# Generalized Logotropic Models and their Cosmological Constraints

H.B. Benaoum,11footnotetext: Corresponding author. Pierre-Henri Chavanis
Hernando Quevedo

###### Abstract

We propose a new class of cosmological unified dark sector models called
“Generalized Logotropic Models". They depend on a free parameter $n$. The
original logotropic model [P.H. Chavanis, Eur. Phys. J. Plus 130, 130 (2015)]
is a special case of our generalized model corresponding to $n=1$. In our
scenario, the Universe is filled with a single fluid, a generalized logotropic
dark fluid (GLDF), whose pressure $P$ includes higher order logarithmic terms
of the rest-mass density $\rho_{m}$. The total energy density $\epsilon$ is
the sum of the rest-mass energy density $\rho_{m}c^{2}$ and the internal
energy density $u$ which play the role of dark matter energy density
$\epsilon_{m}$ and dark energy density $\epsilon_{de}$, respectively. We
investigate the cosmological behavior of the generalized logotropic models by
focusing on the evolution of the energy density, scale factor, equation of
state parameter, decceleration parameter and squared speed of sound. Low
values of $n\leq 3$ are favored. We also study the asymptotic behavior of the
generalized logotropic models. In particular, we show that the model presents
a phantom behavior and has three distinct ways of evolution depending on the
value of $n$. For $n\leq 2$, it leads to a little rip and for $n>2$ to a big
rip. We predict the value of the big rip time as a function of $n$ without any
free (undetermined) parameter.

##  1 Introduction

Cosmological observations from various independent research teams show that
the current Universe is accelerating [1, 2, 5, 3, 4]. This accelerating
expansion is due to an unknown component called “dark energy" which works
against gravity. The nature of the dark sector is still unknown and several
alternative models of dark matter (DM) and dark energy (DE) have been proposed
to account for the observation of the present cosmic acceleration. The
standard cold dark matter ($\Lambda$CDM) model is the simplest DE model and
relies on a cosmological constant to drive the current acceleration of the
Universe and on the existence of a pressureless DM to explain the observed
properties of the large-scale structures of the Universe [6, 7]. However, the
$\Lambda$CDM model suffers from the cosmological constant problem [8, 9],
namely why the value of the cosmological constant is so tiny, and the cosmic
coincidence problem [10, 11], namely why DM and DE are of similar magnitudes
today although they scale differently with the universe’s expansion. The CDM
model also faces important problems at the scale of DM halos such as the core-
cusp problem [12], the missing satellite problem [13, 14, 15], and the “too
big to fail” problem [16]. This leads to the so-called small-scale crisis of
CDM [17]. Among the wide range of alternative DE models that have been
proposed in the literature (quintessence, k-essence, Chaplygin gas, tachyons,
phantom fields, holography…), an interesting class of dynamical DE models
considers DM and DE as different manifestations of a single-component
underlying fluid, often assumed to be a perfect fluid. The unified dark matter
and dark energy (UDM) models have the remarkable feature of describing the
dark sector of the Universe as a single component which behaves as DM at early
times and as DE at late times [18, 19, 24, 25, 35, 23, 20, 22, 21, 30, 31, 33,
32, 36, 37, 34, 26, 27, 28, 29] . The cosmological aspects of these UDM models
have been studied recently in refs. [36, 37, 34, 38, 39, 40, 42]. In
particular, the logotropic model is a candidate for unifying DE and DM [30].
The logotropic model is able to account for the transition between a DM era
and a DE era and is indistinguishable from the $\Lambda$CDM model, for what
concerns the evolution of the cosmological background, up to $25$ billion
years in the future when it becomes phantom [30, 31, 39, 33]. Remarkably, the
logotropic model implies that DM halos should have a constant surface density
and it predicts its universal value $\Sigma_{0}^{\rm
th}=0.01955c\sqrt{\Lambda}/G=133\,M_{\odot}/{\rm pc}^{2}$ [30, 31, 39, 33]
without adjustable parameter. This theoretical value is in good agreement with
the value $\Sigma_{0}^{\rm obs}=141_{-52}^{+83}\,M_{\odot}/{\rm pc}^{2}$
obtained from the observations [41]. The logotropic model also predicts the
values of the present proportion of DM and DE
$\Omega_{dm,0}=\frac{1}{1+e}(1-\Omega_{b,0})=0.256$ and
$\Omega_{de,0}=\frac{e}{1+e}(1-\Omega_{b,0})=0.695$ [33], in good agreement
with the observed values $\Omega_{dm,0}=0.259$, $\Omega_{de,0}=0.691$ and
$\Omega_{b,0}=0.0486$.

In this work, we introduce a new class of UDM models called “Generalized
Logotropic Models" characterized by a single fluid equation of state (EoS) of
the form

$\displaystyle
P=\sum_{i=0}^{N}A_{i}\ln^{i}\left(\frac{\rho_{m}}{\rho_{P}}\right),$ (1.1)

where $P$ is the pressure, $\rho_{m}$ is the rest-mass density, $A_{i}$ are
constants with the dimension of an energy density and $\rho_{P}$ is a constant
with the dimension of a mass density. The above EoS returns the standard
$\Lambda$CDM model for $N=0$ and the original logotropic model [30] for $N=1$.
Following [30], we shall identify $\rho_{P}$ with the Planck density
$\rho_{P}={c^{5}}/({\hbar G^{2}})=5.16\times 10^{99}\,{\rm g\,m^{-3}}$. The
single fluid which obeys Eq. (1.1) will be called the generalized logotropic
dark fluid (GLDF).

In this paper, we are interested in studying the dynamical evolution of
various generalized logotropic models. In particular, we examine in detail
various forms of generalized logotropic EoS and investigate how the Universe
evolution is affected by the corresponding EoS. In section 2, we provide an
approach to motivate the logotropic model. In section 3, we consider an
isotropic, homogeneous and spatially flat Universe and introduce the main
equations characterizing generalized logotropic models. In section 4, we
consider a subclass of models with $A_{i}=A_{n}\delta_{i,n}$ that are more
easily tractable. In section 5, we investigate the cosmological behavior of
the generalized logotropic models and focus on the evolution of the energy
density, scale factor, EoS parameter, decceleration parameter and squared
speed of sound. We also derive analytically the asymptotic behavior of the
generalized logotropic models and distinguish three types of evolution
depending on the value of $n$. Finally, section 6 is devoted to remarks and
conclusions. In Appendix A, we show that the GLDF is asymptotically equivalent
to a form of Modified Chaplygin Gas (MCG). In Appendix B, we consider the two-
fluid model associated with the GLDF and determine the EoS of DE. In Appendix
C, we derive the present proportion of DM and DE by taking into account the
presence of baryons.

##  2 Generalized Logotropic Equation of State

We consider the scenario where both DM and DE originate from a single dark
fluid. We first recall the justification of the standard logotropic equation
of state given in [30]. In the Newtonian regime, the condition of hydrostatic
equilibrium, which describes the balance between the gravitational force and
the pressure gradient, is given by

$\displaystyle\nabla P+\rho\nabla\Phi={\bf 0}.$ (2.1)

Here $P$ and $\rho$ denote the pressure and mass density of the fluid,
respectively. Moreover, $\Phi$ is the gravitational potential. We assume one
of the simplest direct relations between the mass density $\rho$ and the
pressure $P$ of the fluid given by the polytropic EoS

$\displaystyle P=K\rho^{\gamma},$ (2.2)

where $\gamma$ is the polytropic index and $K$ is a constant. Using the above
equation, the condition of hydrostatic equilibrium becomes

$\displaystyle K\gamma\rho^{\gamma-1}\nabla\rho+\rho\nabla\Phi={\bf 0}.$ (2.3)

Considering the limit $\gamma\rightarrow 0$, $K\rightarrow+\infty$ with
$A=K\gamma$ fixed [30], we get

$\displaystyle A\frac{\nabla\rho}{\rho}+\rho\nabla\Phi={\bf 0}.$ (2.4)

By comparing equations (2.4) and (2.1), we find that the EoS of the logotropic
gas reads [30]

$\displaystyle P=A\ln\left(\frac{\rho}{\rho_{*}}\right),$ (2.5)

where $\rho_{*}$ is an integration constant. In this paper, we consider a
generalization of this EoS of the form

$\displaystyle
P=\sum_{i=0}^{N}A_{i}\ln^{i}\left(\frac{\rho}{\rho_{*}}\right),$ (2.6)

where $A_{i}$ are arbitrary constants. This generalized logotropic EoS is
interesting in its own right and, as we shall see, possesses interesting
properties. However, the above derivation suggests that the standard
logotropic EoS (2.5) plays a special role in the problem.

##  3 Generalized Logotropic Cosmology

In this section, we consider a flat homogeneous and isotropic Friedmann-
Lemaître-Robertson-Walker (FLRW) universe filled with a perfect single fluid,
having an energy density $\epsilon$, rest-mass density $\rho_{m}$, and
pressure $P$. The expansion dynamics is governed by the Friedmann equations
[43]

$\displaystyle H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3c^{2}}\epsilon,$ (3.1)
$\displaystyle\dot{H}+H^{2}=\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3c^{2}}\left(\epsilon+3P\right),$ (3.2)

where $a(t)$ is the scale factor and $H=\dot{a}/a$ is the Hubble parameter.
Combining Eqs. (3.1) and (3.2), we obtain the energy conservation equation

$\frac{d\epsilon}{dt}+3H\left(\epsilon+P\right)=0.$ (3.3)

For a relativistic fluid with an adiabatic evolution (or at $T=0$), the first
law of thermodynamics reduces to

$\displaystyle d\epsilon=\frac{P+\epsilon}{\rho_{m}}d\rho_{m}.$ (3.4)

By integrating this equation for a given EoS, $P=P(\rho_{m})$, a relation
between the energy density and the rest-mass density can be obtained as [30]

$\displaystyle\epsilon=\rho_{m}c^{2}+\rho_{m}\int^{\rho_{m}}\frac{P\left(\rho^{\prime}\right)}{\rho{{}^{\prime}}^{2}}d\rho^{\prime}=\rho_{m}c^{2}+u(\rho_{m}),$
(3.5)

where $\rho_{m}c^{2}$ is the rest-mass energy density and $u$ is the internal
energy density. For our generalized logotropic model with an EoS given by Eq.
(1.1), we get

$\displaystyle\epsilon=\rho_{m}c^{2}-\sum_{i=0}^{N}A_{i}I_{i}\left(\rho_{m}\right)$
(3.6)

with the integral $I_{i}\left(\rho_{m}\right)$ given by

$\displaystyle
I_{i}\left(\rho_{m}\right)=\rho_{m}\int_{\rho_{m}}^{+\infty}\ln^{i}\left(\frac{\rho^{\prime}_{m}}{\rho_{P}}\right)\,\frac{d\rho^{\prime}_{m}}{{\rho^{\prime}}^{2}}.$
(3.7)

With the change of variables $t=\ln(\rho^{\prime}_{m}/\rho_{P})$, we obtain

$\displaystyle
I_{i}\left(\rho_{m}\right)=\frac{\rho_{m}}{\rho_{P}}\int_{\ln(\rho_{m}/\rho_{P})}^{+\infty}t^{i}e^{-t}\,dt.$
(3.8)

In terms of the incomplete gamma function

$\displaystyle\Gamma(\alpha+1,x)=\int_{x}^{+\infty}t^{\alpha}e^{-t}\,dt$ (3.9)

the integral $I_{i}\left(\rho_{m}\right)$ can be rewritten as

$\displaystyle
I_{i}\left(\rho_{m}\right)=\frac{\rho_{m}}{\rho_{P}}~{}\Gamma\left[i+1,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]=i!\sum_{k=0}^{i}\frac{1}{k!}\ln^{k}\left(\frac{\rho_{m}}{\rho_{P}}\right).$
(3.10)

To get the second equality, we have used the relation

$\displaystyle\Gamma\left(n+1,x\right)=n!\,e^{-x}\sum_{k=0}^{n}\frac{1}{k!}x^{k},$
(3.11)

which can be obtained by computing $\partial\Gamma(n+1,x)/\partial x$ from
Eqs. (3.9) and (3.11). For future purposes, we note the asymptotic
behavior222We have assumed that $n$ is an integer. However, it can be shown
that this asymptotic behavior remains valid when $n$ is a real number.

$\displaystyle\Gamma\left(n+1,x\right)\sim
x^{n}e^{-x}\qquad(x\rightarrow\infty).$ (3.12)

The energy density can then be written as

$\displaystyle\epsilon=\rho_{m}c^{2}-\sum_{i=0}^{N}A_{i}\frac{\rho_{m}}{\rho_{P}}\Gamma\left[i+1,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right].$
(3.13)

Equations (1.1) and (3.13) determine the EoS $P(\epsilon)$ in parametric form.
By combining the energy conservation equation (3.3) with the first law of
thermodynamics (3.4) one finds that the rest-mass density evolves as [30]

$\displaystyle\frac{d\rho_{m}}{dt}+3H\rho_{m}=0\qquad\Rightarrow\qquad\rho_{m}=\frac{\rho_{m,0}}{a^{3}},$
(3.14)

where $\rho_{m,0}=\rho_{m}\left(a_{0}=1\right)$ is the present value of the
rest-mass density and $a_{0}=1$ is the present value of the scale factor.
Equation (3.14) expresses the conservation of the rest-mass. On the other
hand, the internal energy is given by

$\displaystyle
u=-\sum_{i=0}^{N}A_{i}I_{i}\left(\rho_{m}\right)=-\sum_{i=0}^{N}A_{i}I_{i}\left(\frac{\rho_{m,0}}{a^{3}}\right).$
(3.15)

As argued in [30], the rest-mass energy density $\rho_{m}c^{2}$ plays the role
of pressureless DM ($\epsilon_{m}=\rho_{m}c^{2}$) and the internal energy $u$
plays the role of DE ($\epsilon_{de}=u$).333The notation $\rho_{m}$ stands
either for pressureless DM density or for rest-mass density. The energy
density $\epsilon=\epsilon_{m}+\epsilon_{de}$ of the generalized logotropic
model is therefore the sum of two terms where the first term $\epsilon_{m}$
can be interpreted as the DM energy density given by

$\displaystyle\epsilon_{m}=\frac{\epsilon_{m,0}}{a^{3}}$ (3.16)

and the second term $\epsilon_{de}$ can be interpreted as the DE energy
density given by

$\displaystyle\epsilon_{de}=-\frac{e^{-1-1/B}}{a^{3}}\sum_{i=0}^{N}A_{i}~{}\Gamma\left(i+1,-1-\frac{1}{B}-3\ln
a\right),$ (3.17)

where, following [30], we have defined the dimensionless parameter $B$ through
the relation

$\displaystyle\frac{\rho_{P}}{\rho_{m,0}}=e^{1+1/B}.$ (3.18)

The Friedmann equation (3.1) can be written as

$\displaystyle
H^{2}=H_{0}^{2}\left(\frac{\epsilon_{m}}{\epsilon_{0}}+\frac{\epsilon_{de}}{\epsilon_{0}}\right),$
(3.19)

where $\epsilon_{0}={3H_{0}^{2}c^{2}}/{8\pi G}$ is the critical energy density
and $H_{0}$ is the value of the Hubble parameter at the present time.

The pressure and the energy density of the generalized logotropic model can be
expressed in terms of the scale factor as

$\displaystyle P=\sum_{i=0}^{N}A_{i}\left(-1-\frac{1}{B}-3\ln a\right)^{i},$
(3.20)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}-\frac{e^{-1-1/B}}{a^{3}}\sum_{i=0}^{N}A_{i}~{}\Gamma\left(i+1,-1-\frac{1}{B}-3\ln
a\right).$ (3.21)

The present proportion of DM and DE, denoted $\Omega_{m,0}$ and
$\Omega_{de,0}$, are given by

$\displaystyle\Omega_{m,0}=\frac{\epsilon_{m,0}}{\epsilon_{0}},$ (3.22)
$\displaystyle\Omega_{de,0}=\frac{\epsilon_{de,0}}{\epsilon_{0}}=-\frac{e^{-1-1/B}}{\epsilon_{0}}\sum_{i=0}^{N}A_{i}~{}\Gamma\left(i+1,-1-\frac{1}{B}\right)=1-\Omega_{m,0}.$
(3.23)

For given values of $\Omega_{m,0}$ and $\epsilon_{0}$ (through $H_{0}$), which
can be obtained from the observations, Eq. (3.18) determines $B$ and Eq.
(3.23) provides a constraint on the coefficients $A_{i}$. Using
$\rho_{P}={c^{5}}/({\hbar G^{2}})=5.16\times 10^{99}\,{\rm g\,m^{-3}}$ and
$\rho_{m,0}=\Omega_{m,0}\epsilon_{0}/c^{2}$ with $\Omega_{m,0}=0.309$ and
$\epsilon_{0}/c^{2}=8.62\times 10^{-24}\,{\rm g\,m^{-3}}$, we get
$B=3.53\times 10^{-3}$. We have taken the values of $\Omega_{m,0}$ and
$\epsilon_{0}$ from the $\Lambda$CDM model but since $B$ is defined by a
logarithm, its value is very insensitive to the precise values of
$\Omega_{m,0}$ and $\epsilon_{0}$. Therefore, the value $B=3.53\times 10^{-3}$
is very robust [31].

In the late Universe ($a\rightarrow\infty$ and $\rho\rightarrow 0$), the DE
dominates and, using Eq. (3.12), we get $\epsilon\sim-A_{N}(-3\ln a)^{N}$ and
$P\sim A_{N}(-3\ln a)^{N}$, which implies that the EoS $P(\epsilon)$ behaves
asymptotically as $P/\epsilon\rightarrow-1$. We note that in order to have
$\epsilon>0$ when $a\rightarrow+\infty$, the parameter $A_{N}$ must be
negative if $N$ is even and positive if $N$ is odd. Below we present some
interesting UDM models derived from the generalized logotropic model, focusing
on the EoS $P(\epsilon)$, the energy density $\epsilon$ and the cosmological
implications of these models. We also investigate the era of DE dominance that
drives the accelerated expansion of the Universe.

##  4 Particular models

###  4.1 The case $A_{i}=A_{n}~{}\delta_{i,n}$

This case focuses on just the n-th order term of the finite series leading to
the EoS

$\displaystyle P=A_{n}\ln^{n}\left(\frac{\rho_{m}}{\rho_{P}}\right).$ (4.1)

The energy density is given by

$\displaystyle\epsilon=\rho_{m}c^{2}-A_{n}\frac{\rho_{m}}{\rho_{P}}\Gamma\left[n+1,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]=\rho_{m}c^{2}+u=\epsilon_{m}+\epsilon_{de},$
(4.2)

where the first term is the rest-mass energy density (DM) and the second term
is the internal energy density (DE). The pressure and total energy density as
a function of the scale factor reduce to

$\displaystyle P=A_{n}\left(-1-\frac{1}{B}-3\ln a\right)^{n},$ (4.3)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}-A_{n}\frac{e^{-1-1/B}}{a^{3}}~{}\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a\right).$ (4.4)

At the present time (i.e. $a=1$), the above equation leads to

$\displaystyle
A_{n}=-\frac{\left(1-\Omega_{m,0}\right)\epsilon_{0}e^{1+1/B}}{\Gamma\left(n+1,-1-\frac{1}{B}\right)},$
(4.5)

which determines $A_{n}$ as a function of the measured values of
$\Omega_{m,0}$ and $\epsilon_{0}$. The dimensionless constant $B$ is also
determined by the measured values of $\Omega_{m,0}$ and $\epsilon_{0}$ (see
above). As a result, there is no free parameter in this model. Note that

$\displaystyle\frac{A_{n}}{\left(1-\Omega_{m,0}\right)\epsilon_{0}}=-\frac{e^{1+1/B}}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}$
(4.6)

is a dimensionless constant depending only on $n$ and $B$. Furthermore,
$\rho_{\Lambda}\equiv\left(1-\Omega_{m,0}\right)\epsilon_{0}/c^{2}=5.96\times
10^{-24}\,{\rm g\,m^{-3}}$ is the present value of the DE density (it is equal
to the cosmological density $\rho_{\Lambda}=\Lambda/(8\pi G)$ in the
$\Lambda$CDM model). From Eq. (4.6), we find that $A_{n}<0$ for $n$ even and
$A_{n}>0$ for $n$ odd.

On the other hand, from Eq. (4.1), the pressure $P$ vanishes when
$\rho_{m}=\rho_{P}$. If $n$ is even, the pressure $P$ is always negative. If
$n$ is odd, the pressure $P$ is positive for $\rho_{m}>\rho_{P}$ and negative
for $\rho_{m}<\rho_{P}$. In practice, we consider a regime where
$\rho_{m}\ll\rho_{P}$ because the logotropic model does not describe the early
inflation. In that case, the pressure is always negative. The EoS
$P(\epsilon)$ is given in the reversed form $\epsilon(P)$ by

$\displaystyle\epsilon=e^{\mp\left|\frac{P}{A_{n}}\right|^{1/n}}\rho_{P}c^{2}-A_{n}e^{\mp\left|\frac{P}{A_{n}}\right|^{1/n}}\Gamma\left(n+1,\mp\left|\frac{P}{A_{n}}\right|^{1/n}\right),$
(4.7)

where the upper sign corresponds to the most relevant case $\rho_{m}<\rho_{P}$
and the lower sign corresponds to $\rho_{m}>\rho_{P}$. Substituting Eq. (4.5)
into Eqs. (4.3) and (4.4), we obtain after simple manipulations

$\displaystyle
P=-\frac{\left(1-\Omega_{m,0}\right)\epsilon_{0}}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}e^{1+1/B}~{}\left(-1-\frac{1}{B}-3\ln
a\right)^{n},$ (4.8)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+\frac{\left(1-\Omega_{m,0}\right)\epsilon_{0}}{a^{3}}~{}\frac{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a\right)}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}.$ (4.9)

In the early Universe ($a\rightarrow 0$, $\rho_{m}\rightarrow+\infty$) the DM
$\epsilon_{m}$ dominates and we have
$\epsilon\sim\Omega_{m,0}\epsilon_{0}/a^{3}$, $P\sim A_{n}(-3\ln a)^{n}$ such
that $P/\epsilon\rightarrow 0$ whereas, in the late Universe
($a\rightarrow\infty$, $\rho_{m}\rightarrow 0$), the DE $\epsilon_{de}$
dominates and we have $\epsilon\sim-A_{n}(-3\ln a)^{n}$, $P\sim A_{n}(-3\ln
a)^{n}$ such that $P/\epsilon\rightarrow-1$. Furthermore, in order to have
$\epsilon>0$ for $a\rightarrow+\infty$, the parameter $A_{n}$ must be negative
if $n$ is even and positive if $n$ is odd. As we have seen, this is guaranteed
by Eq. (4.5). Finally, it is worth mentioning that for $B\rightarrow 0$
(corresponding to a form of semiclassical limit $\rho_{P}\rightarrow+\infty$
or $\hbar\rightarrow 0$), the $\Lambda$CDM model is recovered [31]. Indeed,
using Eq. (3.12), we find that Eqs. (4.8) and (4.9) reduce to

$\displaystyle P=-\left(1-\Omega_{m,0}\right)\epsilon_{0},$ (4.10)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+(1-\Omega_{m,0})\epsilon_{0}.$
(4.11)

###  4.2 The case $A_{i}=A_{0}~{}\delta_{i,0}$

For $n=0$, we recover the $\Lambda$CDM model (interpreted as a UDM model)
corresponding to the EoS

$\displaystyle P=A_{0}.$ (4.12)

The energy density is given by

$\displaystyle\epsilon=\rho_{m}c^{2}-A_{0},$ (4.13)

where the first term is DM and the second term is DE. The total energy density
as a function of the scale factor reads

$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}-A_{0}.$ (4.14)

At the present time (i.e. $a=1$), Eq. (4.14) leads to

$\displaystyle A_{0}=-\left(1-\Omega_{m,0}\right)\epsilon_{0}.$ (4.15)

We note that $A_{0}<0$. Numerically, $A_{0}/c^{2}=-\rho_{\Lambda}=-5.96\times
10^{-24}\,{\rm g\,m^{-3}}$, where $\rho_{\Lambda}$ is the cosmological
density. The pressure $P$ and the energy density $\epsilon$ are then given by
Eqs. (4.10) and (4.11). The pressure $P=-\rho_{\Lambda}c^{2}$ is constant and
negative. As the universe expands, starting from $+\infty$, the energy density
$\epsilon$ decreases and tends to a constant value $\rho_{\Lambda}c^{2}$. The
DE density $\epsilon_{de}=\rho_{\Lambda}c^{2}$ is constant.

###  4.3 The case $A_{i}=A_{1}~{}\delta_{i,1}$

For $n=1$, we recover the original logotropic model [30] corresponding to the
EoS

$\displaystyle P=A\ln\left(\frac{\rho_{m}}{\rho_{P}}\right),$ (4.16)

where we have noted $A_{1}=A$. The energy density is given by

$\displaystyle\epsilon=\rho_{m}c^{2}-A\left[1+\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right],$
(4.17)

where the first term is DM and the second term is DE. The EoS $P(\epsilon)$ is
given in the reversed form $\epsilon(P)$ by

$\displaystyle\epsilon=e^{P/A}\rho_{P}c^{2}-A-P.$ (4.18)

The pressure and total energy density as a function of the scale factor read

$\displaystyle P=-A\left(1+\frac{1}{B}+3\ln a\right),$ (4.19)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+A\left(\frac{1}{B}+3\ln
a\right).$ (4.20)

At the present time (i.e. $a=1$), Eq. (4.20) leads to

$\displaystyle A=B\left(1-\Omega_{m,0}\right)\epsilon_{0}.$ (4.21)

We note that $A>0$. Numerically, $A/c^{2}=B\rho_{\Lambda}=3.53\times
10^{-3}\rho_{\Lambda}=2.10\times 10^{-26}\,{\rm g\,m^{-3}}$. The pressure $P$
and the energy density $\epsilon$ become

$\displaystyle P=-\left(1-\Omega_{m,0}\right)\epsilon_{0}\left(B+1+3B\ln
a\right),$ (4.22)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+\left(1-\Omega_{m,0}\right)\epsilon_{0}\left(1+3B\ln
a\right).$ (4.23)

The pressure $P$ is positive when $\rho_{m}>\rho_{P}$ and negative when
$\rho_{m}<\rho_{P}$. It vanishes at $\rho_{m}=\rho_{P}$. As the universe
expands, the pressure decreases from $+\infty$ to $-\infty$. Starting from
$+\infty$, the energy density $\epsilon$ first decreases, reaches a minimum
$\epsilon_{\rm min}=A\ln(\rho_{P}c^{2}/A)>0$ at $\rho_{m}=A/c^{2}$, then
increases to $+\infty$. The DE density $\epsilon_{de}$ increases from
$-\infty$ to $+\infty$. The DE is negative when $\rho_{m}>\rho_{P}/e$ and
positive when $\rho_{m}<\rho_{P}/e$ (its value at $\rho_{m}=\rho_{P}$ is
$\epsilon_{de}=-A<0$). In the logotropic model, since the DE density
corresponds to the internal energy density $u$ of the LDF, it can very well be
negative as long as the total energy density $\epsilon$ is positive. Note,
however, that in the regime of interest $\rho_{m}\ll\rho_{P}$, the DE density
$\epsilon_{de}$ is positive.

###  4.4 The case $A_{i}=A_{2}~{}\delta_{i,2}$

For $n=2$, we obtain the EoS

$\displaystyle P=A_{2}\ln^{2}\left(\frac{\rho_{m}}{\rho_{P}}\right).$ (4.24)

The energy density is given by

$\displaystyle\epsilon=\rho_{m}c^{2}-2A_{2}\left[1+\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)+\frac{1}{2}\ln^{2}\left(\frac{\rho_{m}}{\rho_{P}}\right)\right],$
(4.25)

where the first term is DM and the second term is DE. The EoS $P(\epsilon)$ is
given in the reversed form $\epsilon(P)$ by

$\displaystyle\epsilon=e^{\mp\sqrt{\frac{P}{A_{2}}}}\rho_{P}c^{2}-2A_{2}\left(1\mp\sqrt{\frac{P}{A_{2}}}+\frac{1}{2}\frac{P}{A_{2}}\right),$
(4.26)

where the upper sign corresponds to the most relevant case $\rho_{m}<\rho_{P}$
and the lower sign corresponds to $\rho_{m}>\rho_{P}$. The pressure and total
energy density as a function of the scale factor read

$\displaystyle P=A_{2}\left(1+\frac{1}{B}+3\ln a\right)^{2},$ (4.27)
$\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}-2A_{2}\left[-\frac{1}{B}-3\ln
a+\frac{1}{2}\left(1+\frac{1}{B}+3\ln a\right)^{2}\right].$ (4.28)

At the present time (i.e. $a=1$), Eq. (4.28) leads to

$\displaystyle A_{2}=-\frac{1-\Omega_{m,0}}{1+\frac{1}{B^{2}}}\epsilon_{0}.$
(4.29)

We note that $A_{2}<0$. Numerically,
$A_{2}/c^{2}=-B^{2}/(1+B^{2})\rho_{\Lambda}=-1.25\times
10^{-5}\rho_{\Lambda}=7.43\times 10^{-29}\,{\rm g\,m^{-3}}$. The pressure $P$
and the energy density $\epsilon$ become

$\displaystyle
P=-\frac{\left(1-\Omega_{m,0}\right)\epsilon_{0}}{1+B^{2}}~{}\left(1+B+3B\ln
a\right)^{2},$ (4.30)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+\left(1-\Omega_{m,0}\right)\epsilon_{0}~{}\frac{1+B^{2}+6B\ln
a+9B^{2}\ln^{2}a}{1+B^{2}}.$ (4.31)

The pressure $P$ is always negative and vanishes at $\rho_{m}=\rho_{P}$. As
the universe expands, starting from $-\infty$, the pressure first increases,
vanishes at $\rho=\rho_{m}$, then decreases to $-\infty$. Starting from
$+\infty$, the energy density $\epsilon$ first decreases, reaches a minimum
$\epsilon_{\rm min}=-A_{2}(\rho_{m}c^{2}/2A_{2}-1)^{2}>0$ at $\rho_{m}$
solution of $\rho_{m}c^{2}=2A_{2}[1+\ln(\rho_{m}/\rho_{P})]$, then increases
to $+\infty$. Starting from $+\infty$, the DE density $\epsilon_{de}$ first
decreases, reaches a minimum $\epsilon_{de}^{\rm min}=-A_{2}>0$ at
$\rho_{m}=\rho_{P}/e$, then increases to $+\infty$ (its value at
$\rho_{m}=\rho_{P}$ is $\epsilon_{de}=-2A_{2}>0$).

Cases with $n=1,2$ are the simplest logotropic models. Since $n$ is a free
parameter, cases with $n>2$ lead to a collection of generalized logotropic
models.

###  4.5 The case $N=2$

This generalized logotropic model is obtained by considering the first three
terms $A_{0}$, $A_{1}$ and $A_{2}$. This leads to the EoS

$\displaystyle
P=A_{0}+A_{1}\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)+A_{2}\ln^{2}\left(\frac{\rho_{m}}{\rho_{P}}\right).$
(4.32)

The energy density is given by

$\displaystyle\epsilon=\rho_{m}c^{2}-A_{0}-A_{1}\left[1+\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]-2A_{2}\left[1+\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)+\frac{1}{2}\ln^{2}\left(\frac{\rho_{m}}{\rho_{P}}\right)\right],$
(4.33)

where the first term is DM and the other terms are DE. The EoS $P(\epsilon)$
can be obtained in the reversed form $\epsilon(P)$ by eliminating $\rho_{m}$
between Eqs. (4.32) and (4.33).444We can note that Eq. (4.32) is a second
degree equation in $\ln({\rho_{m}}/{\rho_{P}})$. The pressure and the energy
density evolve with the scale factor as

$\displaystyle P=A_{0}+A_{1}\left(-1-\frac{1}{B}-3\ln
a\right)+A_{2}\left(-1-\frac{1}{B}-3\ln a\right)^{2},$ (4.34)
$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}-A_{0}+A_{1}\left(\frac{1}{B}+3\ln
a\right)-2A_{2}\left[-\frac{1}{B}-3\ln a+\frac{1}{2}\left(1+\frac{1}{B}+3\ln
a\right)^{2}\right].$ (4.35)

At the present time (i.e. $a=1$), the above equation gives

$\displaystyle(1-\Omega_{m,0})~{}\epsilon_{0}=-A_{0}+\frac{A_{1}}{B}-A_{2}\left(1+\frac{1}{B^{2}}\right),$
(4.36)

which constrains one of the three free parameters. As a result, the model has
only two free parameters. As expected, in the late Universe
($a\rightarrow\infty$, $\rho\rightarrow 0$), the DE density $\epsilon_{de}$
dominates and we have $P/\epsilon\rightarrow-1$. It is worth mentioning that
in order to have $\epsilon>0$, $A_{2}$ must be negative.

##  5 Evolution of the generalized logotropic model

In the following we consider models with $A_{i}=A_{n}~{}\delta_{i,n}$. The
evolution of the energy density as a function of the scale factor is plotted
in Fig. 1,

Figure 1: Energy density $\epsilon/\epsilon_{0}$ as a function of the scale
factor $a$ for $\Omega_{m,0}=0.309$, $B=3.53\times 10^{-3}$ and
$n=0,1,2,3,5,10$

where we have used the best-fit values of the parameters $\Omega_{m,0}=0.309$
and $B=3.53\times 10^{-3}$ obtained from the $\Lambda$CDM model (they are
consistent with the cosmological analysis of the original logotropic model
[39, 40]). It shows that the Universe starts at $a=0$ with an infinite energy
density.555We recall that the logotropic model is not aimed at describing the
early inflation. The energy density $\epsilon$ first decreases with the
increase of the scale factor $a$, reaches a minimum, then increases with the
scale factor characterizing a phantom Universe [44, 45].

In the generalized logotropic model, the Friedmann equation (3.19) takes the
form

$\displaystyle
H=\frac{\dot{a}}{a}=H_{0}\sqrt{\frac{\Omega_{m,0}}{a^{3}}+\frac{1-\Omega_{m,0}}{a^{3}}~{}\frac{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a\right)}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}}.$ (5.1)

The evolution of the scale factor as a function of the time is given by

$\displaystyle
H_{0}t=\int_{0}^{a}\frac{da^{\prime}}{a^{\prime}~{}\sqrt{\frac{\Omega_{m,0}}{a^{\prime
3}}+\frac{1-\Omega_{m,0}}{a^{\prime
3}}~{}\frac{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a^{\prime}\right)}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}}}.$ (5.2)

The $\Lambda$CDM model is recovered from Eq. (5.2) for $n=0$ or $B=0$. In that
case, Eq. (5.2) can be integrated analytically yielding

$\displaystyle
a=\left(\frac{\Omega_{m,0}}{1-\Omega_{m,0}}\right)^{1/3}\sinh^{2/3}\left(\frac{3}{2}\sqrt{1-\Omega_{m,0}}H_{0}t\right),$
(5.3)
$\displaystyle\frac{\epsilon}{\epsilon_{0}}=\frac{1-\Omega_{m,0}}{\tanh^{2}\left(\frac{3}{2}\sqrt{1-\Omega_{m,0}}H_{0}t\right)},$
(5.4)

whereas for $B\neq 0$ it can only be integrated numerically. Figure 2 shows
the behavior of the scale factor $a$ as a function of $H_{0}t$ for
$n=0,1,2,3,5,10$.

Figure 2: Scale factor $a$ as a function of the cosmic time $t$ for
$\Omega_{m,0}=0.309$, $B=3.53\times 10^{-3}$ and $n=0,1,2,3,5,10$

The age of the universe is given by

$\displaystyle t_{\rm
age}=\frac{1}{H_{0}}\int_{0}^{1}\frac{da^{\prime}}{a^{\prime}~{}\sqrt{\frac{\Omega_{m,0}}{a^{\prime
3}}+\frac{1-\Omega_{m,0}}{a^{\prime
3}}~{}\frac{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a^{\prime}\right)}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}}}$ (5.5)

with $H_{0}=2.195\times 10^{-18}\,{\rm s}^{-1}$, i.e., $H_{0}^{-1}=14.4\,{\rm
Gyrs}$. For the $\Lambda$CDM model ($n=0$) we recover the well-known result
$t_{\rm age}=0.956\,H_{0}^{-1}=13.8\,{\rm Gyrs}$. A difference larger than
$0.1\,\%$ (the typical error bar on the age of the universe) occurs in models
with $n>3$ so these models should be rejected. For example, we get $t_{\rm
age}=0.966\,H_{0}^{-1}=13.9\,{\rm Gyrs}$ for $n=20$, $t_{\rm
age}=0.998\,H_{0}^{-1}=14.4\,{\rm Gyrs}$ for $n=100$, and $t_{\rm
age}=1.12\,H_{0}^{-1}=16.1\,{\rm Gyrs}$ for $n=1000$.

The asymptotic behavior of the generalized logotropic model can be obtained
analytically. For $a\rightarrow 0$, we have a matter dominated Universe which
corresponds to Einstein-deSitter solution given by

$\displaystyle a\sim\left(\frac{3}{2}\sqrt{\Omega_{m,0}}H_{0}t\right)^{2/3},$
(5.6)
$\displaystyle\frac{\epsilon}{\epsilon_{0}}\sim\frac{4}{9H_{0}^{2}t^{2}}.$
(5.7)

For $a\rightarrow+\infty$, the energy density behaves as

$\displaystyle\epsilon\sim 3^{n}|A_{n}|(\ln a)^{n}.$ (5.8)

In this asymptotic regime, the Friedmann equation can be written as

$\displaystyle\frac{\dot{a}}{a}=\sqrt{\frac{8\pi G}{3c^{2}}\epsilon}\sim
K_{n}~{}(\ln a)^{n/2},$ (5.9)

where we define the constant $K_{n}$ as

$\displaystyle
K_{n}=3^{n/2}H_{0}\sqrt{\frac{\left(1-\Omega_{m,0}\right)e^{1+1/B}}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}}.$
(5.10)

The above equation can be integrated into

$\displaystyle K_{n}~{}t\sim\int^{a}\frac{da^{\prime}}{a^{\prime}(\ln
a^{\prime})^{n/2}}\sim\int^{\ln a}\frac{ds}{s^{n/2}},$ (5.11)

where we have made the change of variables $s=\ln a^{\prime}$ to get the
second equality. It is interesting to note that the asymptotic evolution of
the scale factor as a function of time depends mainly on the value of the
parameter $n$. We can distinguish three relevant types of evolution which
correspond to $n<2,n=2$ and $n>2$.

(i) For $n<2$: this solution, which includes the original logotropic model
[30, 31, 39], describes a super de Sitter evolution of the form

$\displaystyle a\propto{\rm
exp}\left\\{{\left[\frac{1}{2}(2-n)~{}K_{n}~{}t\right]^{\frac{2}{2-n}}}\right\\},\qquad\epsilon\propto
t^{n},$ (5.12)

where the scale factor grows super exponentially rapidly with cosmic time
causing an algebraic divergence of the energy density. For $n=1$ we recover
the original logotropic model where $a\sim e^{t^{2}}$ and $\epsilon\sim t$
[30, 31, 39]. Since the scale factor and the density increase indefinitely
with time, this is called Little Rip [46]. For $n=0$ we recover the
$\Lambda$CDM model presenting an exponential (de Sitter) expansion $a\sim
e^{t}$ and a constant energy density $\epsilon\sim 1$.

(ii) For $n=2$: we get a double exponential evolution of the form

$\displaystyle a\propto{\rm exp}{[{{\rm
exp}(K_{2}~{}t)]}},\qquad\epsilon\propto{\rm exp}(2K_{2}t),$ (5.13)

where the scale factor grows hyper exponentially rapidly with cosmic time
causing an exponential divergence of the energy density (Little Rip).

(iii) For $n>2$: this case represents a situation in which the Universe ends
up with a finite-time future singularity. One finds

$\displaystyle a\propto{\rm
exp}\left\\{{\left[\frac{1}{2}(n-2)~{}K_{n}~{}(t_{s}-t)\right]^{-\frac{2}{n-2}}}\right\\},\qquad\epsilon\propto\left[\frac{1}{2}(n-2)~{}K_{n}~{}(t_{s}-t)\right]^{-\frac{2n}{n-2}}.$
(5.14)

The singularity at $t=t_{s}$ corresponds to a Big Rip [47] characterized by
the divergence of the scale factor and energy density in finite time. The big
rip time is given by

$\displaystyle t_{\rm
BR}=\frac{1}{H_{0}}\int_{0}^{+\infty}\frac{da^{\prime}}{a^{\prime}~{}\sqrt{\frac{\Omega_{m,0}}{a^{\prime
3}}+\frac{1-\Omega_{m,0}}{a^{\prime
3}}~{}\frac{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a^{\prime}\right)}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}}}.$ (5.15)

For measured values of $\Omega_{m,0}$ and $\epsilon_{0}$ (hence $H_{0}$), this
is just a function of $n$. There is no other free (undetermined) parameter in
our model. We get $t_{\rm BR}=3240\,{\rm Gyrs}$ for $n=3$, $t_{\rm
BR}=1650\,{\rm Gyrs}$ for $n=4$, $t_{\rm BR}=422\,{\rm Gyrs}$ for $n=10$,
$t_{\rm BR}=47.1\,{\rm Gyrs}$ for $n=100$, and $t_{\rm BR}=19.2\,{\rm Gyrs}$
for $n=1000$. However, we recall that the models with $n>3$ are excluded from
the observations.

Using Eqs. (4.8) and (4.9), the EoS parameter $w=P/\epsilon$ can be expressed
in terms of the scale factor as

$\displaystyle
w(a)=\frac{-\left(1-\Omega_{m,0}\right)e^{1+1/B}a^{3}\left(-1-\frac{1}{B}-3\ln
a\right)^{n}}{\Omega_{m,0}\Gamma\left(n+1,-1-\frac{1}{B}\right)+\left(1-\Omega_{m,0}\right)\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a\right)}.$ (5.16)

For $a=1$, we get

$\displaystyle
w_{0}=-\frac{\left(1-\Omega_{m,0}\right)e^{1+1/B}\left(-1-\frac{1}{B}\right)^{n}}{\Gamma\left(n+1,-1-\frac{1}{B}\right)}.$
(5.17)

For the $\Lambda$CDM ($n=0$) we recover the well-known value
$w_{0}=-\left(1-\Omega_{m,0}\right)=-0.691$. We get $w_{0}=-0.693$ for $n=1$,
$w_{0}=-0.696$ for $n=2$, $w_{0}=-0.698$ for $n=3$, $w_{0}=-0.701$ for $n=4$,
$w_{0}=-0.715$ for $n=10$, $w_{0}=-0.935$ for $n=100$, and $w_{0}=-3.12$ for
$n=1000$. For $n>3$ we are out of the error bars (typically $1\,\%$ for the
value of $w_{0}$) so these models should be rejected. The behavior of the EoS
parameter $w(a)$ as function of the scale factor $a$ is plotted in Fig. 3. It
shows that the GLDF behaves asymptotically as a superposition of two non-
interacting fluids (i.e. DM and DE) with the total pressure and total energy
given by the sum

$\displaystyle P=P_{m}+P_{de},\qquad\epsilon=\epsilon_{m}+\epsilon_{de}.$
(5.18)

Their EoS parameters are

$\displaystyle w_{m}=\frac{P_{m}}{\epsilon_{m}}=0,\qquad
w_{de}=\frac{P_{de}}{\epsilon_{de}}=-\frac{e^{1+1/B}a^{3}\left(-1-\frac{1}{B}-3\ln
a\right)^{n}}{\Gamma\left(n+1,-1-\frac{1}{B}-3\ln a\right)}.$ (5.19)

Figure 3: EoS parameter $w(a)$ as a function the scale factor $a$ for
$\Omega_{m,0}=0.309$, $B=3.53\times 10^{-3}$ and $n=0,1,2,3,5,10$

In a flat Universe, the deceleration parameter
$q=-{\ddot{a}a}/{\dot{a}^{2}}=-{\dot{H}}/{H^{2}}-1$ is related to the EoS
parameter by

$\displaystyle q=\frac{1+3w}{2}.$ (5.20)

The Universe is undergoing an accelerating expansion if $q<0$ (i.e. $w<-1/3$).
In Fig. 4, we show the behavior of $q(a)$ as a function of the scale factor
for different values of $n$.

Figure 4: Acceleration $q(a)$ as a function of the scale factor $a$ for
$\Omega_{m,0}=0.309$, $B=3.53\times 10^{-3}$ and $n=0,1,2,3,5,10$

From Eq. (4.9) the transition scale factor $a_{t}$, corresponding to
$\epsilon_{m}=\epsilon_{de}$, is obtained by solving the transcendental
equation

$\displaystyle\Gamma\left(n+1,-1-\frac{1}{B}-3\ln
a_{t}\right)=\frac{\Omega_{m,0}}{1-\Omega_{m,0}}~{}\Gamma\left(n+1,-1-\frac{1}{B}\right).$
(5.21)

Another parameter to study is the squared speed of sound $c_{s}^{2}$ which is
a key ingredient to investigate the stability of any model. In particular, the
sign of $c_{s}^{2}$ plays a crucial role for determining classical stability.
It is defined by

$\displaystyle
c_{s}^{2}=\frac{dP}{d\epsilon}~{}c^{2}=\frac{dP/d\rho_{m}}{d\epsilon/d\rho_{m}}~{}c^{2}.$
(5.22)

Differentiating Eqs. (1.1) and (3.6) with $A_{i}=A_{n}~{}\delta_{i,n}$, we get

$\displaystyle\frac{c_{s}^{2}}{c^{2}}=\frac{n\ln^{n-1}\left(\frac{\rho_{m}}{\rho_{P}}\right)}{\frac{\rho_{m}c^{2}}{A_{n}}-n\frac{\rho_{m}}{\rho_{P}}\Gamma\left[n,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]}.$
(5.23)

In Fig. 5 we plot the squared speed of sound $c_{s}^{2}/c^{2}$ as a function
of the scale factor for different values of $n$. The causality and classical
stability conditions are satisfied if the speed of sound varies in the range
$0\leq c_{s}^{2}/c^{2}\leq 1$. One sees from Fig. 5 that, in fact, there are
regions where the causality and classical stability conditions are satisfied
but the extent of these regions decreases as $n$ increases.666The speed of
sound is real in the normal regime $d\epsilon/d\rho_{m}>0$ and imaginary in
the phantom regime $d\epsilon/d\rho_{m}<0$. It becomes infinite (before
becoming imaginary) when we enter into the phantom regime, i.e., when
$d\epsilon/d\rho_{m}=0$.

Figure 5: Squared speed of sound $c_{s}^{2}(a)/c^{2}$ as a function of the
scale factor $a$ for $\Omega_{m,0}=0.309$, $B=3.53\times 10^{-3}$ and
$n=0,1,2,3,5,10$

##  6 Conclusions

In this work, we have proposed a new class of cosmological unified dark sector
models “Generalized Logotropic Models". These models are a generalization of
the logotropic model [30] by considering the pressure $P$ as a sum of higher
logarithmic terms of the rest-mass density $\rho_{m}$. The pressure can
naturally take negative values in these cosmological models. In this scenario,
the Universe is filled with a single fluid without the need of a cosmological
constant. Our generalized model depends on a set of free parameters $A_{i}$
with $i=1,2,\cdots,N$. In particular, we have considered a special class of
generalized logotropic models where $A_{i}=A_{n}\delta_{in}$, which depends on
two free parameters $A_{n}$ and $n$. The usual logotropic model [30]
corresponds to $n=1$ and $A_{1}$. We have also presented the model with $N=2$,
which contains the first two terms $A_{1}$ and $A_{2}$ of the series. We have
highlighted the most relevant properties of these generalized logotropic
models. To fix bounds on the free parameters of our models, we employed the
best fit of the parameters $\Omega_{m,0}$ and $B$ obtained from the
cosmological analysis carried on the original logotropic model (i.e. $n=1$)
[39]. After fixing the free parameters, we investigated the cosmological
behavior of the generalized logotropic models by focusing on the evolution of
the DE density, scale factor, EoS parameter, decceleration parameter and
squared speed of sound. We showed the asymptotic behavior of these models and
noticed three distinct ways of evolution depending on the value of $n$. In all
the analyzed cases, we established that generalized logotropic models lead to
realistic cosmological models in which the dark sector is represented by a
unique fluid. The deviation of the generalized models from the standard
$\Lambda$CDM model depends on the value of the parameter $n$. At later times,
higher values of $n$ lead to higher deviations from $\Lambda$CDM. This implies
that the generalized logotropic models can be used as realistic background
cosmological models to describe our Universe with a free parameter $n$. We
estimated that only models with $n\leq 3$ are consistent with the
observations. To find out the most suitable values of $n$, it will be
necessary to perform a fitting procedure by using, for instance, the Monte
Carlo method and a detailed comparison with cosmological data like SN, BAO,
and CMB surveys. We expect to perform this analysis in future works.

A further interesting generalization of the logotropic model is to consider a
single fluid described by the EoS

$\displaystyle P=A\ln^{\alpha}\left(\frac{\rho_{m}}{\rho_{P}}\right),$ (6.1)

where $A$ is a real number given by

$\displaystyle
A=-\frac{\left(1-\Omega_{m,0}\right)\epsilon_{0}e^{1+1/B}}{\Gamma\left(\alpha+1,-1-\frac{1}{B}\right)},$
(6.2)

and $\alpha$ is a real positive parameter which can be constrained by the
cosmological data. There are two ways to recover the $\Lambda$CDM, either by
taking $\alpha=0$ or $B=0$ which makes the model rich and particularly more
appealing in describing DE especially for $0<\alpha<1$. Such a model
interpolates between the $\Lambda$CDM for values of $\alpha$ close to zero and
to the usual logotropic model for values of $\alpha$ close to one.

In summary, single fluids with generalized logotropic EoS may have interesting
cosmological features and, thus, they represent a good candidate to describe
the DE sector. The investigation and analysis of these models will be carried
out in detail in future works.

Finally, we would like to mention that the logotropic model and the
generalization presented in this work are not free of intrinsic problems, as
all the cosmological models known in the literature. In fact, the speed of
sound in logotropic models has the unpleasant property of increasing with the
scale factor, leading, like for the Chaplygin gas model, to oscillations in
the mass power-spectrum that are not detected in observations at the
cosmological level [48]. However, there are several possibilities to solve
this problem by considering additional effects such as non-linear and non-
adiabatic perturbations, among others (see the discussion in [49]). In any
case, UDM models constitute a subject of intensive research as possible
alternative scenarios to the popular and generally accepted $\Lambda$CDM
model. Our paper provides a class of models exhibiting a transition between a
normal behavior and a phantom behavior governed by a single equation of state.
In addition, depending on the value of the parameter $n$, we can have
different types of late evolution: no singularity, little rip, big rip… It is
very interesting to note that all these models are consistent with the
$\Lambda$CDM model up to the present time but will differ in the future. A
virtue of our model is to show that it is very difficult to predict the future
evolution of the universe based on present observations.

## Acknowledgements

We would like to thank Prof. O. Luongo for helpful comments. HBB gratefully
acknowledges the financial support from University of Sharjah (grant number
V.C.R.G./R.438/2020). The work of HQ was partially supported by UNAM-DGAPA-
PAPIIT, Grant No. 114520, and Conacyt-Mexico, Grant No. A1-S-31269.

##  Appendix A Asymptotic equation of state

In this Appendix, we establish the asymptotic EoS $P(\epsilon)$ of the GLDF
and make the connection with the MCG model [19, 26, 27, 28].

For $a\rightarrow+\infty$, using Eqs. (3.12) and (3.21), we find that the
energy density evolves with the scale factor as

$\epsilon\sim|A_{N}|\left(1+\frac{1}{B}+3\ln a\right)^{N}.$ (A.1)

Let us determine the corresponding asymptotic EoS from the energy conservation
equation (3.3) which can be rewritten as

$\frac{d\epsilon}{da}+\frac{3}{a}(\epsilon+P)=0.$ (A.2)

From Eqs. (A.1) and (A.2), we get

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\epsilon-\frac{a}{3}\frac{d\epsilon}{da}$ (A.3)
$\displaystyle=$ $\displaystyle-\epsilon-N|A_{N}|\left(1+\frac{1}{B}+3\ln
a\right)^{N-1}$ $\displaystyle=$
$\displaystyle-\epsilon-N|A_{N}|\left(\frac{\epsilon}{|A_{N}|}\right)^{(N-1)/N}$
$\displaystyle=$
$\displaystyle-\epsilon\left[1+N\left(\frac{|A_{N}|}{\epsilon}\right)^{1/N}\right].$

Therefore, the asymptotic EoS of the GLDF reads

$\displaystyle P=-\epsilon-N|A_{N}|^{1/N}\epsilon^{1-1/N}.$ (A.4)

This is a particular case of the generalized polytropic EoS (or MCG model)
[19, 26, 27, 28]

$\displaystyle P=\alpha\epsilon+K\left(\frac{\epsilon}{c^{2}}\right)^{\gamma}$
(A.5)

corresponding to $\alpha=-1$, $K=-N(|A_{N}|c^{2})^{1/N}$ and $\gamma=1-1/N$.
Therefore, the GLDF is asymptotically equivalent to the MCG with $\alpha=-1$.
Since $w=P/\epsilon<-1$, the EoS (A.4) leads to a phantom behavior in
agreement with the results of Sec. 5. For $N=1$, the EoS (A.4) reduces to

$\displaystyle P=-\epsilon-A.$ (A.6)

For $N=2$, it reduces to

$\displaystyle P=-\epsilon-2\sqrt{|A_{2}|}\epsilon^{1/2}.$ (A.7)

Let us check that we recover the asymptotic EoS (A.4) directly from the
generalized logotropic EoS defined by Eqs. (3.20) and (3.21). For
$a\rightarrow+\infty$, we have

$P\simeq A_{N}x^{N}+A_{N-1}x^{N-1}+...,$ (A.8) $\epsilon\simeq-
A_{N}x^{N}-A_{N}Nx^{N-1}-A_{N-1}x^{N-1}+...,$ (A.9)

where

$x\equiv-1-\frac{1}{B}-3\ln a\rightarrow-\infty.$ (A.10)

We stress that it is necessary to account for the first order correction to
the leading term $x^{N}$ in Eqs. (A.8) and (A.9). From

$P\simeq A_{N}x^{N}\left(1+\frac{A_{N-1}}{A_{N}}\frac{1}{x}\right),$ (A.11)
$\epsilon\simeq-
A_{N}x^{N}\left(1+N\frac{1}{x}+\frac{A_{N-1}}{A_{N}}\frac{1}{x}\right),$
(A.12)

we get

$\displaystyle\frac{P}{\epsilon}$ $\displaystyle\simeq$
$\displaystyle-\left(1+\frac{A_{N-1}}{A_{N}}\frac{1}{x}\right)\left(1-N\frac{1}{x}-\frac{A_{N-1}}{A_{N}}\frac{1}{x}\right)$
(A.13) $\displaystyle\simeq$ $\displaystyle-\left(1-\frac{N}{x}\right)$
$\displaystyle=$
$\displaystyle-\left[1+N\left(\frac{|A_{N}|}{\epsilon}\right)^{1/N}\right],$

which returns Eq. (A.3).

##  Appendix B The two-fluid model

In this Appendix, we determine the two-fluid model corresponding to the GLDF
with $A_{i}=A_{n}\delta_{i,n}$. In particular, we establish the EoS of the DE
in the two-fluid model.

The GLDF is a one-fluid model (i.e. a UDM model) unifying DM and DE. The
pressure and the energy density are given by

$\displaystyle P=A_{n}\ln^{n}\left(\frac{\rho_{m}}{\rho_{P}}\right),$ (B.1)
$\displaystyle\epsilon=\rho_{m}c^{2}-A_{n}\frac{\rho_{m}}{\rho_{P}}\Gamma\left[n+1,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]=\rho_{m}c^{2}+u=\epsilon_{m}+\epsilon_{de}.$
(B.2)

Concerning the evolution of the homogeneous background, this one-fluid model
is equivalent777The equivalence between the one-fluid model and the two-fluid
model is lost when we consider the formation of structures. to a two-fluid
model made of pressureless DM with an EoS $P_{m}=0$ giving
$\epsilon_{m}=\Omega_{m,0}\epsilon_{0}/a^{3}$ and DE with an EoS
$P_{de}(\epsilon_{de})$ giving $\epsilon_{de}=u(a)$. Noting that
$P=P_{m}+P_{de}=P_{de}$, the EoS $P_{de}(\epsilon_{de})$ of DE is determined
in parametric form by the equations

$\displaystyle P_{de}=A_{n}\ln^{n}\left(\frac{\rho_{m}}{\rho_{P}}\right),$
(B.3)
$\displaystyle\epsilon_{de}=-A_{n}\frac{\rho_{m}}{\rho_{P}}\Gamma\left[n+1,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right].$
(B.4)

Eliminating $\rho_{m}$ between these two expressions we get the EoS of DE
under the reversed form $\epsilon_{de}(P_{de})$ as

$\displaystyle\epsilon_{de}=-A_{n}e^{\mp\left|\frac{P_{de}}{A_{n}}\right|^{1/n}}\Gamma\left[n+1,\mp\left|\frac{P_{de}}{A_{n}}\right|^{1/n}\right],$
(B.5)

where the upper sign corresponds to the most relevant case $\rho_{m}<\rho_{P}$
and the lower sign corresponds to $\rho_{m}>\rho_{P}$.

Let us check that the two-fluid model returns the results of the one-fluid
model for the homogeneous background. The evolution $\epsilon_{de}(a)$ of the
DE density with the scale factor can be obtained from the energy conservation
equation

$\frac{d\epsilon_{de}}{da}+\frac{3}{a}(\epsilon_{de}+P_{de})=0\qquad\Leftrightarrow\qquad\int\frac{d\epsilon_{de}}{\epsilon_{de}+P_{de}}=-3\ln
a$ (B.6)

with the EoS $P_{de}(\epsilon_{de})$ defined by Eqs. (B.3) and (B.4). At this
stage, $\rho_{m}$ is just a dummy variable. It is easy to establish that

$\displaystyle\epsilon_{de}^{\prime}(\rho_{m})=-A_{n}n\frac{\rho_{P}}{\rho_{m}^{2}}\Gamma\left[n,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right]$
(B.7)

and

$\displaystyle
P_{de}(\rho_{m})+\epsilon_{de}(\rho_{m})=-A_{n}n\frac{\rho_{P}}{\rho_{m}}\Gamma\left[n,\ln\left(\frac{\rho_{m}}{\rho_{P}}\right)\right].$
(B.8)

Therefore, we have the identity

$\displaystyle
P_{de}(\rho_{m})+\epsilon_{de}(\rho_{m})=\rho_{m}\epsilon_{de}^{\prime}(\rho_{m}),$
(B.9)

and the energy conservation equation (B.6) becomes

$\frac{d\rho_{m}}{da}+\frac{3}{a}\rho_{m}=0\qquad\Leftrightarrow\qquad\int\frac{d\rho_{m}}{\rho_{m}}=-3\ln
a,$ (B.10)

implying that $\rho_{m}\propto a^{-3}$. This leads to results consistent with
those of Sec. 4. However, we cannot establish that $\rho_{m}$ is the rest-mass
(or DM) density (they could differ by a multiplicative constant). This implies
that we cannot determine $A_{n}$ in the two-fluid model contrary to the one-
fluid model. More explicit results are given below.

###  B.1 The case $n=1$

For $n=1$, Eq. (B.5) reduces to the affine EoS

$\displaystyle P_{de}=-\epsilon_{de}-A$ (B.11)

with $A>0$. This is the EoS of DE corresponding to the original logotropic gas
[39]. It coincides with the asymptotic EoS (A.6) of the LDF seen as a one-
fluid (UDM) model. The affine EoS (B.11) was first introduced and studied in
[28].

Let us check that the two-fluid model returns the results of the one-fluid
model. Integrating the energy conservation equation (B.6) for DE with the EoS
(B.11) we obtain

$\displaystyle\epsilon_{de}=3A\ln\left(\frac{a}{a_{*}}\right),$ (B.12)

where $a_{*}$ is a constant of integration. If we add the contribution of
pressureless DM, we find that the total energy density is given by

$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+3A\ln\left(\frac{a}{a_{*}}\right).$
(B.13)

Using the condition (at $a=1$)

$\displaystyle\epsilon_{0}(1-\Omega_{m,0})=-3A\ln a_{*},$ (B.14)

we can rewrite Eq. (B.13) under the form

$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+3A\ln
a+\epsilon_{0}(1-\Omega_{m,0}).$ (B.15)

This expression is consistent with Eq. (4.23) of the one-fluid model if we set
$A=B\epsilon_{0}(1-\Omega_{m,0})$, except that the two-fluid model does not
determine the value of the constant $A$, contrary to the one-fluid model.
Indeed, the Planck scale $\rho_{P}$ does not occur in Eq. (B.11), unlike Eq.
(4.16). This is a huge advantage of the one-fluid model [30].

###  B.2 The case $n=2$

For $n=2$, Eq. (B.5) reduces to

$\displaystyle\epsilon_{de}=-2A_{2}\left(1\mp\sqrt{\frac{P_{de}}{A_{2}}}+\frac{P_{de}}{2A_{2}}\right)$
(B.16)

with $A_{2}<0$. This yields a second degree equation for $\sqrt{P_{de}/A_{2}}$
of the form

$\displaystyle\frac{P_{de}}{A_{2}}\mp
2\sqrt{\frac{P_{de}}{A_{2}}}+2+\frac{\epsilon_{de}}{A_{2}}=0.$ (B.17)

When $\rho_{m}>\rho_{P}$, Eq. (B.17) with the lower sign has just one solution

$\displaystyle\sqrt{-\frac{P_{de}}{|A_{2}|}}=-1+\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1}.$
(B.18)

As $\rho_{m}$ decreases from $+\infty$ to $\rho_{P}$, the DE density
$\epsilon_{de}$ decreases from $+\infty$ to $2|A_{2}|$ and the pressure
$P_{de}$ increases from $-\infty$ to $0$ (see Sec. 4.4). This leads to the EoS

$\displaystyle
P_{de}=-|A_{2}|\left(-1+\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1}\right)^{2}$
(B.19)

or, equivalently,

$\displaystyle
P_{de}=-\epsilon_{de}+2|A_{2}|\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1},$ (B.20)

which is valid for $\epsilon_{de}\geq 2|A_{2}|$. For
$\epsilon_{de}\rightarrow+\infty$, Eq. (B.20) reduces to

$\displaystyle P_{de}\simeq-\epsilon_{de}+2|A_{2}|^{1/2}\sqrt{\epsilon_{de}}.$
(B.21)

When $\rho_{m}<\rho_{P}$, the solutions of Eq. (B.17) with the upper sign are

$\displaystyle\sqrt{-\frac{P_{de}}{|A_{2}|}}=1\pm\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1}.$
(B.22)

As $\rho_{m}$ goes from $\rho_{P}$ to $0$, the DE density $\epsilon_{de}$
first decreases from $2|A_{2}|$ to $|A_{2}|$ then increases to $+\infty$ while
the pressure $P_{de}$ decreases from $0$ to $-\infty$ (see Sec. 4.4). Equation
(B.17) has two solutions for $|A_{2}|\leq\epsilon_{de}\leq 2|A_{2}|$, one
solution (with the upper sign) for $\epsilon_{de}\geq 2|A_{2}|$ and no
solution for $\epsilon_{de}\leq|A_{2}|$. This leads to the EoS

$\displaystyle
P_{de}=-|A_{2}|\left(1\pm\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1}\right)^{2}$
(B.23)

or, equivalently,

$\displaystyle P_{de}=-\epsilon_{de}\mp
2|A_{2}|\sqrt{\frac{\epsilon_{de}}{|A_{2}|}-1},$ (B.24)

which is valid with the two signs for $|A_{2}|\leq\epsilon_{de}\leq 2|A_{2}|$
and with the upper sign for $\epsilon_{de}\geq 2|A_{2}|$. This is the EoS of
DE corresponding to the GLDF in the case $n=2$. For
$\epsilon_{de}\rightarrow+\infty$, Eq. (B.24) reduces to

$\displaystyle P_{de}\simeq-\epsilon_{de}-2|A_{2}|^{1/2}\sqrt{\epsilon_{de}},$
(B.25)

which coincides with the asymptotic EoS (A.7) of the GLDF seen as a one-fluid
(UDM) model.

Let us check that the two-fluid model returns the results of the one-fluid
model. Integrating the energy conservation equation (B.6) for DE with the EoS
(B.20) or (B.24) we obtain

$\displaystyle\frac{\epsilon_{de}}{|A_{2}|}=1+9\ln^{2}\left(\frac{a}{a_{*}}\right),$
(B.26)

where $a_{*}$ is a constant of integration. If we add the contribution of
pressureless DM, we find that the total energy density is given by

$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+|A_{2}|\left[1+9\ln^{2}\left(\frac{a}{a_{*}}\right)\right].$
(B.27)

Using the condition (at $a=1$)

$\displaystyle\epsilon_{0}(1-\Omega_{m,0})=|A_{2}|\left(1+9\ln^{2}a_{*}\right),$
(B.28)

we can rewrite Eq. (B.27) under the form

$\displaystyle\epsilon=\frac{\Omega_{m,0}\epsilon_{0}}{a^{3}}+\epsilon_{0}(1-\Omega_{m,0})\left[1+\frac{9|A_{2}|\ln^{2}a}{\epsilon_{0}(1-\Omega_{m,0})}+\frac{6|A_{2}|\ln
a}{\epsilon_{0}(1-\Omega_{m,0})}\sqrt{\frac{\epsilon_{0}(1-\Omega_{m,0})}{|A_{2}|}-1}\right].$
(B.29)

This expression is consistent with Eq. (4.31) of the one-fluid model if we set
$|A_{2}|/[\epsilon_{0}(1-\Omega_{m,0})]=B^{2}/(1+B^{2})$, except that the two-
fluid model does not determine the value of the constant $A_{2}$, contrary to
the one-fluid model. This is a huge advantage of the one-fluid model [30].

##  Appendix C Present proportion of dark matter and dark energy

In this Appendix we recall the argument of [33] leading to a prediction of the
present proportion of DM and DE in the universe and take into account the
presence of baryons.

The original logotropic model [30, 31, 39, 33] is based on the EoS

$\displaystyle P=A\ln\left(\frac{\rho_{dm}}{\rho_{P}}\right),$ (C.1)

where $\rho_{dm}$ is the rest-mass density of the LDF, $A$ is a new
fundamental constant of physics (superseding the cosmological constant) and
$\rho_{P}$ is the Planck density. This EoS provides a unification of DM and
DE. It is very interesting that the Planck density appears in this EoS in
order to make the variable in the logarithm dimensionless. This implies that
quantum effects play a certain role in the late universe where the logotropic
model applies. The rest-mass density evolves as [see Eq. (3.14)]

$\displaystyle\rho_{dm}=\frac{\Omega_{dm,0}\epsilon_{0}/c^{2}}{a^{3}},$ (C.2)

and it plays the role of DM. As a result $\Omega_{dm,0}$ is interpreted as the
present proportion of DM in the universe. The energy density of the LDF is
[see Eq. (4.17)]

$\displaystyle\epsilon_{df}=\rho_{dm}c^{2}-A\left[1+\ln\left(\frac{\rho_{dm}}{\rho_{P}}\right)\right]=\epsilon_{dm}+\epsilon_{de},$
(C.3)

where the first term (rest-mass) is interpreted as DM and the second term
(internal energy) as DE. We must also include the contribution of baryons
which form a pressureless gas ($P_{b}=0$). Their energy density evolves as

$\displaystyle\epsilon_{b}=\frac{\Omega_{b,0}\epsilon_{0}}{a^{3}},$ (C.4)

where $\Omega_{b,0}$ is the present proportion of baryons in the universe. The
total energy density is therefore

$\displaystyle\epsilon=\rho_{dm}c^{2}-A\left[1+\ln\left(\frac{\rho_{dm}}{\rho_{P}}\right)\right]+\epsilon_{b}.$
(C.5)

Substituting Eq. (C.2) into Eq. (C.5), we get

$\displaystyle\epsilon=\frac{\Omega_{dm,0}\epsilon_{0}}{a^{3}}-A\left[1+\ln\left(\frac{\Omega_{dm,0}\epsilon_{0}}{\rho_{P}c^{2}a^{3}}\right)\right]+\frac{\Omega_{b,0}\epsilon_{0}}{a^{3}}.$
(C.6)

Applying this relation at the present time ($a=1$) and introducing the present
proportion of DE in the universe $\Omega_{de,0}=1-\Omega_{dm,0}-\Omega_{b,0}$
we get

$\displaystyle
A=\frac{\Omega_{de,0}\epsilon_{0}}{\ln\left(\frac{\rho_{P}c^{2}}{\Omega_{dm,0}\epsilon_{0}}\right)-1}.$
(C.7)

Introducing the present DE density
$\rho_{\Lambda}\equiv\Omega_{de,0}\epsilon_{0}/c^{2}$, we can rewrite the
foregoing equation as

$\displaystyle
A=\frac{\rho_{\Lambda}c^{2}}{\ln\left(\frac{\rho_{P}}{\rho_{\Lambda}}\right)+\ln\left(\frac{\Omega_{de,0}}{1-\Omega_{de,0}-\Omega_{b,0}}\right)-1}.$
(C.8)

We now postulate that [33]

$\displaystyle
A=\frac{\rho_{\Lambda}c^{2}}{\ln\left(\frac{\rho_{P}}{\rho_{\Lambda}}\right)}$
(C.9)

or, equivalently, that $B=\ln({\rho_{P}}/{\rho_{\Lambda}})$. This implies

$\displaystyle\frac{\Omega_{de,0}}{1-\Omega_{de,0}-\Omega_{b,0}}=e,$ (C.10)

determining the present proportion of DM and DE [33]

$\displaystyle\Omega_{de,0}^{\rm
th}=\frac{e}{1+e}(1-\Omega_{b,0}),\qquad\Omega_{dm,0}^{\rm
th}=\frac{1}{1+e}(1-\Omega_{b,0}).$ (C.11)

If we neglect baryonic matter $\Omega_{b,0}=0$ we obtain the pure numbers
$\Omega_{de,0}^{\rm th}=\frac{e}{1+e}=0.731059...$ and $\Omega_{dm,0}^{\rm
th}=\frac{1}{1+e}=0.268941...$ which give the correct proportions $70\%$ and
$25\%$ of DE and DM [33]. If we take baryonic matter into account and use the
measured value of $\Omega_{b,0}=0.0486\pm 0.0010$, we get $\Omega_{de,0}^{\rm
th}=0.6955\pm 0.0007$ and $\Omega_{dm,0}^{\rm th}=0.2559\pm 0.0003$ which are
very close to the observed values $\Omega_{de,0}=0.6911\pm 0.0062$ and
$\Omega_{dm,0}=0.2589\pm 0.0057$ within the error bars. The postulate from Eq.
(C.9) means that the fundamental constant $A$ is equal to the present DE
energy density (more precisely
$\rho_{\Lambda}c^{2}/\ln({\rho_{P}}/{\rho_{\Lambda}})$). This can be viewed as
a strong cosmic coincidence [33] giving to our epoch a central place in the
history of the universe.

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