# Evolution of dissipative and non-dissipative universes
in holographic cosmological models with a power-law term

Nobuyoshi Komatsu E-mail<EMAIL_ADDRESS>Department of
Mechanical Systems Engineering, Kanazawa University, Kakuma-machi, Kanazawa,
Ishikawa 920-1192, Japan

###### Abstract

Density perturbations related to structure formations are expected to be
different in dissipative and non-dissipative universes, even if the background
evolution of the two universes is the same. To clarify the difference between
the two universes, first-order density perturbations are studied, using two
types of holographic cosmological models. The first type is a ”$\Lambda(t)$
model” similar to a time-varying $\Lambda(t)$ cosmology for the non-
dissipative universe. The second type is a ”BV model” similar to a bulk
viscous cosmology for the dissipative universe. To systematically examine the
two different universes, a power-law term proportional to $H^{\alpha}$ is
applied to the $\Lambda(t)$ and BV (bulk-viscous-cosmology-like) models,
assuming a flat Friedmann–Robertson–Walker model for the late universe. Here,
$H$ is the Hubble parameter and $\alpha$ is a free parameter whose value is a
real number. The $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models are used
to examine first-order density perturbations for matter, in which the
background evolution of the two models is equivalent. In addition,
thermodynamic constraints on the two models are discussed, with a focus on the
maximization of entropy on the horizon of the universe, extending previous
analyses [Phys. Rev. D 100, 123545 (2019); 102, 063512 (2020)]. Consequently,
the $\Lambda(t)$-$H^{\alpha}$ model for small $|\alpha|$ values is found to be
consistent with observations and satisfies the thermodynamic constraints,
compared with the BV-$H^{\alpha}$ model. The results show that the non-
dissipative universe described by the $\Lambda(t)$-$H^{\alpha}$ model similar
to lambda cold dark matter models is likely favored.

###### pacs:

98.80.-k, 95.30.Tg, 98.80.Es

##  I Introduction

A paradigm for the cosmic expansion history, that is, an accelerated expansion
of the late universe PERL1998_Riess1998 ; Suzuki_2012 ; Hubble2017 ;
Planck2018 ; Riess20162019 , can be explained by lambda cold dark matter
($\Lambda$CDM) models Weinberg1 ; Bamba1 . The $\Lambda$CDM model assumes an
extra driving term related to the cosmological constant $\Lambda$ and an
additional energy component called ”dark energy.” However, it is well known
that the $\Lambda$CDM model suffers from several theoretical difficulties,
including the cosmological constant problem and the cosmic coincidence problem
Weinberg1989 . To resolve these difficulties, various cosmological models have
been suggested, such as $\Lambda(t)$CDM models [i.e., a time-varying
$\Lambda(t)$ cosmology] Freese ; Sola_2009 ; Nojiri2006 ; Richarte2013a ;
Basilakos2018 ; Sola_2015L14 ; Valent2015 ; Sola_2017-2018 ; Valent2018a ;
Sola2019 ; Rezaei2019 ; Sola_2013b ; Sola2020_2 ; Sola2020_3 , bulk viscous
models Weinberg0 ; Murphy1 ; Barrow11-Zimdahl1 ; Brevik1-Nojiri1 ; Barrow21 ;
Piattella1 ; Meng3 ; Avelino2etc ; Barbosa2017 ; Sasidharan ; Odintsov2020 ,
creation of CDM (CCDM) models Prigogine_1988-1989 ; Lima1992-2016 ; Freaza2002
; Others2001-2016 ; Nunes2016B ; Barrow20172019 ; Lima2011 ; Ramos_2014 ;
Ramos_2014b ; Sola2020 ; Cardenas2020 , and thermodynamic scenarios Easson ;
Basilakos1 ; Basilakos2014 ; Gohar_a_b ; Nunes2016 ; Lepe1 ; Padma2012AB ;
Cai2012-Tu2013 ; Chakraborty-Morad ; Neto2018a ; Koma457 ; Koma8 ; Koma10 ;
Koma11 ; Koma12 ; 2020Morad ; Koma6 ; Koma9 ; Koma14 ; Koma15 .

Formulations of these cosmological models can be categorized into several
types, and their theoretical backgrounds are different Koma6 ; Koma9 ; Koma14
; Koma15 . For example, from a dissipative viewpoint, the formulation should
be categorized into two types in a Friedmann–Robertson–Walker (FRW) universe.
The first type is $\Lambda(t)$, which is similar to $\Lambda(t)$CDM models
Koma6 ; Koma9 ; Koma14 ; Koma15 . In $\Lambda(t)$ models, an extra driving
term is added to both the Friedmann equation and the Friedmann–Lemaître
acceleration equation. The second type is BV, which is similar to both bulk
viscous models and CCDM models Koma6 ; Koma9 ; Koma14 ; Koma15 . In bulk-
viscous-cosmology-like (BV) models, the acceleration equation includes an
extra driving term, whereas the Friedmann equation does not. It is possible to
consider that the $\Lambda(t)$ model is related to ”reversible entropy,” due
to, for example, the exchange of matter (energy) Barrow22 ; Prigogine_1998 ,
whereas the BV model is related to ”irreversible entropy,” due to, for
example, gravitationally induced particle creation Prigogine_1988-1989 ;
Lima1992-2016 . In this sense, the $\Lambda(t)$ and BV models can describe
non-dissipative and dissipative universes, respectively Koma15 .

The background evolution of the universe in the $\Lambda(t)$ and BV models
becomes the same when an equivalent driving term is assumed Koma15 . However,
even in this case, density perturbations related to structure formations are
expected to be different Koma6 ; Koma9 . In fact, the influences of several
driving terms, such as constant, Hubble parameter ($H$), and $H^{2}$ terms,
have been examined in the $\Lambda(t)$ and BV models. For the $\Lambda(t)$
model, Basilakos et al. reported that the $H^{2}$ terms in $\Lambda(t)$CDM
models do not describe structure formations properly Basilakos1 ; Sola_2009 .
In contrast, Solà et al. showed that a combination of the constant and $H^{2}$
terms is favored Sola_2015L14 . Such a power series of $H$ has been examined
in, for example, the works of Gómez-Valent et al. Valent2015  and Rezaei et
al. Sola2019 . For the BV model, Li and Barrow have reported that bulk viscous
models with the $H$ term are inconsistent with observations of structure
formations Barrow21 . Barbosa et al. used bulk and shear viscous models to
point out a similar inconsistency Barbosa2017 . In addition, Jesus et al.
Lima2011  and Ramos et al. Ramos_2014  showed that CCDM models with a constant
term are inconsistent with the observed growth rate for clustering. (In the
CCDM model, a negative sound speed Lima2011  and the existence of clustered
matter Ramos_2014  are necessary to properly describe the growth rate Koma8 .)

In those works, the $\Lambda(t)$ and BV models are separately discussed and,
therefore, the dissipative and non-dissipative universes have not yet been
examined systematically. Accordingly, it is worthwhile to clarify the
difference between the dissipative and non-dissipative universes. To
systematically study the two universes, two types of holographic cosmological
models that include $H^{\alpha}$ terms Koma14 ; Koma15  are suitable. Here,
$\alpha$ is a dimensionless constant whose value is a real number. The power-
law term is obtained from, for example, Padmanabhan’s holographic
equipartition law Padma2012AB  with a power-law-corrected entropy
Das2008Radicella2010 , as examined in a previous work Koma11 .

In this context, we study the evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models and discuss observational
constraints on the two models. The universe is expected to behave as an
ordinary isolated macroscopic system Mimoso2013 , where the entropy of the
universe does not decrease and approaches a certain maximum value in the last
stage Pavon2013 ; Krishna20172019 ; Pavon2019 . In fact, thermodynamic
constraints on the two models have been separately discussed in previous works
Koma14 ; Koma15 . Accordingly, we examine the observational constraints in
combination with the thermodynamic constraints. The observational and
thermodynamic constraints should provide new insights into a discussion of the
dissipative and non-dissipative universes. Note that this discussion is
focused on the late universe and, therefore, the inflation of the early
universe is not discussed here.

The remainder of the present paper is organized as follows. In Sec. II,
cosmological equations and first-order density perturbations for the
$\Lambda(t)$ and BV models are reviewed. In Sec. III, a power-law term
proportional to $H^{\alpha}$ is applied to the $\Lambda(t)$ and BV models. In
Sec. III.1, the $\Lambda(t)$ and BV models that include the $H^{\alpha}$ term,
that is, the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models, are
formulated. In Sec. III.2, density perturbations in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models are derived. Sec. III.3
reviews thermodynamic constraints on the two models. In Sec. IV, the evolution
of the universe in the two models is examined. In addition, the observational
and thermodynamic constraints on the two models are investigated. Finally,
Sec. V presents the conclusions of the study.

##  II Cosmological equations and first-order density perturbations

We consider a homogeneous, isotropic, and spatially flat universe and examine
the scale factor $a(t)$ at time $t$ in the FRW metric. An expanding universe
is assumed from observations Hubble2017 .

In this section, we review cosmological equations and density perturbations in
dissipative and non-dissipative universes, according to previous works Koma6 ;
Koma9 ; Koma14 ; Koma15 . In Sec. II.1, cosmological equations for the
$\Lambda(t)$ and BV models are presented. Sec. II.2 reviews first-order
density perturbations in the two models.

###  II.1 Cosmological equations

We present formulations of cosmological equations for $\Lambda(t)$ and BV
models in a flat FRW universe Koma6 ; Koma9 ; Koma14 ; Koma15 . The Friedmann,
acceleration, and continuity equations are written as Koma6 ; Koma9 ; Koma14 ;
Koma15

$H(t)^{2}=\frac{8\pi G}{3}\rho(t)+f_{\Lambda}(t),$ (1)
$\displaystyle\frac{\ddot{a}(t)}{a(t)}$ $\displaystyle=-\frac{4\pi
G}{3}(1+3w)\rho(t)+f_{\Lambda}(t)+h_{\textrm{B}}(t),$ (2)
$\dot{\rho}+3H(1+w)\rho=-\frac{3}{8\pi G}\dot{f}_{\Lambda}(t)+\frac{3}{4\pi
G}Hh_{\textrm{B}}(t),$ (3)

with the Hubble parameter $H(t)$ defined as

$H(t)\equiv\frac{da/dt}{a(t)}=\frac{\dot{a}(t)}{a(t)},$ (4)

where $G$, $\rho(t)$, and $p(t)$ are the gravitational constant, the mass
density of cosmological fluids, and the pressure of cosmological fluids,
respectively Koma9 . Also, $w$ represents the equation of the state parameter
for a generic component of matter, which is given as Koma14

$w=\frac{p(t)}{\rho(t)c^{2}},$ (5)

where $c$ represents the speed of light. For a matter-dominated universe and a
radiation-dominated universe, the values of $w$ are $0$ and $1/3$,
respectively. Here, we consider a matter-dominated universe, that is, $w=0$,
and we neglect the influence of radiation in the late universe.

In the above formulation, two extra driving terms, $f_{\Lambda}(t)$ and
$h_{\textrm{B}}(t)$, are phenomenologically assumed Koma14 . Specifically,
$f_{\Lambda}(t)$ is used for $\Lambda(t)$ models and $h_{\textrm{B}}(t)$ is
used for BV models. Accordingly, we set $h_{\textrm{B}}(t)=0$ for the
$\Lambda(t)$ model and $f_{\Lambda}(t)=0$ for the BV model Koma14 ; Koma15 .

In a matter-dominated universe ($w=0$), coupling Eq. (1) with Eq. (2) yields
Koma14 ; Koma15

$\dot{H}=-\frac{3}{2}H^{2}+\frac{3}{2}f_{\Lambda}(t)+h_{\textrm{B}}(t),$ (6)

or equivalently,

$\dot{H}=\begin{cases}-\frac{3}{2}H^{2}+\frac{3}{2}f_{\Lambda}(t)&(\Lambda(t)\hskip
2.84526pt\rm{model}),\\\ -\frac{3}{2}H^{2}+h_{\textrm{B}}(t)&(\rm{BV}\hskip
2.84526pt\rm{model}).\\\ \end{cases}$ (7)

These equations indicate that the background evolution of the universe in the
$\Lambda(t)$ and BV models is equivalent when the driving terms are equal
Koma15 :

$\frac{3}{2}f_{\Lambda}(t)=h_{\textrm{B}}(t).$ (8)

In this study, we set $\frac{3}{2}f_{\Lambda}(t)=h_{\textrm{B}}(t)$, as shown
in Eq. (8), and therefore, the background evolution of the universe in the two
models is the same. In addition, a power-law term proportional to $H^{\alpha}$
is used for the driving term. The power-law term is discussed in Sec. III.

###  II.2 First-order density perturbations

In this subsection, we present two formulations to examine density
perturbations in the $\Lambda(t)$ and BV models. In Secs. II.2.1 and II.2.2,
we review density perturbations in the $\Lambda(t)$ and BV models,
respectively. The formulation for the $\Lambda(t)$ and BV models used in this
study is based on the works of Basilakos et al. Sola_2009  and Jesus et al.
Lima2011 , respectively. The two formulations were summarized in a previous
report Koma6 , using the neo-Newtonian approach proposed by Lima et al.
Lima_Newtonian_1997 . A unified formulation based on the neo-Newtonian
approach is summarized in Appendix A.

In this paper, we examine first-order density perturbations in the linear
approximation by assuming a matter-dominated universe ($w=0$), which is
equivalent to a fluid without pressure ($p=0$) Koma6 . In other words, we
focus on density perturbations for matter and neglect other perturbations, as
discussed below.

####  II.2.1 Formulations for the $\Lambda(t)$ model

Density perturbations in $\Lambda(t)$CDM models have been examined in various
studies, including Basilakos et al. Sola_2009 , Solà et al. Sola_2015L14 ,
Gómez-Valent et al. Valent2015 , and Rezaei et al. Sola2019 . The formulation
for the $\Lambda(t)$ model discussed here is essentially equivalent to that
for the $\Lambda(t)$CDM model, although the theoretical backgrounds are
different. Therefore, we review density perturbations in the $\Lambda(t)$
model, according to Ref. Sola_2009  and a previous work Koma6 .

In a matter-dominated universe, substituting $w=0$ into Eq. (3) yields

$\dot{\rho}+3H\rho=-\frac{3}{8\pi G}\dot{f}_{\Lambda}(t),$ (9)

where ${f}_{\Lambda}(t)$ is a general driving term and $h_{\textrm{B}}(t)=0$
is used for the $\Lambda(t)$ model. The right-hand side of Eq. (9) is zero
when ${f}_{\Lambda}(t)$ is constant. (A power-law term is discussed in Sec.
III.)

In Ref. Sola_2009 , Basilakos et al. focused on models in which the time
dependence of $\Lambda(t)$ appears always at the expense of an interaction
with matter Koma6 . The model is considered to be an energy exchange cosmology
that assumes the transfer of energy (matter) between two fluids Barrow22 .
Similarly, in holographic cosmological models, we assume an interchange of
energy between the bulk (the universe) and the boundary (the horizon of the
universe) Lepe1 , as if it is an energy exchange cosmology Koma6 . (For
example, cosmological equations can be derived from the expansion of cosmic
space due to the difference between the degrees of freedom on the boundary and
in the bulk by applying the holographic equipartition law with an associated
entropy on the horizon Padma2012AB . Interacting holographic dark energy
models were examined in Ref. Richarte2013a .)

Consequently, the time evolution equation for the matter density contrast
$\delta\equiv\delta\rho_{m}/\rho_{m}$, namely the perturbation growth factor,
is given by Waga1994

$\ddot{\delta}+\left(2H+Q\right)\dot{\delta}-\left[4\pi
G\rho-2HQ-\dot{Q}\right]\delta=0,$ (10)

where

$\rho=\frac{3}{8\pi G}[H^{2}-{f}_{\Lambda}(t)],\quad Q=-\frac{3}{8\pi
G}\frac{\dot{f}_{\Lambda}(t)}{\rho},$ (11)

and $\rho$ is the mass density of matter. Specifically, $\rho_{m}$ is replaced
by $\rho$ because a matter-dominated universe is considered Koma6 . In
addition, $\rho$ in Eq. (10) represents $\bar{\rho}$, corresponding to a
homogenous and isotropic solution for the unperturbed equations Koma6 .
Substituting Eq. (11) into Eq. (9) yields

$\dot{\rho}+3H\rho=Q\rho.$ (12)

In this study, for numerical purposes, we use an independent variable Lima2011
that is defined as

$\eta\equiv\ln[\tilde{a}(t)],$ (13)

where $\tilde{a}(t)$ is the normalized scale factor given by

$\tilde{a}=\frac{a}{a_{0}}$ (14)

and $a_{0}$ is the scale factor at the present time Koma15 . From this
definition, $\dot{\delta}$ and $\ddot{\delta}$ are written as

$\dot{\delta}=H\frac{d\delta}{d\eta}=H\delta^{\prime}\quad\textrm{and}\quad\ddot{\delta}=H^{2}\delta^{\prime\prime}+H^{\prime}H\delta^{\prime},$
(15)

where ′ represents the differential with respect to $\eta$, namely $d/d\eta$.
Equation (15) can be written as $\dot{x}=Hx^{\prime}$ and
$\ddot{x}=H^{2}x^{\prime\prime}+H^{\prime}Hx^{\prime}$ by using an arbitrary
variable $x$. Applying these equations to Eq. (10) and performing several
calculations, the differential equation is written as

$\delta^{\prime\prime}+F_{\Lambda}(\eta)\delta^{\prime}+G_{\Lambda}(\eta)\delta=0,$
(16)

where $F_{\Lambda}(\eta)$ and $G_{\Lambda}(\eta)$ are given by

$F_{\Lambda}(\eta)=2+\frac{Q+H^{\prime}}{H}$ (17)

and

$\displaystyle G_{\Lambda}(\eta)=$
$\displaystyle-\frac{3}{2}+\frac{2Q+Q^{\prime}}{H}+\frac{3f_{\Lambda}(t)}{2H^{2}},$
(18)

where $H\neq 0$ and $4\pi G\rho=\frac{3}{2}(H^{2}-{f}_{\Lambda}(t))$, as given
by Eq. (1). We use Eqs. (16)–(18) to examine density perturbations in the
$\Lambda(t)$ model.

It should be noted that it is necessary to define explicitly the functional
form of the ${f}_{\Lambda}(t)$ component to solve the above differential
equation Koma6 . As described in Ref. Sola_2009 , the approach based on Eq.
(10) implies that dark energy perturbations are assumed to be negligible. This
assumption is generally justified in most cases Sola_2009 ; Sola_2007-2009
and has been most recently examined in the work of Gómez-Valent and Solà
Valent2018a . In this study, we assume that boundary perturbations are
negligible in holographic cosmological models Koma6 .

####  II.2.2 Formulations for the BV model

BV models assume dissipation processes and therefore, the formulation for the
BV model is essentially equivalent to that for both bulk viscous and CCDM
models Koma6 ; Koma9 ; Koma14 ; Koma15 . For example, density perturbations in
the CCDM model were examined by Jesus et al. Lima2011 . In addition, density
perturbations in the BV model can be derived from a neo-Newtonian approach
Koma6 . The neo-Newtonian approach was proposed by Lima et al.
Lima_Newtonian_1997 , following earlier ideas developed by McCrea McCrea_1951
and Harrison Harrison_1965 , to describe a Newtonian universe with pressure
Lima2011 . We apply the ideas in these works to review the density
perturbations in the BV model.

We assume a matter-dominated universe ($w=0$) and model it as a pressureless
fluid ($p=0$). Substituting $w=0$ into Eq. (3) yields Koma6

$\dot{\rho}+3H\rho=\frac{3}{4\pi G}Hh_{\textrm{B}}(t),$ (19)

where $h_{\textrm{B}}(t)$ is a general driving term and ${f}_{\Lambda}(t)=0$
is used for the BV model. (A power-law term is discussed in Sec. III.) The
right-hand side of Eq. (19) is not $0$, even if $h_{\textrm{B}}(t)$ is
constant. Equation (19) is essentially equivalent to the model examined in
Ref. Lima2011 . To confirm this, we consider an effective pressure $p_{e}$ due
to dissipation processes Koma6 . The effective pressure $p_{e}$ is given by
$p_{e}=p+p_{c}$, where $p_{c}$ is the creation pressure for constant specific
entropy in CCDM models Lima2011 . In this study, $p_{e}$ is equivalent to
$p_{c}$, because $p=0$. In addition, as examined in Ref. Lima2011 , $p_{c}$
for a CDM component can be written as

$p_{c}=-\frac{\rho c^{2}\Gamma}{3H},$ (20)

where $\Gamma$ is given by

$\Gamma=\frac{3}{4\pi G}\frac{Hh_{\textrm{B}}(t)}{\rho}.$ (21)

Therefore, Eq. (19) is written as

$\dot{\rho}+3H\rho=\Gamma\rho,$ (22)

where $\Gamma$ is a parameter related to entropy production processes Koma15 .
In the CCDM model, $\Gamma$ is considered to be the creation rate of CDM
particles Lima2011 .

We note that a perturbation analysis in cosmology generally requires a full
relativistic description, as examined in Jesus et al. Lima2011 . This is
because the standard non-relativistic (Newtonian) approach works well only
when the scale of perturbation is much less than the Hubble radius and the
velocity of peculiar motions is small in comparison with the Hubble flow
Lima2011 . In fact, Jesus et al. proposed a neo-Newtonian approximation that
circumvents such difficulties. In this study, we applied a neo-Newtonian
approximation to the BV model, as discussed in Ref. Koma6 .

In our units, $c=1$ and the time evolution equation for the matter density
contrast $\delta$ is given by Lima2011

$\displaystyle\ddot{\delta}$
$\displaystyle+\left[2H+\Gamma+3c_{\rm{eff}}^{2}H-\frac{\Gamma\dot{H}-H\dot{\Gamma}}{H(3H-\Gamma)}\right]\dot{\delta}$
$\displaystyle+\Bigg{\\{}3(\dot{H}+2H^{2})\left(c_{\rm{eff}}^{2}+\frac{\Gamma}{3H}\right)$
$\displaystyle+3H\left[\dot{c}_{\rm{eff}}^{2}-(1+c_{\rm{eff}}^{2})\frac{\Gamma\dot{H}-H\dot{\Gamma}}{H(3H-\Gamma)}\right]$
$\displaystyle-4\pi
G\rho\left(1-\frac{\Gamma}{3H}\right)(1+3c_{\rm{eff}}^{2})+\frac{k^{2}c_{\rm{eff}}^{2}}{a^{2}}\Bigg{\\}}\delta=0.$
(23)

In this study, the effective sound speed, $c_{\rm{eff}}^{2}\equiv\delta
p_{c}/\delta\rho$, is set to

$c_{\rm{eff}}^{2}\equiv\frac{\delta p_{c}}{\delta\rho}=0$ (24)

to ensure equivalence between the neo-Newtonian and general relativistic
approaches Reis_2003 . In fact, the neo-Newtonian equation given by Eq. (23)
is equivalent to the general relativistic equation for a single-fluid-
dominated universe only when $c_{\rm{eff}}^{2}=0$, as examined by Reis
Reis_2003 . The equivalence is discussed by Ramos et al. Ramos_2014 .

For numerical purposes, we use an independent variable
$\eta\equiv\ln[\tilde{a}(t)]$, which is defined by Eq. (13). Therefore,
$\dot{\delta}$ and $\ddot{\delta}$ are given by Eq. (15). Also, the Friedmann
equation for the BV model is written as $4\pi G\rho=3H^{2}/2$. We apply these
equations and $c_{\rm{eff}}^{2}=0$ to Eq. (23) and perform several operations
to obtain

$\delta^{\prime\prime}+F_{\textrm{B}}(\eta)\delta^{\prime}+G_{\textrm{B}}(\eta)\delta=0,$
(25)

where $F_{\textrm{B}}(\eta)$ and $G_{\textrm{B}}(\eta)$ are given by Lima2011

$F_{\textrm{B}}(\eta)=2+\frac{\Gamma+H^{\prime}}{H}-\frac{\Gamma
H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)},$ (26)
$G_{\textrm{B}}(\eta)=\left(\frac{\Gamma}{H}-1\right)\left(\frac{\Gamma}{2H}+\frac{3}{2}\right)-\frac{3(\Gamma
H^{\prime}-H\Gamma^{\prime})}{H(3H-\Gamma)}.$ (27)

Equations (25)–(27) are used to examine density perturbations in the BV model.

In this section, we reviewed density perturbations in the $\Lambda(t)$ and BV
models without using a power-law term proportional to $H^{\alpha}$. In the
next section, we apply the power-law term and derive density perturbations in
the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models.

##  III $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models

This section discusses two types of cosmological models with a power-law term
proportional to $H^{\alpha}$. In Sec. III.1, the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models are formulated. In Sec. III.2, density perturbations in
the two models are derived. Finally, Sec. III.3 uses Refs. Koma14 ; Koma15  to
review thermodynamic constraints on the two models.

###  III.1 Cosmological equations for the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models

General formulations of cosmological equations for the $\Lambda(t)$ and BV
models are described in the previous section. In this section, a power-law
term proportional to $H^{\alpha}$ is applied to the $\Lambda(t)$ and BV
models.

In fact, cosmological equations can be derived from the expansion of cosmic
space due to the difference between the degrees of freedom on the surface and
in the bulk, using Padmanabhan’s holographic equipartition law with an
associated entropy on the horizon Padma2012AB . As examined in a previous work
Koma11 , an acceleration equation that includes $H^{\alpha}$ terms is derived
using the holographic equipartition law with entropy corrected by a power law
Das2008Radicella2010 . For the derivation, see Ref. Koma11 .

The power-law term, namely the $H^{\alpha}$ term, was investigated for a non-
dissipative universe based on $\Lambda(t)$ models Koma11 ; Koma14  and a
dissipative universe based on BV models Koma15 . In this paper, the power-law
term is applied to the $\Lambda(t)$ and BV models, which we call the
$\Lambda(t)$-$H^{\alpha}$ model and the BV-$H^{\alpha}$ model, respectively.

For the $\Lambda(t)$-$H^{\alpha}$ model, a driving term $f_{\Lambda}(t)$ is
given by

$f_{\Lambda}(t)=\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha},$
(28)

and for the BV-$H^{\alpha}$ model, the driving term $h_{\textrm{B}}(t)$ is

$h_{\textrm{B}}(t)=\frac{3}{2}\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha},$
(29)

where $H_{0}$ represents the Hubble parameter at the present time and $\alpha$
is a dimensionless constant whose value is a real number Koma14 ; Koma15 .
Also, $\Psi_{\alpha}$ is a density parameter for effective dark energy and is
assumed to be

$0\leq\Psi_{\alpha}\leq 1.$ (30)

In this paper, $\alpha$ and $\Psi_{\alpha}$ are considered to be independent
free parameters Koma14 ; Koma15 . This means that we phenomenologically assume
the power-law term for the two models. In addition, as shown in Eqs. (28) and
(29), the two driving terms are set to
$\frac{3}{2}f_{\Lambda}(t)=h_{\textrm{B}}(t)$ so that Eq. (8) can be
satisfied. Substituting Eq. (28) for the $\Lambda(t)$-$H^{\alpha}$ model into
Eq. (7) yields

$\displaystyle\dot{H}$
$\displaystyle=-\frac{3}{2}H^{2}\left[1-\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2}\right].$
(31)

An equivalent equation is obtained by substituting Eq. (29) for the
BV-$H^{\alpha}$ model into Eq. (7). In this way, the same background evolution
of the universe is established for both the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models.

Equation (31) has been examined previously Koma14 ; Koma15 . The solution for
$\alpha\neq 2$ is written as

$\left(\frac{H}{H_{0}}\right)^{2-\alpha}=(1-\Psi_{\alpha})\tilde{a}^{-\frac{3(2-\alpha)}{2}}+\Psi_{\alpha},$
(32)

and the solution for $\alpha=2$ is

$\frac{H}{H_{0}}=\tilde{a}^{-\frac{3(1-\Psi_{\alpha})}{2}},$ (33)

where $\tilde{a}$ is the normalized scale factor given by Eq. (14). These
solutions can be applied to the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$
models. The derivation is summarized in Ref. Koma14 . (When $\alpha=0$,
replacing $\Psi_{\alpha}$ by $\Omega_{\Lambda}$, the density parameter for
$\Lambda$, gives a background evolution that is equivalent to that in
$\Lambda$CDM models Koma15 . We neglect the influence of radiation in a late,
flat FRW universe.) In addition, the temporal deceleration parameter $q$,
defined by $q\equiv-\left(\frac{\ddot{a}}{aH^{2}}\right)$, can be calculated
from the above equations. Using the result of Refs. Koma14 ; Koma15 , the
deceleration parameter $q$ for the two models is written as

$\displaystyle q$
$\displaystyle=\frac{1}{2}-\frac{3}{2}\Psi_{\alpha}\left[(1-\Psi_{\alpha})\tilde{a}^{-\frac{3(2-\alpha)}{2}}+\Psi_{\alpha}\right]^{-1},$
(34)

where a positive and negative $q$ represent deceleration and acceleration,
respectively.

In the present study, the background evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models is set to be the same, as
mentioned previously. However, even in this case, the evolution of density
perturbations is expected to be different. We discuss density perturbations in
the next subsection. Hereafter, we consider $\alpha\neq 2$ because the result
for $\alpha\neq 2$ reduces to that for $\alpha=2$ when $\alpha\rightarrow 2$.

It should be noted that a power series of $H$ for $\Lambda(t)$ models was
examined in works such as Solà et al. Sola_2015L14 , Gómez-Valent et al.
Valent2015 , and Rezaei et al. Sola2019 . For BV models, a power-law term was
examined in, for example, the works of Freaza et al. Freaza2002 , Ramos et al.
Ramos_2014 , and Cárdenas et al. Cardenas2020 . From a microscopic viewpoint,
the driving term for running vacuum models [related to $\Lambda(t)$ models]
can be obtained from various concepts, including renormalization group
equations Sola_2013b , quantum field theory in curved spacetime Sola2020_2 ,
and string theory Sola2020_3 . In contrast, the driving term for bulk viscous
and CCDM models (related to BV models) is phenomenologically assumed based on
macroscopic properties, such as bulk viscosity of cosmological fluids
Weinberg0 ; Murphy1  and matter creation Prigogine_1988-1989 ; Lima1992-2016 ,
respectively. In this paper, we phenomenologically assume the power-law term
for the $\Lambda(t)$ and BV models and therefore, the theoretical backgrounds
are different from those of the above models. However, the formulations used
here are essentially equivalent to those for the models.

###  III.2 First-order density perturbations in the $\Lambda(t)$-$H^{\alpha}$
and BV-$H^{\alpha}$ models

In this section, we examine first-order density perturbations in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. To calculate density
perturbations, $H^{\prime}/H$ is required. Accordingly, we write the
background evolution again. From Eq. (32), the evolution of the Hubble
parameter for $\alpha\neq 2$ can be written as

$\displaystyle\frac{H}{H_{0}}$
$\displaystyle=\left[(1-\Psi_{\alpha})\tilde{a}^{-\frac{3(2-\alpha)}{2}}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}}$
$\displaystyle=\left[(1-\Psi_{\alpha})\tilde{a}^{-\beta}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}}$
$\displaystyle=\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}},$
(35)

where $\tilde{a}^{-\beta}$ is replaced by $e^{-\beta\eta}$ using
$\eta\equiv\ln[\tilde{a}(t)]$ given by Eq. (13). A dimensionless parameter
$\beta$ is used for simplicity and given by

$\beta=\frac{3(2-\alpha)}{2}.$ (36)

Differentiating Eq. (35) with respect to $\eta$ yields

$\displaystyle\frac{H^{\prime}}{H_{0}}$
$\displaystyle=\frac{d}{d\eta}\left(\frac{H}{H_{0}}\right)=\frac{d}{d\eta}\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}}$
$\displaystyle=\frac{(-\beta)(1-\Psi_{\alpha})e^{-\beta\eta}}{2-\alpha}\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}-1}$
$\displaystyle=\frac{-3(1-\Psi_{\alpha})e^{-\beta\eta}}{2}\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{\alpha-1}{2-\alpha}}.$
(37)

Dividing Eq. (37) by Eq. (35) yields

$\displaystyle\frac{H^{\prime}}{H}$
$\displaystyle=\frac{\frac{-3(1-\Psi_{\alpha})e^{-\beta\eta}}{2}\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{\alpha-1}{2-\alpha}}}{\left[(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}\right]^{\frac{1}{2-\alpha}}}$
$\displaystyle=\frac{-\frac{3}{2}(1-\Psi_{\alpha})e^{-\beta\eta}}{(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}}=\frac{-\frac{3}{2}(1-\Psi_{\alpha})}{(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}}.$
(38)

Then, we use the obtained $H^{\prime}/H$ to derive density perturbations in
the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. Note that
$\alpha\neq 2$ is considered here because the result for $\alpha\neq 2$
reduces to that for $\alpha=2$ when $\alpha\rightarrow 2$.

####  III.2.1 Density perturbations in the $\Lambda(t)$-$H^{\alpha}$ model

We examine density perturbations in the $\Lambda(t)$-$H^{\alpha}$ model. From
Eq. (28), the driving term $f_{\Lambda}(t)$ is written as

$f_{\Lambda}(t)=\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha}.$
(39)

We write Eqs. (16)–(18) again as

$\delta^{\prime\prime}+F_{\Lambda}(\eta)\delta^{\prime}+G_{\Lambda}(\eta)\delta=0,$
(40) $F_{\Lambda}(\eta)=2+\frac{Q+H^{\prime}}{H},$ (41) $\displaystyle
G_{\Lambda}(\eta)=$
$\displaystyle-\frac{3}{2}+\frac{2Q+Q^{\prime}}{H}+\frac{3f_{\Lambda}(t)}{2H^{2}},$
(42)

where $Q$ and $\rho$ given by Eq. (11) are written as

$Q=-\frac{3}{8\pi G}\frac{\dot{f}_{\Lambda}(t)}{\rho},$ (43)
$\rho=\frac{3}{8\pi G}[H^{2}-{f}_{\Lambda}(t)].$ (44)

We now calculate $F_{\Lambda}(\eta)$ and $G_{\Lambda}(\eta)$. To this end, we
require three terms, namely $Q/H$, $Q^{\prime}/H$, and $f_{\Lambda}(t)/H^{2}$.
First, we calculate $Q$ to obtain $Q/H$. Substituting Eq. (44) into Eq. (43)
and applying Eq. (39) yields

$\displaystyle Q$ $\displaystyle=-\frac{3}{8\pi
G}\frac{\dot{f}_{\Lambda}(t)}{\rho}=-\frac{3}{8\pi
G}\frac{\dot{f}_{\Lambda}(t)}{\frac{3}{8\pi G}(H^{2}-{f}_{\Lambda}(t))}$
$\displaystyle=-\frac{\alpha\Psi_{\alpha}H_{0}\left(\frac{H}{H_{0}}\right)^{\alpha-1}\dot{H}}{H^{2}-\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha}}=-\frac{\alpha\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2}\frac{\dot{H}}{H}}{1-\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2}}$
$\displaystyle=-\frac{\alpha\Psi_{\alpha}H^{\prime}}{\left(\frac{H}{H_{0}}\right)^{2-\alpha}-\Psi_{\alpha}},$
(45)

where $\dot{H}=HH^{\prime}$ is also used. In addition, substituting Eq. (35)
into Eq. (45) yields

$\displaystyle Q$
$\displaystyle=-\frac{\alpha\Psi_{\alpha}H^{\prime}}{(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}-\Psi_{\alpha}}=-\frac{\alpha\Psi_{\alpha}e^{\beta\eta}H^{\prime}}{1-\Psi_{\alpha}}.$
(46)

Dividing Eq. (46) by $H$ allows us to write $Q/H$ as

$\displaystyle\frac{Q}{H}$
$\displaystyle=-\frac{\alpha\Psi_{\alpha}e^{\beta\eta}}{1-\Psi_{\alpha}}\frac{H^{\prime}}{H}.$
(47)

Second, we calculate $Q^{\prime}$ to obtain $Q^{\prime}/H$. Differentiating
Eq. (46) with respect to $\eta$ yields

$\displaystyle Q^{\prime}$
$\displaystyle=\frac{dQ}{d\eta}=\left(-\frac{\alpha\Psi_{\alpha}}{1-\Psi_{\alpha}}\right)(\beta
e^{\beta\eta}H^{\prime}+e^{\beta\eta}H^{\prime\prime})$
$\displaystyle=\left(-\frac{\alpha\Psi_{\alpha}}{1-\Psi_{\alpha}}\right)e^{\beta\eta}(\beta
H^{\prime}+H^{\prime\prime}).$ (48)

Dividing Eq. (48) by $H$ gives

$\displaystyle\frac{Q^{\prime}}{H}$
$\displaystyle=\left(-\frac{\alpha\Psi_{\alpha}}{1-\Psi_{\alpha}}\right)e^{\beta\eta}\left(\beta\frac{H^{\prime}}{H}+\frac{H^{\prime\prime}}{H}\right).$
(49)

To calculate this equation, $H^{\prime}/H$ and $H^{\prime\prime}/H$ are
required. The first term $H^{\prime}/H$ is given by Eq. (38), and the second
term $H^{\prime\prime}/H$ can be calculated as follows. After differentiating
$H^{\prime}/H_{0}$ [Eq. (37)] with respect to $\eta$, dividing the resultant
equation by $H/H_{0}$ [Eq. (35)], and performing several calculations, we have

$\displaystyle\frac{H^{\prime\prime}}{H}$
$\displaystyle=\frac{9(1-\Psi_{\alpha})[(1-\Psi_{\alpha})+(2-\alpha)\Psi_{\alpha}e^{\beta\eta}]}{4\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]^{2}}.$
(50)

Thus, $Q^{\prime}/H$ is calculated from Eq. (49) by applying Eqs. (38) and
(50). Third, we calculate $f_{\Lambda}(t)/H^{2}$. Substituting Eq. (39) into
$f_{\Lambda}(t)/H^{2}$ and applying Eq. (35) gives

$\displaystyle\frac{f_{\Lambda}(t)}{H^{2}}$
$\displaystyle=\frac{\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha}}{H^{2}}=\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2}$
$\displaystyle=\frac{\Psi_{\alpha}}{(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}}=\frac{\Psi_{\alpha}e^{\beta\eta}}{(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}}.$
(51)

In this way, the three terms $Q/H$, $Q^{\prime}/H$, and $f_{\Lambda}(t)/H^{2}$
are obtained.

Substituting Eqs. (38) and (47) into Eq. (41) yields

$F_{\Lambda}(\eta)=\frac{(1-\Psi_{\alpha})+(4+3\alpha)\Psi_{\alpha}e^{\beta\eta}}{2\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]}.$
(52)

Substituting Eqs. (47), (49), and (51) into Eq. (42) and applying Eqs. (38)
and (50), we have

$\displaystyle G_{\Lambda}(\eta)=$
$\displaystyle-\frac{3}{2}+\frac{3[\alpha(2+\beta)+1]\Psi_{\alpha}e^{\beta\eta}}{2\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]}$
$\displaystyle-\frac{9\alpha\Psi_{\alpha}e^{\beta\eta}[(1-\Psi_{\alpha})+(2-\alpha)\Psi_{\alpha}e^{\beta\eta}]}{4\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]^{2}}.$
(53)

Here, $\beta$ is $\frac{3(2-\alpha)}{2}$ given by Eq. (36) and $\alpha$ is
treated as a real number. When $\alpha$ was an integer, such as $0$ or $1$,
$\Lambda(t)$-$H^{\alpha}$ models were examined although they were considered
to be different models. For example, $\Lambda(t)$-$H^{0}$ models (i.e.,
$\Lambda$CDM models) and $\Lambda(t)$-$H^{1}$ models were examined as two
different models in a previous work Koma6 . In the present study, we
systematically examine the $\Lambda(t)$-$H^{\alpha}$ model through the free
parameter $\alpha$. (We have confirmed that the above equations are equivalent
to results examined in the previous work, when $\alpha=0$ and $\alpha=1$.)

Using $F_{\Lambda}(\eta)$ and $G_{\Lambda}(\eta)$, we numerically solve the
differential equation [Eq. (40)] for the $\Lambda(t)$-$H^{\alpha}$ model. To
solve this, we use the initial conditions of the Einstein–de Sitter growing
model Lima2011 . The initial conditions are given by

$\displaystyle\delta(\tilde{a}_{i})=\tilde{a}_{i}\quad\textrm{and}\quad\delta^{\prime}(\tilde{a}_{i})=\tilde{a}_{i},$
(54)

where $\tilde{a}_{i}$ is set to $10^{-3}$ Lima2011 ; Koma6 . Note that the
initial conditions are applied to BV-$H^{\alpha}$ models as well.

####  III.2.2 Density perturbations in the BV-$H^{\alpha}$ model

Here, we examine density perturbations in the BV-$H^{\alpha}$ model. The
driving term $h_{\textrm{B}}(t)$ given by Eq. (29) is written as

$h_{\textrm{B}}(t)=\frac{3}{2}\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha}.$
(55)

Then, we rewrite Eqs. (25)–(27).

$\delta^{\prime\prime}+F_{\textrm{B}}(\eta)\delta^{\prime}+G_{\textrm{B}}(\eta)\delta=0,$
(56) $F_{\textrm{B}}(\eta)=2+\frac{\Gamma+H^{\prime}}{H}-\frac{\Gamma
H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)},$ (57)
$G_{\textrm{B}}(\eta)=\left(\frac{\Gamma}{H}-1\right)\left(\frac{\Gamma}{2H}+\frac{3}{2}\right)-\frac{3(\Gamma
H^{\prime}-H\Gamma^{\prime})}{H(3H-\Gamma)},$ (58)

where, from Eq. (21), $\Gamma/H$ can be written as

$\frac{\Gamma}{H}=\frac{3}{4\pi G}\frac{h_{\textrm{B}}(t)}{\rho},$ (59)

and $\rho$ is given by

$\rho=\frac{3H^{2}}{8\pi G}.$ (60)

Equation (60) is obtained from the Friedmann equation by substituting
$f_{\Lambda}(t)=0$ into Eq. (1).

We now calculate $F_{\textrm{B}}(\eta)$ and $G_{\textrm{B}}(\eta)$. To this
end, we require three terms, namely $\Gamma/H$, $\Gamma^{\prime}/H$, and
$\frac{\Gamma H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)}$. The second term is
required for calculating the third term. First, we calculate $\Gamma/H$.
Substituting Eqs. (55) and (60) into Eq. (59) yields

$\displaystyle\frac{\Gamma}{H}=\frac{3}{4\pi
G}\frac{\frac{3}{2}\Psi_{\alpha}H_{0}^{2}\left(\frac{H}{H_{0}}\right)^{\alpha}}{\frac{3H^{2}}{8\pi
G}}$ $\displaystyle=3\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2},$
(61)

or equivalently,

$\displaystyle\Gamma$
$\displaystyle=3\Psi_{\alpha}H_{0}\left(\frac{H}{H_{0}}\right)^{\alpha-1}.$
(62)

Substituting Eq. (35) into Eq. (61) yields

$\displaystyle\frac{\Gamma}{H}$
$\displaystyle=3\Psi_{\alpha}\left(\frac{H}{H_{0}}\right)^{\alpha-2}=\frac{3\Psi_{\alpha}}{(1-\Psi_{\alpha})e^{-\beta\eta}+\Psi_{\alpha}}$
$\displaystyle=\frac{3\Psi_{\alpha}e^{\beta\eta}}{(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}}.$
(63)

Second, we calculate $\Gamma^{\prime}/H$, which is used for determining the
third term. After differentiating Eq. (62) with respect to $\eta$, applying
Eq. (63), and dividing the resultant equation by $H$, we write
$\Gamma^{\prime}/H$ as

$\frac{\Gamma^{\prime}}{H}=(\alpha-1)\frac{\Gamma}{H}\frac{H^{\prime}}{H}.$
(64)

Third, we calculate $\frac{\Gamma H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)}$.
Reformulating this term and substituting Eq. (64) into the resultant equation
yields

$\displaystyle\frac{\Gamma H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)}$
$\displaystyle=\frac{\frac{\Gamma}{H}\frac{H^{\prime}}{H}-\frac{\Gamma^{\prime}}{H}}{3-\frac{\Gamma}{H}}=\frac{\frac{\Gamma}{H}\frac{H^{\prime}}{H}-(\alpha-1)\frac{\Gamma}{H}\frac{H^{\prime}}{H}}{3-\frac{\Gamma}{H}}$
$\displaystyle=\frac{(2-\alpha)\frac{\Gamma}{H}\frac{H^{\prime}}{H}}{3-\frac{\Gamma}{H}}.$
(65)

Substituting Eqs. (38) and (63) into Eq. (65) gives

$\displaystyle\frac{\Gamma H^{\prime}-H\Gamma^{\prime}}{H(3H-\Gamma)}$
$\displaystyle=\frac{-\frac{3}{2}(2-\alpha)\Psi_{\alpha}e^{\beta\eta}}{(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}}.$
(66)

From these results, we can calculate $F_{\textrm{B}}(\eta)$ and
$G_{\textrm{B}}(\eta)$. Substituting Eqs. (38), (63), and (66) into Eq. (57)
yields

$F_{\textrm{B}}(\eta)=\frac{(1-\Psi_{\alpha})+(16-3\alpha)\Psi_{\alpha}e^{\beta\eta}}{2\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]}.$
(67)

Substituting Eqs. (63) and (66) into Eq. (58) yields

$\displaystyle G_{\textrm{B}}(\eta)=$
$\displaystyle-\frac{3}{2}+\frac{(24-9\alpha)\Psi_{\alpha}e^{\beta\eta}}{2\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]}$
$\displaystyle+\frac{9\Psi_{\alpha}^{2}e^{2\beta\eta}}{2\left[(1-\Psi_{\alpha})+\Psi_{\alpha}e^{\beta\eta}\right]^{2}}.$
(68)

Here, $\beta$ is $\frac{3(2-\alpha)}{2}$ given by Eq. (36), and $\alpha$ is
treated as a real number. Using $F_{\textrm{B}}(\eta)$,
$G_{\textrm{B}}(\eta)$, and the initial conditions given by Eq. (54), we can
numerically solve the differential equation [Eq. (56)] for the BV-$H^{\alpha}$
model.

When $\alpha$ was an integer, such as $0$ or $1$, the BV-$H^{\alpha}$ models
were examined, although they were considered to be different models. For
example, BV-$H^{0}$ and BV-$H^{1}$ models were examined as two different
models Koma6 . We have confirmed that Eqs. (67) and (68) are equivalent to
those in Ref. Koma6  when $\alpha=0$ and $\alpha=1$. Cosmological models
similar to the BV-$H^{0}$ and BV-$H^{1}$ models were investigated in, for
example, Refs. Lima2011  and Barrow21 , respectively. In this study, $\alpha$
is a free parameter and a real number. Therefore, we can systematically
examine the BV-$H^{\alpha}$ model, which was not possible in previous works.

###  III.3 Thermodynamic constraints

Ordinary, isolated macroscopic systems spontaneously evolve to equilibrium
states that maximize the entropy consistent with their constraints Callen . In
other words, the entropy of such systems does not decrease (i.e., the second
law of thermodynamics) and approaches a certain maximum value at the last
stage (i.e., the maximization of entropy) Koma14 . In fact, a certain type of
universe is expected to be constrained by thermodynamics as if it behaves as
the macroscopic system Mimoso2013 .

In this subsection, we use the results of previous works Koma14 ; Koma15  to
review such thermodynamic constraints on the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models. In fact, the two models always satisfy the second law
of thermodynamics, whereas they satisfy the maximization of entropy only under
specific conditions Koma14 ; Koma15 . In particular, the maximization of
entropy depends almost entirely on the constraints on $\ddot{S}_{\rm{BH}}<0$,
where $S_{\rm{BH}}$ is the Bekenstein–Hawking entropy Koma15 . Therefore, to
discuss the thermodynamic constraints, we examine $\ddot{S}_{\rm{BH}}<0$. For
this purpose, we present the Bekenstein–Hawking entropy on the horizon of the
universe.

The Bekenstein–Hawking entropy $S_{\rm{BH}}$ is written as

$S_{\rm{BH}}=\frac{k_{B}c^{3}}{\hbar G}\frac{A_{H}}{4},$ (69)

where $\hbar$ is the reduced Planck constant, defined as $\hbar\equiv
h/(2\pi)$ using the Planck constant $h$ Koma11 ; Koma12 ; Koma14  and $A_{H}$
is the surface area of a sphere with a Hubble horizon $r_{H}$, which is given
by $c/H$. In a flat FRW universe, the Hubble horizon is equivalent to the
apparent horizon. Substituting $A_{H}=4\pi r_{H}^{2}$ into Eq. (69) and
applying $r_{H}=c/H$ yields

$S_{\rm{BH}}=\left(\frac{\pi k_{B}c^{5}}{\hbar G}\right)\frac{1}{H^{2}}.$ (70)

This equation indicates that $S_{\rm{BH}}$ depends on the background evolution
of the universe. Therefore, the evolution of $S_{\rm{BH}}$ in the two models
is equivalent when their background evolution is the same.

Substituting Eq. (32) into Eq. (70) and performing several calculation, we can
obtain $S_{\rm{BH}}$, $\dot{S}_{\rm{BH}}$, and $\ddot{S}_{\rm{BH}}$. The
result of Ref. Koma15  is used to write the normalized $\ddot{S}_{\rm{BH}}$ as

$\frac{\ddot{S}_{\rm{BH}}}{S_{\rm{BH},0}H_{0}^{2}}=\frac{9}{2}\frac{(1-\Psi_{\alpha})\tilde{a}^{-\beta}\left[(1-\Psi_{\alpha})\tilde{a}^{-\beta}+(\alpha-2)\Psi_{\alpha}\right]}{\left[(1-\Psi_{\alpha})\tilde{a}^{-\beta}+\Psi_{\alpha}\right]^{2}},$
(71)

where $S_{\rm{BH},0}$ is the current Bekenstein–Hawking entropy and $\beta$ is
$\frac{3(2-\alpha)}{2}$ given by Eq. (36). Equation (71) indicates that
$\ddot{S}_{\rm{BH}}<0$ should be satisfied at least in the last stage, that
is, $\tilde{a}\rightarrow\infty$, when $\alpha<2$ Koma14 . In other words, a
region that satisfies $\ddot{S}_{\rm{BH}}<0$ in the $(\Psi_{\alpha},\alpha)$
plane varies with time before the last stage. To study such a relaxation-like
process, we use the boundary required for $\ddot{S}_{\rm{BH}}=0$, which is
given by Koma14

$\displaystyle\Psi_{\alpha}=\frac{\tilde{a}^{-\beta}}{2-\alpha+\tilde{a}^{-\beta}}.$
(72)

From this equation, we can plot the boundary of $\ddot{S}_{\rm{BH}}=0$ in the
$(\Psi_{\alpha},\alpha)$ plane. Here $\Psi_{\alpha}$ corresponds to a density
parameter for effective dark energy.

Figure 1:  (Color online). Thermodynamic constraints on the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models in the
$(\Psi_{\alpha},\alpha)$ plane. The boundary of $\ddot{S}_{\rm{BH}}=0$ for
$\tilde{a}=0.5$, $1$, and $10$ is shown. The boundary for the two models is
the same because the same background evolution is used for both models. The
arrow on each boundary indicates a region that satisfies
$\ddot{S}_{\rm{BH}}<0$. All the boundaries intersect at the point
$(\Psi_{\alpha},\alpha)=(1,2)$. The horizontal dashed line represents
$\alpha=2$. The region below the dashed line should satisfy
$\ddot{S}_{\rm{BH}}<0$, at least in the last stage Koma14 . In the last stage,
the region should also satisfy observational constraints on an initially
decelerating and then accelerating universe Koma15 . Similar thermodynamic
constraints have been examined in Refs. Koma14 ; Koma15 .

To examine the evolution of the boundary, typical boundaries for
$\tilde{a}=0.5$, $1$, and $10$ are shown in Fig. 1. The boundary for the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models is the same because the
background evolution of the universe in both models is equivalent. The arrow
on each boundary indicates a region that satisfies $\ddot{S}_{\rm{BH}}<0$.
(Similar boundaries are examined in Refs. Koma14 ; Koma15 .) As shown in Fig.
1, this region gradually extends and approaches $\alpha=2$ with increasing
$\tilde{a}$. When $\alpha<2$, maximization of the entropy,
$\ddot{S}_{\rm{BH}}<0$, should be satisfied, at least in the last stage of the
evolution of an expanding universe Koma14 . In this way, thermodynamic
constraints on the two models can be discussed in the $(\Psi_{\alpha},\alpha)$
plane. In the next section, we examine observational constraints in
combination with the thermodynamic constraints shown here.

It should be noted that cosmological adiabatic particle creation results in
the generation of irreversible entropy Prigogine_1988-1989 . The irreversible
entropy $S_{m}$ due to adiabatic particle creation was examined in a previous
work Koma15 . Consequently, $\ddot{S}_{m}<0$ is found to be always satisfied
when $\alpha<2$. That is, constraints on $\ddot{S}_{m}<0$ are slightly looser
than those on $\ddot{S}_{\rm{BH}}<0$. In addition, it is well known that
$S_{\rm{BH}}$ is extremely large in comparison with the other entropies Egan1
. These results indicate that the maximization of entropy depends almost
entirely on the constraints on $\ddot{S}_{\rm{BH}}<0$, as examined in Ref.
Koma15 . Accordingly, we use $\ddot{S}_{\rm{BH}}<0$ to discuss the
thermodynamic constraints in this study.

##  IV Evolution of the universe in the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models

In this section, we examine the evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. In Sec. IV.1, the
evolution of the universe for $\Psi_{\alpha}=0.685$ is discussed as a specific
case. In Sec. IV.2, the observational and thermodynamic constraints on the two
models are investigated with chi-squared functions in the
$(\Psi_{\alpha},\alpha)$ plane. Here, $\Psi_{\alpha}$ and $\alpha$ are treated
as free parameters. Note that we do not discuss the significant tension
between the Planck results Planck2018  and the local (distance ladder)
measurement from the Hubble Space Telescope Riess20162019 .

###  IV.1 Evolution of the universe for $\Psi_{\alpha}=0.685$

We examine the typical evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. To this end,
$\Psi_{\alpha}$ is set to $0.685$, which is equivalent to $\Omega_{\Lambda}$
for the standard $\Lambda$CDM model from the Planck 2018 results Planck2018 .
($\Psi_{\alpha}$ corresponds to a density parameter for effective dark
energy.) Therefore, the following result for $\alpha=0$ of the
$\Lambda(t)$-$H^{\alpha}$ model is equivalent to that for the $\Lambda$CDM
model.

####  IV.1.1 Background evolution of the universe for $\Psi_{\alpha}=0.685$

Figure 2:  (Color online). Background evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models for
$\Psi_{\alpha}=0.685$. (a) Luminosity distance $d_{L}$. (b) Deceleration
parameter $q$. The background evolution of the universe in the two models is
the same. In (a), the symbols with error bars are observed supernova data
points taken from Ref. Suzuki_2012 . To normalize the data points, $H_{0}$ is
set to $67.4$ km/s/Mpc from the Planck 2018 results Planck2018 . In (b), the
horizontal dashed line represents $q=0$.

To examine the background evolution of the universe, we use the luminosity
distance $d_{L}$ Sato1 ; Carroll01 , which is written as

$\left(\frac{H_{0}}{c}\right)d_{L}=(1+z)\int_{1}^{1+z}\frac{dy}{F(y)},$ (73)

where the redshift $z$ is given by

$z=\tilde{a}^{-1}-1.$ (74)

The integrating variable $y$ and the function $F(y)$ are given by

$y=\tilde{a}^{-1}\quad\textrm{and}\quad F(y)=\frac{H}{H_{0}}.$ (75)

For the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models, $H/H_{0}$ is
given by Eq. (32). The background evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models is equivalent and
therefore, $d_{L}$ for the two models is the same. Similarly, the temporal
deceleration parameter $q$ for the two models is the same, where $q$ is given
by Eq. (34).

Figure 2 shows the background evolution of the universe for
$\Psi_{\alpha}=0.685$. To examine typical results, $\alpha$ is set to $0$,
$0.5$, and $1$. In Fig. 2(a), the observed data points are the Union 2.1 set
of $580$ type Ia supernovae Suzuki_2012 . As shown in Fig. 2(a), the
luminosity distance $d_{L}$ for $\alpha=0$, $0.5$, and $1$ is likely
consistent with the supernova data. The luminosity distance $d_{L}$ for
$\alpha=0.5$ and $1$ deviates from $d_{L}$ for $\alpha=0$. Note that the
deviation looks small because a logarithmic scale is used for the vertical
axis in this figure. From Fig. 2(b), we can confirm that $\alpha$ affects the
evolution of the deceleration parameter $q$. Of course, both $\Psi_{\alpha}$
and $\alpha$ affect the background evolution of the universe. We examine this
influence later using chi-squared functions in the $(\Psi_{\alpha},\alpha)$
plane.

####  IV.1.2 Evolution of density perturbations for $\Psi_{\alpha}=0.685$

In this study, the background evolution of the universe is the same in both
models. However, even with this similarity, the density perturbations are
expected to be different. In this subsection, we examine first-order density
perturbations in the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models for
$\Psi_{\alpha}=0.685$.

We first examine the evolution of the perturbation growth factor $\delta$ for
the two models. In this study, $\delta$ is numerically solved using the
initial conditions given by Eq. (54). To examine typical results, $\alpha$ is
set to $0$, $0.5$, and $1$. Accordingly, the background evolution considered
here is the same as that shown in Fig. 2. A previous work Koma6  investigated
similar density perturbations using the $\Lambda(t)$-$H^{0}$,
$\Lambda(t)$-$H^{1}$, BV-$H^{0}$, and BV-$H^{1}$ models. In this study,
$\alpha$ is a free parameter that can be treated as a real number. Therefore,
we can systematically examine the difference between the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models.

For $\tilde{a}\lessapprox 0.1$, $\delta$ increases with $\tilde{a}$, as shown
in Fig. 3. Thereafter, the increase of $\delta$ tends to gradually slow. For
$\tilde{a}\gtrapprox 1$, $\delta$ for $\alpha=0$ of the
$\Lambda(t)$-$H^{\alpha}$ model does not decrease, whereas $\delta$ for the
others decreases. In this way, $\alpha$ and the type of model affect the
density perturbations. In particular, the decrease in $\delta$ for the
BV-$H^{\alpha}$ model is significant, in comparison with the
$\Lambda(t)$-$H^{\alpha}$ model. As expected, density perturbations in the two
models are greatly different even if the background evolution is the same.

Figure 3:  (Color online). Evolution of $\delta$ for the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models with
$\Psi_{\alpha}=0.685$. The bold and thin curves represent the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models, respectively. For the
two models, $\alpha$ is set to $0$, $0.5$, and $1$.

Figure 4:  (Color online). Evolution of $f(z)$ for the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models with
$\Psi_{\alpha}=0.685$. The bold and thin curves represent the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models, respectively. The closed
circles with error bars are observed data points taken from a summary in Jesus
et al. Lima2011 . Each original data point is given in Refs. Colless2001 ;
Guzzo2008 ; Tegmark2006 ; Ross2007 ; Angela2007 ; Viel2004 ; McDonald2005 .

Figure 5:  (Color online). Evolution of $f(z)\sigma_{8}(z)$ for the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models with
$\Psi_{\alpha}=0.685$. The bold and thin curves represent the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models, respectively. The closed
circles with error bars are observed data points taken from a summary in
Nesseris et al. Nesseris2017 . Each original data point is given in Refs. 58 ;
59 ; 60 ; 61 ; 62 ; 63 ; 67 ; 68 ; 69 ; 71 ; 72 ; 73 ; 74 ; 75 .

Next, we use the obtained $\delta$ to calculate an indicator of clustering,
namely the growth rate $f(z)$ of clustering Peebles_1993  given by

$f(z)=\frac{d\ln\delta}{d\ln a}=-(1+z)\frac{d\ln\delta}{dz}.$ (76)

In addition, we calculate a combination value $f(z)\sigma_{8}(z)$. Here
$\sigma_{8}(z)$ is the redshift-dependent root-mean-square (rms) fluctuations
of the linear density field within a sphere of radius $R=8h^{-1}$ Mpc
Nesseris2017  (where $h$ is the reduced Hubble constant defined by
$h=H_{0}/100$). The redshift-dependent rms fluctuations $\sigma_{8}(z)$ can be
written as Basilakos2014

$\sigma_{8}(z)=\sigma_{8}\left[\frac{\delta(z)}{\delta(z=0)}\right],$ (77)

where $\sigma_{8}$ is $\sigma_{8}(z)$ at redshift $z=0$. We set
$\sigma_{8}=0.811$ from the Planck 2018 results Planck2018 . Note that more
exact formalisms are examined elsewhere, such as Refs. Sola_2015L14 ;
Valent2015 ; Sola2019 .

The evolution of $f(z)$ and $f(z)\sigma_{8}(z)$ is shown in Figs. 4 and 5,
respectively. As shown in Fig. 4, $f(z)$ for $\alpha=0$ of the
$\Lambda(t)$-$H^{\alpha}$ model agrees with the observed data points, whereas
$f(z)$ for the other cases does not. Similarly, $f(z)\sigma_{8}(z)$ for
$\alpha=0$ of the $\Lambda(t)$-$H^{\alpha}$ model agrees with the observed
data points (Fig. 5). In contrast, at low $z$, $f(z)$ and $f(z)\sigma_{8}(z)$
for the BV-$H^{\alpha}$ model disagree with the data points, in comparison
with the $\Lambda(t)$-$H^{\alpha}$ model. This is because, as shown in Fig. 3,
$\delta$ for the BV-$H^{\alpha}$ model decays at large $\tilde{a}$,
corresponding to low $z$.

A previous work Koma6  discussed similar results using four different models,
corresponding to the $\Lambda(t)$-$H^{0}$, $\Lambda(t)$-$H^{1}$, BV-$H^{0}$,
and BV-$H^{1}$ models. In the present study, we can systematically examine the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models using the free parameter
$\alpha$. We discuss the systematic study in the next subsection.

###  IV.2 Constraints on the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$
models

Figure 6:  (Color online). Contours of $\chi_{\textrm{SN}}^{2}$ for supernovae
and $\chi_{\textrm{GR}}^{2}$ for the growth rate in the
$(\Psi_{\alpha},\alpha)$ plane. (a) $\Lambda(t)$-$H^{\alpha}$ model. (b)
BV-$H^{\alpha}$ model. The color scale bars for $\chi_{\textrm{SN}}^{2}$ and
$\chi_{\textrm{GR}}^{2}$ are given at the upper and lower sides, respectively.
Small-$\chi_{\textrm{SN}}^{2}$ regions $(\chi_{\textrm{SN}}^{2}<640)$ and
small-$\chi_{\textrm{GR}}^{2}$ regions $(\chi_{\textrm{GR}}^{2}<80)$ are
displayed. The boundary of $\ddot{S}_{\rm{BH}}=0$ for $\tilde{a}=1$ and $10$
is plotted from Fig. 1. The arrow on each boundary indicates a region that
satisfies the maximization of the entropy, that is, $\ddot{S}_{\rm{BH}}<0$.
The contours of $\chi_{\textrm{SN}}^{2}$ and the boundary of
$\ddot{S}_{\rm{BH}}=0$ in (a) are the same as those in (b). The x’s mark the
location of the minimum value of each chi-squared function. The minimum values
are summarized in Table 1.

Figure 7:  (Color online). Contours of $\chi_{\textrm{total}}^{2}$ in the $(\Psi_{\alpha},\alpha)$ plane. (a) $\Lambda(t)$-$H^{\alpha}$ model. (b) BV-$H^{\alpha}$ model. Small-$\chi_{\textrm{total}}^{2}$ regions $(\chi_{\textrm{total}}^{2}<900)$ are displayed. The boundary of $\ddot{S}_{\rm{BH}}=0$ shown in Fig. 6 is also plotted. For the boundary, see the caption of Fig. 6. In (a), the point labeled $\Lambda$CDM represents $(\Psi_{\alpha},\alpha)=(0.685,0)$, corresponding to that model. The x’s mark the locations of the minimum value of $\chi_{\textrm{total}}^{2}$. The minimum values are summarized in Table 1.  Table 1: Minimum values of chi-squared functions for the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. The location for the minimum value, $(\Psi_{\alpha},\alpha)$, is also shown. $\Psi_{\alpha}$ and $\alpha$ are sampled in steps of $0.005$ and $0.025$, respectively. The locations for $\chi_{\textrm{SN},min}^{2}$ and $\chi_{\textrm{GR},min}^{2}$ are plotted in Fig. 6 and the location for $\chi_{\textrm{total},min}^{2}$ is plotted in Fig. 7.  Model | $\chi_{\textrm{SN},min}^{2}$ | $(\Psi_{\alpha},\alpha)$ | $\chi_{\textrm{GR},min}^{2}$ | $(\Psi_{\alpha},\alpha)$ | $\chi_{\textrm{total},min}^{2}$ | $(\Psi_{\alpha},\alpha)$
---|---|---|---|---|---|---
$\Lambda(t)$-$H^{\alpha}$ | $597.2$ | $(0.415,3.000)$ | $11.5$ | $(0.775,0.000)$ | $630.2$ | $(0.580,0.250)$
BV-$H^{\alpha}$ | $597.2$ | $(0.415,3.000)$ | $11.7$ | $(0.160,0.900)$ | $786.8$ | $(0.310,3.000)$

So far, we have considered the specific case of $\Psi_{\alpha}=0.685$. In this
subsection, we examine constraints on the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models in the $(\Psi_{\alpha},\alpha)$ plane and discuss the
question, ”Which model is favored?” To this end, we provide an overview of the
observational and thermodynamic constraints on the two models. In this
analysis, $H_{0}$ is set to $67.4$ km/s/Mpc from the Planck2018 results
Planck2018 .

To examine observational constraints on the two models, we perform a chi-
squared analysis using a distance modulus $\mu$ and a combination value
$f(z)\sigma_{8}(z)$. The distance modulus $\mu$ is defined as

$\mu=5\log d_{L}+25,$ (78)

where the luminosity distance $d_{L}$ is given by Eq. (73). The observed
distance modulus $\mu$ is obtained from the supernova data Suzuki_2012 . The
chi-squared function $\chi_{\textrm{SN}}^{2}$ for the supernovae is given by

$\chi_{\textrm{SN}}^{2}(\Psi_{\alpha},\alpha)=\sum\limits_{i=1}^{580}{\left[\frac{\mu_{\textrm{obs}}(z_{i})-\mu_{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)}{\sigma_{i}^{\textrm{SN}}}\right]^{2}},$
(79)

where $\mu_{\textrm{obs}}(z_{i})$ and
$\mu_{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)$ are the observed and
calculated distance moduli, respectively, and $\sigma_{i}^{\textrm{SN}}$ is
the uncertainty in the observed distance modulus. The Union 2.1 set of $580$
type Ia supernovae Suzuki_2012  is used for the observed data points (numbered
$i=1$ to $580$), which are shown in Fig. 2. Using a combination value
$f(z)\sigma_{8}(z)$, the chi-squared function $\chi_{\textrm{GR}}^{2}$ for the
growth rate is given by

$\displaystyle\chi_{\textrm{GR}}^{2}(\Psi_{\alpha},\alpha)=$
$\displaystyle\sum\limits_{i=1}^{18}{\left[\frac{f_{\textrm{obs}}(z_{i})\sigma_{8}^{\textrm{obs}}(z_{i})-f_{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)\sigma_{8}^{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)}{\sigma_{8,i}^{\textrm{GR}}}\right]^{2}},$
(80)

where $f_{\textrm{obs}}(z_{i})\sigma_{8}^{\textrm{obs}}(z_{i})$ and
$f_{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)\sigma_{8}^{\textrm{cal}}(z_{i},\Psi_{\alpha},\alpha)$
are the observed and calculated values, respectively, and
$\sigma_{8,i}^{\textrm{GR}}$ is the uncertainty in the observed value. The
observed data points (numbered $i=1$ to $18$) are taken from the summary in
Ref. Nesseris2017  and are shown in Fig. 5. Each original data point is given
in Refs. 58 ; 59 ; 60 ; 61 ; 62 ; 63 ; 67 ; 68 ; 69 ; 71 ; 72 ; 73 ; 74 ; 75 .

In addition, a joint chi-squared analysis is performed using the two chi-
squared functions. For the joint chi-squared analysis, the combined chi-
squared function $\chi_{\textrm{total}}^{2}$ is defined by

$\chi_{\textrm{total}}^{2}=\chi_{\textrm{SN}}^{2}+\chi_{\textrm{GR}}^{2}.$
(81)

For these analyses, $\Psi_{\alpha}$ and $\alpha$ are treated as free
parameters. $\Psi_{\alpha}$ is sampled in the range from $0$ to $1$ in steps
of $0.005$ and $\alpha$ is sampled in the range from $-2$ to $3$ in steps of
$0.025$. In the present study, $\alpha=2-\epsilon$ is sampled instead of
$\alpha=2$, to avoid a division by zero, where $\epsilon=10^{-7}$. Also,
$\Psi_{\alpha}=1$ is not sampled.

We now provide an overview of the observational and thermodynamic constraints
on the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. Figure 6 shows
the contours of $\chi_{\textrm{SN}}^{2}$ and $\chi_{\textrm{GR}}^{2}$ in the
$(\Psi_{\alpha},\alpha)$ plane. In addition, the boundary of
$\ddot{S}_{\rm{BH}}=0$ for $\tilde{a}=1$ and $10$ shown in Fig. 1 is plotted
again in this figure. First, we focus on the contours of
$\chi_{\textrm{SN}}^{2}$ for the supernovae, which is related to the
background evolution of the universe. Small-$\chi_{\textrm{SN}}^{2}$ regions
corresponding to $\chi_{\textrm{SN}}^{2}<640$ are displayed in Fig. 6. The
contours of $\chi_{\textrm{SN}}^{2}$ for the two models are the same because
the background evolution for both models is equal. The region surrounded by
the contours indicates a favored region. For $\chi_{\textrm{SN}}^{2}$,
$\Psi_{\alpha}\approx 0.4$–$0.7$ is likely favored. (We note that $\alpha>2$
should not satisfy an initially decelerating and then accelerating universe,
as examined in Ref. Koma15 .)

Next, we focus on the contours of $\chi_{\textrm{GN}}^{2}$ for the growth
rate, which is related to density perturbations.
Small-$\chi_{\textrm{GR}}^{2}$ regions corresponding to
$\chi_{\textrm{GR}}^{2}<80$ are displayed in Fig. 6. The regions surrounded by
contours for the two models are different from each other. For the
$\Lambda(t)$-$H^{\alpha}$ model, $\Psi_{\alpha}\approx 0.1$–$1$ is likely
favored [Fig. 6(a)]. In contrast, for the BV-$H^{\alpha}$ model, a
low-$\Psi_{\alpha}$ region, specifically $\Psi_{\alpha}\approx 0.1$–$0.2$, is
likely favored [Fig. 6(b)]. In particular, for the $\Lambda(t)$-$H^{\alpha}$
model, the regions surrounded by contours of $\chi_{\textrm{GR}}^{2}$ and
$\chi_{\textrm{SN}}^{2}$ partially overlap each other [Fig. 6(a)]. However,
for the BV-$H^{\alpha}$ model, the contours of $\chi_{\textrm{GR}}^{2}$ and
$\chi_{\textrm{SN}}^{2}$ do not overlap [Fig. 6(b)]. Accordingly, the
$\Lambda(t)$-$H^{\alpha}$ model is expected to agree more closely with the
combined observation data, compared with the BV-$H^{\alpha}$ model. In
addition, we discuss thermodynamic constraints on the two models. As shown in
Fig. 6(a), for the $\Lambda(t)$-$H^{\alpha}$ model, the above-mentioned
overlapped region currently satisfies $\ddot{S}_{\rm{BH}}<0$, that is,
$\tilde{a}=1$. In contrast, for the BV-$H^{\alpha}$ model, the region
surrounded by the $\chi_{\textrm{GN}}^{2}$ contours does not currently satisfy
$\ddot{S}_{\rm{BH}}<0$ [Fig. 6(b)]. These results imply that the
$\Lambda(t)$-$H^{\alpha}$ model should agree with observations and satisfy the
thermodynamic constraints. That is, the $\Lambda(t)$-$H^{\alpha}$ model for a
non-dissipative universe is expected to be favored. To confirm this
expectation, we examine a combined chi-squared function
$\chi_{\textrm{total}}^{2}$, which is calculated from Eq. (81).

Figure 7 shows contours of $\chi_{\textrm{total}}^{2}$ in the
$(\Psi_{\alpha},\alpha)$ plane. Small-$\chi_{\textrm{total}}^{2}$ regions
corresponding to $\chi_{\textrm{total}}^{2}<900$ are displayed in this figure.
The region surrounded by contours for the $\Lambda(t)$-$H^{\alpha}$ model is
wide in comparison with that for the BV-$H^{\alpha}$ model. In addition, the
minimum value of $\chi_{\textrm{total}}^{2}$ for the $\Lambda(t)$-$H^{\alpha}$
model is smaller than that for the BV-$H^{\alpha}$ model. The minimum value is
summarized in Table 1. (An overview of the constraints is examined here;
therefore, the best-fit value is not discussed.)

As shown in Fig. 7(b), the region surrounded by contours for the
BV-$H^{\alpha}$ model does not satisfy $\ddot{S}_{\rm{BH}}<0$ for both
$\tilde{a}=1$ and $10$. In fact, even in the last stage, the region for the
BV-$H^{\alpha}$ model does not satisfy $\ddot{S}_{\rm{BH}}<0$ because the
region is outside $\alpha<2$. In contrast, most of the region surrounded by
contours for the $\Lambda(t)$-$H^{\alpha}$ model currently satisfies
$\ddot{S}_{\rm{BH}}<0$, especially for small values of $|\alpha|$, as shown in
Fig. 7(a). Accordingly, the $\Lambda(t)$-$H^{\alpha}$ model in this region
should be favored over the BV-$H^{\alpha}$ model. The small $|\alpha|$ region
includes a point $(\Psi_{\alpha},\alpha)=(0.685,0)$, which corresponds to the
standard $\Lambda$CDM model [Fig. 7(a)]. (The location for
$\chi_{\textrm{total},min}^{2}$ slightly deviates from the point. The small
$|\alpha|$ may be related to a weak entanglement of quantum fields between the
inside and outside of the horizon.)

Figure 8:  (Color online). Contours of the normalized $L_{\textrm{total}}$ in
the $(\Psi_{\alpha},\alpha)$ plane for the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models. The contours of the $1\sigma$, $2\sigma$, and
$3\sigma$ confidence levels are plotted. The horizontal dashed line represents
$\alpha=2$. The boundary of $\ddot{S}_{\rm{BH}}=0$, which is shown in Figs. 6
and 7, is also plotted. For an explanation of this boundary, see the caption
of Fig. 6. The point labeled $\Lambda$CDM represents
$(\Psi_{\alpha},\alpha)=(0.685,0)$ for that model. The x within the contours
marks the location for $L_{\textrm{total},max}$, which is equivalent to that
for $\chi_{\textrm{total},min}^{2}$ shown in Fig. 7 and Table 1. Note that the
region surrounded by contours for the BV-$H^{\alpha}$ model is very small (see
the text for further discussion).

Finally, a joint likelihood analysis is performed. For the joint likelihood
analysis, a combined likelihood function $L_{\textrm{total}}$ is defined by
Koma8

$L_{\textrm{total}}=L_{\textrm{SN}}\times L_{\textrm{GR}},$ (82)

where $L_{\textrm{SN}}$ and $L_{\textrm{GR}}$ are given by

$L_{\textrm{SN}}\propto\exp\left({\frac{-\chi^{2}_{\textrm{SN}}}{2}}\right)\quad\textrm{and}\quad
L_{\textrm{GR}}\propto\exp\left({\frac{-\chi^{2}_{\textrm{GR}}}{2}}\right).$
(83)

In this study, Eq. (82) is normalized by the maximum value of
$L_{\textrm{total}}$, namely $L_{\textrm{total},max}$. The normalized
$L_{\textrm{total}}$ for the $\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$
models can be calculated from the results shown in Figs. 7(a) and (b),
respectively. Using the normalized $L_{\textrm{total}}$, the contours of the
$1\sigma$, $2\sigma$, and $3\sigma$ confidence levels are plotted in Fig. 8.
Here, $1\sigma$, $2\sigma$, and $3\sigma$ correspond to the normalized
$L_{\textrm{total}}=3.17\times 10^{-1}$, $4.60\times 10^{-2}$, and $2.73\times
10^{-3}$, respectively Valent2015 ; Koma8 . Accordingly, the region surrounded
by the contour for the $3\sigma$ is narrow, compared with the region for
$\chi_{\textrm{total}}^{2}<900$ shown in Fig. 7. In particular, the region for
the BV-$H^{\alpha}$ model is very small, as shown in Fig. 8, although the
location for $L_{\textrm{total},max}$ is the same as that for
$\chi_{\textrm{total},min}^{2}$. From Fig. 8, we can confirm that the region
surrounded by the contour for the $\Lambda(t)$-$H^{\alpha}$ model currently
satisfies $\ddot{S}_{\rm{BH}}<0$, whereas the region for the BV-$H^{\alpha}$
model does not.

This section examines the observational and thermodynamic constraints on the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. Consequently, the
$\Lambda(t)$-$H^{\alpha}$ model is found to be favored, compared with the
BV-$H^{\alpha}$ model considered here. In other words, the non-dissipative
universe described by the $\Lambda(t)$-$H^{\alpha}$ model is likely consistent
with our Universe. It should be noted that BV-$H^{\alpha}$ models for
$\alpha=0$, namely the BV-$H^{0}$ models, agree with the observed supernova
and growth-rate data, if negative $c_{\rm{eff}}^{2}$ Lima2011  and clustered
matter Ramos_2014  can be assumed. However, these assumptions were not used in
this study, and detailed analyses are left for future research.

##  V Conclusions

Cosmological models can be categorized according to how they handle energy
dissipation: $\Lambda(t)$ models are used for a non-dissipative universe, and
BV models are used for a dissipative universe. To clarify the difference
between the two universes, we have examined density perturbations using two
types of holographic cosmological models. A power-law term proportional to
$H^{\alpha}$ is applied to the $\Lambda(t)$ and BV models to systematically
examine the two different universes. In this study, an equivalent background
evolution of the universe was set for both the $\Lambda(t)$-$H^{\alpha}$ and
BV-$H^{\alpha}$ models. Based on the background evolution, we derived first-
order density perturbations in the two models. In the derived formulation,
$\alpha$ is a free parameter and therefore, the difference in density
perturbations between the two models can be systematically examined.

Using the formulation, we examined the evolution of the universe in the
$\Lambda(t)$-$H^{\alpha}$ and BV-$H^{\alpha}$ models. A growth rate $f(z)$ and
a combination value $f(z)\sigma_{8}(z)$ for the $\Lambda(t)$-$H^{\alpha}$
model are found to agree with observed data points when $\Psi_{\alpha}=0.685$
(which is equivalent to $\Omega_{\Lambda}$ from the Planck 2018 results), in
contrast with the BV-$H^{\alpha}$ model. In addition, we systematically
examined the observational and thermodynamic constraints on the two models by
using chi-squared functions in the $(\Psi_{\alpha},\alpha)$ plane.
Consequently, the $\Lambda(t)$-$H^{\alpha}$ model for small $|\alpha|$ values
was found to be consistent with the combined observation data, that is, a
distance modulus $\mu$ and $f(z)\sigma_{8}(z)$, and satisfies the maximization
of entropy on the horizon of the universe. This result implies that a
$\Lambda(t)$-$H^{\alpha}$ model similar to $\Lambda$CDM models is favored,
compared with the BV-$H^{\alpha}$ model examined here. In other words, the
non-dissipative universe described by models like the
$\Lambda(t)$-$H^{\alpha}$ model is found to be consistent with our Universe.

Through the present study, we have revealed fundamental properties of the two
types of holographic cosmological models in dissipative and non-dissipative
universes. Similar models, including CCDM models for a dissipative universe
and $\Lambda(t)$CDM models for a non-dissipative universe, have been
separately examined. The present results should promote the development of a
deeper understanding of these cosmological models and bridge the gap between
them.

###### Acknowledgements.

The present study was supported by the Japan Society for the Promotion of
Science, KAKENHI Grant Number JP18K03613.

##  Appendix A Unified formulation for the $\Lambda(t)$ and BV models

In Sec. II.2, first-order density perturbations in the $\Lambda(t)$ and BV
models are separately presented. In this appendix, we review a unified
formulation for the $\Lambda(t)$ and BV models using the neo-Newtonian
approach proposed by Lima et al. Lima_Newtonian_1997 . The unified formulation
has been examined previously Koma6  and should be suitable for describing
density perturbations in both models systematically. We review the unified
formulation in accordance with the previous work Koma6 . In this appendix,
general driving terms for the two models, namely $f_{\Lambda}(t)$ and
$h_{\textrm{B}}(t)$, are considered without using a power-law term.

In this study, the Friedmann, acceleration, and continuity equations are given
by Eqs. (1), (2), and (3), respectively. In a matter-dominated universe, the
continuity equation for a unified formulation can be written as Koma6

$\dot{\rho}+3H\rho=U\rho,$ (84)

where $U$ is given by

$U=\begin{cases}Q&(\Lambda(t)\hskip 2.84526pt\rm{model}),\\\
\Gamma&(\rm{BV}\hskip 2.84526pt\rm{model}).\\\ \end{cases}$ (85)

$Q$ [Eq. (11)] and $\Gamma$ [Eq. (21)] are written as

$Q=-\frac{3}{8\pi G}\frac{\dot{f}_{\Lambda}(t)}{\rho},$ (86)
$\Gamma=\frac{3}{4\pi G}\frac{Hh_{\textrm{B}}(t)}{\rho}.$ (87)

Basic hydrodynamical equations for the neo-Newtonian approach are given in
Refs. Lima_Newtonian_1997 ; Lima2011 . In fact, for the BV model, the
hydrodynamical equations are used in Sec. II.2.2. For the $\Lambda(t)$ model,
we use the fundamental equations examined by Arcuri and Waga Waga1994 .
Consequently, the basic hydrodynamical equations for the unified formulation
can be written as Koma6

$\left(\frac{\partial\mathbf{u}}{\partial
t}\right)_{r}+(\mathbf{u}\cdot\nabla_{r})\mathbf{u}=-\nabla_{r}\Phi-\frac{\nabla_{r}p_{c}}{\rho+\frac{p_{c}}{c^{2}}},$
(88) $\left(\frac{\partial\rho}{\partial
t}\right)_{r}+\nabla_{r}\cdot(\rho\mathbf{u})+\Theta=0,$ (89)
$\nabla_{r}^{2}\Phi=4\pi G\left(\rho+l\right),$ (90)

where $\mathbf{u}$ is the velocity of a fluid element of volume and $\Phi$ is
the gravitational potential. For the unified formulation, $\Theta$ and $l$ are
given as Koma6

$\Theta=\begin{cases}-Q\rho=\frac{3\dot{f}_{\Lambda}(t)}{8\pi
G}&(\Lambda(t)\hskip 2.84526pt\rm{model}),\\\
\frac{p_{c}}{c^{2}}\nabla_{r}\cdot\mathbf{u}&(\rm{BV}\hskip
2.84526pt\rm{model}),\\\ \end{cases}$ (91)
$l=\begin{cases}-\frac{3{f}_{\Lambda}(t)}{4\pi G}&(\Lambda(t)\hskip
2.84526pt\rm{model}),\\\ \frac{3p_{c}}{c^{2}}&(\rm{BV}\hskip
2.84526pt\rm{model}).\\\ \end{cases}$ (92)

Equations (88), (89), and (90) are the Euler, continuity, and Poisson
equations, respectively. Using the basic hydrodynamical equations, we can
calculate the time evolution equation for the matter density contrast
$\delta$. Setting $c=1$, using the linear approximation, and neglecting extra
terms, we can write the time evolution equation for $\delta$ as Koma6

$\displaystyle\ddot{\delta}$
$\displaystyle+\left[H(2+3c_{\rm{eff}}^{2}-3u)-\frac{\dot{w_{c}}}{1+w_{c}}\right]\dot{\delta}$
$\displaystyle+\Bigg{\\{}3(\dot{H}+2H^{2})\left(c_{\rm{eff}}^{2}-u\right)$
$\displaystyle+3H\left[\dot{c}_{\rm{eff}}^{2}-\dot{u}-\frac{\dot{w}_{c}}{1+w_{c}}(c_{\rm{eff}}^{2}-u)\right]$
$\displaystyle-4\pi
G\rho\left(1+w_{c}\right)(1+3c_{\rm{eff}}^{2})+\frac{k^{2}c_{\rm{eff}}^{2}}{a^{2}}\Bigg{\\}}\delta=0,$
(93)

where $u$, $w_{c}$, and $c_{\rm{eff}}^{2}$ are defined by

$u\equiv-\frac{U}{3H}=\begin{cases}-\frac{Q}{3H}&(\Lambda(t)\hskip
2.84526pt\rm{model}),\\\ -\frac{\Gamma}{3H}(=w_{c})&(\rm{BV}\hskip
2.84526pt\rm{model}),\\\ \end{cases}$ (94)
$w_{c}\equiv-\frac{\Gamma}{3H}=\begin{cases}0&(\Lambda(t)\hskip
2.84526pt\rm{model}),\\\ -\frac{\Gamma}{3H}&(\rm{BV}\hskip
2.84526pt\rm{model}),\\\ \end{cases}$ (95) $c_{\rm{eff}}^{2}\equiv\frac{\delta
p_{c}}{\delta\rho}=\begin{cases}0&(\Lambda(t)\hskip 2.84526pt\rm{model}),\\\
\frac{\delta p_{c}}{\delta\rho}&(\rm{BV}\hskip 2.84526pt\rm{model}).\\\
\end{cases}$ (96)

In Eq. (93), $\rho$ represents $\bar{\rho}$, that is, a homogenous and
isotropic solution for the unperturbed equations. The derivation of the above
equation is essentially the same as that shown by Jesus et al. Lima2011 . Note
that they assumed $c_{\rm{eff}}^{2}=c_{\rm{eff}}^{2}(t)$ and that the spatial
dependence of $\delta$ is proportional to $e^{i\bf{k}\cdot\bf{x}}$, where the
comoving coordinates $\bf{x}$ are given by $\bf{x}=\bf{r}$$/a$ using the
proper coordinates $\bf{r}$ Lima2011 .

Equation (93) is the unified equation for the $\Lambda(t)$ and BV models. This
equation reduces to Eq. (10) in the $\Lambda(t)$ model. In the BV model, Eq.
(93) reduces to Eq. (23), as examined in Ref. Koma6 .

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