^{\nu}\phi+g^{\mu\nu}{\mathcal{L}}_{\rm
DE}\,,$

where $g=\det(g_{\mu\nu})$, $g_{\mu\nu}$ are the components of the metric
tensor, and

$T^{\mu\nu}_{\rm
DM*}=\frac{2}{{\sqrt{-g}}}\frac{\delta({\sqrt{-g}}{\mathcal{L}}_{\rm
DM*})}{\delta g_{\mu\nu}}\,.$ (8)

As a consequence of the nonminimal coupling to the quintessence scalar field,
the mass $m$ of a DM particle or a compact DM object is a function of $\phi$
with

$m(\phi)=f(\phi)m_{*}\,,$ (9)

where $m_{*}$ is the mass that the particle would have if $f(\phi)=1$.

###  II.1 Scalar field dynamics in a flat Friedmann-Lemaître-Robertson-Walker
universe

Consider a flat homogeneous and isotropic universe, described by the
Friedmann-Lemaître-Robertson-Walker metric. The corresponding line element is
given by

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt^{2}+a^{2}[t]d\vec{q}\cdot d\vec{q}$ (10) $\displaystyle=$ $\displaystyle-
dt^{2}+a^{2}[t]\left(dq^{2}+q^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right)\,,$

where $a[t]$ is the scale factor, $t$ is the physical time, $\vec{q}$ are
comoving Cartesian coordinates, and $(q,\theta,\phi)$ are comoving spherical
coordinates. If the DM particles are nonrelativistic, then the dynamics of the
nonminimally coupled quintessence scalar field $\phi[t,\vec{q}\,]$ is given by
Avelino (2015)

$\Box\phi=-{\ddot{\phi}}-3H{\dot{\phi}}+\nabla^{2}\phi=\alpha-\beta\sum_{i}m_{i}\delta^{3}[\vec{r}-{\vec{r}}_{i}]\,,$
(11)

where $\Box\equiv\nabla_{\mu}\nabla^{\mu}$ is the d’Alembertian, a dot
represents a derivative with respect to the physical time $t$,
$\vec{r}=a\vec{q}$, $m_{i}$ and $\vec{r}_{i}=a\vec{q}_{i}$ are, respectively,
the masses and positions of the DM particles,
$\nabla^{2}\phi\equiv\nabla_{\vec{q}}^{2}\,\phi/a^{2}$,
$\delta^{3}[\vec{r}\,]$ is the three-dimensional Dirac delta function, and

$\displaystyle\alpha[\phi]$ $\displaystyle\equiv$
$\displaystyle\frac{dV}{d\phi}\,,$ (12) $\displaystyle\beta[\phi]$
$\displaystyle\equiv$ $\displaystyle-\frac{d\ln m}{d\phi}=-\frac{d\ln
f}{d\phi}\,.$ (13)

##  III Quasistatic approximation

Consider a single compact DM object of mass $m$ comoving with the expansion of
the Universe at ${\vec{r}}={\vec{0}}$. In the quasistatic approximation the
field $\phi$ may be written as
$\phi[t,{\vec{r}}\,]={\bar{\phi}}[t]+\delta\phi$, with

$\displaystyle{\ddot{\bar{\phi}}}+3H{\dot{\bar{\phi}}}$ $\displaystyle=$
$\displaystyle-\alpha[\bar{\phi}]\,,$ (14) $\displaystyle\nabla^{2}\delta\phi$
$\displaystyle=$
$\displaystyle-\beta[\bar{\phi}]m[\bar{\phi}]\delta^{3}[\vec{r}\,]\,,$ (15)

where Eq. (14) implies that $\bar{\phi}$ is a function only of the physical
time ($\bar{\phi}=\bar{\phi}[t]$). This is expected to be a good approximation
as long as

$\displaystyle|\dot{\delta\phi}|$ $\displaystyle\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$
$\displaystyle|{\dot{\bar{\phi}}}|\,,$ (16) $\displaystyle|\delta\phi|$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$
$\displaystyle\left|\frac{d\ln\alpha}{d\phi}\right|^{-1}\,,$ (17)
$\displaystyle|\delta\phi|$ $\displaystyle\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$
$\displaystyle\left|\frac{d\ln(\beta m)}{d\phi}\right|^{-1}\,.$ (18)

The nonzero components of the energy-momentum tensor of the DE field are given
approximately by

$\displaystyle{T^{t}}_{t}$ $\displaystyle=$
$\displaystyle{T^{t}}_{t}+\delta{T^{t}}_{t}\,,$ (19)
$\displaystyle{T^{t}}_{q}$ $\displaystyle=$
$\displaystyle{T^{t}}_{q}+\delta{T^{t}}_{q}\,,$ (20)
$\displaystyle{T^{q}}_{q}$ $\displaystyle=$
$\displaystyle{T^{q}}_{q}+\delta{T^{q}}_{q}\,,$ (21)
$\displaystyle{T^{\theta}}_{\theta}$ $\displaystyle=$
$\displaystyle{T^{\theta}}_{\theta}+\delta{T^{\theta}}_{\theta}\,,$ (22)
$\displaystyle{T^{\phi}}_{\phi}$ $\displaystyle=$
$\displaystyle{T^{\theta}}_{\theta}\,,$ (23)

with

$\displaystyle{T^{t}}_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\dot{\bar{\phi}}^{2}-V[\bar{\phi}]\,,$ (24)
$\displaystyle{T^{t}}_{q}$ $\displaystyle=$ $\displaystyle 0\,,$ (25)
$\displaystyle{T^{q}}_{q}$ $\displaystyle=$
$\displaystyle{T^{\theta}}_{\theta}[\bar{\phi}]={T^{\phi}}_{\phi}[\bar{\phi}]=\frac{1}{2}\dot{\bar{\phi}}^{2}-V[\bar{\phi}]\,,$
(26)

and

$\displaystyle\delta{T^{t}}_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\delta\phi)^{\prime
2}-\alpha[\bar{\phi}]\delta\phi\,,$ (27) $\displaystyle\delta{T^{t}}_{q}$
$\displaystyle=$ $\displaystyle-a\dot{\bar{\phi}}\,(\delta\phi)^{\prime}\,,$
(28) $\displaystyle\delta{T^{q}}_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta\phi)^{\prime
2}-\alpha[\bar{\phi}]\delta\phi\,,$ (29)
$\displaystyle\delta{T^{\theta}}_{\theta}$ $\displaystyle=$
$\displaystyle\delta{T^{\phi}}_{\phi}=-\frac{1}{2}(\delta\phi)^{\prime
2}-\alpha[\bar{\phi}]\delta\phi\,,$ (30)

where $(\delta\phi)^{\prime}\equiv a^{-1}\partial\phi/\partial q$.

Let us also compute the proper density and pressure associated with the
background evolution of the scalar field $\bar{\phi}[t]$,

$\displaystyle{\bar{\rho}}_{\rm DE}$ $\displaystyle=$
$\displaystyle-{T^{t}}_{t}=\frac{1}{2}\dot{\bar{\phi}}^{2}+V[\bar{\phi}]\,,$
(31) $\displaystyle{\bar{p}}_{\rm DE}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left({T^{q}}_{q}+{T^{\theta}}_{\theta}+{T^{\phi}}_{\phi}\right)=\frac{1}{2}\dot{\bar{\phi}}^{2}-V[\bar{\phi}]\,,$
(32)

and define the current values of the corresponding equation of state and
fractional energy density parameters:

$\displaystyle\bar{w}$ $\displaystyle\equiv$ $\displaystyle{\bar{p}}_{\rm
DE}/{\bar{\rho}}_{\rm DE}\,,$ (33) $\displaystyle{\bar{\Omega}}_{\rm DE}$
$\displaystyle\equiv$ $\displaystyle\frac{8\pi G{\bar{\rho}}_{\rm
DE}}{3H^{2}}\,.$ (34)

Notice that

$\displaystyle{\dot{\bar{\phi}}}^{\,2}$ $\displaystyle=$
$\displaystyle(1+\bar{w}){\bar{\rho}}_{\rm
DE}=\frac{3H^{2}(1+\bar{w}){\bar{\Omega}}_{\rm DE}}{8\pi G}\,,$ (35)
$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle(1-\bar{w}){\bar{\rho}}_{\rm
DE}=\frac{3H^{2}(1-\bar{w}){\bar{\Omega}}_{\rm DE}}{8\pi G}\,.$ (36)

###  III.1 Quasistatic solution

The quasistatic solution to Eq. (15) outside a compact DM object of radius $R$
is given by

$\delta\phi[t,r]=\delta\phi[t,R]\frac{R}{r}\,,\qquad r=|\vec{r}\,|\geq R\,,$
(37)

where

$\delta\phi[t,R]=\frac{\beta m}{4\pi R}\,.$ (38)

The DE flux towards the interior of a sphere of radius $r\geq R$ centered on
the DM compact object is

$\displaystyle-4\pi
r^{2}{\dot{\bar{\phi}}}\,(\delta\phi)^{\prime}=\beta\,{\dot{\bar{\phi}}}\,m={\dot{m}}\,,$
(39)

thus accounting for the change of the object’s mass.

Calculating the partial derivative of $\delta\phi$ with respect to the
physical time $t$ (at fixed $q=r/a[t]$) one obtains

$\dot{\delta\phi}=\left(\frac{d\ln(\beta
m)}{d\phi}\dot{\bar{\phi}}-H\right)\delta\phi\,,$ (40)

where $H\equiv\dot{a}/a$ is the Hubble parameter. The conditions given in Eqs.
(16) and (18) thus imply that the quasistatic approximation is expected to be
a good approximation for

$(\dot{\delta\phi})^{2}\sim H^{2}\delta\phi^{2}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}{\dot{\bar{\phi}}}^{\,2}\,,$ (41)

or, equivalently,

$\frac{r}{R}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}2\frac{\boldsymbol{\beta}^{2}}{|k|}\varphi\,.$ (42)

Here, $\boldsymbol{\beta}$ is the nonminimal coupling strength parameter
defined by

$\boldsymbol{\beta}\equiv\frac{\beta}{\sqrt{8\pi G}}\,,$ (43)
$k[a]\equiv\frac{d\ln m}{d\ln
a}=-\frac{\beta\dot{\bar{\phi}}}{H}=-\boldsymbol{\beta}\sqrt{3(1+\bar{w}){\bar{\Omega}}_{\rm
DE}}$ (44)

is a related cosmological coupling strength parameter also used often in the
literature, and

$\varphi\equiv\frac{Gm}{R}$ (45)

is the Newtonian gravitational potential at the surface of the compact DM
object. Equation (42) implies that the condition

$|k|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}2\boldsymbol{\beta}^{2}\varphi$ (46)

is required in order to ensure that the quasistatic approximation is valid for
all $r>R$.

##  IV DE halos

The energy density fluctuations of the quintessence scalar field
$\delta\rho=-\delta{T^{t}}_{t}=(\delta\phi)^{\prime
2}/2+\alpha[\bar{\phi}]\delta\phi$ are the sum of two contributions: (1) the
one directly associated with the scalar field gradients $(\delta\phi)^{\prime
2}/2\propto r^{-4}$ and (2) the one associated with spatial variations of the
potential energy $\alpha[\bar{\phi}]\delta\phi\propto r^{-1}$. The first one
provides a contribution that is essentially localized in and around the
particle, while the second one is strongly dependent on how one defines the
halo radius $r_{\rm halo}$. In fact, the energy perturbation inside a sphere
of radius $r_{\rm halo}$ centered on the compact DM object associated with
contribution 2 is given by

$\displaystyle\delta E_{V}[r_{\rm halo}]$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r_{\rm halo}}\alpha[\bar{\phi}]\delta\phi r^{2}dr$ (47)
$\displaystyle=$ $\displaystyle 2\pi\alpha[\bar{\phi}]\delta\phi[t,R]Rr_{\rm
halo}^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left(\frac{\alpha[\bar{\phi}]\delta\phi[t,R]}{V[\bar{\phi}]}\right)\frac{R}{r_{\rm
halo}}E_{V},,$

where $E_{V}=4\pi r_{\rm halo}^{3}V[\bar{\phi}]/3$. Hence, if
$|\alpha[\bar{\phi}]\delta\phi[t,R]|\ll V[\bar{\phi}]$ or $R\ll r_{\rm halo}$,
then this contribution may be neglected. Therefore, in this paper we shall
focus only on the contribution directly associated with the scalar field
gradients.

###  IV.1 Halo energy

The total energy associated with the scalar field gradients outside a sphere
of radius $R$ centered at $r=0$ is given by

$E^{\rm out}_{\rm G}[t]=2\pi\int_{R}^{\infty}(\delta\phi)^{\prime
2}r^{2}dr=\boldsymbol{\beta}^{2}\varphi\,m\,,$ (48)

which will be our best estimate of the total energy of a DE halo ($E_{\rm
halo}\sim E^{\rm out}_{\rm G}$). For $\varphi\equiv Gm/R\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1$
this result should be taken as a rough estimate, since the impact of the local
gravitational field on the dynamics of the quintessence scalar field has not
been taken into account.

###  IV.2 Halo radius

The radius of a DE halo may be determined by defining the halo as the region
where the energy density associated with the scalar field gradients exceeds
the background energy density or, equivalently, $(\delta\phi)^{\prime
2}/2>3H^{2}/(8\pi G)$. This happens for values of $r$ smaller than

$r_{\rm
halo}[t]=\left(\sqrt{\frac{2}{3}}\boldsymbol{\beta}\,\varphi\frac{R}{H}\right)^{1/2}\,.$
(49)

Here we implicitly assume that $R<\sqrt{2/3}\,\boldsymbol{\beta}\varphi
H^{-1}$ or, equivalently, that $r_{\rm halo}>R$.

###  IV.3 Equation of state

Within a DE halo the DE equation of state parameter,

$w_{\rm DE}\equiv\frac{p_{\rm DE}}{\rho_{\rm DE}}=\frac{{\bar{p}}_{\rm
DE}+{\delta p}_{\rm DE}}{{\bar{\rho}}_{\rm DE}+{\delta\rho}_{\rm DE}}\,,$ (50)

is dominated by the contribution of the scalar field gradients to the DE
proper density and pressure. These satisfy $\delta p_{\rm
G}=-(\delta\phi)^{\prime 2}/6=-\delta\rho_{\rm G}/3$ or, equivalently,

$w_{\rm G}\equiv\frac{\delta p_{\rm G}}{\delta\rho_{\rm G}}=-\frac{1}{3}\,.$
(51)

Hence, the DE equation of state parameter is expected to be close to $-1/3$
within an halo, especially in its central regions.

###  IV.4 DE perturbations inside compact DM objects

The knowledge of the density profile inside a compact DM object would be
required in order to accurately compute the energy $E^{\rm in}_{\rm G}$
associated with the scalar field gradients for $r<R$ — notice that this
contribution would vanish if and only if all the mass was located at the
surface of the object, which would be utterly unrealistic. Here we will
provide a rough estimate of $E^{\rm in}_{\rm G}$, assuming that $Gm/r$ is
always significantly smaller than unity and that the DM energy density
distribution for $r<R$ is uniform. In this case, the quasistatic solution to
Eq. (15) is given by

$\delta\phi[t,r]=\frac{3}{2}\delta\phi[t,R]\left(1-\frac{1}{3}\left(\frac{r}{R}\right)^{2}\right)\,.$
(52)

The total energy directly associated to the scalar field gradients inside the
compact DM object would then be equal to

$E^{\rm in}_{\rm G}[t]=2\pi\int_{0}^{R}(\delta\phi)^{\prime
2}r^{2}dr=\frac{1}{5}E^{\rm out}_{\rm G}[t]\,,$ (53)

so that $\sim 17\%$ of the energy associated with the scalar field gradients
would be located inside the DM object. In general one would expect the DM
energy density to be a decreasing function of $r$. Therefore, this should be
regarded as a lower limit on $E^{\rm in}_{\rm G}$.

##  V Discussion and Conclusions

In this paper we characterized the properties of DE concentrations that are
expected to form in and around compact DM objects as a consequence of a
coupling of the mass of DM particles to a standard quintessence scalar field.
We determined the dependence of the energy and radius of a DE halo on the
nonminimal coupling strength and Hubble parameters, and on the mass and radius
of the associated compact DM object. We have also shown that deep inside a DE
halo the equation of state parameter is close to -1/3 and estimated the
contribution of the DE energy perturbations inside DM objects.

Although the results presented in this paper were obtained considering a
single comoving compact DM object, they should also hold in the case of a
network of nonrelativistic DM objects. However, in this case the background
evolution of the DE scalar field will be affected by the energy transfer
between DE to DM. Although this effect needs to be considered for an accurate
characterization of the background dynamics of the DE scalar field, it is not
expected to significantly change our main results. Also, our results were
obtained in the context of the quasistatic approximation, assuming that local
contributions to the evolution of the mass of dark matter particles can be
neglected. If that is not the case a breakdown of the quasistatic
approximation is expected, which can be associated with a significant transfer
of linear-momentum between moving DM particles and the DE scalar field Avelino
(2015).

Scalar field gradients associated with the coupling of the mass of DM
particles to a DE scalar field give rise to attractive fifth forces between DM
particles whose strength is equal to $2\boldsymbol{\beta}^{2}$ times that of
gravitational forces. In addition to these, energy-momentum conservation in
general relativity requires that any change of the proper mass of a compact DM
object associated with the nonminimal coupling to the DE scalar field should
be compensated by a corresponding decrease of its speed with respect to the
local cosmological frame, assuming that backreaction effects associated with a
significant transfer of linear momentum from moving DM particles to the DE
scalar field can be neglected (which is expected to be the case as long as the
quasistatic approximation holds everywhere). This effect essentially changes
the strength of the velocity dependent cosmological damping of the speed of
compact DM objects by a factor of $1+k$.

The velocity dependent forces and fifth forces in coupled DE energy models
have an impact on the growth of cosmic structures which can be constrained
observationally. A tomographic analysis of coupled DE has been performed in
Goh et al. (2022), where a significant redshift dependence of the constraints
on the coupling strength has been found (assuming, for simplicity, a non-
negative coupling strength parameter): $\boldsymbol{\beta}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.1$
at $z<5$, and $\boldsymbol{\beta}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.05$ in the
redshift range $5<z<500$, while $\boldsymbol{\beta}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.02$
for a constant coupling case (at $68\%$ confidence level). This imposes
stringent constraints on the ratio between the energy of a DE halo and the
mass of the associated compact DM object: $E_{\rm halo}/m\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}0.01\varphi$ at $z<5$. It also precludes any substantial
contribution (i.e., exceeding a subpercent level) of DE halos to the DE
equation of state parameter. Whether these limits can be relaxed by
considering broader families of coupled DE models will be the subject of
future work.

###### Acknowledgements.

We thank Lara Sousa, Rui Azevedo, Vasco Ferreira, and David Grüber for many
enlightening discussions. We acknowledge the support by Fundação para a
Ciência e a Tecnologia (FCT) through the research Grants No. UIDB/04434/2020
and No. UIDP/04434/2020. This work was also supported by FCT through the
R$\&$D project 2022.03495.PTDC - Uncovering the nature of cosmic strings.

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# Holographic dark energy with Granda-Oliveros cutoff and ansatz based
approach : An extended look

Oem Trivedi 1 and Maxim Khlopov2,3,4 oem.t<EMAIL_ADDRESS>
(1International Centre for Space Sciences and Cosmology, Ahmedabad University,
Ahmedabad 380009, India
2Research Institute of Physics, Southern Federal University, 344090 Rostov-on-
Don, Russia
3Virtual Institute of Astroparticle Physics, 75018 Paris, France
4Center for Cosmoparticle Physics Cosmion, National Research Nuclear
University “MEPHI”, 115409 Moscow, Russia
)

###### Abstract

Holographic dark energy models have proven to be a very interesting way to
study various aspects of late-time acceleration of the universe. In this work
we extensively study HDE models with the Granda-Oliveros cutoff with an ansatz
based approach. We consider the Tsallis, Barrow and PLEC HDE models in this
regard and consdier simple power law, emergent universe, intermediate and
logamediate forms of for the universe. Studying various cosmologically
interesting parameters alongside the thermodynamical aspects in these models,
we show that the Logamediate models are the best fit out of the other
possibilites, followed by the emergent universe model, intermediate model and
the simple power law models at the very last in terms of feasibility.

##  1 Introduction

The discovery of the Universe’s late-time acceleration marked a significant
breakthrough in cosmology [1]. Since then, extensive research has been devoted
to understanding this expansion phenomenon. Various approaches have been
explored, encompassing conventional methods like the Cosmological constant [2,
3, 4], unconventional theories like Modified gravity [5, 6, 7], and scenarios
involving scalar fields driving late-time cosmic acceleration [8, 9, 10, 11,
12, 13, 14]. Additionally, quantum gravity theories such as Braneworld
cosmology in string theory, loop quantum cosmology, and asymptotically safe
cosmology have contributed to addressing the cosmic-acceleration puzzle [15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. However, discrepancies persist,
exemplified by the famous Hubble tension, reflecting the limitations of our
current understanding of the universe, as evidenced by conflicting
measurements of the Hubble constant [26, 27, 28]. Thus, the present epoch of
the universe poses profound questions, promising advancements in gravitational
physics to deepen our comprehension of cosmology.

Among the various proposed solutions for the issues of late time acceleration
is the idea of appplying the notions of the holographic principle [29, 30]to
cosmology. This principle posits that a system’s entropy is determined not by
its volume but by its surface area [31]. Such an exotic form of DE also piques
interest given the recent results from DESI [32, 33, 34] which suggest a
departure from $\Lambda$CDM cannot be completely ruled out. The initial work
on HDEs carried out in [35] proposed through a QFT approach that a short-
distance cutoff is linked to a long-distance cutoff due to black hole
formation limitations. Specifically, if $\rho$ denotes the quantum zero-point
energy density from a short-distance cutoff, the total energy within a region
of size $L$ should not exceed the mass of a black hole of the same size,
yielding the inequality $L^{3}\rho\leq LM_{\text{pl}}^{2}$. The maximum
allowable value for the infrared cutoff ($L_{{}_{\text{IR}}}$) precisely
satisfies this inequality, leading to the relationship:

$\rho=3c^{2}L_{{}_{\text{IR}}}^{-2},$ (1)

where $c$ is an arbitrary parameter, and we are using $m_{p}=1$ units.

This principle has found widespread application in cosmology, particularly in
elucidating the late-time dark energy era, termed holographic dark energy
(HDE) (for a comprehensive review, see [36]). In this framework, the infrared
cutoff, $L_{{}_{\text{IR}}}$, is cosmologically grounded and serves as the IR
cutoff for a specific HDE model. Numerous studies have explored holographic
dark energy from diverse perspectives in recent years [37, 38, 39, 40, 36, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Over
recent decades, several alternative forms of HDE have been proposed. For
instance, Tsallis HDE models incorporate Tsallis’ corrections to the standard
Boltzmann-Gibbs entropy, resulting in the equation

$\rho_{\Lambda}=3c^{2}L^{-(4-2\sigma)}$ (2)

, where $\sigma$ is the Tsallis parameter, assumed positive [59], with the
simple HDE recovered in the limit $\sigma\to 1$. Conversely, Barrow’s
modification of the Bekenstein-Hawking formula

led to Barrow HDE models described by the energy density

$\rho_{\Lambda}=3c^{2}L^{\Delta-2}$ (3)

, where $\Delta$ is the deformation parameter [60], capped at $\Delta=1$, and
the simple HDE is regained in the limit of $\Delta\to 0$. Another very
promising idea which has its origins in high energy physics is the generalized
uncertainty principle, and taking inspiration from those theories, one can
obtain Power law corrections to the entanglement entropy of black holes which
result in another HDE form given by

$\rho_{\Lambda}=3L^{-2}m_{p}^{2}\left(c^{2}-\frac{\delta}{3}L^{2-\gamma}\right)$
(4)

, with $\gamma$ and $\delta$ representing parameters associated with
underlying corrections [61]. Such models are called as power law entropy
corrected HDE models or PLECHDE. One sees that in the limit of $\delta\to 0$,
one recovers the simple HDE model while $\delta$ is usually constrained to be
as $0<\delta<4$ 111One can also make the case that in the limit of
$\gamma\to\infty$, one also recovers the simple HDE but this is not a physical
limit which can be reached in such models. Indeed it has been successfully
argued that there is a limit to the values that $\gamma$ can take, which has
been found to be $2<\gamma<4$ [61, 62]. In this work we would like to discuss
the various forms of cosmological evolution in the Tsallis, Barrow and PLEC
HDE models with an ansatz based approach. In particular we would like to
discuss the feasibility of a simple power law, intermediate law, emergent
universe and Logamediate law evolutions. In Section II we shall discuss the
basic formulations in this regard, while in section III, in its various
subsections, we discuss the feasibility of the all the evolution schemes
considered. We finally conclude our work in section IV.

##  2 Basic formulations

We would like to consider an interacting dark sector in our work here. One can
explore an interaction between dark energy and dark matter by incorporating
the following continuity equations:

$\dot{\rho}_{\Lambda}+3H\rho_{\Lambda}(1+w_{\Lambda})=-Q$ (5)

$\dot{\rho}_{m}+3H\rho_{m}(1+w_{m})=Q$ (6)

Here, $\rho_{\Lambda}$ and $\rho_{m}$ represent the energy densities of dark
energy and dark matter, respectively. The parameter $w_{\Lambda}$ denotes the
equation of state for dark energy, while $w_{m}$ represents the equation of
state for dark matter. The term $Q$ describes the interaction between the two
components.

For the sake of simplicity, we focus solely on the contributions of dark
energy and dark matter to cosmology. The interaction term $Q$ can take linear
or nonlinear forms. Some examples include:

$Q=3Hb(\rho_{\Lambda}+\rho_{m})$ (7)

$Q=3Hb\frac{\rho_{\Lambda}^{2}}{\rho_{\Lambda}\rho_{m}}$ (8)

In these expressions, $b$ is a coefficient determining the strength of the
interaction. These linear and nonlinear forms of the interaction term offer
different possibilities to explore the dynamics of the interacting dark
sectors. Before considering the ansatz now, it is first important that we do
some analysis which is independent of the ansatz and is general. We also
define the fractional energy density for DE and DM as

$\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{3H^{2}}$ (9)
$\Omega_{m}=\frac{\rho_{m}}{3H^{2}}$ (10)

where $\rho_{\Lambda}$ would depend upon the model of HDE chosen by us. Now
the cutoff we know very well is given by

$L=(\alpha H^{2}+\beta\dot{H})^{-1/2}$ (11)

It is also interesting here to note that in order to have a time varying dark
energy, one does not always need to rely only on exotic models like HDEs,
Scalar field paradigms etc. A fascinating possibility in this regard is the
considertion of unstable dark matter alongside a cosmological constant form of
dark energy [63, 64, 65, 66, 67]. Even such a form of the dark sector can
produce the effect of a time varying dark energy scenario and in future works,
it may be of interest to consider HDEs alongside unstable DM as well.

We start off with the simple HDE model, (1) and take the derivative of the
energy density to have

$\dot{\rho}_{\Lambda}=-2\rho_{\Lambda}\frac{\dot{L}}{L}$ (12)

Here we would now need an expression for $-\frac{\dot{L}}{L}$ to proceed
further, for which we use (11) to write

$-\frac{\dot{L}}{L}=\frac{1}{2}L^{2}(2\alpha H\dot{H}+\beta\ddot{H})$ (13)

Now this is the important part, as from here we can relate the dimensionless
dark energy paramter with the help of (9) to have

$L^{2}=\frac{c^{2}}{\Omega_{\Lambda}H^{2}}$ (14)

Using this we can write

$\dot{\rho_{\Lambda}}=\frac{\rho_{\Lambda}c^{2}}{H^{2}\Omega_{\Lambda}}(2\alpha
H\dot{H}+\beta\ddot{H})$ (15)

This equation takes into account the fractional energy density parameter and
relate it with the time derivative of the energy density. From here we can now
use the continuity equation (5) to have

$w_{\Lambda}=-1-\frac{1}{3H\Omega_{\Lambda}}\Bigg{[}\frac{c^{2}(2\alpha
H\dot{H}+\beta\ddot{H}}{H^{2}}+\frac{Q}{3H^{2}}\Bigg{]}$ (16)

From the EOS parameter (16), we can get the squared sound speed as

$v_{s}^{2}=\frac{\dot{p_{\Lambda}}}{\dot{\rho_{\Lambda}}}=w_{\Lambda}+\frac{\dot{w_{\Lambda}}\rho_{\Lambda}}{\dot{\rho_{\Lambda}}}$
(17)

We can now similarly write these quantities for other HDE models as well.
Let’s consider firstly the Tsallis model with (2), with its derivative being

$\dot{\rho_{\Lambda}}=-(4-2\sigma)\rho_{\Lambda}\frac{\dot{L}}{L}$ (18)

We can now define the fractional energy density parameter for the Tsallis
model with (9), from which we can write

$L^{2}=\left(\frac{c^{2}}{\Omega_{\Lambda}H^{2}}\right)^{\frac{1}{2-\sigma}}$
(19)

Using this in (13), we can write (18) as

$\dot{\rho_{\Lambda}}=\frac{(4-2\sigma)\rho}{2}\left(\frac{c^{2}}{\Omega_{\Lambda}H^{2}}\right)^{\frac{1}{2-\sigma}}(2\alpha
H\dot{H}+\beta\ddot{H})$ (20)

We can then write using (5),

$w_{\Lambda}=-1-\frac{1}{3H\Omega_{\Lambda}}\Bigg{[}\frac{(4-2\sigma)\Omega_{\Lambda}}{2}\left(\frac{c^{2}}{\Omega_{\Lambda}H^{2}}\right)^{\frac{1}{2-\sigma}}(2\alpha
H\dot{H}+\beta\ddot{H})+\frac{Q}{3H^{2}}\Bigg{]}$ (21)

We note that in the limit $\sigma\to 1$, (21) reduces to (16), as would be
expected as in the limit of $\sigma\to 1$, Tsallis thermodynamics reduces to
the usual thermodynamics. In a similar way, we can derive for the Barrow model
(3), to get the evolution of the energy density and the EOS parameter of dark
energy as

$\dot{\rho_{\Lambda}}=\frac{(2-\Delta)\rho}{2}\left(\frac{c^{2}}{\Omega_{\Lambda}H^{2}}\right)^{\frac{2}{2-\Delta}}(2\alpha
H\dot{H}+\beta\ddot{H})$ (22)
$w_{\Lambda}=-1-\frac{1}{3H\Omega_{\Lambda}}\Bigg{[}\frac{(2-\Delta)\Omega_{\Lambda}}{2}\left(\frac{c^{2}}{\Omega_{\Lambda}H^{2}}\right)^{\frac{2}{2-\Delta}}(2\alpha
H\dot{H}+\beta\ddot{H})+\frac{Q}{3H^{2}}\Bigg{]}$ (23)

Where we note that in the limit of the deformation parameter $\Delta\to 0$, we
recover the formulations for the simple HDE as one would expect. Finally, for
the PLECHDE model we get

$\dot{\rho_{\Lambda}}=\left(\frac{(2\rho-(\gamma-2)\delta
L^{-\gamma})}{2}\right)\left(\frac{3c^{2}-\delta
L^{2-\gamma}}{3H^{2}\Omega_{\Lambda}}\right)(2\alpha H\dot{H}+\beta\ddot{H})$
(24)
$w_{\Lambda}=-1-\frac{1}{3H\Omega_{\Lambda}}\Bigg{[}\left(\frac{(2\Omega_{\Lambda}-(\gamma-2)\delta
L^{-\gamma})}{2}\right)\left(\frac{3c^{2}-\delta
L^{2-\gamma}}{3H^{2}\Omega_{\Lambda}}\right)(2\alpha
H\dot{H}+\beta\ddot{H})+\frac{Q}{3H^{2}}\Bigg{]}$ (25)

Again we see that in the limit of $\delta\to 0$, we recover the formulations
of the simple HDE.

It is also important to discuss another point which could help ensure
consistency of holographic dark energy models and that is thermodynamics. The
thermodynamical implications and in particular, examining the validity of
holographic dark energy models in the face of the generalized second law of
thermodynamics was recently explored in [68]. It is well known that
thermodynamical analysis of the gravity theory is an exciting research topic
in the cosmological context and the thermodynamical properties which hold for
a black hole are equally valid for a cosmological horizon. In addition, the
first law of thermodynamics which holds in a black hole horizon can also be
derived from the first Friedmann equation in the FLRW universe when the
universe is bounded by an apparent horizon. This provides well motivation to
select the apparent horizon as the cosmological horizon in order to examine
the thermodynamic properties of any cosmological model. Motivated by the above
arguments, one can consider the universe as a thermodynamic system that is
bounded by the cosmological apparent horizon with the radius [69]

$r_{h}=\left(H^{2}+\frac{k}{a^{2}}\right)^{-1/2}$ (26)

where for the case of the flat universe $(k=0)$, one can then write

$r_{h}=\frac{1}{H}$ (27)

Considering that the major contributions to the entropy of the universe is
from the dark sectors besides the cosmological horizon, we can then write that

$S_{tot}=S_{\Lambda}+S_{m}+S_{h}$ (28)

where $S_{h}$ is the horizon entropy. The time evolution of $S_{\Lambda}$ and
$S_{m}$ can be written using the first law of thermodynamics as

$T\dot{S}_{\Lambda}=\dot{E}_{\Lambda}+p_{\Lambda}\dot{V}$ (29)
$T\dot{S}_{m}=\dot{E}_{m}+p_{m}\dot{V}$ (30)

Where $V=\frac{4\pi r_{h}^{3}}{3}$ is the horizon volume, with

$E_{\Lambda}=\frac{4\pi r_{h}^{3}\rho_{\Lambda}}{3}$

and

$E_{m}=\frac{4\pi r_{h}^{3}\rho_{m}}{3}$

with the time evolutions of the energy densities being given by the continuity
equations (5) and (6). The temperature T of the apparent horizon is given by

$T=\frac{1}{2\pi r_{h}}=\frac{H}{2\pi}$ (31)

The entropy of the horizon would depend on what model of HDE we are
considering. In the case of the simple HDE, we have the usual Benkenstein
hawking entropy at work, which we can write as

$S_{h}=\frac{A}{A_{0}}$ (32)

where $A_{0}=\frac{1}{4G}$ is the Planck area and we can write the above
equation as

$S_{h}=\gamma r_{h}^{2}$ (33)

where $\gamma$ here is a positive constant ( in this case equal to $\pi/G$ but
the utility of such an expression for $S_{h}$ will shortly be clear ). When we
are talking of the Barrow entropy, we have

$S_{h}=\left(\frac{A}{A_{0}}\right)^{1+\Delta/2}$ (34)

where again symbols have similar meanings as before and now we can express
this instead in terms of the horizon radius as

$S_{h}=\gamma_{b}r_{h}^{2+\Delta}$ (35)

where

$\gamma_{b}=\left(\frac{4\pi}{A_{0}}\right)^{1+\Delta/2}$

which will again just be a positive constant. For the Tsallis HDE scenario,
the horizon entropy will be similarly given by

$S_{h}=\gamma r_{h}^{2\sigma}$ (36)

where $\gamma_{t}$ is again a constant in terms of the Planck area and the
Tsallis parameter. We can now write the time evolution of entropy for a
universe with simple HDE using (29)-(30)

$\dot{S}=\frac{2\pi}{H}\Bigg{[}\frac{4\pi}{3H^{3}}(3c^{2}(2\alpha
H\dot{H}+\beta\ddot{H})+Q)-12\Omega_{m}\pi(1+w_{m})\Bigg{]}-\frac{2\gamma\dot{H}}{H^{3}}$
(37)

In the case of the Tsallis scenario, we have the following

$\dot{S}=\frac{2\pi}{H}\Bigg{[}\frac{4\pi}{3H^{3}}\left(\frac{4-2\sigma}{2}3H^{2}\Omega_{\Lambda}\left(\frac{c^{2}}{H^{2}\Omega_{\Lambda}}\right)^{\frac{1}{2-\sigma}}(2\alpha
H\dot{H}+\beta\ddot{H})\right)+\frac{4\pi
Q}{3H^{3}}-12\Omega_{m}\pi(1+w_{m})\Bigg{]}-\frac{2\gamma_{t}\sigma\dot{H}}{H^{2}H^{2\sigma-1}}$
(38)

While in the case of Barrow HDE , we have

$\dot{S}=\frac{2\pi}{H}\Bigg{[}\frac{4\pi}{3H^{3}}\left(\frac{2-\Delta}{2}3H^{2}\Omega_{\Lambda}\left(\frac{c^{2}}{H^{2}\Omega_{\Lambda}}\right)^{\frac{2}{2-\Delta}}(2\alpha
H\dot{H}+\beta\ddot{H})\right)+\frac{4\pi
Q}{3H^{3}}-12\Omega_{m}\pi(1+w_{m})\Bigg{]}-\frac{(2+\Delta)\gamma_{b}\dot{H}}{H^{2}H^{1+\Delta}}$
(39)

Again, we see that the formulas are consistent as (38) and (39) reduce to (37)
in the case of $\sigma\to 1$ and $\Delta\to 0$, respectively. With these
formulations at hand, we can now proceed towards the main plots and results.

##  3 Different evolutions with G-O cutoff

We have now developed enough machinery to tackle the models we want to. This
time we consider not only the simple power law model

$a(t)=a_{0}t^{n}$ (40)

But also the emergent universe scenario with the ansatz

$a(t)=a_{0}(\mu t+\lambda)^{m}$ (41)

And the logamediate scenario with the ansatz

$a(t)=a_{0}e^{A((\log t)^{\alpha_{1}})}$ (42)

where we denote the ansatz in (40), (41) and (42) as $a_{1}$, $a_{2}$ and
$a_{3}$ respectively.

With this, we can start to consider the plots. We have set $c=1$ as that is
fine for this cutoff and have considered a value of $\beta=0.5$, but the value
of $\alpha$ is not constrained and so we can think of ranges of
$(\alpha,\beta)$ where the various holographic scenarios are viable or not
considering $w$ and $v_{s}^{2}$. For the interaction terms, we keep the
parameter $b=0.03$, We take $\Omega_{d}=0.69$ and $\Omega_{m}=0.258$, we take
the various parameters in the ansatz fomrulaes as $n=3$, $m=2$, $\mu=2$,
$\lambda=4$, $\alpha_{1}=4$, $A=0.02$ and $\gamma=3$ ( the parameter in the
PLEC model ). All these values are consistent with the theoretical needs of
these models as seen in the literature.

###  3.1 Simple power law evolution

We start off with the Tsallis model, with the linear interaction (7) and the
ansatz (40). The dark energy EOS and squared sound speed for this for three
different values of the Tsallis parameter $\sigma$ are plotted agains time,
where in each plot we have considered the range of $\alpha$ from $0.7to0.95$.
The plots are displayed in figure 1.

Figure 1: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for simple power law in Tsallis HDE

We see that neither the squared sound speed nor the dark energy EOS are in any
theoretical or observationally sound ranges, as the dark energy EOS takes
absurdly negative or positive values while the squared sound speed is
incredibly negative too. Hence, this scenario is not favoured much.

We then turn our attention to the Barrow model with the linear interaction (7)
and the power law ansatz (40). Figure 2 gives the plot of the DE EOS and
squared sound speed for three different values of the deformation parameter
$\Delta$, again in various ranges of $\alpha$.

Figure 2: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for simple power law in Barrow HDE

We see that neither the squared sound speed nor the dark energy EOS are in any
theoretical or observationally sound ranges, as both the dark energy EOS and
the squared sound speed both take highly negative values. Hence, this scenario
is not favoured much. For the PLEC model with linear interaction and power law
ansatz, we have plotted the EOS in figure 3 and we see that the EOS is not
viable in any case at all and so we have not went in for the sound speed in
the case at all. Hence, here we can conclude that simple power law models with
linear interaction are heavily disfavoured.

Figure 3: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for simple power law in PLEC HDE

We now plot for the non-linear interaction scheme (8), where we firstly plot
for the power law ansatz in the Barrow model. We see in figure 4 that although
in the linear scheme the Barrow model was completely dismissive of the simple
power law model, in the non-linear scheme the dark energy EOS can be in viable
ranges $\Delta=0.4,0.6$. But the squared sound speed is only viable for
$\Delta>0.7$, which makes the model incompatible as a whole again but not as
bad as with the linear scheme.

Figure 4: Plot of the dark energy EOS and squared sound speed against time for
non-linear interaction with simple power law ansatz in Barrow HDE

In figure 5 we have plotted for the simple power law and non linear
interaction in the Tsallis model and we see that the EOS parameter is again
not viable across various ranges of the free parameters and squared sound
speed is only eventually becoming somewhat compatible for $\sigma>2$. But as a
whole, we see that this model is again not very favourable. For the PLEC
model, the EOS parameter follows almost exactly the same plot and conclusions
as in the linear interaction case shown in figure 3 and so we have not plotted
that again. So again we conclude that the simple power law models are quite
disfavoured.

Figure 5: Plot of the dark energy EOS and squared sound speed against time for
non-linear interaction with simple power law ansatz in Tsallis HDE

We also made a 3d plot for dark energy EOS in the Barrow model just for
illustration purposes in figure 6.

Figure 6: 3d Plot of the dark energy EOS against time for linear interaction
in the power law scenario for the Barrow model

###  3.2 Emergent universe evolution

We now consider the second ansatz, the emergent scenario (41) with the linear
interaction in the Barrow model and we have plotted the EOS and sound speed in
figure 7 for this.

Figure 7: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for emergent scenario ansatz in Barrow HDE

We see that in this scenario, the dark energy EOS is in the observationally
viable range for various values of the deformation parameter and a wide range
of values across $\alpha$ too but the squared sound speed is not viable in any
of the range of $\alpha$ and $\Delta$.

Figure 8: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for emergent scenario ansatz in Tsallis HDE

Plotting for the same ansatz in the linear interaction for the Tsallis model
as in figure 8, we see that the EOS is viable for all range of $\alpha$ while
for $\sigma>2$, the squared sound speed is viable for $\alpha$ too and for
$\sigma>1.5$ the sound speed is still viable for $\alpha<0.8$. Hence this
model is quite viable.

Figure 9: Plot of the dark energy EOS and squared sound speed against time for
linear interaction with simple emergent scenario ansatz in PLEC HDE

For the PLEC model with this ansatz in the linear interaction, plots have been
made in figure 9 and we see that the EOS is observationally viable in all
parameter ranges, while the sound speed is viable for $\alpha>0.9$ and
$\delta>3$. So we can say that this is viable as well. Overall we can say that
the emergent scenario models perform quite a lot better with linear
interaction than the simple power law ones.

We have plotted for the non linear interaction with the emergent scenario
ansatz for the Barrow model in figure 10. The plots show that while the EOS is
viable for various ranges of $\Delta$ and $\alpha$, the squared sound speed is
not.

Figure 10: Plots of the Dark energy EOS and squared sound speed against time
for non-linear interaction for emergent scenario ansatz in Barrow HDE

Plotting for the emergent scenario with non linear interaction in the Tsallis
model in figure 11, we see that the EOS is viable for all ranges of $\alpha$
and $\sigma$ and the sound speed is also viable, but only for $\sigma>2$.

Figure 11: Plots of the Dark energy EOS and squared sound speed against time
for non-linear interaction for emergent scenario ansatz in Tsallis HDE

For the PLEC model with emergent ansatz and non linear interaction we see that
while the EOS can be in viable ranges, the squared sound speed cannot as shown
in figure 12. So as a whole while there are minor tweaks, the emergent
universe models perform almost the same as in the non-linear interaction as
they do in the linear interaction.

Figure 12: Plots of the Dark energy EOS and squared sound speed against time
for non-linear interaction for emergent scenario ansatz in PLEC HDE

We have also plotted the entropy for a viable value of the Tsallis parameter
$\sigma>2.2$ ( where it showed both compatible w and $v_{s}^{2}$), in the
emergent scenario.

Figure 13: Plot of the time derivative of the universal entropy against time
for viable value of the Tsallis parameter in emergent scenario

We have plotted this in figure 13 for both linear and non linear interactions,
and we see that the generalized second law of thermodynamics can be satisfied
as even though the plot starts off in the negative regime it eventually stays
on for values $\sim 0$. So the emergent scenario with the Tsallis model for
$\sigma>2$ seems like a good model. Finally we have also made 3d plots of the
dark energy EOS for the Barrow and Tsallis models in the emergent scenario in
the linear interaction case in figures 14 and 15 just for illustrations.

Figure 14: 3d Plot of the dark energy EOS against time for linear interaction
in the emergent scenario for the Tsallis model Figure 15: 3d Plot of the dark
energy EOS against time for linear interaction in the emergent scenario for
the Barrow model

###  3.3 Intermediate law evolution

Plotting firstly for the Tsallis HDE scenario with linear interaction as in
figure 16, we see that model gives viable values for the EOS parameter for
$\sigma<1$ and also for $\sigma>2$. But the squared sound speed is not viable
in any case as it is either too positive or too negative, across all ranges of
the Tsallis parameter. So it seems that this scenario is not very favourable.

Figure 16: Plots of the EOS parameter of dark energy and squared sound speed
against time for various values of $\sigma$ for the intermediate scenario with
Tsallis HDE and linear interaction. The plot is the same for all viable values
of $\alpha$

Plotting for the Tsallis HDE scenario with non linear interaction as in figure
17, we see that model gives viable values for the EOS parameter again for
$\sigma<1$ and also for $\sigma>2$. But the squared sound speed is not viable
once again in any case as it is either having very large positive or negative
values for all ranges of the Tsallis parameter. So it seems that this scenario
is not very favourable too. One thing we would notice in both these plots is
that we have only plotted for one value of $\alpha$, the reason behind that is
that interestingly the results are the same for all viable values of $\alpha$
(which corresponds to $\alpha\sim\mathcal{O}(1)$

Figure 17: Plots of the EOS parameter of dark energy and squared sound speed
against time for various values of $\sigma$ for the intermediate scenario with
Tsallis HDE and non linear interaction.The plot is the same for all viable
values of $\alpha$

Plotting for the Barrow model, we observed that the plots are quite the same
for both linear and non linear interactions so we have just plotted for the
non linear case in figure 18. We see that while the EOS parameter gives us
viable values for all ranges of the deformation paraemter, the squared sound
speed is incredibly negative for all ranges of $\Delta$ as well hereby making
this scenario very unrealistic too. As the plots here for these parameters are
the almost the same for the linear interaction too, we conclude similarly for
that as well.

Figure 18: Plots of the EOS parameter of dark energy and squared sound speed
against time for various values of $\Delta$ for the intermediate scenario with
Barrow HDE and non linear interaction.The plot is the same for all viable
values of $\alpha$ and also same for linear interaction too.

Plotting for the PLEC HDE scenario we again noticed that both the linear and
non linear cases of interaction were giving almost the same plots and so we
have plotted here in figure 19 for the linear interaction only. We see here
that both the EOS parameter and the squared sound speed are incredibly
unrealistic, for all ranges of the model parameter $\delta$ and so the
scenario seems very unfavourable as well. Similar conclusions hold for the non
linear interaction case here.

Figure 19: Plots of the EOS parameter of dark energy and squared sound speed
against time for various values of $\delta$ for the intermediate scenario with
PLEC HDE and linear interaction.The plot is the same for all viable values of
$\alpha$ and also same for non linear interaction too.

###  3.4 Logamediate law evolution

We have plotted similarly for Logamediate scenario (42) with linear
interaction, where we plotted for the Barrow model as shown in figure 20 and
for the Tsallis model in figure 21.

Figure 20: Plots of the Dark energy EOS and squared sound speed against time
for linear interaction for Logamediate scenario ansatz in Barrow HDE Figure
21: Plots of the Dark energy EOS and squared sound speed against time for
linear interaction for Logamediate scenario ansatz in Tsallis HDE

What is interesting to note is that for both these models, the EOS of dark
energy has the same behaviour, where it eventually hovers to value near -1
very quickly and then stays that way. Similarly, squared sound speed has
similar behaviour too where we see that the sound speed takes a few jumps in
both cases initially but then stays on to values $\sim 0$. Interestingly, the
behaviour is exactly the same for the PLEC models and we have not plotted for
that here because of that. So the logamediate ansatz with linear interaction
produces viable models in all of Barrow, Tsallis and PLEC models, making it
the most viable model we have seen so far.

For the Logamediate case in the non-linear interactions, we find that the
plots follow and are almost exactly the same as in the linear interaction case
for all of Barrow, Tsallis and PLEC models. For example, we show the plot for
the Tsallis scenario in this case as in figure 22, which evidently shows
almost the same behaviour as in figure 21 for the linear case.

Figure 22: Plot of the dark energy EOS and squared sound speed against time
for non-linear interaction with logamediate scenario ansatz in Tsallis HDE

So we again find that the EOS parameter takes up values near -1 and stays on
it for the eventuality, while the squared sound speed takes a few small jumps
initially and then stays on values $\sim 0$. Hence, we again find that the
logamediate models are suitable for all the holographic scenarios discussed.

As we find the Logamediate scenario to also have the distinction of having
both viable EOS parameter and squared speed sound values in various cases, we
will also check the validity of the generalized second law $\dot{S}\geq 0$ for
them. We find that the logamediate models, in all cases of Barrow, Tsallis and
PLEC have a similar trend to each other.

Figure 23: Plot of the time derivative of the universal entropy against time
in Logamediate scenario for Tsallis and Barrow models

In all cases, they start off from negative values but then eventually stay on
for values $\sim 0$ and we show the plot for the same for Tsallis and Barrow
scenarios in the linear interaction in figure 23. The trend is the same for
the nonlinear interaction too and also the same for the PLEC models. So we can
say that the logamediate models are the best fit seeing all this, as they
produce viable EOS parameter and squared sound speed values in multiple models
and are also fit with generalized second law of thermodynamics.

##  4 Conclusions

In this work we have performed an extensive and detailed analysis of ansatz’
based approach to holographic dark energy models. We considered three widely
studied forms of extended HDE models, namely the Tsallis, Barrow and PLEC HDE
models. We also considered 4 different forms of cosmological evolutions in our
work which were namely the simple power law form, logamediate form,
intermediate form and emergent universe form. We firstly developed the
formulation for various key cosmological parametes in this case and also the
key thermodynamical asepcts of these paradigms.

The main takeaways from this are simple power law models perform horribly in
all the HDE models, for both the linear and non-linear interaction regime as
in all universal evolution scenarios, they do not give compatible models
Logamediate models perform the best Emergent scenario models do reasonably
well too, a lot better than simple power law Intermediate scenario models are
better than simple power law but are still quite far behind the emergent
scenario and logamediate models While they produce subtle and minor
difference, the results are very much equivalent for both linear and non-
linear interaction schemes of DE and DM The generalized second law of
thermodynamics is satisfied in the viable HDE scenarios ( Logamediate models
for all HDE scenarios and emergent universe models for certain Tsallis
scenarios), but only just as the time derivative of entropy maintains values
close to 0 in universal evolution making it just about satisfying the law.

##  5 Acknowledgements

The research by M.K. was carried out at Southern Federal University with
financial support from the Ministry of Science and Higher Education of the
Russian Federation (State contract GZ0110/23-10-IF). The authors would like to
thank Alexander Oliveros for various helpful discussions.

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