# Painlevé analysis for the cosmological field equations in Weyl Integrable
Spacetime

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Instituto de
Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia
5090000, Chile

###### Abstract

We apply singularity analysis to investigate the integrability properties of
the gravitational field equations in Weyl Integrable Spacetime for a spatially
flat Friedmann–Lemaître–Robertson–Walker background spacetime induced with an
ideal gas. We find that the field equations possess the Painlevé property in
the presence of the cosmological constant and the analytic solution is given
by a left Laurent expansion.

Painlevé analysis; Weyl integrable space; scalar field; integrability

###### pacs:

98.80.-k, 95.35.+d, 95.36.+x

##  I Introduction

The Weyl Integrable Spacetime (WIS) is a natural way to extend Einstein’s
General Relativity, in which a scalar field is introduced in the natural space
by geometrical degrees of freedom salim96 . Scalar fields play an important
role in the description of gravitational phenomena at large scales obs1 ; obs2
. Indeed, the late time or early time acceleration phases of the universe have
been proposed to be attributed to scalar fields sf1 ; sf2 ; sf3 ; sf5 ; sf6 .
Another novelty of WIS is that interaction is introduced geometrically between
the matter components ch1 ; ch2 ; ch3 . Cosmological models with interaction
in the dark sector of the universe have been widely studied before.
Furthermore, that interacting models survive the constraints follow by the
cosmological observations os1 ; os2 ; os3 . Specifically, the interaction in
the dark sector of the universe is a mechanism to solve the cosmic coincidence
problem and it has been used for the explanation of the discrepancy in the
cosmological constant Amendola-ide1 ; Amendola-ide2 ; Pavon:2005yx ;
delCampo:2008jx ; Wetterich-ide1 . In addition, WIS is related with the
chameleon cosmology cham1 .

There are various studies in the literature on cosmological models on WIS.
Multi-dimensional spacetimes were investigated in wm1 , while lower-
dimensional gravitational models were studied inwm1a . Inflation is WIS
investigated in wm2 ; wm3 , where it was found that nonsingular inflationary
solutions are provided by the specific theory. In wm4 ; ww4 ; ww5  WIS has
been considered as the dark energy mechanism which drives the present
acceleration of the universe, while recently an extension of WIS in Einstein-
Aether theory was studied in ww3 . The dynamical analysis of the field
equations with different fluid sources for the construction of the
cosmological history was the subject of study in wm6 .

The field equations in WIS are nonlinear ordinary differential equations of
second-order. Because of the new degrees of freedom provided by the scalar
field there are not many known solutions in the literature. Note that the
cosmological field equations in WIS admit a minisuperspace description the
Noether symmetry analysis for which was applied in wm7  for the construction
of conservation laws and the derivation of analytic solutions. The Noether
symmetry analysis is a powerful and systematic approach for the study of the
integrability properties of dynamical systems which follow from a variational
principle with many applications in cosmological studies ns1 ; ns2 ; ns3 ; ns4
; ns5 ; ns6 . For the cosmological model in WIS in which the matter source is
an ideal gas it was found that the introduction of the cosmological constant
term into the gravitational Action Integral leads to a dynamical system with
fewer Noetherian conservation laws from which we are not able to infer the
integrability properties. In this work we investigate the same problem with
the use of another important tool of analysis for the construction of
conservation laws and analytic solutions, known as Painlevé analysis or
singularity analysis buntis . The singularity analysis for the analysis of the
field equations in gravity covers a wide range of applications; see for
instance ftAn ; ksAn ; cots11 ; cots2 ; Feix97 ; pcosm ; bun1 ; cots . The
determination of the integrability properties of dynamical systems is
essential in physics and in all areas of applied mathematics. The novelty when
a physical system is described by an integrable dynamical system is that we
know that an actual solution exists when we apply numerical methods for the
study of the system, or there exists a closed-form function which solves the
dynamical system. For a recent discussion on the integrable cosmological
models we refer the reader to tw1 . In our study the singularity analysis
leads to the construction of analytic solutions expressed as Laurent
expansions around a movable singularity. The plan of the paper is as follows.

In Section II we present the cosmological model of our consideration which is
that of WIS in an homogeneous and isotropic universe in which the matter
source is that of an ideal gas. In Section III we review previous results
wherein we show the existence of additional conservation laws for the field
equations in the absence of the cosmological constant term. Section IV
includes the main analysis of this study, in which the singularity analysis is
applied for the study of the integrability properties for the field equations.
We find that the field equations possess the Painlevé property with or without
a cosmological constant term and the analytic solution is expressed by Laurent
expansions. Finally, in Section V we summarize our results and we draw our
conclusions.

##  II Field equations

In WIS or Weyl Integrable Geometry (WIG), the modified Einstein-Hilbert Action
is expressed as

$S_{W}=\int
dx^{4}\sqrt{-g}\left(\tilde{R}+\xi\left(\tilde{\nabla}_{\nu}\left(\tilde{\nabla}_{\mu}\phi\right)\right)g^{\mu\nu}-\Lambda+\mathcal{L}_{m}\right),$
(1)

in which $g_{\mu\nu}$ is the metric tensor for the physical space,
$\tilde{\nabla}_{\mu}$ denotes covariant derivative defined by the symbols
$\tilde{\Gamma}_{\mu\nu}^{\kappa}$, where $\tilde{\Gamma}_{\mu\nu}^{\kappa}$
are the Christoffel symbols for the conformally related metric
$\tilde{g}_{\mu\nu}=\phi g_{\mu\nu}$. The scalar field $\phi$ is the coupling
function, $\Lambda$ is the cosmological constant term, the parameter $\xi$ is
an arbitrary coupling constant and $\mathcal{L}_{m}$ is the Lagrangian
function for the matter source.

When $\mathcal{L}_{m}$ describes a perfect fluid with energy density $\rho$
and pressure component $p$, the field equations in the Einstein-Weyl theory
are derived to be

$\tilde{G}_{\mu\nu}+\tilde{\nabla}_{\nu}\left(\tilde{\nabla}_{\mu}\phi\right)-\left(2\xi-1\right)\left(\tilde{\nabla}_{\mu}\phi\right)\left(\tilde{\nabla}_{\nu}\phi\right)+\xi
g_{\mu\nu}g^{\kappa\lambda}\left(\tilde{\nabla}_{\kappa}\phi\right)\left(\tilde{\nabla}_{\lambda}\phi\right)-\Lambda
g_{\mu\nu}=-\left(\tilde{\rho}+\tilde{p}\right)u_{\mu}u_{\nu}-\tilde{p}g_{\mu\nu},$
(2)

where $\tilde{G}_{\mu\nu}$ is the Einstein tensor with respect the metric
$\tilde{g}_{\mu\nu}$ and $u^{\mu}$ is the comoving observer. Parameters
$\tilde{\rho},~{}\tilde{p}\,\ $are the energy density and pressure components
for the matter source multiplied by the factor $e^{-\frac{\phi}{2}}$

The field equations (2) can be written in the equivalent form

$G_{\mu\nu}-\lambda\left(\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}g_{\mu\nu}\phi^{,\kappa}\phi_{,\kappa}\right)-\Lambda
g_{\mu\nu}=-\left(\tilde{\rho}+\tilde{p}\right)u_{\mu}u_{\nu}-\tilde{p}g_{\mu\nu},$
(3)

where $G_{\mu\nu}$ is the Einstein tensor for the background space
$g_{\mu\nu}$ and $\lambda=2\xi-\frac{3}{2}$.

For the homogeneous and isotropic spatially flat
Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime,

$ds^{2}=-N^{2}\left(t\right)dt^{2}+a^{2}\left(t\right)\left(dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right),$ (4)

the modified Friedmann’s equations are as follow

$3H^{2}-\frac{\lambda}{2N^{2}}\dot{\phi}^{2}-\Lambda-e^{-\frac{\phi}{2}}\rho=0,$
(5)
$\dot{H}+H^{2}+\frac{1}{6}e^{-\frac{\phi}{2}}\left(\rho+3p\right)+\frac{\lambda}{3N^{2}}\dot{\phi}^{2}-\frac{\Lambda}{3}=0,$
(6)
$\ddot{\phi}-\frac{\dot{N}}{N}\dot{\phi}+3H\dot{\phi}+\frac{1}{2\lambda}N^{2}e^{-\frac{\phi}{2}}\rho=0$
(7)

and

$\dot{\rho}+3NH\left(\rho+p\right)-\rho\dot{\phi}=0$ (8)

in which we have considered the comoving observer,
$u^{\mu}=\frac{1}{N}\delta_{t}^{\mu}$, and $H=\frac{1}{N}\frac{\dot{a}}{a}$ is
the Hubble function.

In this study we consider that the matter source is that of an ideal gas, that
is $p=\left(\gamma-1\right)\rho,~{}\gamma<2$. For $\gamma=2$ a two scalar
field model is recovered tw2 .From equation (8) it follows that
$\rho=\rho_{0}a^{-3\gamma}e^{\phi}$. Thus we end with the set of differential
equations (5), (6) and (7).

The cosmological history for this specific cosmological model investigated
before in wm6 . From the analysis of the dynamics it was found that the
cosmological history is consisted by two matter epochs and two acceleration
phases. The matter epochs correspond to unstable asymptotic solutions while
for the two accelerated phases the one is always unstable which can be
corresponded to the early acceleration phase of the universe, and the second
accelerated asymptotic solution describes the future de Sitter attractor.

##  III Lagrangian description

In the case of an ideal gas, the cosmological field equations (5), (6) and (7)
can be reproduced by the variation of the point-like Lagrangian

$\mathcal{L}\left(N,a,\dot{a},\phi,\dot{\phi}\right)=\frac{1}{N}\left(-3a\dot{a}^{2}+\frac{\lambda}{2}a^{3}\dot{\phi}^{2}\right)-N\left(a^{3}\Lambda+\rho_{m0}e^{\frac{\phi}{2}}a^{3-3\gamma}\right).$
(9)

The Lagrangian function (9) is a singular Lagrangian because
$\frac{\partial\mathcal{L}}{\partial\dot{N}}=0.$ The constraint equation (5)
is the Euler-Lagrange equation with respect to the lapse function
$N\left(t\right)$, i.e. $\frac{\partial\mathcal{L}}{\partial N}=0$, while the
second-order differential equations, (6) and (7), are the Euler-Lagrange
equations with respect to the variables $a\left(t\right)$ and
$\phi\left(t\right)$ respectively, i.e.
$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{a}}\right)-\frac{\partial\mathcal{L}}{\partial
a}=0$ ;
$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\right)-\frac{\partial\mathcal{L}}{\partial\phi}=0$.

The existence of the singular Lagrangian (9) and the minisuperspace
description of the gravitational model are important characteristics for the
study of the integrability properties of the field equations. Without loss of
generality the lapse function can be assumed to be
$N\left(t\right)=N\left(a\left(t\right),\phi\left(t\right)\right)$. In such a
consideration, the Lagrangian (9) is regular while equation (5) is the
conservation law of “energy”, that is, the Hamiltonian of the dynamical
system.

###  III.1 Normal coordinates

We follow wm6  and without loss of generality we select
$N\left(t\right)=a\left(t\right)^{3}$. Furthermore, we define the new field
$\Phi$,

$\phi=\frac{1}{\lambda\left(2-\gamma\right)}\left(2\lambda\ln\Phi-\ln
a\right).$ (10)

In the new variables $\left\\{a,\Phi\right\\}$ the Lagrangian of the field
equations is

$\mathcal{L}\left(a,\dot{a},\Phi,\dot{\Phi}\right)=\frac{1}{2}\left(6-\frac{1}{\left(\gamma-2\right)^{2}\lambda}\right)\left(\frac{\dot{a}}{a}\right)^{2}+\frac{2}{\left(\gamma-2\right)^{2}}\frac{\dot{a}\Phi}{a\Phi}-\frac{2\lambda}{\left(\gamma-2\right)^{2}}\left(\frac{\dot{\Phi}}{\Phi}\right)^{2}+\Lambda
a^{6}+a^{3\left(2-\gamma\right)-\frac{1}{2\lambda\left(2-\gamma\right)}}\Phi^{\frac{1}{2-\gamma}}\text{~{}}.$
(11)

The field equations are

$\frac{1}{2}\left(6-\frac{1}{\left(\gamma-2\right)^{2}\lambda}\right)\left(\frac{\dot{a}}{a}\right)^{2}+\frac{2}{\left(\gamma-2\right)^{2}}\frac{\dot{a}\Phi}{a\Phi}-\frac{2\lambda}{\left(\gamma-2\right)^{2}}\left(\frac{\dot{\Phi}}{\Phi}\right)^{2}-\Lambda
a^{6}-a^{3\left(2-\gamma\right)-\frac{1}{2\lambda\left(2-\gamma\right)}}\Phi^{\frac{1}{2-\gamma}}=0,$
(12)
$\ddot{a}+\frac{1}{2}a^{7-3\gamma-\frac{1}{2\lambda\left(2-\gamma\right)}}\Phi^{\frac{1}{2-\gamma}}-\frac{\dot{a}^{2}}{a}-\Lambda
a^{7}=0$ (13)

and

$\ddot{\Phi}-\frac{\dot{\Phi}^{2}}{\Phi}-\frac{\Lambda}{2\lambda}a^{6}\Phi=0.$
(14)

When $\Lambda=0$, equation (14) provides the conservation law
$I_{0}=\frac{d}{dt}\ln\Phi$. This conservation law was derived before in wm6 ;
wm7 . The set of variables $\left\\{a,\Phi\right\\}$ constitutes the normal
coordinates for the field equations. However, in the presence of the
cosmological constant the conservation law does not exist. The existence of
the second conservation law for the field equations is essential in order to
be able to conclude about the integrability of the field equations. Indeed, in
wm7  the Hamilton-Jacobi equation was solved and the field equations were
reduced into a system of two first-order ordinary differential equation.

In the presence of $\Lambda$, we applied the symmetry analysis for point and
contact symmetries and we find that the field equations do not admit any
Noether symmetry. Furthermore, we considered polynomial functions of
$I\left(a,\dot{a},\Phi,\dot{\Phi}\right)$ to be a conservation law. However,
we were not able to determine any function $I$ with this specific requirement.

The concept of integrability is not limited in the existence of invariant
functions and conservation laws. According to the Painlevé approach, a
dynamical system is integrable if admits a movable pole and its solution is
expressed in terms of a Laurent expansion around the movable pole. In the
following we apply the singularity analysis in order to investigate the
integrability properties for the dynamical system of our consideration.

##  IV Singularity analysis

The modern treatment of singularity analysis is described by the ARS
algorithm. The algorithm has three main steps. They are (a) derivation of the
leading-order behaviour, (b) derivation of the resonances and (c) the
consistency test . For more details and examples on the application of the ARS
algorithm we refer the reader to buntis . In the first step of the algorithm
we should show that there exists a movable singularity for the dynamical
system at which the solution is approximated by a singular expression. For
instance, for power-law expressions $\left(\tau-\tau_{0}\right)^{p}$, the
exponent $p$ should be negative number. However, it has been shown that $p$
can be also a rational number. The resonances should be the same in number as
the degrees of freedom of the problem and one of the resonances has to be
$-1$, while the resonances should be rational numbers. If the requirements of
the steps (a) and (b) are satisfied, we write the solution in a Laurent
expansion, in our case in Puiseux series, and we replace in the original
equation to check if it is an actual solution for the original dynamical
system.

We follow cots1  and we apply the singularity analysis for the equivalent
dynamical system in the dimensionless variables. Indeed, we follow the
$H-$normalization approach and we define the new variables

$x=\sqrt{\frac{1}{6}}\frac{\dot{\phi}}{H}~{},~{}\Omega_{\Lambda}=\frac{\Lambda}{3H^{2}}~{},~{}\Omega_{m}=\frac{\rho_{m}}{3H^{2}}e^{-\frac{\phi}{2}}~{},~{}d\tau=NHdt.$
(15)

where the latter parameter $\tau$ is the new independent variable.

Consequently, the field equations (5)-(8) for the ideal gas are written as the
following equivalent algebraic-differential system

$\frac{d\Omega_{\Lambda}}{d\tau}=-\Omega_{\Lambda}\left(\left(\gamma-2\right)\lambda
x^{2}+\gamma\left(\Omega_{\Lambda}-1\right)\right)~{}~{}$ (16)

and

$\frac{dx}{d\tau}=\frac{1}{12\lambda}\left(\left(\lambda
x^{2}-1\right)\left(\sqrt{6}-6\left(\gamma-2\right)\lambda
x\right)+\left(\sqrt{6}-6\gamma\lambda x\right)\Omega_{\Lambda}\right)~{}$
(17)

with constraint equation

$\Omega_{m}=1-\lambda x^{2}-\Omega_{\Lambda}.$ (18)

Therefore we continue with the investigation of the singularity analysis for
the system of first-order ordinary differential equations (16) and (17).

###  IV.1 Case $\Lambda=0$

We focus now on the case $\Lambda=0$. Equation (17) reduces to the simple form

$\frac{dx}{d\tau}=\frac{1}{12\lambda}\left(\left(\lambda
x^{2}-1\right)\left(\sqrt{6}-6\left(\gamma-2\right)\lambda
x\right)\right)~{}.$ (19)

The closed-form solution of the this equation is

$\tau-\tau_{0}=\frac{6\lambda\left(\gamma-2\right)}{1-6\lambda\left(\gamma-2\right)^{2}}\ln\frac{\left(\lambda
x^{2}-1\right)}{\left(\sqrt{6}-6\lambda\left(\gamma-2\right)x\right)^{2}}-\frac{2\sqrt{\lambda}}{1-6\lambda\left(\gamma-2\right)^{2}}\arctan
h\left(\lambda x\right).$ (20)

However, we are interested to investigate if equation (19) possesses the
Painlevé property and if the analytic solution can be expressed in a Laurent
expansion around a movable pole.

According to the ARS algorithm, we replace
$x\left(t\right)=x_{0}\left(\tau-\tau_{0}\right)^{p}$ in (19), that is,

$px_{0}\left(\tau-\tau_{0}\right)^{-1+p}-\frac{1}{12\lambda}\left(\left(\lambda
x_{0}^{2}\left(\tau-\tau_{0}\right)^{2p}-1\right)\left(\sqrt{6}-6\left(\gamma-2\right)\lambda
x_{0}\left(\tau-\tau_{0}\right)^{p}\right)\right).$ (21)

Hence, the dominant terms provides the algebraic equation $-1+p=3p$, that is,
$p=-\frac{1}{2}$, and $\lambda\left(\gamma-2\right)x_{0}^{2}-1=0$.

For the determination of the resonances we substitute

$x\left(t\right)=x_{0}\left(\tau-\tau_{0}\right)^{-\frac{1}{2}}+m\left(\tau-\tau_{0}\right)^{-\frac{1}{2}+S}~{},~{}\lambda
x_{0}^{2}+1=0,$ (22)

into (19) and we expand around $m^{2}\rightarrow 0$. Hence, we end with the
algebraic equation $S+1=0$, which means that $S=-1$. That is in agreement with
the singularity analysis, because one of the resonances should be $-1$. This
resonance indicates that the leading-order behaviour describes a movable
singularity and $t_{0}$ is an integration constant.

For the third step of the ARS algorithm, known as the consistency test, we
substitute

$x\left(\tau\right)=x_{0}\left(\tau-\tau_{0}\right)^{-\frac{1}{2}}+x_{1}\left(\tau-\tau_{0}\right)^{0}+x_{2}\left(\tau-\tau_{0}\right)^{\frac{1}{2}}+x_{3}\left(\tau-\tau_{0}\right)+...~{},$
(23)

into (19) where we find

$x_{1}=\frac{x_{0}^{2}}{3\sqrt{6}}~{},~{}x_{2}=\frac{x_{0}\left(x_{0}^{2}+18\left(\gamma-2\right)\right)}{72}~{},~{}...~{}.$

We conclude that equation (19) possesses the Painlevé property and the
analytic solution is described by the Puiseux expansion.

###  IV.2 Case $\Lambda\neq 0$

We focus now on the case for which there exists a nonzero cosmological
constant term. In order to determine the leading-order term for the system
(16), (17) we substitute

$\displaystyle\Omega_{\Lambda}\left(\tau\right)$
$\displaystyle=\Omega_{\Lambda 0}\left(\tau-\tau_{0}\right)^{q}~{},$ (24)
$\displaystyle x\left(\tau\right)$
$\displaystyle=x_{0}\left(\tau-\tau_{0}\right)^{p}\text{ }$ (25)

and as above we determine $p=-\frac{1}{2}$ and $p=-1$ with constraint equation
$\lambda\left(\gamma-2\right)x_{0}^{2}-1=\gamma\Omega_{\Lambda 0}$. Easily we
see that for $\Omega_{\Lambda}=0$, the previous leading-order term is
recovered.

For the resonances we replace

$\displaystyle\Omega_{\Lambda}\left(\tau\right)$
$\displaystyle=\Omega_{\Lambda
0}\left(\tau-\tau_{0}\right)^{-1}+n\left(\tau-\tau_{0}\right)^{-1+S},$ (26)
$\displaystyle x\left(\tau\right)$
$\displaystyle=x_{0}\left(\tau-\tau_{0}\right)^{-\frac{1}{2}}+m\left(\tau-
t_{0}\right)^{-\frac{1}{2}+S}\text{ }$ (27)

in (16), (17).

We expand around $m^{2}\rightarrow 0$, $n^{2}\rightarrow 0$ and $mn\rightarrow
0$. The first-order perturbed terms provide the matrix

$A=\begin{pmatrix}-\frac{2}{x_{0}}\left(\Omega_{\Lambda
0}\left(\gamma\Omega_{\Lambda
0}-1\right)\right)&-\left(S+\gamma\Omega_{\Lambda 0}\right)\\\
\left(\left(1+S\right)-\gamma\Omega_{\Lambda
0}\right)&\frac{1}{2}x_{0}\gamma\end{pmatrix},$ (28)

where the zeros of the determinant $\det\left(A\right)=0,~{}$give the
resonances; they are

$S_{1}=-1,~{}S_{2}=0.$ (29)

The resonances indicate that the there exists a movable singularity and that
one of the coefficients, $x_{0}$ or $\Omega_{\Lambda 0}$, is arbitrary and is
the second integration constant of the dynamical system.

We write the Puiseux series

$\Omega_{\Lambda}\left(\tau\right)=\Omega_{\Lambda
0}\left(\tau-\tau_{0}\right)^{-1}+\Omega_{\Lambda
1}\left(\tau-\tau_{0}\right)^{-\frac{1}{2}}+\Omega_{\Lambda
2}\left(\tau-\tau_{0}\right)^{0}+\Omega_{\Lambda
3}\left(\tau-\tau_{0}\right)^{\frac{1}{2}}+...~{},$ (30)
$x\left(\tau\right)=x_{0}\left(\tau-\tau_{0}\right)^{-\frac{1}{2}}+x_{1}\left(\tau-\tau_{0}\right)^{0}+x_{2}\left(\tau-\tau_{0}\right)^{\frac{1}{2}}+x_{3}\left(\tau-\tau_{0}\right)+...~{}$
(31)

which we replace in (16), (17). Hence, the latter Puiseux series solves the
dynamical system when

$\Omega_{\Lambda 1}=\frac{4x_{1}\Omega_{\Lambda 0}\left(\gamma\Omega_{\Lambda
0}-1\right)}{x_{0}\left(1+2\gamma\Omega_{\Lambda
0}\right)},~{}x_{1}=\frac{x_{0}\left(\sqrt{6}x_{0}\left(1-2\Omega_{\Lambda
0}\right)+6\gamma\Omega_{\Lambda 1}\left(\gamma\Omega_{\Lambda
0}-1\right)\right)}{6\left(\gamma\Omega_{\Lambda
0}-1\right)\left(2\gamma\Omega_{\Lambda 0}-3\right)}~{},~{}...~{}.~{}$ (32)

We summarize our results in the following proposition.

Proposition 1: The cosmological field equations in Weyl Integrable theory for
a spatially flat FLRW background geometry induced with a cosmological constant
term and an ideal gas form a dynamical system which possesses the Painlevé
property that is, the field equations are integrable and the analytic solution
is expressed by the Puiseux expansions (30) and (31).

##  V Conclusions

In this piece of study we investigated the integrability properties of the
field equations in Weyl Integrable theory. In particular we studied the
existence of analytic solutions of the cosmological scenario for a universe
with matter source that of an ideal gas in a spatially flat FLRW geometry with
a nonzero cosmological constant term. For our analysis we applied the
singularity analysis which has been widely applied in gravitational physics
with many interesting results.

The field equations for the model of our consideration have the property to
admit a minisuperspace. That means that the field equations follow from the
variation for the dynamical variables of a point-like Lagrangian. For the case
in which there is no cosmological constant term, from previous results we know
that there exists a second conservation law which indicates the integrability
properties for the dynamical system. However, in the presence of the
cosmological constant, this conservation law does not exist. Thus we were not
able to infer about the integrability of the field equations.

For simplicity of our analysis we wrote the field equations into the
equivalent system with the use of dimensionless variables. For the equivalent
system we applied the singularity analysis and we found that the field
equations possess the Painlevé property, that is, the cosmological model of
our analysis is integrable with or without the cosmological constant term.
This is a very interesting result because after the loss of the additional
integration constant provided by Noether’s analysis we were not able to make
an inference about the integrability and someone could infer that the
introduction of the cosmological constant into the field equations may violate
the integrability property.

As we have discussed before, this cosmological model describes important eras
of the cosmological history. The existence of the analytic solution indicates
that there exist an actual real solutions which correspond to the numerical
simulations of the field equations. This is an important feature because we
can understand the evolution of the dynamical system according to the free
parameters. In addition the above analysis can be used for the analytic
reconstruction of the cosmographic parameters com1 ; com2  and the relation of
these parameters with the initial value problem.

This work contributes to the subject of the derivation of exact and analytic
solutions in gravitational physics, while it shows that Noether symmetry
analysis and the singularity analysis are two complementary methods which can
provide interesting results.

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# A unified model of inflation and dark energy based on the holographic
spacetime foam

Daniel Jiménez-Aguilar<EMAIL_ADDRESS>Department of Physics, UPV/EHU,
48080, Bilbao, Spain

###### Abstract

I present a model of inflation and dark energy in which the inflaton potential
is constructed by imposing that a scalar field representing the classical
energy of the spacetime foam inside the Hubble horizon is an exact solution to
the cosmological equations. The resulting potential has the right properties
to describe both the early and late expansion epochs of the universe in a
unified picture.

##  I 1\. Introduction

Recent cosmological observations indicate that the universe is currently
experiencing an accelerated expansion Riess et al. (1998); Perlmutter et al.
(1999); Spergel et al. (2003); Tegmark et al. (2004). This fact can be
attributed to the existence of some form of energy (dubbed dark energy) with
negative pressure. Our unawareness of the nature of dark energy is certainly
at the root of one of the greatest unknowns in theoretical physics: the
cosmological constant problem Weinberg (1989). On the other hand, it is widely
accepted today that the very early universe underwent a period of quasi-
exponential expansion called inflation, which can address many of the problems
of the standard Big Bang cosmology Guth (1981); Starobinsky (1979); Kazanas
(1980); Sato (1981); Linde (1982); Albrecht and Steinhardt (1982).

Both expansion epochs may be traced to a common cause by interpreting dark
energy as a dynamical scalar field (the inflaton) that slowly rolls down its
potential to drive inflation in the early universe and finally resembles a
cosmological constant at the present time, at a much lower energy scale.
Depending on the shape of the potential, the field will end up oscillating
about a minimum (as in the model presented in this paper) or rolling down an
infinite tail as quintessence. The latter scenario is called quintessential
inflation Peebles and Vilenkin (1999).

Another chance of uncovering the nature of dark energy may be found in the
structure of spacetime itself. Due to its quantum nature, spacetime is foamy
on scales of the order of the Planck length Misner  _et al._ (1973). The
quantum fluctuations of the metric are responsible for a perpetual change in
the geometry of spacetime, and consequently, any measurement of space and time
intervals acquires some uncertainty. In a region of spatial extent $L$, this
uncertainty is given by the Károlyházy relation Karolyhazy (1966), which was
derived independently by other authors Sasakura (1999); Ng and Van Dam (1994);
Christiansen  _et al._ (2006); Arzano  _et al._ (2007):

$\delta L\sim L_{p}^{2/3}L^{1/3},$ (1)

where $L_{p}$ is the Planck length. Note that $\delta L\sim L$ precisely at
the Planck scale. This picture of the small scale structure of spacetime is
called spacetime foam. An appealing idea that has been proposed is that the
energy density associated to the spacetime foam is the one that drives both
the early and late expansions of the universe Ng (2021).

The Károlyházy uncertainty relation (1) is closely related to the holographic
principle ’t Hooft (1993); Susskind (1995), as it establishes a connection
between the ultraviolet ($\delta L$) and infrarred ($L$) cut-off scales of the
system. Indeed, expression (1) is also known as the holographic spacetime foam
model, as it suggests that the number of degrees of freedom or bits of
information in that region is proportional to its surface area:
$\left(L/\delta L\right)^{3}\propto L^{2}$. The ultraviolet cut-off scale is
related to the energy density of the vacuum, and one is led to a holographic
dark energy of the form

$\rho\sim\frac{1}{L_{p}^{2}L^{2}}\,.$ (2)

The usual approach in holographic dark energy models is to propose an ansatz
for the infrarred cut-off $L$ (some natural choices are the Hubble radius Hsu
(2004); Zimdahl and Pavon (2007), the event horizon Li (2004), the age of the
universe Cai (2007) and the Ricci length Duran and Pavon (2011)) and combine
equation (2) with the Friedmann equation in order to extract the Hubble rate
as a function of time. Then, one can see whether this particular choice leads
to an accelerated expansion, and also constrain the model with the
observational data. While these models have been intensively applied to the
late universe (see del Campo  _et al._ (2011) for a review), only a few
authors have extended these ideas to the early universe (for instance, see
Nojiri  _et al._ (2019)).

Finally, and more on the line of thought of this work, there have been several
studies aimed at establishing a connection between holographic dark energy and
scalar field models Zhang (2010); Karami and Fehri (2010); Cui  _et al._
(2010); Sheykhi  _et al._ (2010), although the explicit reconstruction of the
scalar field potential has proved to be challenging (however, see Sheykhi
(2011); Granda and Oliveros (2009)).

In this paper, I present a unified model of inflation and dark energy in which
the scalar potential is reconstructed by making a specific ansatz for the
field in the cosmological equations. This ansatz will be rooted at the
holographic model of spacetime foam.

This letter is structured as follows: in section 2, I expose the main idea
behind the construction of the inflaton potential. This idea is applied in
section 3 under the assumption that the scalar field is real. In this case,
the potential obtained is not satisfactory. The calculation is redone in
section 4 for a complex scalar field, obtaining a family of acceptable
potentials. In section 5, I pick up the potential that has direct connection
with the spacetime foam and I discuss some of its properties. Finally, some
concluding remarks are made in section 6.

Natural units ($c=\hbar\equiv 1$) are used throughout the letter. In
particular, this implies that Newton’s gravitational constant is given by
$G=L_{p}^{2}=M_{p}^{-2}$, where $M_{p}$ is the Planck mass.

##  II 2\. The main idea

Regardless of the initial field configuration in the patch of the universe
that is going to inflate, the field evolves under the influence of a potential
$V\left(\phi\right)$. The Friedmann and Klein-Gordon equations in a spatially
flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe establish a
correspondence between the potential and a homogeneous function of time
$\phi\left(t\right)$. Therefore, one can obtain the scalar potential by
imposing that a particular $\phi_{s}\left(t\right)$ is an exact solution to
these equations (the subscript $s$ stands for _solution_). For instance,
$V\left(\phi\right)$ could be determined by imposing that
$\phi_{s}\left(t\right)\propto H\left(t\right)$ is a solution to the equations
(here and henceforth, $H$ denotes the Hubble rate). Whatever our ansatz is,
$\phi_{s}\left(t\right)$ is not the actual inflaton field
$\phi\left(t,\vec{x}\right)$ in the universe, but just a tool to determine the
potential. Indeed, a natural initial state for the field is arbitrary and
inhomogeneous. Since
$\phi\left(t=0,\vec{x}\right)\neq\phi_{s}\left(t=0\right)$ and
$\dot{\phi}\left(t,\vec{x}\right)\neq\dot{\phi}_{s}\left(t=0\right)$, the
inflaton field would simply end up settling in the vacuum of the potential,
possibly resembling the cosmological constant.

Here we will obtain a family of scalar field potentials by considering the
simplest ansatz with dimensions of energy that one can make out of $M_{p}$ and
$H$:

$\phi_{s}\left(H\right)=AM_{p}^{\gamma}H^{1-\gamma},$ (3)

where $A$ is a dimensionless constant and $\gamma$ is a generic exponent that
will label the different elements of the family of solutions. Some of them
will not only have the right shape to allow for inflation in the early
universe and cosmological constant at the present time, but also a connection
with the spacetime foam, as we point out now.

Note that ansatz (3) for $\gamma=2/3$ is essentially the inverse of the
Károlyházy uncertainty relation (1) evaluated at the Hubble scale. One can
show that $1/\delta L$ is proportional to the energy of the spacetime foam
inside the Hubble horizon. The classical energy density associated to the
metric fluctuations is given by Maziashvili (2006)

$\rho_{foam}\sim\frac{1}{L_{p}^{2/3}L^{10/3}},$ (4)

and although this expression was derived in Minkowski spacetime, we will
assume that the powers of $L_{p}$ and $L$ remain unchanged in a spatially flat
FLRW universe. If the scale $L$ in equation (4) is chosen to be the Hubble
radius, the classical energy of the spacetime foam in that volume is

$E_{foam}\sim M_{p}^{2/3}H^{1/3}.$ (5)

This expression for the energy can also be derived from the Margolus-Levitin
theorem in quantum computation Margolus and Levitin (1998). The dynamical
evolution of any physical system can be thought of as a succession of
orthogonal quantum states, and each step between consecutive states can be
understood as an elementary operation. It can be easily shown that each of
these steps takes at least time $\delta t\sim E^{-1}$, where $E$ is the
average energy of the system. Therefore, $E\sim\delta t^{-1}$, which yields
expression (5) if we impose that $\delta t$ corresponds to the uncertainty
given by the Károlyházy relation (1) evaluated at the Hubble scale.

##  III 3\. Real scalar field

The simplest possibility is to consider a real scalar field. The first
Friedmann equation and the Klein-Gordon equation read

$H^{2}=\frac{8\pi}{3M_{p}^{2}}\left[\frac{\dot{\phi}^{2}}{2}+V\left(\phi\right)\right],$
(6) $\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0.$ (7)

Taking the time derivative of (6) and combining it with (7) yields

$\dot{H}=-\frac{4\pi}{M_{p}^{2}}\dot{\phi}^{2}.$ (8)

Since $\phi=\phi\left(t\right)$ and $H=H\left(t\right)$, one can take
$H=H\left(\phi\right)$. Taking this into account, and using equations (6) and
(8), one can express $V\left(\phi\right)$ as

$V\left(\phi\right)=\frac{M_{p}^{2}}{8\pi}\left[3H^{2}\left(\phi\right)-\frac{M_{p}^{2}}{4\pi}\left(\frac{dH}{d\phi}\right)^{2}\right]\,.$
(9)

Now we determine $V\left(\phi\right)$ by imposing that
$\phi=\phi_{s}\left(H\right)$, given by equation (3). Inverting to get
$H\left(\phi\right)$ and substituting in (9) yields

$V\left(\phi\right)=\frac{M_{p}^{\frac{2\left(1-2\gamma\right)}{1-\gamma}}}{8\pi
A^{\frac{2}{1-\gamma}}}\left[3\phi^{\frac{2}{1-\gamma}}-\frac{M_{p}^{2}}{4\pi\left(1-\gamma\right)^{2}}\phi^{\frac{2\gamma}{1-\gamma}}\right]\,.$
(10)

We further assume that $V\left(\phi\right)$ takes finite values for finite
$\phi$. Together with the fact that the potential is real, this implies that
the exponent $\frac{2\gamma}{1-\gamma}\equiv n$ is a natural number (if this
is the case, the first exponent is also a natural number:
$\frac{2}{1-\gamma}=n+2$). This means that $\gamma$ is quantized:
$\gamma=\frac{n}{2+n}$. As shown in figures 1 and 2, the potential is either
symmetric (if $n$ is even) or antisymmetric (if $n$ is odd). For $\gamma=0$,
the potential is a parabola with its minimum at $\phi=0$, at a negative energy
density.

Figure 1: Potential (10) for $\gamma=1/2$ (blue), $\gamma=2/3$ (orange) and
$\gamma=3/4$ (green). In this plot, we have set $A=1$. Figure 2: Potential
(10) for $\gamma=1/3$ (blue), $\gamma=3/5$ (orange) and $\gamma=5/7$ (green).
In this plot, we have set $A=1$.

In the case of the symmetric double-well potentials, note that the flatness
around $\phi=0$ is controlled by $\gamma$. Note also that the minima are
placed at negative values of the potential energy density. Therefore, none of
them can be the actual potential for the inflaton. However, if one considers a
complex scalar field, the potential contains an extra term related to the
phase. This extra term can act as an uplifting term, so one can obtain vacua
with positive cosmological constant in that case.

##  IV 4\. Complex scalar field

If we consider a complex field
$\phi=\frac{\varphi}{\sqrt{2}}e^{i\frac{\beta}{\varphi_{v}}}$, where $\beta$
is the angular part of the field and $\varphi_{v}$ is the vacuum expectation
value of the radial part, equations (6) and (8) read

$H^{2}=\frac{8\pi}{3M_{p}^{2}}\left[\frac{\dot{\varphi}^{2}}{2}+\frac{\varphi^{2}\dot{\beta}^{2}}{2\varphi_{v}^{2}}+V\left(\varphi\right)\right],$
(11)
$\dot{H}=-\frac{4\pi}{M_{p}^{2}}\left(\dot{\varphi}^{2}+\frac{\varphi^{2}\dot{\beta}^{2}}{\varphi_{v}^{2}}\right).$
(12)

Again, we assume that $\varphi$ is a function of $H$ and hence
$H=H\left(\varphi\right)$. Combining equations (11) and (12), one gets

$V\left(\varphi\right)=\frac{M_{p}^{2}}{8\pi}\left[3H^{2}\left(\varphi\right)-\frac{M_{p}^{2}}{4\pi}\left(\frac{dH}{d\varphi}\right)^{2}-\frac{\varphi^{2}\dot{\beta^{2}}}{\varphi_{v}^{2}\dot{\varphi}}\frac{dH}{d\varphi}\right]\,.$
(13)

This potential has the same form as (9), except for the presence of a new term
that contains the time derivative of the angular part of the field. Note that
this extra term can be written as
$-\frac{\varphi^{2}\dot{\beta^{2}}}{\varphi_{v}^{2}\dot{\varphi}^{2}}\dot{H}$,
and this is always positive as $\dot{H}<0$ (see equation (12)). If the phase
$\beta$ is chosen in such a way that

$\dot{\beta}^{2}\propto\frac{dH^{-1}}{dt}\,,$ (14)

that is, proportional to the slow-roll parameter $\epsilon$, the extra term is
a constant that uplifts the potential found in the previous section. Note that
this choice of $\beta$ is made at the same level as ansatz (3) for $\varphi$.
The phase determined by equation (14), as well as any other $\beta$ that one
could have come up with, is an exact solution of the equations of motion
provided that the potential energy density is given by (13). Other choices of
$\beta$ could also do the job of uplifting the potential by locally deforming
it, but in these cases the deformation would not be uniform in $\varphi$. This
uniformity condition is obviously not necessary to obtain a potential with the
right properties to account for the expansion history of the universe, but let
us focus on this possibility for simplicity. Making use of ansatz (14), the
extra term in the potential is a positive constant with dimensions of energy
squared:
$-\frac{\varphi^{2}\dot{\beta^{2}}}{\varphi_{v}^{2}\dot{\varphi}}\frac{dH}{d\varphi}\equiv
BM_{p}^{2}$, where $B>0$ is a dimensionless constant.

Now we impose that
$\varphi=\varphi_{s}\left(H\right)=AM_{p}^{\gamma}H^{1-\gamma}$. Finding
$H\left(\varphi\right)$ and substituting in (13), one gets the following
potential:

$V\left(\varphi\right)=\frac{M_{p}^{\frac{2\left(1-2\gamma\right)}{1-\gamma}}}{8\pi
A^{\frac{2}{1-\gamma}}}\left[3\varphi^{\frac{2}{1-\gamma}}-\frac{M_{p}^{2}}{4\pi\left(1-\gamma\right)^{2}}\varphi^{\frac{2\gamma}{1-\gamma}}\right]+\frac{BM_{p}^{4}}{8\pi}\,.$
(15)

For the values of $\gamma$ corresponding to even $n=\frac{2\gamma}{1-\gamma}$,
this is a Mexican-hat potential with the circle of minima at
$\varphi=\varphi_{v}=M_{p}\sqrt{\frac{\gamma}{12\pi\left(1-\gamma\right)^{2}}}$.
The vacuum energy density can be shown to be

$V\left(\varphi_{v}\right)=-\frac{M_{p}^{2}m^{2}}{48\pi}+\frac{BM_{p}^{4}}{8\pi}\,,$
(16)

where $m=f\left(\gamma\right)M_{p}$ is the mass of small radial fluctuations
about the vacuum, i.e., the mass of the inflaton field. The function
$f\left(\gamma\right)$ is given by

$f\left(\gamma\right)=\left[\frac{\gamma^{\gamma}}{8^{1-\gamma}\times
12^{2\gamma-1}\pi\left(1-\gamma\right)^{1+\gamma}A^{2}}\right]^{\frac{1}{2\left(1-\gamma\right)}}.$
(17)

We want the vacuum energy density (16) to be equal to the dark energy density
today, namely,
$\rho_{\Lambda}^{0}=\Omega_{\Lambda}^{0}\rho_{c}^{0}=\frac{3\Omega_{\Lambda}^{0}}{8\pi}H_{0}^{2}M_{p}^{2}$.
This implies that

$B=3\Omega_{\Lambda}^{0}\left(\frac{H_{0}}{M_{p}}\right)^{2}+\frac{1}{6}\left(\frac{m}{M_{p}}\right)^{2}\approx\frac{1}{6}\left(\frac{m}{M_{p}}\right)^{2}\,.$
(18)

The first term is negligible, but its presence makes this choice of $B$
extremely fine-tuned. Evidently, since the model does not provide a
fundamental explanation for it, the cosmological constant problem is not
resolved. However, once this term is neglected, the approximate value of $B$
is arguably natural: condition (18) implies that the Hubble rate during
inflation ($H_{I}$) is given by

$H_{I}\approx\sqrt{\frac{8\pi}{3M_{p}^{2}}V\left(0\right)}=M_{p}\sqrt{\frac{B}{3}}\approx\frac{m}{3\sqrt{2}}\,.$
(19)

This means that the Hubble radius during inflation is of the same order of
magnitude as the Compton wavelength of the inflaton, which is the only
fundamental length scale in the problem apart from the Planck length.
Therefore, although the model requires fine-tuning to explain the exact value
of the cosmological constant, it may not need it to account merely for its
smallness.

##  V 5\. The case $\gamma=2/3$

As previously stated, the ansatz for the radial part of the scalar field can
be interpreted as the energy of the spacetime foam inside the Hubble horizon
if $\gamma=2/3$. Defining $\tilde{\varphi}\equiv\varphi/M_{p}$ and
$\tilde{V}\equiv V/M_{p}^{4}$, and making use of (17) and (18), the potential
(15) for $\gamma=2/3$ can be written as

$\tilde{V}\left(\tilde{\varphi}\right)\approx\left(\frac{m}{M_{p}}\right)^{2}\left[\frac{\pi^{2}}{3}\left(\tilde{\varphi}^{6}-\frac{3}{4\pi}\tilde{\varphi}^{4}\right)+\frac{1}{48\pi}\right]\,.$
(20)

With this potential, plotted in figure 3, ansatz (3) with $\gamma=2/3$ for
$\varphi$ and ansatz (14) for $\beta$ are exact solutions of the equations of
motion. This means that, until radiation and matter become important and
influence the dynamics of the scalar field, $\varphi$ and $\beta$ would evolve
as dictated by their corresponding ansätze as long as the initial conditions
for both of them are exactly given by these expressions. However, we find it
more natural to consider the possibility that the initial state is arbitrary
and inhomogeneous, and that the scalar field is in different regions of the
potential in different Hubble patches. In this case, $\varphi$ and $\beta$
will definitely not follow (3) and (14), as the evolution of the scalar field
will proceed essentially as in a cosmological phase transition. As we will
shortly see, a patch of the universe where the radial part of the scalar field
starts out sufficiently close to $\varphi\left(t=0,\vec{x}\right)=0$ in
potential (20) will undergo inflation. After that, the field will reach the
circle of minima at $\varphi=\varphi_{v}$ and will start oscillating as
dictated by the Friedmann and Klein-Gordon equations. Production of the
standard model particles will then take place and the universe will be
subsequently dominated by radiation and matter. In the long term, the residual
vacuum energy (16) of the field will start dominating as a cosmological
constant.

Figure 3: Potential (20) for $m=10^{-4}M_{p}$. $\tilde{\phi}_{1}$ and
$\tilde{\phi}_{2}$ correspond to the real and imaginary parts of
$\tilde{\phi}=\phi/M_{p}$, respectively.

Let us note that, even if the initial conditions were those determined by the
ansätze (3) and (14), quantum fluctuations on top of that state may spoil the
subsequent expected evolution and yield a picture which would be similar to
the one presented above.

The properties of the $\gamma=2/3$ potential (20) are studied quantitatively
in the following subsections.

###  V.1 5.1. Mass of the inflaton

As it is explicit in equation (17), the mass of the inflaton depends on
$\gamma$. Since $\gamma$ is quantized, the ratio $m/M_{p}$ can only take
discrete values. Interestingly, out of all the possibilities, the minimum
inflaton mass is obtained precisely for $\gamma=2/3$ (see figure 4).

Figure 4: Inflaton mass (rescaled by the Planck mass) as a function of
$\gamma$ (equation (17)). In this plot, we have set $A=1$.

Our model gives a prediction for the inflaton mass (equation (17)), but the
parameter $A$ is uncertain as we do not know the exact proportionality factor
for the energy of the spacetime foam (equation (5)). Considering a
conservative choice like $A<20$, we get $m>10^{-5}M_{p}$ as an order of
magnitude estimate for the minimum inflaton mass. On the other hand, according
to Planck 2018 results Akrami et al. (2020), the Hubble rate during inflation
has to verify $H_{I}<2.5\times 10^{-5}M_{p}$. Combining this observational
constraint with equation (19), we obtain $m<1.1\times 10^{-4}M_{p}$.

###  V.2 5.2. Number of e-folds

Given the inflaton potential (equation (20)), and assuming slow-roll
conditions, one can calculate the number of e-folds of inflation as

$N\left(\varphi\right)=\frac{8\pi}{M_{p}^{2}}\int_{\varphi}^{\varphi_{i}}\frac{V\left(\hat{\varphi}\right)}{V^{\prime}\left(\hat{\varphi}\right)}\,d\hat{\varphi},$
(21)

where $\varphi_{i}\sim H_{I}<2.5\times 10^{-5}M_{p}$ is the initial value of
the inflaton field. Its value at the end of inflation, $\varphi_{end}\approx
0.15M_{p}$, is chosen in such a way that the greatest of the two slow-roll
parameters $\epsilon$ and $|\eta|$ is approximately $1$:

$\epsilon\left(\varphi\right)=\frac{M_{p}^{2}}{16\pi}\left[\frac{V^{\prime}\left(\varphi\right)}{V\left(\varphi\right)}\right]^{2}\,,$
(22)
$|\eta\left(\varphi\right)|=\left|\frac{M_{p}^{2}}{8\pi}\frac{V^{\prime\prime}\left(\varphi\right)}{V\left(\varphi\right)}\right|\,.$
(23)

The result for $\gamma=2/3$ is $N\left(\varphi_{end}\right)\sim 10^{8}$.
Therefore, the $\gamma=2/3$ potential allows for more than enough expansion.

The number of e-folds has been also computed for the cases $\gamma=1/2$ and
$\gamma=3/4$. The results are $N\left(\varphi_{end}\right)\sim 1$ and
$N\left(\varphi_{end}\right)\sim 10^{16}$, respectively.

Note that the previous results are independent of the choice of parameters in
the model: after adjusting the potential uplift with (18), they only enter via
$m$ in (20), which cancels out exactly in the ratios appearing in (21), (22)
and (23).

###  V.3 5.3. Tensor-to-scalar ratio and scalar tilt

Let $N_{*}$ be the number of e-folds between horizon crossing for observable
wavelengths and the end of inflation:
$N_{*}=N\left(\varphi_{end}\right)-N\left(\varphi_{hc}\right)$. Here, the
subscript $hc$ stands for _horizon crossing_. The tensor-to-scalar ratio ($r$)
and scalar tilt ($n_{s}$) can be computed as

$r=16\epsilon\left(\varphi_{hc}\right)\,,$ (24)
$n_{s}=1-6\epsilon\left(\varphi_{hc}\right)+2\eta\left(\varphi_{hc}\right)\,.$
(25)

Assuming $50<N_{*}<60$, we get $5.96\times 10^{-6}<r<1.02\times 10^{-5}$ and
$0.941<n_{s}<0.951$ for the $\gamma=2/3$ case. According to Planck 2018
results, $r<0.064$ and $0.957<n_{s}<0.968$.

##  VI 6\. Concluding remarks

I have presented a novel approach to describe early inflation and the late
time acceleration of the universe in a unified picture, in the sense that a
single scalar potential can account for both cosmic eras. In this proposal,
the potential is determined by imposing that a scalar field with a particular
meaning is an exact solution to the Friedmann and Klein-Gordon equations in a
spatially flat FLRW universe.

The ansatz for the scalar field has been associated with the energy of the
spacetime foam inside the Hubble horizon. This correspondence leads to a
potential that allows for more than enough inflation in the early universe,
and it yields acceptable values for the inflationary observables. Demanding
consistency with the observational data and imposing natural bounds on the
parameters of the model, one gets the following estimate for the mass of the
inflaton: $m\sim 10^{-5}M_{p}$. Moreover, the potential can also describe the
current expansion of the universe as the field ends up at a circle of vacua at
positive energy density given by (16), thus acting effectively as a
cosmological constant. Although the model cannot justify the exact value of
the vacuum energy, it predicts that it has to be much smaller than $M_{p}^{4}$
if one assumes that the Hubble radius during inflation is of the order of the
Compton wavelength of the inflaton.

Remarkably, the requirement that the potential does not blow up for finite
values of the field yields the quantization of $\gamma$. The particular
solution $\gamma=2/3$ turns out to be special, because it is the one that
minimizes the mass of the inflaton. Since this is precisely the solution with
direct connection with the holographic model of spacetime foam, this might be
regarded as an argument to strengthen our faith on the Károlyházy uncertainty
relation.

##  VII Acknowledgements

I would like to express my sincere gratitude to Y. Jack Ng and Diego Pavón for
their support and insightful words. I am also grateful to L.N. Granda, Thomas
W. Kephart and Ahmad Sheykhi for useful comments.

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