# Kinetic dominance and the wavefunction of the universe

Gabriel Álvarez  Luis Martínez Alonso  Departamento de Física Teórica,
Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain
Elena Medina  Departamento de Matemáticas, Facultad de Ciencias, Universidad
de Cádiz, 11510 Puerto Real, Cádiz, Spain

###### Abstract

We analyze the emergence of classical inflationary universes in a kinetic-
dominated stage using a suitable class of solutions of the Wheeler-De Witt
equation with a constant potential. These solutions are eigenfunctions of the
inflaton momentum operator that are strongly peaked on classical solutions
exhibiting either or both a kinetic dominated period and an inflation period.
Our analysis is based on semiclassical WKB solutions of the Wheeler-De Witt
equation interpreted in the sense of Borel (to perform a correct connection
between classically allowed regions) and on the relationship of these
solutions to the solutions of the classical model. For large values of the
scale factor the WKB Vilenkin tunneling wavefunction and the Hartle-Hawking
no-boundary wavefunctions are recovered as particular instances of our class
of wavefunctions.

##  I INTRODUCTION

Inflationary cosmology is a successful theoretical framework to understand the
evolution and structure of our Universe [1, 2, 3, 4, 5, 6, 7]. In particular,
the study of the appropriate initial conditions for the emergence of an
inflationary universe requires the use of quantum cosmology. Thus, in the
standard descriptions of the early Universe (like those based on the Hartle-
Hawking no-boundary wavefunction [8, 9] or on the Vilenkin tunneling
wavefunction [10, 11, 12, 13, 14]), the interest is focused on the Universe
nucleation at the appropriate initial condition for inflation.

Nevertheless, a series of recent results [15, 16, 17, 18] (see also [19]) show
that for broad classes of solutions of classical inflationary (INF) models the
resulting early Universe is in a state of _kinetic dominance_ (KD). The KD
period of inflationary models is a pre-inflationary stage [5, 20, 21, 22, 23,
24, 14, 25, 26, 27, 28, 15, 16] which occurs when the kinetic energy of the
inflaton field dominates over its potential energy, a stage that quickly
evolves to the INF stage. Moreover, it has been proved [16] that KD initial
conditions are a consistent alternative to the standard INF, or slow-roll,
initial conditions.

Moreover, due to the fact that during the KD stage the comoving Hubble horizon
grows, it follows that initial KD conditions give rise to oscillations and to
a cutof at large scales in the cosmic microwave angular power spectra.

Periods of KD behavior occur near a singularity $a=0$, and since a
neighborhood of this singularity is outside the range of validity of the
classical treatment, it is natural to expect that the transition to the
quantum regime of INF models might be of a KD character and that it might be
studied with semiclassical approximations. In the present work we analyze the
emergence of KD classical universes using semiclassical solutions of the
Wheeler-De Witt (WDW) equation [29].

More concretely, we consider single inflaton models in a closed Friedmann
universe with unit curvature [8, 9, 30, 10, 11, 12, 13]. The standard analysis
of the well-known Hartle-Hawking no-boundary [8, 9] and Vilenkin tunneling
[10, 11] semiclassical wavefunctions does not lead to classically allowed
regions of KD type, because these wavefunctions are independent of the
inflaton field for small values of the scale factor. Some semiclassical
solutions of the WDW equation whose corresponding classical solutions exhibit
KD behavior have indeed been considered in Refs. [10, 11], but we could not
find in the literature a more extensive exploration of these solutions.

The usual hypothesis on the inflaton potential $V(\phi)$ of the WDW equation
is that it is slowly varying on a region around a certain value $\phi_{0}$ of
the inflaton [11, 31]. In the present work we consider wavefunctions that are
eigenfunctions of the momentum conjugate to the inflaton variable, and show
that under the assumption of a constant inflaton potential $V(\phi_{0})$, the
set of these wavefunctions include a variety of families with WKB
approximations strongly peaked about classical solutions on either or both KD
and INF states. Our discussion is guided by the properties of the solutions of
the classical version of the inflationary model with a constant potential and
establishes a close correspondence between its solutions and appropriate
wavefunctions of the WDW equation. Moreover, our family of wavefunctions
includes functions which for large values of the scale factor $a$ reduce to
products of phase factor $\exp(ik\phi/\hbar)$ times Hartle-Hawking no-boundary
wavefunctions [8, 9] and Vilenkin tunneling wavefunctions [10, 11], while for
$a$ near zero they manifest classically allowed regions of KD type. We
formulate an approximation scheme to obtain the explicit form of the
associated WKB wavefunctions and we obtain the corresponding connection rules
between classically allowed regions $\mbox{INF}\rightarrow\mbox{KD}$ and
$\mbox{KD}\rightarrow\mbox{INF}$. We also discuss the emergence of classical
universes in KD and INF periods in terms of the solutions of the classical
version of the model. We also consider the probability distributions provided
by the wavefunctions of the model representing expanding universes with an
inflation period. Then we prove that the WKB approximations for $a$ near zero
(KD region) and for large $a$ (INF region) lead to the same result.

The paper is organized as follows. In Section II we describe the basic aspects
of the KD period of solutions of classical inflationary models, and in
particular we introduce the solutions of the classical inflaton model with a
constant inflaton potential and analyze their KD and INF periods. In Sec. III
we review the properties of the WDW equation and of its WKB solutions, paying
special attention to the concept of WKB solutions peaked about classical
solutions in classically allowed regions. In Sec. IV we discuss the WDW
equation with a constant potential. We introduce our family of wavefunctions
and determine the explicit form of their associated WKB wavefunctions. We also
obtain the corresponding WKB connection rules. Section V is devoted to the
classification of emergent universes which arise from our wavefunctons, and in
Sec. VI we discuss probability distributions. Finally, we defer to an Appendix
the exact solution of the WDW near $a=0$ that provides expressions of the
corresponding WKB wavefunctions on the KD region in terms of first-kind
modified Bessel functions.

##  II CLASSICAL INFLATON MODELS

We consider classical single-field inflaton models in a closed Friedmann
universe with unit curvature [4, 5, 6, 7],

$ds^{2}=-dt^{2}+a(t)^{2}\left[\frac{dr^{2}}{1-r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right].$ (1)

The dynamical variables are the scale factor $a(t)\geq 0$ and the real field
$-\infty<\phi(t)<\infty$, which are functions of the cosmic time $t$. They
satisfy the evolution equation,

$\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,$ (2)

and the Friedmann equation,

$H^{2}=\frac{1}{3M_{\rm
pl}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right)-\frac{1}{a^{2}},$ (3)

where dots denote derivatives with respect to $t$, $H=\dot{a}/a$ is the Hubble
parameter, $V(\phi)$ is a given positive smooth potential, and $M_{\rm
pl}=\sqrt{\hbar c/8\pi G}$ is the reduced Planck mass.

The energy density is

$\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (4)

which using Eq. (3) can be expressed as

$\rho=3M_{\rm pl}^{2}\left(H^{2}+\frac{1}{a^{2}}\right),$ (5)

while its cosmic-time derivative, which follows from Eqs. (2) and (3), is

$\dot{\rho}=-3H\dot{\phi}^{2}.$ (6)

Note also that for the classical inflaton models to be reliable, the energy
density $\rho$ must be smaller than the Planck density $\rho_{\rm pl}=M_{\rm
pl}^{4}$, i.e.,

$\rho<M_{\rm pl}^{4},$ (7)

which combined with Eq. (5) give a classical lower limit for the scale factor

$a>\frac{\sqrt{3}}{M_{\rm pl}}.$ (8)

###  II.1 Inflationary regime and kinetic dominance

The acceleration equation,

$\frac{\ddot{a}}{a}=-\frac{1}{3M_{\rm pl}^{2}}(\dot{\phi}^{2}-V(\phi)),$ (9)

which also follows from Eqs. (2) and (3), shows that the inflationary regime
$\ddot{a}>0$ is determined by the condition

$\dot{\phi}^{2}<V(\phi),$ (10)

or, in terms of the energy density,

$\rho<\frac{3}{2}V.$ (11)

Therefore, as a consequence of Eq. (5), during the inflationary period

$a>\sqrt{\frac{2}{V}}M_{\rm pl}.$ (12)

Kinetic dominance is the opposite regime, wherein the energy density is
dominated by the kinetic energy of the inflaton field,

$\dot{\phi}^{2}\gg V(\phi),$ (13)

or, equivalently,

$\rho\gg\frac{3}{2}V.$ (14)

In the KD regime we may neglect $V$, $V^{\prime}$ and $1/a^{2}$ in Eqs. (2)
and (3), which then decouple into an equation for the inflaton field,

$\ddot{\phi}\pm\sqrt{\frac{3}{2}}\frac{1}{{M_{\rm
pl}}}|\dot{\phi}|\,\dot{\phi}\sim 0,$ (15)

and an equation to calculate $H=\dot{a}/a$ given any solution $\phi(t)$ of the
former,

$\frac{\dot{a}}{a}\sim\pm\frac{1}{\sqrt{6}\,{M_{\rm pl}}}|\dot{\phi}|.$ (16)

Thus we obtain the following two families of asymptotic solutions of Eqs.
(2)–(3) near a singularity $a=0$: for the plus sign in Eq. (15), solutions
expanding from the singularity,

$\phi(t)\sim\pm\sqrt{\frac{2}{3}}{M_{\rm pl}}\log(t-t^{*})+\phi^{*},\quad
a(t)\sim a^{*}(t-t^{*})^{1/3},\mbox{ as }t\rightarrow(t^{*})^{+},$ (17)

and for the minus sign in Eq. (15), solutions collapsing towards the
singularity,

$\phi(t)\sim\pm\sqrt{\frac{2}{3}}{M_{\rm pl}}\log(t^{*}-t)+\phi^{*},\quad
a(t)\sim a^{*}(t^{*}-t)^{1/3},\mbox{ as }t\rightarrow(t^{*})^{-},$ (18)

where $t^{*}$, $\phi^{*}$ and $a^{*}>0$ are arbitrary integration constants.
Note also that these asymptotic solutions are independent of the inflaton
potential $V(\phi)$. This behavior is illustrated in Fig. 1, which shows the
results of numerical integrations of Eqs. (2)–(3) for the quadratic potential
$V(\phi)=m^{2}\phi^{2}$ and for the Starobinski potential
$V(\phi)=\Lambda^{2}[1-\exp(-\sqrt{2/3}\phi/{M_{\rm pl}})]^{2}$. (The KD
regime may have different behaviors if the potential is not everywhere
nonnegative [32]. We do not consider those potentials in this paper.)

Figure 1:  Scale factor as a function of cosmic time obtained by numerical
integration of Eqs. (2)–(3) (continuous lines) for: (a) the quadratic
potential $V(\phi)=m^{2}\phi^{2}$ with initial values $t_{0}=0$,
$\phi_{0}=M_{\rm pl}$, $\dot{\phi}_{0}=10mM_{\rm pl}$, and $a_{0}=m^{-1}$; (b)
the Starobinski potential $V(\phi)=\Lambda^{2}[1-\exp(-\sqrt{2/3}\phi/{M_{\rm
pl}})]^{2}$ with initial values $t_{0}=0$, $\phi_{0}=M_{\rm pl}$,
$\dot{\phi}_{0}=10\Lambda$, $a_{0}=M_{\rm pl}/\Lambda$. Both models exhibit
the KD behavior (dashed lines) $a(t)\sim a^{*}(t-t^{*})^{1/3}$ as
$t\rightarrow(t^{*})^{+}$.

###  II.2  The classical inflaton model with a constant potential.

Figure 2: Trajectories in the $(a,\dot{a})$ phase space of the solutions of
the classical model with constant inflaton potential given by Eq. (21) in the
four possible cases: (a) $(4/27)a_{+}^{4}<b_{+}$; (b)
$0<b_{+}<(4/27)a_{+}^{4}$; (c) $b_{+}=(4/27)a_{+}^{4}$; and (d) $b_{+}=0$.

We consider inflaton models with a constant potential as their solutions
exhibit one or both periods of kinetic dominance and of inflationary
expansion. These models are relevant on the domain of interest of models with
slowly-varying potentials around a value $\phi_{0}$, where the potential may
be approximated by a constant $V_{0}$,

$V(\phi)\approx V(\phi_{0})=V_{0},$ (19)

For example, the Starobinski model with potential
$V(\phi)=\Lambda^{2}[1-\exp(-\sqrt{2/3}\phi/{M_{\rm pl}})]^{2}$ is slowly-
varying around any large value of $\Phi$, where $V(\phi)\approx\Lambda^{2}$.

For a constant potential Eq (2) reduces to

$\dot{\phi}(t)a(t)^{3}=\dot{\phi}(t_{0})a(t_{0})^{3},$ (20)

where $t_{0}$ is a given value of the cosmic time, and Eq. (3) can be written
as

$\dot{a}^{2}=\frac{a^{2}}{a_{+}^{2}}+\frac{b_{+}}{a^{4}}-1,$ (21)

where

$a_{+}=M_{\rm pl}\sqrt{3/V_{0}},$ (22)

and

$b_{+}=\frac{(\dot{\phi}(t_{0})a(t_{0})^{3})^{2}}{6{M_{\rm pl}}^{2}}.$ (23)

Equation (21) can be solved by separation of variables,

$t-t_{0}=\pm\int_{a(t_{0})}^{a(t)}\frac{da}{\sqrt{\frac{a^{2}}{a_{+}^{2}}+\frac{b_{+}}{a^{4}}-1}}.$
(24)

These solutions can be classified into four types depending on the number of
zeros of the right-hand side of Eq. (21) on the domain $a>0$. For convenience,
in the following discussion we denote this right-hand side by $f(a)$, i.e.,
$f(a)=a^{2}/a_{+}^{2}+b_{+}/a^{4}-1$.

  * (a)

If $(4/27)a_{+}^{4}<b_{+}$, then there are no zeros of $f(a)$ in the domain
$a>0$, and the solutions give rise to the two families of trajectories in the
$(a,\dot{a})$ phase map shown in Fig. 2(a). The upper trajectory $A_{1}$
describes an expanding universe starting from a KD stage near a singularity
$a=0$ at cosmic time

$t^{*}=t_{0}+\int_{a(t_{0})}^{0}\frac{da}{\sqrt{\frac{a^{2}}{a_{+}^{2}}+\frac{b_{+}}{a^{4}}-1}},$
(25)

such that $\dot{a}$ first decreases towards a positive minimum value at

$a_{\text{in}}=(2b_{+})^{1/6}\,a_{+}^{1/3},$ (26)

an then enters an unlimited period of inflationary expansion. The solutions
giving rise to the trajectory $A_{2}$ describe the corresponding contracting,
time-reversed evolution.

  * (b)

If $0<b_{+}<(4/27)a_{+}^{4}$, then $f(a)$ has two positive zeros
$0<a_{1}<a_{2}$,

$a_{k}=a_{+}\sqrt{\frac{1}{3}+\frac{2}{3}\cos\left(\frac{1}{3}\arccos\left(1-\frac{27\,b_{+}}{2\,a_{+}^{4}}\right)-(k-1)\frac{2\pi}{3}\right)},\quad
k=1,2,$ (27)

and there are the two classes of trajectories shown in Fig. 2(b). The
trajectory $B_{1}$ represents solutions with $a\leq a_{1}$ that describe
universes which expand from a KD stage near a singularity $a=0$ until they
reach a maximum $a(t)=a_{1}$, and then contract towards another KD stage near
$a=0$. These solutions do not have an INF period. The trajectory $B_{2}$
represents solutions with $a\geq a_{2}$ which describe universes without any
KD period: they contract until they bounce at $a=a_{2}$ and begin a period of
unlimited INF expansion.

  * (c)

If $b_{+}=(4/27)a_{+}^{4}$, $f(a)$ has a unique positive zero at

$a_{0}=\sqrt{\frac{2}{3}}\,a_{+},$ (28)

but $f(a)\geq 0$ for $a>0$. The corresponding trajectories are shown in Fig.
2(c), which includes the constant solution $a=a_{0}$ as well as four families
of trajectories, two of which ($C_{1}$ and $C_{2}$) tend asymptotically to
$a_{0}$ in the future, and the other two ($C_{3}$ and $C_{4}$) in the past.

  * (d)

Finally, if $b_{+}=0$, then $f(a)$ has a unique positive zero at $a=a_{+}$,
but $f(a)\geq 0$ only for $a\geq a_{+}$. The corresponding trajectories are
shown in Fig. 2(d), which features the constant solutions $a=a_{+}$ as well as
two families of solutions ($D_{1}$, expanding, and $D_{2}$, contracting),
without KD periods, and which tend asymptotically to $a_{+}$ in the past and
in the future, respectively. The solutions corresponding to $D_{1}$ are in an
unlimited period of INF expansion. The particular case $a(t_{0})=a_{+}$ can be
explicitly integrated to give the de Sitter form,

$a(t)=a_{+}\cosh\left(\frac{1}{a_{+}}(t-t_{0})\right).$ (29)

##  III The WDW equation and its WKB solutions

In this Section we briefly recall the derivation of the WDW equation, the
structure of its WKB solutions, and the relation of the latter to the
classical inflaton Eqs. (2)–(3). This relation will be particularized in the
following Section to the case of a constant inflaton potential, which in turn
will be used to discuss possible types of emerging universes.

###  III.1 The WDW equation

The momenta conjugate to the variables $a$ and $\phi$ in the Hamiltonian
formulation of the inflaton model turn out to be [11]

$P_{a}=-12\pi^{2}{M_{\rm pl}}^{2}a\dot{a},$ (30)

and

$P_{\phi}=2\pi^{2}a^{3}\dot{\phi},$ (31)

respectively. If we substitute these values in Eq. (3) rewritten in the form

$6{M_{\rm pl}}^{2}a\,\dot{a}^{2}-a^{3}\,\dot{\phi}^{2}+6{M_{\rm
pl}}^{2}\,\left(a-\frac{V(\phi)}{3{M_{\rm pl}}^{2}}a^{3}\right)=0,$ (32)

we arrive at

$P_{a}^{2}-\frac{6{M_{\rm pl}}^{2}}{a^{2}}\,P_{\phi}^{2}+U(a,\phi)=0,$ (33)

where $U(a,\phi)$ is the _superpotential_ function

$U(a,\phi)=\frac{1}{\lambda^{2}}\left(a^{2}-\frac{V(\phi)}{3{M_{\rm
pl}}^{2}}a^{4}\right),$ (34)

and where

$\lambda=\frac{1}{12\pi^{2}{M_{\rm pl}}^{2}}.$ (35)

Incidentally, since the left-hand side of Eq. (33) is proportional to
$a\mathcal{H}$, where $\mathcal{H}$ is the Hamiltonian of the inflaton model
[11], Eq. (33) shows that the solutions of Eqs. (2)–(3) satisfy the zero-
energy constraint $\mathcal{H}=0$.

Quantization is performed by the rules,

$P_{a}\rightarrow-i\,\hbar\frac{\partial}{\partial a},\quad
P_{\phi}\rightarrow-i\,\hbar\frac{\partial}{\partial\phi},$ (36)

with the proviso of an ambiguity in the ordering of $a$ and $\partial/\partial
a$ in the quantization of $P_{a}^{2}$, which in effect leads to the
quantization rule

$P_{a}^{2}\rightarrow-\hbar^{2}\frac{1}{a^{p}}\frac{\partial}{\partial
a}a^{p}\frac{\partial}{\partial a},$ (37)

where $p$ is an (in principle) arbitrary parameter. Thus we arrive at the
celebrated WDW equation [29],

$\left(\frac{\partial^{2}}{\partial a^{2}}+\frac{p}{a}\frac{\partial}{\partial
a}-\frac{6{M_{\rm
pl}}^{2}}{a^{2}}\frac{\partial^{2}}{\partial\phi^{2}}-\frac{1}{\hbar^{2}}U(a,\phi)\right)\Psi(a,\phi)=0.$
(38)

###  III.2 General remarks on the WKB solutions of the WDW equation

WKB solutions of Eq. (38) have the usual exponential form

$\Psi(a,\phi)=\exp\left(i\,\mathbb{S}(a,\phi)/\hbar\right),$ (39)

where $\mathbb{S}$ is expanded as an asymptotic power series in $\hbar$,

$\mathbb{S}=\mathbb{S}_{0}+\frac{\hbar}{i}\mathbb{S}_{1}+\left(\frac{\hbar}{i}\right)^{2}\mathbb{S}_{2}+\cdots,\quad\hbar\rightarrow
0.$ (40)

By substituting Eqs. (39)–(40) into the WDW Eq. (38) and setting to zero terms
with the same power of $\hbar$ we obtain a sequence of two-dimensional WKB
equations for the $\mathbb{S}_{n}$, which have to be solved recursively.

The first equation, for $\mathbb{S}_{0}$, is the Hamilton-Jacobi equation,

$\left(\frac{\partial\mathbb{S}_{0}}{\partial
a}\right)^{2}-\frac{1}{2\pi^{2}\lambda
a^{2}}\left(\frac{\partial\mathbb{S}_{0}}{\partial\phi}\right)^{2}+U=0,$ (41)

and the second equation, for $\mathbb{S}_{1}$, is

$\frac{\partial^{2}\mathbb{S}_{0}}{\partial a^{2}}-\frac{6{M_{\rm
pl}}^{2}}{a^{2}}\frac{\partial^{2}\mathbb{S}_{0}}{\partial\phi^{2}}+2\frac{\partial\mathbb{S}_{0}}{\partial
a}\frac{\partial\mathbb{S}_{1}}{\partial a}-\frac{12{M_{\rm
pl}}^{2}}{a^{2}}\frac{\partial\mathbb{S}_{0}}{\partial\phi}\frac{\partial\mathbb{S}_{1}}{\partial\phi}+\frac{p}{a}\frac{\partial\mathbb{S}_{0}}{\partial
a}=0.$ (42)

Note that the Hamilton-Jacobi Eq. (41) does not depend on the ordering
parameter $p$, while Eq. (42) does.

WKB solutions of the WDW equation in the classically allowed regions are
oscillatory, and these are the regions where the nucleation of classical
universes with specific properties may emerge [13, 31]. We also recall that
the potential $V(\phi)$ must be positive and smaller than the Planck density
${M_{\rm pl}}^{4}$ for the semiclassical solutions obtained from Eqs.
(41)–(42) to be valid [10]. The identification between the classical momenta
as defined in Eqs. (30)–(31) and the corresponding derivatives of
$\mathbb{S}_{0}$,

$P_{a}=\frac{\partial\mathbb{S}_{0}}{\partial a},\quad
P_{\phi}=\frac{\partial\mathbb{S}_{0}}{\partial\phi},$ (43)

leads to the system of first-order differential equations,

$\dot{a}=-\frac{\lambda}{a}\frac{\partial\mathbb{S}_{0}}{\partial
a},\quad\dot{\phi}=\frac{1}{2\pi^{2}a^{3}}\frac{\partial\mathbb{S}_{0}}{\partial\phi}.$
(44)

Thus, each solution of Eq. (41) corresponds to a biparametric (the initial
conditions for $a$ and $\phi$) family of solutions of Eqs. (2)–(3), and this
family in turn characterizes the properties of the emergent classical
universes in the classically allowed regions. It is then said [13, 31] that
the wavefunction $\Psi(a,\phi)$ is _peaked_ about the associated solutions of
the system (44).

##  IV THE WDW EQUATION WITH A CONSTANT POTENTIAL

In this Section we look for solutions of the WDW equation with a constant
potential using the ansatz

$\Psi(k,a,\phi)=\psi(k,a)e^{ik\phi/\hbar},$ (45)

where $k$ is a real parameter. Variables in the WDW equation separate and we
get the following equation for $\psi(k,a)$,

$\left(\frac{d^{2}}{da^{2}}+\frac{p}{a}\frac{d}{da}-\hbar^{-2}\mathrm{U}(k,a)\right)\psi(k,a)=0,$
(46)

where

$\mathrm{U}(k,a)=-\frac{6{M_{\rm
pl}}^{2}k^{2}}{a^{2}}+U(a,\phi_{0})=-\frac{6{M_{\rm
pl}}^{2}k^{2}}{a^{2}}+\frac{a^{2}}{\lambda^{2}}\left(1-\frac{a^{2}}{a_{+}^{2}}\right).$
(47)

The first term in the modified superpotential $\mathrm{U}(k,a)$ is a
consequence of the phase factor $\exp(ik\phi/\hbar)$ in the wavefunction
ansatz Eq. (45), and induces a dependence of the solutions of the WDW equation
on the scale factor different from those in Refs. [11, 31]. This term is
dominant as $a\to 0^{+}$, and in the next Sections we will discuss its effect
on the overall behavior of the WKB solutions via connection formulas.

###  IV.1 WKB solutions

The actions $\mathbb{S}$ and $S$ of the WKB solutions for $\Psi(k,a,\phi)$ and

$\psi(k,a)=e^{iS(k,a)/\hbar},$ (48)

in Eq. (45) are related by $\mathbb{S}=S+k\phi$, which after expansion in
powers of $\hbar$ leads to $\mathbb{S}_{0}=S_{0}+k\phi$ and
$\mathbb{S}_{n}=S_{n}$ for $n\geq 1$ (with $S_{n}$ independent of $\phi$).
Therefore, the classical system Eq. (44) on whose solutions the wavefunctions
$\Psi(k,a,\phi)$ are strongly peaked is,

$\dot{a}^{2}=-\frac{\lambda^{2}}{a^{2}}\mathrm{U}(k,a),\quad\dot{\phi}=\frac{k}{2\pi^{2}a^{3}},$
(49)

which using Eq. (47) reduces to Eqs. (20)–(21) with

$b_{+}=\frac{k^{2}}{24\pi^{4}{M_{\rm pl}}^{2}}.$ (50)

Likewise, Eqs. (41) and (42) reduce to equations for $S_{0}$ and $S_{1}$,

$\left(\frac{dS_{0}}{da}\right)^{2}+\mathrm{U}(k,a)=0,$ (51)
$\frac{d^{2}S_{0}}{da^{2}}+2\frac{dS_{0}}{da}\frac{dS_{1}}{da}+\frac{p}{a}\frac{dS_{0}}{da}=0,$
(52)

which can be readily solved. On the classically allowed regions
$\mathrm{U}(k,a)<0$ we find

$S_{0}(k,a)=\int_{a_{r}}^{a}\mathrm{p}(k,a^{\prime})da^{\prime},$ (53)

where

$\mathrm{p}(k,a)=\left(-\mathrm{U}(k,a)\right)^{1/2},$ (54)

$a_{r}$ is an appropriate reference point, and

$S_{1}(k,a)=-\frac{1}{2}\log(a^{p}\mathrm{p}(k,a)).$ (55)

Therefore, the basic WKB solutions of Eq. (46) on the classically allowed
regions are

$\psi_{\pm}(k,a)=\frac{1}{(a^{p}\mathrm{p}(k,a))^{1/2}}\exp\left(\pm\frac{i}{\hbar}\int_{a_{r}}^{a}\mathrm{p}(k,a^{\prime})da^{\prime}\right).$
(56)

Eq. (47) shows that for $k\neq 0$ the modified superpotential
$\mathrm{U}(k,a)$ is strictly negative both for large and small values of
$a>0$, and therefore these regions are classically allowed regions for
nonvanishing values of $k$, values to which we restrict hereafter. The
classical system Eq. (44) on whose solutions the wavefunctions $\psi$ are
strongly peaked is

$\dot{a}=\mp\frac{\lambda}{a}\mathrm{p}(k,a),\quad\dot{\phi}=\frac{k}{2\pi^{2}a^{3}},$
(57)

which reduces to Eqs. (20)–(21) with $b_{+}$ defined in Eq. (50). Note in
particular that the first equation of this system shows that $\psi_{+}$ and
$\psi_{-}$ describe contracting and expanding universes, respectively.

###  IV.2 Leading behavior of the WKB solutions at large scale factor

As we mentioned earlier, at large $a$ we may neglect the $k$-dependent term in
the modified superpotential Eq. (47),

$\mathrm{U}(k,a)\sim
U_{+}(a)=\frac{a^{2}}{\lambda^{2}}\left(1-\frac{a^{2}}{a_{+}^{2}}\right),\quad\mbox{
as }a\to\infty,$ (58)

in effect recovering the well-known model for studying the emergence of
inflation from the WDW equation discussed, e.g., in Refs. [11, 31],

$\left(\frac{d^{2}}{da^{2}}+\frac{p}{a}\frac{d}{da}-\hbar^{-2}U_{+}(a)\right)\psi^{(\infty)}(a)=0.$
(59)

Since in our case $k\neq 0$ these results apply only to large $a$, we have
labeled the wavefunction as $\psi^{(\infty)}(a)$. We summarize here the
relevant notations and results.

The generic shape of the superpotential $U_{+}(a)$ in shown in Fig. 3, and
corresponds to a quantum tunneling problem with a classically forbidden region
$0<a<a_{+}$ and a classically allowed region $a>a_{+}$.

Figure 3: Generic shape of the superpotential $U_{+}(a)$ defined in Eq. (58).

Since $a_{+}$ is the unique positive zero of $U_{+}(a)$, it is natural to take
it as the reference point for the WKB solutions. By using Eq. (56) with
$\mathrm{U}(k,a)$ replaced by $U_{+}(a)$ and $a_{r}=a_{+}$, we obtain the
following ($k$-independent) approximations $\psi_{\pm}^{(\infty)}(a)$ to the
($k$-dependent) functions $\psi_{\pm}(k,a)$ valid for large $a$,

$\psi_{\pm}^{(\infty)}(a)=\lambda^{1/2}a^{-p/2}\left(\frac{a^{4}}{a_{+}^{2}}-a^{2}\right)^{-1/4}\exp\left[\pm\frac{i}{\hbar}\,\frac{{M_{\rm
pl}}^{2}}{\lambda
V_{0}}\left(\frac{a^{2}}{a_{+}^{2}}-1\right)^{3/2}\right],\quad a_{+}<a.$ (60)

The wavefunctions $\psi_{+}^{(\infty)}(a)$ and $\psi_{-}^{(\infty)}(a)$
represent contracting and expanding universes in the inflationary period,
respectively.

Two distinguished wavefunctions are usually defined in terms of the
$\psi_{\pm}^{(\infty)}(a)$. The Vilenkin _tunneling_ wavefunction
$\psi_{\mathrm{T}}(a)$ is defined as [11, 31],

$\psi_{\mathrm{T}}(a>a_{+})=e^{i\pi/4}A\psi_{-}^{(\infty)}(a),$ (61)

where

$A=e^{-{M_{\rm pl}}^{2}/(\hbar\lambda V_{0})}.$ (62)

The function $\psi_{\mathrm{T}}(a)$ represents an expanding universe in the
region $a>a_{+}$, and the corresponding complete wavefunction in that region
is

$\Psi_{\mathrm{T}}(k,a,\phi)=\psi_{\mathrm{T}}(a)e^{ik\phi/\hbar}.$ (63)

Likewise, the Hartle-Hawking _no-boundary_ wavefunction [8, 9] is defined by
the following superposition of wavefunctions representing contracting and
expanding universes,

$\psi_{\mathrm{NB}}(a>a_{+})=A^{-1}(e^{-i\pi/4}\psi_{+}^{(\infty)}(a)+e^{+i\pi/4}\psi_{-}^{(\infty)}(a)),$
(64)

and the corresponding complete wavefunction is

$\Psi_{\mathrm{NB}}(k,a,\phi)=\psi_{\mathrm{NB}}(a)e^{ik\phi/\hbar}.$ (65)

Note that if $V_{0}\ll{M_{\rm pl}}^{4}$, then $a_{+}$ is much larger than the
classical lower limit for the scale factor given by Eq. (8),

$a_{+}\gg\frac{\sqrt{3}}{{M_{\rm pl}}},$ (66)

and from Eq. (12) it follows that during the classical inflation regime

$a>\sqrt{\frac{2}{3}}a_{+}.$ (67)

Therefore, $a_{+}$ is an appropriate lower limit for a classical inflationary
stage.

###  IV.3 Leading behavior of the WKB solutions at small scale factor

For $a$ near zero we neglect the term proportional to $a^{4}$ in
$\mathrm{U}(k,a)$,

$\mathrm{U}(k,a)\sim
U_{-}(k,a)=\frac{a^{2}}{\lambda^{2}}\left(1-\frac{a_{-}(k)^{4}}{a^{4}}\right),\quad\mbox{
as }a\to 0^{+},$ (68)

where

$a_{-}(k)=(\sqrt{6}{M_{\rm pl}}\lambda|k|)^{1/2},$ (69)

is the unique positive zero of $U_{-}(k,a)$. The superpotential $U_{-}(k,a)$,
illustrated in Fig. 4, corresponds to quantum tunneling between the
classically allowed region $0<a<a_{-}(k)$ and the classically forbidden region
$a_{-}(k)<a$, The analog of Eq. (59) is

$\left(\frac{d^{2}}{da^{2}}+\frac{p}{a}\frac{d}{da}-\frac{1}{\hbar^{2}}U_{-}(k,a)\right)\psi^{(0)}(k,a)=0.$
(70)

Figure 4: Generic shape of the superpotential $U_{-}(k,a)$ (with $k\neq 0$)
defined in Eqs. (68)–(69).

Again, since $a_{-}(k)$ is the unique positive zero of $U_{-}(k,a)$, it is
natural to take it as the reference point for the WKB solutions. By using Eq.
(56) with $\mathrm{U}(k,a)$ replaced by $U_{-}(k,a)$ and $a_{r}=a_{-}(k)$, we
obtain the following approximations $\psi_{\pm}^{(0)}(k,a)$ to the functions
$\psi_{\pm}(k,a)$ valid for small $a$,

$\psi_{\pm}^{(0)}(k,a)=\lambda^{1/2}a^{-\frac{p+1}{2}}\left(\frac{a_{-}(k)^{4}}{a^{4}}-1\right)^{-1/4}\exp\left(\pm\frac{i}{\hbar}\int_{a_{-}(k)}^{a}\mathrm{p}_{-}(k,a^{\prime})da^{\prime}\right),$
(71)

where

$\displaystyle\int_{a_{-}(k)}^{a}\mathrm{p}_{-}(k,a^{\prime})da^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\left[\sqrt{a_{-}(k)^{4}-a^{4}}+a_{-}(k)^{2}\log
a^{2}\right.$ (72)
$\displaystyle\qquad{}\left.-a_{-}(k)^{2}\log\left(a_{-}(k)^{2}+\sqrt{a_{-}(k)^{4}-a^{4}}\right)\right].$

Since for $a\ll a_{-}(k)$

$\int_{a_{-}(k)}^{a}\mathrm{p}_{-}(k,a^{\prime})da^{\prime}\sim\frac{a_{-}(k)^{2}}{\lambda}\log
a,$ (73)

Eq. (71) implies

$\psi_{\pm}^{(0)}(k,a)\sim\frac{\lambda^{1/2}}{a_{-}(k)}a^{(1-p)/2}\exp\left(\pm\frac{i}{\hbar}\sqrt{6}\,M_{\rm
pl}|k|\log a\right),\quad a\ll a_{-}(k).$ (74)

Thus the complete wavefunctions can be approximated as

$\Psi_{\pm}^{(0)}(k,a,\phi)\sim\frac{\lambda^{1/2}}{a_{-}(k)}a^{2\nu}\exp\left[\pm\frac{i}{\hbar}(\sqrt{6}\,M_{\rm
pl}|k|\log a\pm k\phi)\right],\quad a\ll a_{-}(k),$ (75)

where $\nu=(1-p)/4$. The admissible wavefunctions should be regular at $a=0$
and this is satisfied only if $\nu\geq 0$ or, equivalently, if $p\leq 1$.

The wavefronts of these approximations are given by the classical trajectories
of $(a,\phi)$ in the KD period,

$\sqrt{6}\,M_{\rm pl}\log a\pm\phi=const,$ (76)

and the wavefunctions $\Psi_{+}^{(0)}(k,a,\phi)$ and
$\Psi_{-}^{(0)}(k,a,\phi)$ represent contracting and expanding universes in
the KD period, respectively.

Finally, taking into account Eq. (35), from Eq. (69) we find that

$a_{-}^{2}=\frac{|k|}{2\sqrt{6}\pi^{2}M_{\rm pl}}.$ (77)

Therefore, $a_{-}$ satisfies the classical limit condition Eq. (8) for the
scale factor if

$|k|>\frac{6\sqrt{6}\pi^{2}}{M_{\rm pl}}.$ (78)

##  V TYPES OF EMERGENT UNIVERSES

In this Section we will describe the different types of classical universes
which may emerge from the solutions Eq. (45) of the WDW equation with a
constant potential.

###  V.1 Type-$A$ universes

Classical universes evolving according to solutions of type $A_{1}$
(expanding) and $A_{2}$ (contracting) of Eqs. (20)–(21) with both KD and INF
periods arise provided that $(4/27)a_{+}^{4}<b_{+}$, which restated in terms
of the parameters appearing in the corresponding semiclassical wavefunctions
is equivalent to (see Eq. (50)),

$k^{2}>32\pi^{4}\frac{{M_{\rm pl}}^{6}}{V_{0}^{2}}.$ (79)

In this case $\mathrm{U}(k,a)$ is strictly negative on the domain $a>0$, and
the wavefunctions

$\Psi_{\pm}(k,a,\phi)=\psi_{\pm}(k,a)e^{ik\phi/\hbar},$ (80)

where

$\psi_{\pm}(k,a)=\frac{1}{(a^{p}\mathrm{p}(k,a))^{1/2}}\exp\left(\pm\frac{i}{\hbar}\int_{a_{+}}^{a}\mathrm{p}(k,a^{\prime})da^{\prime}\right),$
(81)

are well-defined oscillatory functions on the whole classically allowed domain
$a>0$.

In Sec. IV-B we showed that for large $a$ the function $\Psi_{-}(k,a,\phi)$
describes expanding INF universes. If we factorize $\psi_{-}(k,a)$ in the form

$\psi_{-}(k,a)=\frac{1}{(a^{p}\mathrm{p}(k,a))^{1/2}}\exp\left(-\frac{i}{\hbar}\int_{a_{+}}^{a_{-}(k)}\mathrm{p}(k,a^{\prime})da^{\prime}\right)\exp\left(-\frac{i}{\hbar}\int_{a_{-}(k)}^{a}\mathrm{p}(k,a^{\prime})da^{\prime}\right),$
(82)

we see that near $a=0$ the last factor behaves as $\psi_{-}^{(0)}(k,a)$ and,
from the analysis of Sec. IV-C, in this small $a$ region $\Psi_{-}(k,a,\phi)$
describes expanding universes in the KD regime. Therefore $\Psi_{-}(k,a,\phi)$
is indeed peaked about a type-$A_{1}$ solution.

A similar argument shows that $\Psi_{+}(k,a,\phi)$ is peaked about a
type-$A_{2}$ solution.

###  V.2 Type-$B$ universes

Classical universes evolving according to solutions of type $B_{1}$ or $B_{2}$
of Eqs. (20)–(21) arise if $0<b_{+}<(4/27)a_{+}^{4}$, or, equivalently,

$0<k^{2}<32\pi^{4}\frac{{M_{\rm pl}}^{6}}{V_{0}^{2}}.$ (83)

In this case $\mathrm{U}(k,a)$ has two positive zeros $0<a_{1}<a_{2}$ given by
Eq. (27), and corresponds to the potential barrier illustrated in Fig. 5.

Figure 5: Generic shape of the superpotential $\mathrm{U}(k,a)$ for
$0<k^{2}<32\pi^{4}{M_{\rm pl}}^{6}/V_{0}^{2}$ with the two classically allowed
regions $0<a<a_{1}$ and $a_{2}<a$.

There are two classically allowed regions, $0<a<a_{1}$ and $a_{2}<a$, with
respective basic WKB wavefunctions given by,

$\psi_{\pm}^{(r)}(k,a)=\frac{1}{(a^{p}\mathrm{p}(k,a))^{1/2}}\exp\left(\pm\frac{i}{\hbar}\int_{a_{r}}^{a}\mathrm{p}(k,a^{\prime})da^{\prime}\right),\quad
r=1,2,$ (84)

which in turn can be approximated by $\psi_{\pm}^{(0)}(k,a)$ and
$\psi_{\pm}^{(\infty)}(a)$ for $r=1$ and $r=2$ respectively.

Using the standard WKB connection formulas (wherein the semiclassical
expansions are interpreted as asymptotic expansions in the sense of Poincaré)
to connect these two sets of WKB solutions across the barrier leads to a well-
known loss of unitarity [33, 34]. To circumvent this problem, we interpret the
asymptotic expansions in the sense of Borel, and use the ensuing connections
formulas of Silverstone [35]. Since the classically forbidden region is a
Stokes line, it is necessary to “pick sides,” and we choose to take
$\mathrm{Im}\,a\to 0^{+}$. The calculation is strictly analogous to the
calculation corresponding to the piecewise linear potential in Fig. 1(b) of
Ref. [36], where all the intermediate steps are carried out in detail. Thus we
find,

$\psi_{+}^{(2)}(k,a)\rightarrow
e^{K(k)}(\psi_{+}^{(1)}(k,a)-i\psi_{-}^{(1)}(k,a)),$ (85)
$\psi_{-}^{(2)}(k,a)\rightarrow
e^{K(k)}(\psi_{-}^{(1)}(k,a)+i\,\psi_{+}^{(1)}(k,a))-ie^{-K(k)}\psi_{+}^{(1)}(k,a),$
(86)

and, conversely,

$\psi_{+}^{(1)}(k,a)\rightarrow
e^{K(k)}(\psi_{+}^{(2)}(k,a)+i\psi_{-}^{(2)}(k,a)),$ (87)
$\psi_{-}^{(1)}(k,a)\rightarrow
e^{K(k)}(\psi_{-}^{(2)}(k,a)-i\psi_{+}^{(2)}(k,a))+ie^{-K(k)}\psi_{+}^{(2)}(k,a),$
(88)

where

$K(k)=\frac{1}{\hbar}\int_{a_{1}}^{a_{2}}|\mathrm{p}(k,a)|\,da.$ (89)

As a consequence it follows that

$\psi_{+}^{(1)}(k,a)-i\psi_{-}^{(1)}(k,a)\rightarrow
e^{-K(k)}\psi_{+}^{(2)}(k,a),$ (90)

and

$\psi_{+}^{(2)}(k,a)+i\psi_{-}^{(2)}(k,a)\rightarrow
e^{-K(k)}\psi_{+}^{(1)}(k,a).$ (91)

(The standard WKB connection formulas would have led to Eqs. (90) and (91)
with their right-hand sides equal to zero, with the apparent result that a
nonzero oscillatory wavefunction at one side of the barrier connects to a zero
wavefunction on the other side of the barrier. These connection formulas do
not capture the physically intuitive result that the wavefunction on the other
side of the barrier is also oscillatory but suppressed by an exponentially
small factor of the order of $\exp(-K(k))$.)

Therefore, the wavefunction corresponding to the WKB approximation given by

$\Psi^{(1)}(k,a,\phi)=\left(\psi_{+}^{(1)}(k,a)-i\psi_{-}^{(1)}(k,a)\right)e^{ik\phi/\hbar},\quad
0<a<a_{1},$ (92)

is a superposition of an expanding component and a contracting component
peaked about classical solutions of Eqs. (20)–(21) in a KD period. In
addition, this WKB wavefunction is suppressed by an exponentially small factor
for $a>a_{2}$ and, consequently, $\Psi^{(1)}(k,a,\phi)$ is peaked about
$B_{1}$ solutions of Eqs. (20)–(21).

Likewise, the wavefunction corresponding to the WKB approximation

$\Psi^{(2)}(k,a,\phi)=\left(\psi_{+}^{(2)}(k,a)+i\psi_{-}^{(2)}(k,a)\right)e^{ik\phi/\hbar},\quad
a_{2}<a<\infty,$ (93)

is a superposition of expanding and contracting components peaked about
classical solutions of Eqs. (20)–(21) in an inflationary period, and is
suppressed by an exponentially small factor for $0<a<a_{1}$ . Therefore
$\Psi^{(2)}(k,a,\phi)$ is peaked about the $B_{2}$ type solutions of Eqs.
(20)–(21). Note that for large $a$ the no-boundary wavefunction
$\psi_{\mathrm{NB}}(k,a,\phi)$ and $\Psi^{(2)}(k,a,\phi)$ are related by

$\Psi_{\mathrm{NB}}(k,a,\phi)=A^{-1}e^{-i\pi/4}\Psi^{(2)}(k,a,\phi).$ (94)

###  V.3 Type-$C$ and type-$D$ universes

Type-$C$ universes emerge as semiclassical limits of wavefunctions for
$k^{2}=32\pi^{4}{M_{\rm pl}}^{6}/V_{0}^{2}$, and are limiting cases of
type-$B$ universes as $a_{1}\rightarrow a_{2}$.

Finally, type-$D$ universes are associated with wavefunctions with $k=0$, and
are the same as those studied in Ref. [11]. The subtype $D_{1}$ describes
expanding inflationary universes without a KD period and corresponds to
$\psi_{-}^{(\infty)}(a)$ wavefunctions in Eq. (60). Subtype $D_{2}$ describes
contracting universes without a KD period and corresponds to wavefunctions
$\psi_{+}^{(\infty)}(a)$ in Eq. (60).

##  VI PROBABILITY DISTRIBUTIONS

The probabilistic interpretation of the solutions of the WDW equation is
formulated in terms of the components of the current [29, 10, 11, 31]

$j^{a}=\frac{i}{2}\hbar\,a^{p}(\Psi^{*}\partial_{a}\Psi-\Psi\partial_{a}\Psi^{*}),$
(95) $j^{\phi}=-i\,3{M_{\rm
pl}}^{2}\,\hbar\,a^{p-2}(\Psi^{*}\partial_{\phi}\Psi-\Psi\partial_{\phi}\Psi^{*}),$
(96)

which satisfy the continuity equation

$\partial_{a}j^{a}+\partial_{\phi}j^{\phi}=0.$ (97)

In general, these components are not positive definite and their
interpretation requires the analysis of an appropriate time variable [10, 11].
If $a$ is taken as a time variable, then $\rho_{\phi_{0}}(a)=j^{a}$ is
interpreted as the probability density for $\phi=\phi_{0}$ at a given value of
$a$. The quantity $\partial_{\phi}j^{\phi}$ for the solutions $\Psi(k,a,\phi)$
of the form of Eq. (45) vanishes identically and, consequently, the density
$\rho_{\phi_{0}}(a)$ is $a$-independent. It turns out that the
$\rho_{\phi_{0}}(a)$ for a generic WKB wavefunction of the WDW equation is
positive (negative) if the wavefunction corresponds to expanding (contracting)
universes [10].

Let us consider the expanding universes with an inflationary period
$A_{1},B_{2},C_{3}$ and $D_{1}$. Their wavefunctions for large $a$ are
approximated by $\psi_{-}^{(\infty)}(a)$, and from Eq. (56) it follows that

$\psi_{-}^{(\infty)*}\partial_{a}\psi_{-}^{(\infty)}-\psi_{-}^{(\infty)}\partial_{a}\psi_{-}^{(\infty)*}=-\frac{2i}{\hbar
a^{p}},$ (98)

which leads to a constant density

$\rho_{\phi_{0}}(a)=1.$ (99)

As a consequence of Eq. (99), Vilenkin’s tunneling wavefunction
$\psi_{\mathrm{T}}$ on the region $a>a_{+}$ satisfies

$\rho_{T,\phi_{0}}(a)=\exp{\left(-\frac{24\pi^{2}{M_{\rm pl}}^{4}}{\hbar
V(\phi_{0})}\right)}=\exp{\left(-\frac{2}{3\mathbb{V}(\phi_{0})}\right)},$
(100)

where $\mathbb{V}=\hbar V/(6\pi{M_{\rm pl}}^{2})^{2}$ is the potential
function with the Vilenkin normalization (see Eq. (4.15) in Ref. [11]). This
expression is the standard result for the probability density for the Vilenkin
tunneling function [11, 31].

Of the four types of universes $A_{1},B_{2},C_{3}$ and $D_{1}$ under
consideration, only the type $A_{1}$ has a KD period in addition to the
inflationary period. Moreover, from Eq. (82) we have that near $a=0$ the
wavefunction corresponding to the $A_{1}$ type is approximated by

$\psi_{-}(k,a)=\exp\left(-\frac{i}{\hbar}\int_{a_{+}}^{a_{-}(k)}\mathrm{p}(k,a^{\prime})da^{\prime}\right)\psi_{-}^{(0)}(k,a).$
(101)

Then, using Eqs. (75) and (69) we obtain

$\rho_{\phi_{0}}(a)=\frac{\lambda}{a_{-}^{2}}\sqrt{6}{M_{\rm pl}}|k|=1$ (102)

which is the same constant density as that calculated for large $a$ in Eq.
(99).

##  VII CONCLUSIONS

In the present work we have discussed the emergence of KD classical universes
in closed Friedmann universes with unit curvature from a certain family of WKB
solutions of the WDW equation with a constant inflaton potential. These WKB
solutions are semiclassical approximations to eigenfunctions of the momentum
operator conjugate to the inflaton, i.e., they depend on the inflaton field
$\phi$ through a phase factor $\exp{i(k\phi/\hbar)}$. The standard treatment
of this problem has been restricted to $k=0$, and leads to a large-$a$,
classically allowed, INF region, and to a small-$a$, classically forbidden
region. Allowing for $k\neq 0$ replaces this scenario with a new one, in which
two classically allowed regions separated by a potential barrier via two
turning points may arise. In the innermost region there are semiclassical
solutions peaked about KD classical solutions, and in the outermost region we
find, among others, the well-known Vilenkin tunneling and Hartle-Hawking no-
boundary solutions times the $\phi$-dependent phase factor. We have also
discussed how these solutions connect in our setting to superpositions of
semiclassical wavefunctions in the KD regime. We have recovered within our
approach the standard results on the INF region. Finally, in the Appendix we
use the explicit integrability of the WDW equation near $a=0$ to provide
expressions for the wavefunctions with WKB approximations on the KD regions in
terms of first-kind modified Bessel functions.

###### Acknowledgements.

The financial support of the Spanish Ministerio de Economía y Competitividad
under Project No. PGC2018-094898-B-I00 is gratefully acknowledged.

##  Appendix A Solution of the WDW equation near $a=0$ in terms of Bessel
functions

The general solution of Eq. (70) is a linear combination of the functions

$f_{\pm}(k,a)=I_{\pm\omega(k)}(a^{2}/2\hbar\lambda)a^{2\nu},$ (103)

where $I_{\pm\omega}$ are the first-kind modified Bessel functions [37] and

$\omega(k)=\frac{1}{2}\left[(2\nu)^{2}-6{M_{\rm
pl}}^{2}k^{2}/\hbar^{2}\right]^{1/2}.$ (104)

The generic shape of the real and imaginary parts of Eq. (103) are illustrated
in Fig. 6. From the asymptotic formula [37]

$I_{\omega}(x)\sim\frac{1}{\Gamma(1+\omega)}\left(\frac{x}{2}\right)^{\omega},\quad\mbox{
as }x\rightarrow 0,$ (105)

it follows that

$f_{\pm}(k,a)\sim C_{\pm}a^{2(\nu\pm\omega(k))},\quad\mbox{ as }a\rightarrow
0,$ (106)

where

$C_{\pm}=\frac{1}{\Gamma(1\pm\omega(k))(4\hbar\lambda)^{\pm\omega(k)}}.$ (107)

A necessary condition for the validity of the WKB approximation Eq. (71) is

$\frac{6M_{\rm pl}^{2}k^{2}}{\hbar^{2}}\gg 4\nu^{2},$ (108)

in which case $\omega(k)$ is purely imaginary and in fact $\omega(k)\sim
i\sqrt{6}M_{\rm pl}|k|/(2\hbar)$. Therefore, using Eq. (106) for functions
$f_{\pm}(k,a)$ with values of $k$ satisfying Eq. (108), it follows that

$\displaystyle F_{\pm}(k,a,\phi)$ $\displaystyle=$ $\displaystyle
f_{\pm}(k,a)e^{ik\phi/\hbar}$ (109) $\displaystyle\sim$ $\displaystyle
C_{\pm}a^{2\nu}\exp\left[\pm\frac{i}{\hbar}\,(\sqrt{6}\,M_{\rm pl}|k|\log a\pm
k\phi)\right],\quad\mbox{ as }a\rightarrow 0.$ (110)

Therefore, in view of Eqs. (75) and (110) it is clear that as $a\rightarrow 0$
the wavefunctions $\Psi_{\mp}^{(0)}(k,a,\phi)$ are proportional to the
semiclassical expansions of the explicit functions

$F_{\pm}(k,a,\phi)=I_{\pm\omega(k)}(a^{2}/2\hbar\lambda)a^{2\nu}e^{ik\phi/\hbar}.$
(111)

Figure 6: The functions (a)
$(2\hbar\lambda)^{\nu}\mathrm{Re}(\psi_{\pm}^{(0)}(k,a))$, and (b)
$\pm(2\hbar\lambda)^{\nu}\mathrm{Im}(\psi_{\pm}^{(0)}(k,a))$ on the
oscillatory region near $a=0$ for $\nu=1/6$ and $k=2\hbar/{M_{\rm pl}}$.

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# Interacting tachyonic scalar field 2

V K<EMAIL_ADDRESS>Adithya A<EMAIL_ADDRESS>and S D Pathak2<EMAIL_ADDRESS>1 Department of Physics, Sardar
Vallabhbhai National Institute of Technology, Surat, 395 007, India.
2Department of Physics, Lovely Professional University, Phagwara, Punjab, 144
411, India.

###### Abstract

The existence of dark energy is essential to explain the cosmic accelerated
expansion. We consider a homogenous interacting tachyonic scalar field as a
possible candidate for the dynamical dark energy. The interaction between the
tachyonic field and matter can be gauged to be linear in the energy density of
matter (or the tachyonic field) and Hubble’s parameter. We estimate the rate
of expansion, the age of the universe, the evolution of energy density of
matter and tachyonic field, and the coupling strength of the interaction for a
spatially flat ($k=0$) universe. We observed that the upper limit of coupling
strength is 1, and it is the same whether the interaction term depends on the
energy density of matter or the energy density of tachyonic scalar field.

##  I Introduction

Type Ia supernovae Riess_1998 ; Perlmutter_1999  observation confirms the
universe’s accelerated expansion. This observation demands the existence of a
medium with negative pressure, usually called dark energy. Different scalar
fields are observed to satisfy this negative pressure property and have been
used extensively to study dark energy copeland2006dynamics ;
padmanabhan2002can ; verma2014cosmic ; verma2014bicep2 . Some popular choices
of scalar fields are phantom cruz2022phantom ; narawade2022phantom ;
tripathy2020phantom ; johri2004phantom ; dkabrowski2006quantum ;
dabrowski2003phantom , quintessence adil2023quintessential ;
yang2019constraints ; wetterich2022quantum ; piedipalumbo2020noether , and
tachyon felegary2020evolution ; singh2019low ; koussour2022interacting ;
sadeghi2014phenomenological ; gorini2004tachyons ; kundu2021interacting . All
these scalar fields exhibit negative pressure, making each a possible
candidate for dark energy. In fact in Kumar:2023qum , it has been shown that
these three scalar fields are indistinguishable under the slow roll
approximation. In this article, we consider the tachyonic scalar field as a
candidate for dark energy, making it the source of accelerated expansion of
the universe.

The tachyonic scalar field was first defined in string theory Sen_2002 ;
Sen1_2002 ; Sen2_2002  and later used as a possible candidate for dark energy
padmanabhan2002can ; sadeghi2014phenomenological . The equation of state for a
tachyonic scalar field, $p=-\rho$, satisfies the negative pressure condition
for a universe component necessary to be termed as dark energy. Without
interactions, the tachyonic scalar field exhibits behavior similar to the
cosmological constant. But a static model of the universe, where the tachyonic
field does not interact with other universe components, suffers from two major
problems, the coincidence problem, and the Cosmological Constant problem. An
interacting model of dark energy resolves this problem Chimento_2010 ;
Chimento_2008 ; Bertolami_2012 ; Wang_2007 ; lu2012investigate ;
Farajollahi_2012 ; Zimdahl_2012 ; Yang_2018 ; Cao_2011 ; Di_Valentino_2020 ;
Verma_2012 ; Verma_2013 ; V_liviita_2010 ; Pan_2018 ; Amendola_2018 ;
B_gu__2019 ; Pan_2019 ; Papagiannopoulos_2020 ; Savastano_2019 ;
von_Marttens_2019 ; yang2019reconstructing ; Asghari_2019 . In such a model,
the dark energy field is considered dynamic, with energy exchange between the
matter and the universe’s dark energy content. A major challenge in
considering the interacting model is fixing the interactions’ functional
forms. Different forms of interactions have been proposed by several authors
based on dimensional and phenomenological arguments over the years
Chimento_2010 ; Chimento_2008 ; Bertolami_2012 ; Wang_2007 ; lu2012investigate
; Farajollahi_2012 ; Zimdahl_2012 ; Yang_2018 ; Di_Valentino_2020 ;
shahalam2017dynamics ; shahalam2015dynamics .

We consider an interacting model of dark energy, in which the dark energy is
modeled by a tachyonic scalar field that interacts with the universe’s matter
content. Specifically, we consider that the interaction between the tachyonic
field and matter depends linearly on the energy density of matter. Using this
interacting model, we investigate the behavior of the energy density, scale
factor, and age of the universe and the possible values of the coupling
strength. Previously we had considered the interaction to be linearly
dependent on the tachyonic field and found that the coupling strength of
interaction can not exceed the value 1 kundu2021interacting .

The article is organized as follows. We start with some theoretical background
in the section II, and then briefly discuss the interacting tachyonic scalar
field model in section III. Section III.1 and III.2 consist of derivation and
analysis of the evolution of energy density for matter and field. The
evolution of scale factor and the age of the universe are discussed in section
IV, and V respectively. A comparison of coupling strength when interaction
depends linearly on matter density and tachyonic field is presented in section
VI. Finally, we conclude in section VII.

##  II Theoretical Background

The FLRW metric in natural unit is given by

$g^{\mu\nu}=\text{diag}\left(-1,~{}\frac{a^{2}}{1-\displaystyle\frac{\kappa
r^{2}}{R_{0}^{2}}},~{}a^{2}r^{2},~{}a^{2}r^{2}\sin^{2}\theta\right)$ (1)

where $a$ is the time-dependent scale factor, and $\kappa$ is the global
curvature of the universe. This metric follows from the cosmological
principle, which implies that for our universe, the stress-energy tensor takes
on a relatively simple form

$T^{\mu\nu}=\text{diag}(\rho(t)^{2},~{}p(t),~{}p(t),~{}p(t))$ (2)

For a universe with the FLRW metric and the stress energy tensor given above,
the Friedmann equation follows from the Einstein field equations and is given
as

$\left(\frac{\dot{a}(t)}{a(t)}\right)^{2}=\frac{8\pi
G}{3c^{2}}\rho(t)-\frac{\kappa c^{2}}{R_{0}^{2}}a(t)^{2}$ (3)

For a flat universe, $\kappa=0$, and the Friedmann equation takes on a simpler
form

$\left(\frac{\dot{a}(t)}{a(t)}\right)^{2}=\frac{8\pi G}{3c^{2}}\rho(t)$ (4)

The principle of conservation of energy, when applied to the energy component
of the universe gives the fluid equation

$\dot{\rho}(t)+3\frac{\dot{a}(t)}{a(t)}\left(\rho(t)+p(t)\right)=0$ (5)

The last two equations, along with the equation of state,
$p(t)=\omega\rho(t)$, specify the dynamics of the universe.

##  III Interacting Tachyonic Scalar Field

We consider the universe with two dominant components: matter and dark energy,
with dark energy modeled by a spatially homogenous tachyonic scalar field
(TSF). The Lagrangian density for TSF is Sen_2002 ; Sen1_2002

$\mathcal{L}=-V(\phi)\sqrt{1-\partial^{\mu}\partial_{\mu}\phi}.$ (6)

For such a TSF, the stress-energy tensor is

$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-g^{\mu\nu}\mathcal{L},$
(7)

and the energy and pressure density follows directly from the stress-energy
tensor as

$\rho=\frac{V(\phi)}{\sqrt{1-\partial^{\mu}\partial_{\mu}\phi}},$ (8)

and

$p=-V(\phi)\sqrt{1-\partial^{\mu}\partial_{\mu}\phi}.$ (9)

Since the TSF is spatially homogeneous, the spatial derivatives vanish, while
the time derivative survives. Thus

$p=-(1-\dot{\phi}^{2})\rho$ (10)

giving $\omega_{\phi}=-(1-\dot{\phi}^{2})$.

We consider the interaction of the TSF with matter via the transfer of energy.
The two components exchange energy and hence violate individual energy
conservation. But overall, the total energy of the universe is conserved. Thus
for such a model, the continuity equation for the energy density of TSF
($\rho_{\phi}$) and matter ($\rho_{m}$) gets modified as

$\dot{\rho_{\phi}}+3\frac{\dot{a}}{a}(1+\omega_{\phi})\rho_{\phi}=-Q,$ (11)
$\dot{\rho_{m}}+3\frac{\dot{a}}{a}(1+\omega_{m})\rho_{m}=Q,$ (12)

where $\omega_{\phi}=p_{\phi}/\rho_{\phi}$, and $\omega_{m}=p_{m}/\rho_{m}$.
Based on the phenomenological argument, the functional form of the interaction
term can be guessed to be linear in the energy density of the matter or field
Pav_n_2008 ; DeArcia:2022nps . Previously we investigated the case when $Q$
depends linearly on the energy density of TSF ($\rho_{\phi}$)
kundu2021interacting . In this work, we investigate the case when $Q$ depends
linearly on the energy density of matter ($\rho_{m}$). In particular, we
choose $Q\propto\rho_{m}$, with the proportionality constant being
$3\displaystyle\beta\frac{\dot{a}}{a}$, where $\beta$ is a dimensionless
coupling constant specifying the strength of interaction. Thus the interaction
term takes on the form

$Q=3\beta\frac{\dot{a}}{a}\rho_{m}$ (13)

###  III.1 Evolution of Energy Densities

With the above-said interaction term, the continuity equations read

$\dot{\rho_{\phi}}+3\frac{\dot{a}}{a}(1+\omega_{\phi})\rho_{\phi}=-3\beta\frac{\dot{a}}{a}\rho_{m},$
(14)

and

$\dot{\rho_{m}}+3\frac{\dot{a}}{a}(1+\omega_{m})\rho_{m}=3\beta\frac{\dot{a}}{a}\rho_{m}.$
(15)

These can be solved to get the equation governing the evolution of
$\rho_{\phi}$ and $\rho_{m}$ with scale factor $a$.
$\bullet$ Solving for $\rho_{m}$

$\dot{\rho_{m}}+3\frac{\dot{a}}{a}(1+\omega_{m})=3\beta\displaystyle\frac{\dot{a}}{a}\rho_{m},$
$\implies\dot{\rho_{m}}+3\frac{\dot{a}}{a}(1+\omega_{m}-\beta)\rho_{m}=0$ (16)

This differential equation has a simple solution

$\frac{\rho_{m}}{\rho_{m}^{0}}=\left(\frac{a}{a^{0}}\right)^{-3(1+\omega_{m}-\beta)}=\left(\frac{a}{a^{0}}\right)^{-\gamma}$
(17)

where $\rho_{m}^{0}$ is the energy density of matter at the present time, and
$a^{0}$ is the scale factor at present time.

$\bullet$ Solving for $\rho_{\phi}$

$\dot{\rho_{\phi}}+3\frac{\dot{a}}{a}(1+\omega_{\phi})\rho_{\phi}=-3\beta\displaystyle\frac{\dot{a}}{a}\rho_{m}^{0}\left(\frac{a}{a^{0}}\right)^{-\gamma}$
(18)

Changing variables,
$a=xa^{0},~{}\rho_{\phi}=R\rho_{\phi}^{0}\implies\dot{a}=\dot{x}a^{0},~{}\dot{\rho_{\phi}}=\dot{R}\rho_{\phi}^{0}$

$\dot{R}+3\frac{\dot{x}}{x}(1+\omega_{\phi})R=-3\beta\dot{x}\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}x^{-\gamma-1}$
(19)

The equation can be rewritten as

$\frac{dR}{dx}+\frac{3(1+\omega_{\phi})}{x}R=-3\beta\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}x^{-\gamma-1}$
(20)

Solving, we get

$R=-x^{-3(1+\omega_{\phi})}\left(3\beta\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}\frac{x^{3(1+\omega_{\phi})-\gamma}}{3(1+\omega_{\phi})-\gamma}-1\right)-3\beta\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}\frac{-x^{-3(1+\omega_{\phi})}}{3(1+\omega_{\phi})-\gamma}$
(21)

For the TSF to mimic the cosmological constant, $\omega_{\phi}=-1$ (implying
that the TSF is a constant over time also, i.e. $\dot{\phi}=0$), thus the
above equation becomes

$\frac{\rho_{\phi}}{\rho_{\phi}^{0}}=3\beta\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}\frac{1}{\gamma}\left(\left(\frac{a}{a^{0}}\right)^{-\gamma}-1\right)+1$
(22)

The variation of the energy densities of matter and dark energy with the scale
factor for different values of $\beta$ is plotted in Fig.(1).

$\displaystyle\frac{\rho}{\rho^{0}}$ vs scale factor $a$

(a) $\beta=-1$

(b) $\beta=0$

(c) $\beta=0.25$

(d) $\beta=0.9$

Figure 1: Plot of the energy densities of matter and TSF against the scale
factor for different values of coupling constant $\beta$.

It is worth noting that $\beta$ determines the coupling strength between
matter and the scalar field $\phi$. For $\beta>0$, dark energy decreases as
matter density increases. On the other hand, for $\beta<0$, dark energy
increases as matter density increases. When $\beta=0$, the interaction
vanishes, and the two sectors evolve independently, i.e., the cosmological
constant scenario.

###  III.2 Evolution of $\Omega$ with $\text{ln}(a)$

The equations of cosmology can be written in terms of another variable -
density parameter, defined as the ratio of the energy density and the critical
density $\rho_{c}$,

$\Omega=\frac{\rho}{\rho_{c}}=\frac{8\pi Ga^{2}}{3\dot{a}^{2}}\rho=\frac{8\pi
G}{3H^{2}}\rho,~{}\text{with}~{}H=\frac{\dot{a}}{a}.$ (23)

The numerical value of $\Omega$ indicates the nature of the universe locally.
For $\Omega=1$, the universe is spatially flat; for $\Omega<1$, it is
negatively curved; and for $\Omega>1$, it is positively curved.
Since, for any function $f$

$\frac{df}{d\text{ln}(a)}=\frac{df}{da}/\frac{d\text{ln}(a)}{da}=a.(\frac{df}{dt}/\frac{da}{dt}),$

therefore, for $H=\displaystyle\frac{\dot{a}}{a}$, we have

$\frac{df}{d\text{ln}(a)}=\frac{1}{H}\frac{df}{dt}.$ (24)

Defining $\Omega_{m}=\displaystyle\frac{\rho_{m}}{\rho_{c}}$,
$\Omega_{\phi}=\displaystyle\frac{\rho_{\phi}}{\rho_{c}}$, we can write

$\frac{d\Omega_{m}}{d\text{ln}(a)}=\frac{8\pi
G}{3}\frac{1}{H}\left(\frac{\dot{\rho_{m}}}{H^{2}}-2\frac{\rho_{m}}{H^{3}}\dot{H}\right)$
(25)

and

$\frac{d\Omega_{\phi}}{d\text{ln}(a)}=\frac{8\pi
G}{3}\frac{1}{H}\left(\frac{\dot{\rho_{\phi}}}{H^{2}}-2\frac{\rho_{\phi}}{H^{3}}\dot{H}\right)$
(26)

From the continuity equations (Eq.(14,15)), we get

$\dot{\rho}_{\phi}=-3\beta
H\rho_{m}~{}~{}\&~{}~{}\dot{\rho}_{m}=-3H\rho_{m}+3\beta H\rho_{m}$ (27)

and from the Friedmann equation, we get

$2H\dot{H}=\frac{8\pi G}{3}(\dot{\rho}_{m}+\dot{\rho}_{\phi})=\frac{8\pi
G}{3}(-3\rho_{m})$ $\implies\dot{H}=\frac{-8\pi G\rho_{m}}{2}$ (28)

Using Eqs.(27,III.2), and the fact that for a spatially flat universe such as
ours, $\Omega_{m}+\Omega_{\rho}=1$, the differential Eqs.(25,26) become

$\frac{d\Omega_{m}}{d\text{ln}(a)}=3\beta\Omega_{m}+3\Omega_{m}^{2}-3\Omega_{m},$
(29)

and

$\frac{d\Omega_{\phi}}{\text{ln}(a)}=3(1-\Omega_{\phi})\Omega_{\phi}-3\beta(1-\Omega_{\phi}).$
(30)

These equations are solved, using conditions $\Omega_{m}|_{a=1}=0.3$ and
$\Omega_{\phi}|_{a=1}=0.7$ for different values of $\beta$ and the evolution
of $\Omega_{m}$ and $\Omega_{\phi}$ with ln$(a)$ is shown in Fig.(2).

For $\beta=0$ we recover the result for our universe, the matter density is
dominant initially, and in the future, the dark energy density will be
dominant. Also we can see that we are at the epoch of equality of $\Omega_{m}$
and $\Omega_{\phi}$. For negative $\beta$, $\Omega_{\phi}$ goes to negative
values as expected from the behavior of $\rho_{\phi}$. For universes with
positive $\beta$, the initial difference between $\Omega_{m}$ and
$\Omega_{\phi}$ is reduced as $\beta$ is increased, and for $\beta\gtrapprox
0.5$, dark energy becomes the dominant constituent at all times.

$\Omega~{}\text{vs ln}(a)$ for different values of coupling constant $\beta$

(a) $\beta=-0.2$

(b) $\beta=0$

(c) $\beta=0.2$

(d) $\beta=0.6$

Figure 2: Plot of the evolution of density parameter $\Omega$ for matter and
TSF as the function of ln$(a)$ for different values of coupling constant
$\beta$.

##  IV Evolution of scale factor

Another important factor to consider is how the scale factor varies with time
in the universe. This not only quantifies the expansion of the universe but
also allows for the conversion between functions of $a$ and $t$. The Friedmann
equation gives a relation between $a$ and $\rho$ and this can be used to
obtain the form of scale factor as a function of time. The Friedmann equation
can be written as

$\frac{\dot{a}}{a}=\sqrt{\frac{8\pi
G}{3c^{2}}}~{}\sqrt{\rho_{\phi}+\rho_{m}}.$ (31)

Using the equation for the evolution of energy densities (Eqs.(17,22)), and
$a=xa^{0}$, we get

$\frac{\dot{x}}{x}=\sqrt{\frac{8\pi
G}{3c^{2}}}\sqrt{\rho_{m}^{0}x^{-\gamma}+\rho_{\phi}^{0}\left(3\beta\frac{\rho_{m}^{0}}{\rho_{\phi}^{0}}\frac{1}{\gamma}\left(x^{-\gamma}-1\right)+1\right)},$

which on further simplification becomes

$\frac{\dot{x}}{x}=\sqrt{\frac{8\pi
G\rho_{m}^{0}}{3c^{2}}}\sqrt{x^{-\gamma}+3\beta\frac{1}{\gamma}\left(x^{-\gamma}-1\right)+\frac{\rho_{\phi}^{0}}{\rho_{m}^{0}}}.$
(32)

From this, we can write

$t=\sqrt{\frac{3c^{2}}{8\pi
G\rho_{m}^{0}}}\bigint\frac{dx}{x\sqrt{\displaystyle\frac{1}{x^{\gamma}}\left(1+\frac{3\beta}{\gamma}\right)+\left(\frac{\rho_{\phi}^{0}}{\rho_{m}^{0}}-\frac{3\beta}{\gamma}\right)}~{}},$
(33)

and using $\omega_{m}=0$, $\rho_{c}^{0}=8.7\times
10^{-27}~{}\mathrm{kg/m^{3}}$, $\rho_{m}^{0}=\Omega_{m}^{0}\rho_{c}^{0}$,
$\rho_{\phi}^{0}=\Omega_{\phi}^{0}\rho_{c}^{0}$ and integrating over $x$ gives

$H^{0}t=-\frac{6.28701\sinh^{-1}\left(\displaystyle\frac{1}{3}\sqrt{21-30\beta}\sqrt{x^{3-3\beta}}\right)}{\sqrt{21-30\beta}\sqrt{3-3\beta}},$

implying that the scale factor $x$ evolves as

$x=\frac{a}{a^{0}}=\left(3\frac{\sinh\left(0.159\sqrt{21-30\beta}\sqrt{3-3\beta}~{}tH^{0}\right)}{\sqrt{21-30\beta}}\right)^{\displaystyle\frac{2}{3-3\beta}}.$
(34)

The expansion of the universe, i.e., the evolution of the scale factor with
time for different values of coupling constant $\beta$ is plotted in FIG. 3.
For our universe, with $\beta=0$, the plot shows that the universe’s expansion
is decelerating in the initial stage and then accelerating at the later stage.
For negative $\beta$, the initial expansion rate of the universe is magnified,
making the deceleration curve more prominent. In contrast, for positive
$\beta$, the initial expansion rate is very small, and the universe’s
deceleration does not occur.

$a~{}\text{vs}~{}t$ for different values of coupling constant $\beta$

(a) $\beta=-1$

(b) $\beta=0$

(c) $\beta=0.25$

(d) $\beta=0.5$

(e) $\beta=0.75$

(f) $\beta=0.95$

Figure 3: Plot of the evolution of scale factor $a$ of the universe for
different values of coupling constant $\beta$.

##  V Age of the Universe

The Age of the Universe (AOU) is the difference between the times when the
scale factor is 1 (i.e., the present day) and when the scale factor was 0 (the
Big Bang). This time difference which gives the AOU can be directly calculated
as the definite integral of the equation

$t_{AOU}=(H^{0})^{-1}H^{0}\sqrt{\frac{3c^{2}}{8\pi
G\rho_{m}^{0}}}\bigints_{0}^{1}\frac{dx}{x\sqrt{\displaystyle\frac{1}{x^{\gamma}}\left(1+\frac{3\beta}{\gamma}\right)+\left(\frac{\rho_{\phi}^{0}}{\rho_{m}^{0}}-\frac{3\beta}{\gamma}\right)}~{}}$

Using the values of Hubble’s constant $H^{0}$, and the matter and dark energy
density from cosmological observations, and integrating the equation over $x$,
we obtain the age of the universe as a function of the coupling constant
$\beta$

$t_{AOU}(\beta)=(H^{0})^{-1}\frac{1.21\,_{2}F_{1}(0.5,0.5;1.5;3.33\beta-2.33)}{\sqrt{1-\beta}},$
(35)

for $\beta<1$. This equation is plotted to visualize the variation of the age
of the universe with the coupling constant $\beta$ in Fig.(4).

For $\beta\geq 1$ the definite integral does not converge to a single value
and hence the age of the universe becomes indeterminate. Therefore such a
model cannot represent any real universe, and hence we obtain an upper limit
on the coupling constant $\beta$ for a real universe, i.e. $\beta<1$.

Figure 4: The Age of the Universe (AOU) as the function of coupling constant.

| $\beta$ | AOU
---|---
-1. | 0.575731
-0.9 | 0.598269
-0.8 | 0.622909
-0.7 | 0.649977
-0.6 | 0.679872
-0.5 | 0.713089
-0.4 | 0.75025
-0.3 | 0.792149
-0.2 | 0.839813
-0.1 | 0.894606
| $\beta$ | AOU
---|---
0. | 0.958375
0.1 | 1.03369
0.2 | 1.12427
0.3 | 1.23568
0.4 | 1.37672
0.5 | 1.56232
0.6 | 1.82015
0.7 | 2.20903
0.8 | 2.88417
0.9 | 4.47667
Table 1: The Age of the Universe (in the unit of $(H^{0})^{-1}$) for different
values of $\beta$.

There does not exist a lower limit on the coupling constant, as there are no
breaks or discontinuities in the age of the universe as a function of $\beta$.
The $t_{AOU}$ is a smooth and continuous function of $\beta$, with a lower
limit $-\infty$ and an upper limit $1$ on the parameter $\beta$. The age of
the universe converges to 0 as the coupling constant goes to $-\infty$, i.e.
for a universe with coupling constant $\beta\rightarrow-\infty$, the evolution
of the universe from $a=0$ to $a=1$ takes place instantaneously. As
$\beta\rightarrow 1$, the AOU tends to $\infty$, which implies that the
dynamics of such a universe are very slow on cosmic scales. This behavior is
also visible in the evolution of scale factor, where the scale factor tends to
0 even at large t as $\beta\rightarrow 1$. The numerical value of the Age of
the Universe in the terms of inverse Hubble’s constant for different values of
$\beta$ have also been tabulated in Table 1.

##  VI Comparison of coupling strength: $3\beta\rho_{m}$ vs
$\alpha\rho_{\phi}$

The interaction term $Q$ can have two possible forms: $Q=\alpha\rho_{\phi}$,
or $Q=3\beta\rho_{m}$. The complete analysis for the former form has been done
in kundu2021interacting  with a significant conclusion that even though the
coupling strength has no lower bound, the upper bound on the coupling constant
$\alpha$ must be 1. In this article, we did the entire analysis by taking the
later form of the interaction. We obtain a similar conclusion for the coupling
strength $\beta$, i.e., it has no lower bound, but the upper bound must be 1.

As the possible range of both $\alpha$ and $\beta$ are the same $(-\infty,1)$,
the question arises: are they different, or are they just the same thing with
different notations? Further investigations in this line can be interesting
and significant if we can replace the multiple coupling constant with just
one. The age of the universe as a function of the coupling constant obtained
from both form of interaction is compared in Fig.(5).

Both the coupling constants give rise to the universe, whose age decreases
with the value of the coupling constant and goes to infinity as it approaches
1. Both are continuously increasing functions, and for the interaction-less
universe, both converge to the same value: our universe’s age.

Figure 5: The age of the universe as a function of the coupling constant in
the two different models considered. Here $\zeta$ is used as a common coupling
constant in place of $\alpha$ and $\beta$.

##  VII Conclusion

We have investigated the dynamics of a universe with an interacting tachyonic
scalar field as a possible source of dark energy. The interaction between the
matter and dark energy component is modeled as an energy exchange, with the
energy exchange depending linearly on the energy density of the matter. We
also presented the evolution of energy densities with respect to scale factor,
as well as the evolution of density parameter ($\Omega$) with the natural
logarithm of the scale factor and derived the Age of the Universe was derived
as the function of the coupling constant $\beta$.

Our analysis reveals that the case where $\beta=0$, (no interaction between
matter and TSF) corresponds to the dynamics of our universe, as expected. We
have also found that for $\beta\geq 1$, the age of the universe does not
converge and hence becomes indeterminate. This puts an upper bound on the
possible values of the coupling constant $\beta$ for real universes. This
result is similar to the case when interaction term depend linearly on the
energy density of the TSF kundu2021interacting . A comparison with the work
kundu2021interacting  shows that in both the cases, the constraints on the
coupling constant are the same, implying that the bound on the coupling
constant is the same for both the case, irrespective of whether the
interaction term depends on the energy density of matter or the energy density
of dark energy (tachyonic field). Since the bound on both $\alpha$ and $\beta$
are the same, i.e. $(-\infty,1)$, it would be interesting to further
investigate whether the two can be unified and replaced by a single coupling
constant.

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