#  Noether Symmetry analysis in Chameleon Field Cosmology

Roshni<EMAIL_ADDRESS>Sourav
<EMAIL_ADDRESS>Subenoy<EMAIL_ADDRESS>1Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India
2Department of Mathematics, Dr. Meghnad Saha College, Itahar, Uttar
Dinajpur-733128, West Bengal, India.

###### Abstract

This work deals with chameleon field cosmology (a scalar field nonminimally
coupled to cold dark matter) in the background of flat Friedmann-Lemaitre-
Robertson-Walker (FLRW) space-time. Both classical and quantum cosmology have
been investigated using Noether symmetry analysis of the underlying physical
system. The Wheeler-DeWitt (WD) equation has been constructed on the
minisuperspace and solutions have been obtained using conserved charge.

Keywords: Noether Symmetry; quantum cosmology; chameleon scalar field.

##  I Introduction

Standard cosmology has been facing a great challenge since the end of the last
century. The observational evidences since 1998 r1 ; r2 ; r3 ; r4 ; r5  are
not in favour of decelerated expansion (prediction of standard cosmology)
rather they are in favour of accelerated expansion. So far there are two
options proposed by the cosmologists to accomodate these observational
evidences. One of the possibilities is to introduce some exotic matter (known
as dark energy (DE)) within the framework of Einstein gravity. This mysterious
matter component is totally of unknown nature except its large negative
pressure. At first cosmologists took cosmological constant as the DE
candidate. But due to two severe drawbacks (namely discrepancy in its
predicted and observed value and coincidence problem r6 ) the cosmological
constant is not a well accepted DE model rather dynamical DE models r7 ; r8 ;
r9 ; r10  are widely used in the literature. This work is an example of using
dynamical DE model. Usually, a scalar field having potential $V(\phi)$ is
chosen as DE candidate so that the pressure component
$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$ can evolve to have the required
negative value for observed accelerated expansion. Here, the scalar field
(chosen as dynamical DE) is nonminimal coupled to dark matter (DM) through an
interference term in the action r11 . As a result, there is a new source term
in the matter conservation equation. This kind of DE model is termed as
chameleon field. This model is quite useful to obtain accelerated expansion of
the universe and other interesting cosmological consequences r12  (for details
see the Ref. r13 ).

On the other hand, since the last century, symmetry analysis has a significant
role in studying global continuous symmetries (i.e, translation and rotation)
as well as in local gauge symmetries, internal symmetries of the space-time
(in cosmology) and permutation symmetry in quantum field theory r14 ; r15 . In
particular, geometrical symmetries of the space-time and symmetries of the
physical system have great role in analyzing any physical motion. From the
perspective of Noether symmetry, the conserved charge has been used to
identify the actual one among similar physical processes. Furthermore, in
Noether symmetry approach, the Noether integral (i.e, the first integral) has
been chosen as a tool for simplification of a system of differential equations
or for the integrability of the system r16 ; r17 ; r18 ; r19 ; r20 ; r21 .

In addition, an advantage of using Noether symmetry to any physical system
involving arbitrary physical parameters or some arbitrary functions of the
field variables is that symmetry analysis uniquely determines these physical
parameters or arbitrary functions involved (for details see Ref.r22 ). Also
since recent past symmetry analysis has been used for physical systems in
Riemannian spaces r23 ; r24 ; r25 ; r26 ; r27 ; r28 ; r29 ; r1.3 .

Moreover, Noether symmetry analysis has opened new windows in studying quantum
cosmology with suitable operator ordering, and the Wheeler DeWitt (WD)
equation so constructed on the minisuperspace is associated with Lie point
symmetries. It is possible to have a subset of the general solutions of the WD
equation having oscillatory behaviours r30 ; r31  by imposing Noether
symmetries. The Noether symmetries with Hartle criterion can identify those
classical trajectories in minisuperspace r32 ; r33  which are solutions of the
cosmological evolution equations i.e, one may consider Noether symmetries as a
bridge between quantum cosmology and classical observable universe.

This work is another example of extensive use of Noether symmetry analysis to
both classical and quantum cosmology for chameleon field DE model. By imposing
Noether symmetry to the Lagrangian and making canonical transformation of the
dynamical variables it is possible to have classical solutions of the coupled
nonlinear Einstein field equations. WD equation is constructed for the present
chameleon DE cosmological model in the background of FLRW space-time and
Noether symmetry is used as a tool to solve the WD equation. The plan of the
paper is as follows: a brief overview of Noether symmetry is described in
Section–II whereas Section– III presents the Noether symmetry and cosmological
solutions to chameleon field DE model and Section-IV deals with quantum
cosmology in the minisuperspace approach: a general prescription and the
formation of WD equation in the present cosmological model and possible
solution with Noether symmetry are presented in Section-V; finally the paper
ends with a conclusion in Section-VI.

##  II A Brief Overview of Noether Symmetry Approach

Noether’s first theorem states that any physical system is associated with
some conserved quantities provided the Lagrangian of the system is invariant
with respect to the Lie derivative r34 ; r35  along an appropriate vector
field ($\mathcal{L}_{\overrightarrow{V}}f=\overrightarrow{V}(f)$). By imposing
these symmetry constraints, the evolution equations of the physical system can
either be solvable or simplified to a great extent r36 ; r37 .

For a point like canonical Lagrangian
$L[q^{\alpha}(x^{i}),\dot{q}^{\alpha}(x^{i})]$, the Euler-Lagrange equations

$\partial_{j}\left(\frac{\partial
L}{\partial\partial_{j}q^{\alpha}}\right)=\frac{\partial L}{\partial
q^{\alpha}}$ (1)

can be contracted with some unknown functions $\lambda^{\alpha}(q^{\beta})$ as
follows:

$\lambda^{\alpha}\bigg{[}\partial_{j}\left(\frac{\partial
L}{\partial\partial_{j}q^{\alpha}}\right)-\frac{\partial L}{\partial
q^{\alpha}}\bigg{]}=0$ (2)

i.e,

$\lambda^{\alpha}\frac{\partial L}{\partial
q^{\alpha}}+(\partial_{j}\lambda^{\alpha})\left(\frac{\partial
L}{\partial\partial_{j}q^{\alpha}}\right)=\partial_{j}\left(\lambda^{\alpha}\frac{\partial
L}{\partial\partial_{j}q^{\alpha}}\right)$

Thus

$\mathcal{L}_{\overrightarrow{X}}L=\lambda^{\alpha}\frac{\partial L}{\partial
q^{\alpha}}+(\partial_{j}\lambda^{\alpha})\frac{\partial
L}{\partial\left(\partial_{j}q^{\alpha}\right)}=\partial_{j}\left(\lambda^{\alpha}\frac{\partial
L}{\partial\partial_{j}q^{\alpha}}\right)$ (3)

So according to Noether theorem the vector field r38 ; r39

$\overrightarrow{X}=\lambda^{\alpha}\frac{\partial}{\partial
q^{\alpha}}+\left(\partial_{j}q^{\alpha}\right)\frac{\partial}{\partial\left(\partial_{j}q^{\alpha}\right)}$
(4)

can be chosen appropriately so that the Lagrangian of the system is invariant
along the vector field i.e, $\mathcal{L}_{\overrightarrow{X}}L=0$ and
consequently, the physical system is called invariant under Noether symmetry
with $\overrightarrow{X}$, the infinitesimal generator of the symmetry. It is
to be noted that the above symmetry vector as well as the Lagrangian is
defined on the tangent space of configurations:
$TQ\\{q^{\alpha},\dot{q}^{\alpha}\\}$. In general Noether symmetry approach is
very much relevant to identify conserved quantities of a physical system. The
above symmetry condition is associated with a constant of motion for the
Lagrangian having conserved phase flux along the vector field
$\overrightarrow{X}$. Furthermore, from Eq.(3) this symmetry criteria is
associated with a constant of motion of the system r16 ; r17 ; r36

$Q^{i}=\lambda^{\alpha}\frac{\partial
L}{\partial\left(\partial_{i}q^{\alpha}\right)}$ (5)

satisfying

$\partial_{i}Q^{i}=0$ (6)

So $Q^{i}$ is identified as Noether current or conserved current. Furthermore,
the energy function associated with system is

$E=\dot{q}^{\alpha}\frac{\partial L}{\partial\dot{q}^{\alpha}}-L$ (7)

The energy function (also known as Hamiltonian of the system) is a constant of
motion provided there is no explicit time dependence in the Lagrangian r16 ;
r17 ; r36 . Moreover, if the conserved current due to Noether symmetry has
some physical meaning, r16 ; r17 ; r36  then symmetry analysis can identify
reliable models. In the following, we shall show how symmetry analysis will
simplify the present coupled cosmological model and as a result classical
cosmological solutions can be obtained easily.

In the context of quantum cosmology, Hamiltonian formulation is very useful
and Noether symmetry condition is rewritten as follows: r37

$\mathcal{L}_{\overrightarrow{X}_{H}}H=0$ (8)

with

${\overrightarrow{X}_{H}}=\dot{q}\frac{\partial}{\partial
q}+\ddot{q}\frac{\partial}{\partial\dot{q}}$

In minisuperspace models of quantum cosmology, symmetry analysis determines
appropriate interpretation of the wave function. The conserved canonically
conjugate momenta due to Noether symmetry can be written as follows:

$\Pi_{l}=\frac{\partial L}{\partial q^{l}}={\sum}_{l}$ (9)

$l=1,2,...,m,$ where ‘$m$’ denotes the number of symmetries. Also, the
operator version (i.e, quantization) of Eq. (9) i.e,

$-i\partial_{q^{l}}\ket{\psi}={\sum}_{l}\ket{\psi}$ (10)

identifies a translation along $q^{l}$-axis through symmetry analysis. Also
Eq. (10) has oscillatory solution for real conserved quantity $\sum_{l}$ i.e,

$\ket{\psi}=\sum_{l=1}^{m}e^{i\sum_{l}q^{l}}\ket{\phi(q^{k})},k<n,$ (11)

where the index ‘$k$’ stands for directions along which there is no symmetry
with $n$ the dimension of the minisuperspace. Thus oscillatory part of the
wave function implies existence of Noether symmetry and the conjugate momenta
along the symmetry directions should be conserved and vice-versa r41 . Due to
symmetries the first integrals of motion identify the classical trajectories.
In fact, for 2D minisuperspace, it is possible to have complete solution of
the system by Noether symmetry.

##  III Noether Symmetry and Cosmological Solutions to Chameleon Field DE
Model

This section is devoted to study chameleon field DE cosmological model. This
model consists of a canonical scalar field (having self-interaction potential)
nonminimally coupled to DM. So the potential function and the coupling
function are the two unknown functions of the scalar field. The action
integral of the model has the explicit form r42 ; r43

$I=\int\left[\frac{R}{16\pi
G}+\frac{1}{2}\phi_{,\mu}\phi^{,\mu}-V(\phi)+f(\phi)L_{m}\right]\sqrt{-g}d^{4}x$
(12)

where as usual $R$ is the Ricci scalar, $G$ is the Newtonian gravitational
constant and $\phi$ is the chameleon scalar field having potential $V(\phi)$.
Here, $L_{m}$ is the Lagrangian for DM which is nonminimally coupled to the
chameleon scalar field with $f(\phi)$ (an analytic function), the coupling
function. By choosing the DM to be an ideal gas, the matter Lagrangian can be
chosen as $L_{m}\simeq\rho_{m}$ r1.1 .

In the background of flat FLRW space-time the point-like Lagrangian for the
above cosmological model takes the following form:

$L(a,\dot{a},\phi,\dot{\phi})=3a\dot{a}^{2}-a^{3}\left(\frac{\dot{\phi}^{2}}{2}-V(\phi)\right)-\rho_{m}f(\phi)a^{3}$
(13)

Now the Euler-Lagrange equations (i.e, the Einstein field equations) for the
Lagrangian (13) are given by

$\displaystyle 3H^{2}=\rho_{m}f(\phi)+\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (14)
$\displaystyle{2\dot{H}+3H^{2}=-\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\rho_{m}\omega
f(\phi)},$ (15)

where an over dot indicates differentiation with respect to the cosmic time
‘$t$’. Furthermore, the equation of motion $T^{\mu\nu}_{;\nu}=0$ for the
cosmological fluid with energy momentum tensor
$T_{\mu\nu}=T^{(\phi)}_{\mu\nu}+T^{(m)}_{\mu\nu}$ is given by

${\dot{\phi}\ddot{\phi}+3H\dot{\phi}^{2}+v(\phi)\dot{\phi}+\rho_{m}f^{\prime}(\phi)\dot{\phi}+\dot{\rho_{m}}f(\phi)+3H(1+\omega)\rho_{m}f(\phi)=0}$
(16)

One may note that among these three evolution equations (14)-(16), only two
are independent while (14) is termed as constraint equation.

As in the present cosmological model there is the interaction term $f(\phi)$
so one has $\left(T^{(\phi)\mu\nu}\right)_{;\nu}=-Q$,
$\left(T^{(m)\mu\nu}\right)_{;\nu}=Q$ or eqivalently

${\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)+f^{\prime}(\phi)\rho_{m}=-\frac{Q}{\dot{\phi}}}$
(17)

and

${\dot{\rho_{m}}f(\phi)+3H(1+\omega)\rho_{m}f(\phi)=Q}$ (18)

As the present model reduces to that of Weyl integrable gravity with
$f(\phi)=f_{0}e^{\lambda\phi}$ (r1.2 ) and setting
$Q=\alpha\rho_{m}f^{\prime}(\phi)\dot{\phi}$ (where $\alpha$ is a non zero
constant), Eqs. (17) and (18) become

${\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)+(1+\alpha)f^{\prime}(\phi)\rho_{m}=0}$
(19)

and

${\dot{\rho_{m}}f(\phi)+3H(1+\omega)\rho_{m}f(\phi)-\alpha
f^{\prime}(\phi)\dot{\phi}\rho_{m}=0}$ (20)

The matter conservation equation (20) can be integrated to have

${\rho_{m}(t)=\rho_{0}a^{-3(1+\omega)}\\{f(\phi)\\}^{\alpha}}$ (21)

Thus the scalar field evolution equation (i.e, the modified Klein-Gordon
equation) (19) becomes

${\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=-\rho_{0}a^{-3(1+\omega)}\left[\\{f(\phi)\\}^{\alpha+1}\right]}.$
(22)

The configuration space for the present model is a 2D space $(a,\phi)$ and the
infinitesimal generator for the Noether symmetry takes the form

$\overrightarrow{X}=p\frac{\partial}{\partial
a}+q\frac{\partial}{\partial\phi}+\dot{p}\frac{\partial}{\partial\dot{a}}+\dot{q}\frac{\partial}{\partial\dot{\phi}},$
(23)

where $p=p(a,\phi)$ and $q=q(a,\phi)$ are the unknown coefficients with
$\dot{p}=\frac{\partial p}{\partial a}\dot{a}+\frac{\partial
p}{\partial\phi}\dot{\phi}$ and similarly for $\dot{q}$.

These coefficients of the Noether symmetry vector are determined from an
overdetermined system of partial differential equation, obtained by imposing
Noether symmetry to the Lagrangian i.e,

$\mathcal{L}_{\overrightarrow{X}}L=0$

i.e,

$\displaystyle p+2a\frac{\partial p}{\partial a}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle 3p+2a\frac{\partial q}{\partial\phi}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle 6\frac{\partial
p}{\partial\phi}-a^{2}\frac{\partial q}{\partial a}$ $\displaystyle=$
$\displaystyle 0$ (24)

with a differential equation for the potential and coupling function as
follows:

${3\rho_{0}\omega
pa^{-3\omega-1}F(\phi)+3pa^{2}V(\phi)+qa^{3}V^{\prime}(\phi)-\rho_{0}a^{-3\omega}qF^{\prime}(\phi)=0}$
(25)

where ${F(\phi)=\\{f(\phi)\\}^{\alpha+1}}$.

The above set of partial differential equations (24) are solvable using the
method of separation of variables i.e, $p(a,\phi)=p_{1}(a)p_{2}(\phi)$,
$q(a,\phi)=q_{1}(a)q_{2}(\phi)$ as

$p=a^{-\frac{1}{2}}\left(c_{p}e^{m\phi}+c_{q}e^{-m\phi}\right)$
${q=-4ma^{-\frac{3}{2}}\left(c_{p}e^{m\phi}-c_{q}e^{-m\phi}\right)}$ (26)

where $m^{2}=\frac{3}{8}$, $c_{p}$, $c_{q}$ and $q_{0}$ are arbitrary
constants. Using the above solutions (26) into (25), the solutions for
$V(\phi)$ and $f(\phi)$ can take the form (with $\omega=-1$)

${V(\phi)-\rho_{0}F(\phi)=k\left(c_{p}e^{m\phi}-c_{q}e^{-m\phi}\right)^{2}}$
(27)

where $k$ is a positive integration constant.

Thus, the infinitesimal generator of the Noether symmetry is determined
(except for arbitrary integration constants) by imposing symmetry condition
which in turn determines a relation between the potential function and the
coupling function.

Another important issue related to Noether symmetry is the conserved
quantities associated with it. In general for a field theory in curved space
there is no well-defined notion of energy. However, the conserved quantity
derived from Noether’s theorem is the energy-momentum tensor. In particular,
when the system has time-like killing vector then there is an associated
conserved energy. Though FLRW space-time has no time-like killing vector
field, but the Lagrangian density is explicit time independent. Hence in
analogy with point-like Lagrangian, it is possible to define an energy which
will be conserved in nature. Thus in the context of Noether symmetry to the
present cosmological model one can have two conserved quantities, namely
conserved charge (defined in Eq. (5)) and conserved energy (defined in Eq.
(7)) having explicit form

${Q=6\dot{a}a^{\frac{1}{2}}\left(c_{p}e^{m\phi}+c_{q}e^{-m\phi}\right)+a^{3}\dot{\phi}\bigg{\\{}4ma^{-\frac{3}{2}}\left(c_{p}e^{m\phi}-c_{q}e^{-m\phi}\right)\bigg{\\}}}$
${E=3a\dot{a}^{2}-\frac{1}{2}a^{3}\dot{\phi}^{2}-a^{3}V(\phi)+\rho_{0}F(\phi)a^{-3\omega}}$
(28)

Usually, associated with Noether symmetry there is a conserved current
(defined in Eq. (5)), whose time component integrating over spatial volume
gives a conserved charge. But in the present context as all the variables are
time dependent only so $Q$ defined in (28) is the Noether charge. Moreover,
the above conserved charge can be expressed geometrically as the inner product
of the infinitesimal generator with cartan one form r44  as follows:

$Q=i_{\overrightarrow{X}}\theta_{L}$ (29)

where $i_{\overrightarrow{X}}$ denotes the inner product with the vector field
$\overrightarrow{X}$ and

$\theta_{L}=\frac{\partial L}{\partial a}da+\frac{\partial
L}{\partial\phi}d\phi$ (30)

is termed as cartan one form.

On the other hand, this geometric inner product representation is useful to
find out cyclic variables in the Lagrangian. In context of solving coupled
nonlinear evolution equations, determination of cyclic variables will be very
useful as not only the Lagrangian but also the evolution equations will be
simplified to a great extent.

In the present context the transformation of the 2D augmented space:
$(a,\phi)\rightarrow(u,v)$ transform the symmetry vector as

$\overrightarrow{X_{T}}=\left(i_{\overrightarrow{X}}du\right)\frac{\partial}{\partial
u}+\left(i_{\overrightarrow{X}}dv\right)\frac{\partial}{\partial
v}+\left\\{\frac{d}{dt}\left(i_{\overrightarrow{X}}du\right)\right\\}\frac{d}{d\dot{u}}+\left\\{\frac{d}{dt}\left(i_{\overrightarrow{X}}dv\right)\right\\}\frac{d}{d\dot{v}}$
(31)

Geometrically, $\overrightarrow{X_{T}}$ may be interpreted as the lift of a
vector field on the augmented space. Now, without any loss of generality we
restrict the above point transformation to r44

$i_{\overrightarrow{X}}du=1~{}~{}\mbox{and}~{}~{}i_{\overrightarrow{X}}dv=0$
(32)

so that

$\overrightarrow{X_{T}}=\frac{\partial}{\partial
u}~{}~{}\mbox{and}~{}~{}\frac{\partial L_{T}}{\partial u}=0$ (33)

i.e, $u$ is the cyclic variable. The above geometric process of identification
of cyclic variables can be interpreted so as to choose the transformed
infinitesimal generator along any co-ordinate line (identified as the cyclic
variable) r45 .

Now the explicit form of the above point transformation (32) is the first-
order linear partial differential equations having solution as follows:

Case I: $c_{p}=c_{q}$

$\displaystyle u$ $\displaystyle=$
$\displaystyle\frac{2}{3}a^{\frac{3}{2}}\cosh{m\phi},$ $\displaystyle v$
$\displaystyle=$ $\displaystyle a^{\frac{3}{2}}\sinh{m\phi}.$ (34)

Case II: $c_{p}\neq c_{q}$

$\displaystyle u$ $\displaystyle=$
$\displaystyle\frac{1}{6c_{p}c_{q}}a^{\frac{3}{2}}\left(c_{p}e^{m\phi}+c_{q}e^{-m\phi}\right),$
$\displaystyle v$ $\displaystyle=$ $\displaystyle
a^{\frac{3}{2}}\left(c_{p}e^{m\phi}-c_{q}e^{-m\phi}\right).$ (35)

The simplified Lagrangian in the new variables has the following forms:

$\displaystyle L$ $\displaystyle=$ $\displaystyle
3\dot{u}^{2}-\frac{4}{3}\dot{v}^{2}+4kc_{p}^{2}v^{2},~{}~{}~{}~{}~{}(\mbox{Case
I})$ (36) $\displaystyle=$ $\displaystyle
12c_{p}c_{q}\dot{u}^{2}-\frac{1}{3c_{p}c_{q}}\dot{v}^{2}+kv^{2}~{}~{}~{}~{}~{}(\mbox{Case
II}).$ (37)

The conserved quantities in the new variables can be expressed as follows:

$\displaystyle Q$ $\displaystyle=$ $\displaystyle 6\dot{u}$ $\displaystyle E$
$\displaystyle=$ $\displaystyle
3\dot{u}^{2}-\frac{4}{3}\dot{v}^{2}-4kc_{p}^{2}v^{2}~{}~{}~{}~{}~{}(\mbox{Case
I})$

and

$\displaystyle Q$ $\displaystyle=$ $\displaystyle 24c_{p}c_{q}\dot{u}$
$\displaystyle E$ $\displaystyle=$ $\displaystyle
12c_{p}c_{q}\dot{u}^{2}-\frac{1}{3c_{p}c_{q}}\dot{v}^{2}-kv^{2}~{}~{}~{}~{}~{}(\mbox{Case
II})$

Now solving the Euler-Lagrange equations for the new Lagrangian, the new
augmented variables have the following forms:

$\displaystyle u$ $\displaystyle=$ $\displaystyle At+B$ $\displaystyle v$
$\displaystyle=$ $\displaystyle
k_{1}\cos\sqrt{3k}c_{p}t+k_{2}\sin\sqrt{3k}c_{p}t~{}~{}~{}~{}~{}~{}(\mbox{Case
I})$

and

$\displaystyle u$ $\displaystyle=$ $\displaystyle rt+s$ $\displaystyle v$
$\displaystyle=$ $\displaystyle
k^{\prime}_{1}\cos\sqrt{3c_{p}c_{q}k}t+k^{\prime}_{2}\sin\sqrt{3c_{p}c_{q}k}~{}~{}~{}~{}~{}(\mbox{Case
II})$

Hence, the cosmic scale factor and the chameleon scalar field have the
following explicit expressions:

$\displaystyle a(t)$ $\displaystyle=$
$\displaystyle\left[\frac{9}{4}\left(At+B\right)^{2}-\left(k_{1}\cos\sqrt{3k}c_{p}t+k_{2}\sin\sqrt{3k}c_{p}t\right)^{2}\right]^{\frac{1}{3}}$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{2}}{3}\tanh^{-1}\left[\frac{2\left(k_{1}\cos\sqrt{3k}c_{p}t+k_{2}\sin\sqrt{3k}c_{p}t\right)}{3(At+B)}\right]~{}~{}~{}~{}~{}(\mbox{Case
I})$

and

$\displaystyle a(t)$ $\displaystyle=$
$\displaystyle\left[9c_{p}c_{q}\left(rt+s\right)^{2}-\frac{1}{4c_{p}c_{q}}\left(k^{\prime}_{1}\cos\sqrt{3c_{p}c_{q}k}t+k^{\prime}_{2}\sin\sqrt{3c_{p}c_{q}k}t\right)^{2}\right]^{\frac{1}{3}}$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{2}}{\sqrt{3}}\ln\frac{6c_{p}c_{q}(rt+s)+\left(k^{\prime}_{1}\cos\sqrt{3c_{p}c_{q}k}t+k^{\prime}_{2}\sin\sqrt{3c_{p}c_{q}k}t\right)}{2c_{p}\left[9c_{p}c_{q}\left(rt+s\right)^{2}-\frac{1}{4c_{p}c_{q}}\left(k^{\prime}_{1}\cos\sqrt{3c_{p}c_{q}k}t+k^{\prime}_{2}\sin\sqrt{3c_{p}c_{q}k}t\right)^{2}\right]^{\frac{1}{2}}}~{}~{}~{}~{}~{}(\mbox{Case
II})$

In the above solutions $(A,B,k_{1},k_{2})$ and
$(r,s,k^{\prime}_{1},k^{\prime}_{2})$ are arbitrary integration constants.

Figure 1: Graphical representation of $\frac{\ddot{a}}{a}$ with respect to
cosmic time $t$ when $c_{p}=c_{q}$.
Figure 2: Represents $\frac{\ddot{a}}{a}$ with respect to cosmic time $t$ when
$c_{p}\neq c_{q}$.

##  IV Quantum Cosmology in the Minisuperspace Approach: A General
Prescription

Minisuperspaces are considered as restrictions of geometrodynamics of the
superspace and physically important and interesting models are defined on
minisuperspaces. In cosmology, the simplest and widely used minisuperspace
models are homogeneous and isotropic merics and matter fields and consequently
the lapse function is homogeneous (i.e, $N=N(t)$) while shift function
vanishes identically. So in 4D manifold, using $(3+1)$-decomposition the
metric can be written as follows:

$ds^{2}=-N^{2}(t)dt^{2}+h_{ab}(x,t)dx^{a}dx^{b}$ (38)

and the Einstein-Hilbert action can be written as follows:

$I(h_{ab},N)=\frac{{m^{2}}_{p}}{16\pi}\int
dt~{}d^{3}xN\sqrt{h}\left[k_{ab}k^{ab}-k^{2}+(3)_{R}-2\Lambda\right],$ (39)

where $k_{ab}$ is the extrinsic curvature of the $3$ space, $k=k_{ab}h^{ab}$
is the trace of the extrinsic curvature, $(3)_{R}$ is the curvature scalar of
the three space and $\Lambda$ is the cosmological constant.

Now due to homogeneity of the three space, the metric $h_{ab}$ is
characterized by a finite number of time functions $q^{\alpha}(t)$,
$\alpha=0,1,2,...,n-1$ and the above action can be written in the form of a
relativistic point particle with self-interacting potential in a $n$D curved
space-time as r42 ; r43

$I\left(q^{\alpha}(t),N(t)\right)=\int_{0}^{1}dtN\left[\frac{1}{2N^{2}}f_{\alpha\beta}(q)\dot{q}^{\alpha}\dot{q}^{\beta}-V(q)\right]$
(40)

So the equation of motion of the (equivalent) relativistic particle can be
written as (considering variation of the action with respect to the field
variables $q^{\alpha}(t)$)

$\frac{1}{N}\frac{d}{dt}\left(\frac{\dot{q}^{\alpha}}{N}\right)+\frac{1}{N^{2}}\Gamma^{\alpha}_{\mu\nu}\dot{q}^{\mu}\dot{q}^{\nu}+f^{\alpha\beta}\frac{\partial\nu}{\partial
q^{\beta}}=0$ (41)

with $\Gamma^{\alpha}_{\beta\gamma}$ being the christoffel symbols in the
minisuperspace. Also there is a constraint equation obtained by variation with
respect to the lapse function as follows:

$\frac{1}{2N^{2}}f_{\alpha\beta}\dot{q}^{\alpha}\dot{q}^{\beta}+V(q)=0$ (42)

For canonical quantization scheme one has to switch over to Hamiltonian
formulation. The momenta canonical to $q^{\alpha}$ are given by

$p_{\alpha}=\frac{\partial L}{\partial
q^{\alpha}}=f_{\alpha\beta}\frac{\dot{q}^{\beta}}{N},$ (43)

so the Hamiltonian is defined as follows:

$H=p_{\alpha}\dot{q}^{\alpha}-L=N\left[\frac{1}{2}f^{\alpha\beta}p_{\alpha}p_{\beta}+V(q)\right]=N\mathcal{H}$
(44)

with $f^{\alpha\beta}$ being the inverse metric. Using the definition of
$p_{\alpha}$ (i.e, equation (43)) into the constraint equation (42) one
obtains

$\mathcal{H}(q^{\alpha},p_{\alpha})\equiv\frac{1}{2}f^{\alpha\beta}p_{\alpha}p_{\beta}+V(q)=0$
(45)

Now, writing $p_{\alpha}$ as $-i\hbar\dfrac{\partial}{\partial q_{\alpha}}$ in
quantization scheme, the operator version of the above constraint equation
(45) on a time-independent function (the wave function of the universe), one
gets the WD equation in quantum cosmology as follows:

$\mathcal{H}\left(q^{\alpha},-i\hbar\frac{\partial}{\partial
q^{\alpha}}\right)\psi(q^{\alpha})=0$ (46)

In general, the minisuperspace metric depends on $q^{\alpha}$, so the above WD
equation has operator ordering problem. However by imposing the quantization
in minisuperspace to be covariant in nature, one may resolve the above
operator ordering problem. Furthermore, in the context of quantam cosmology
for probability measure, $\exists$ a conserved current for hyperbolic type of
partial differential equation is as follows:

$\overrightarrow{J}=\frac{i}{2}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})$ (47)

with $\overrightarrow{\nabla}.\overrightarrow{J}=0$. Here, $\psi$ is the
solution of the hyperbolic-type WD differential equation. Thus it is possible
to define the probability measure on the minisuperspace as follows:

$dp=|\psi(q^{\alpha})|^{2}dV$ (48)

where $dV$ is a volume element on minisuperspace.

##  V Formation of WD Equation in the Present Cosmological Model and possible
solution with Noether Symmetry

In the present cosmological model, the 2D configuration space $\\{a,\phi\\}$
is associated with conjugate momenta is given by

$\displaystyle p_{a}$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{a}}=6a\dot{a}$ $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial L}{\partial\dot{\phi}}=-a^{3}\dot{\phi}$ (49)

So the Hamiltonian of the system (also known as Hamiltonian constraint) can be
expressed as follows:

${\mathcal{H}=\frac{1}{12a}p_{a}^{2}-\frac{1}{2a^{3}}p_{\phi}^{2}-a^{3}V(\phi)+\rho_{0}F(\phi)a^{-3\omega}}$
(50)

with equivalent Hamilton’s equations of motion

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle\frac{1}{6a}p_{a}$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{a^{3}}p_{\phi}$ $\displaystyle{\dot{p_{a}}}$
$\displaystyle=$
$\displaystyle{\frac{1}{12a^{2}}p_{a}^{2}-\frac{3}{2a^{4}}p_{\phi}^{2}+3a^{2}V(\phi)+3\rho_{0}\omega
F(\phi)a^{-3\omega-1}}$ $\displaystyle{\dot{p_{\phi}}}$ $\displaystyle=$
$\displaystyle{a^{3}V^{\prime}(\phi)-\rho_{0}F^{\prime}(\phi)a^{-3\omega}}$
(51)

Furthermore, the Lagrangian (i.e, Eq. (13)) of the system can be interpreted
geometrically, dividing it into two parts. The first two terms are known as
kinetic part and the remaining two terms constitute the dynamic part. Also the
kinetic part may be viewed as a 2D pseudo- Riemannian space with line element

$ds^{2}=-6ada^{2}+a^{2}d\phi^{2}$ (52)

This 2D Lorentzian manifold $(a,\phi)$ is known as minisuperspace (in quantum
cosmology). The wave function of the universe in quantum cosmology is a
solution of the WD equation, a second-order hyperbolic partial differential
equation defined over minisuperspace and it is the operator version of the
Hamiltonian constraint.

Furthermore, in the context of WKB approximation one can write the wave
function as $\psi(x^{k})\sim e^{i\delta(x^{k})}$ and hence the WD equation
(46) becomes first-order nonlinear partial differential equation which is
nothing but (null) Hamilton-Jacobi (H-J) equation in the same geometry.

In quantization of the model one has to construct the WD equation
$\hat{\mathcal{H}}\psi(u,v)=0$, with $\hat{\mathcal{H}}$ the operator version
of the Hamiltonian (50) and $\psi(u,v)$, the wave function of the universe. In
course of conversion to the operator version there is a problem related to the
ordering of a variable and its conjugate momentum r46 . In the first term of
the Hamiltonian (50) there is a product of ‘$a$’ and ‘$p_{a}$’, so one has to
consider the ordering consideration: $p_{a}\rightarrow-i\partial_{a}$,
$p_{\phi}\rightarrow-i\partial_{\phi}$. As a result there is a two-parameter
family of WD equation

${\bigg{[}-\frac{1}{12}\frac{1}{a^{l}}\frac{\partial}{\partial
a}\frac{1}{a^{m}}\frac{\partial}{\partial
a}\frac{1}{a^{n}}+\frac{1}{a^{3}}\frac{\partial^{2}}{\partial\phi^{2}}-a^{3}V(\phi)+\rho_{0}F(\phi)a^{-3\omega}\bigg{]}\psi(a,\phi)=0}$
(53)

with the triplet of real numbers $(l,m,n)$ satisfying $l+m+n=1$. Due to
infinite possible choices for $(l,m,n)$ one may have infinite number of
possible ordering. Also the semi-classical limit, namely, the Hamilton Jacobi
equation (obtained by substituting $\psi=\exp(is))$ does not regard to the
above triplet. In fact, the following choices are commonly used:

i) $l=2,m=-1,n=0$ : D’Alembert operator ordering.

ii) $l=0=n,m=1$: Vilenkin operator ordering.

iii) $l=1,m=0=n$: no ordering.

Thus factor ordering affects the behaviour of the wave function while semi
classical results will not be influenced by the above ordering problem. Now
choosing the third option (i.e., no ordering) the WD equation for the present
model has the following explicit form:

${\bigg{[}-\frac{1}{12a}\frac{\partial^{2}}{\partial
a^{2}}+\frac{1}{2a^{3}}\frac{\partial^{2}}{\partial\phi^{2}}-a^{3}V(\phi)+\rho_{0}F(\phi)a^{-3\omega}\bigg{]}\psi(a,\phi)=0}$
(54)

The general solution of the above second-order hyperbolic partial differential
equation is known as the wave function of the universe. This solution can be
constructed from the separation of the eigen functions of the above WD
operator as follows:r41

$\psi(a,\phi)=\int W(Q)\psi(a,\phi,Q)~{}dQ$ (55)

with $\psi$ being an eigen function of the WD operator, $W(Q)$ being a weight
function and $Q$ being the conserved charge. Now it is desirable to have wave
function in quantum cosmology that is consistent with classical theory. In
other words, one has to construct a coherent wave packet having good
asymptotic behaviour in the minisuperspace and maximize around the classical
trajectory. As the minisuperspace variables $\\{a,\phi\\}$ are highly coupled
in the WD operator so it is not possible to have any explicit solution of the
WD equation even with separation of variable method. Thus one may analyze the
present model in the context of quantum cosmology using the new variables
$(u,v)$ (obtained by point transformation) in the augmented space

Case-I: $c_{p}=c_{q}$

In this case the Lagrangian is given by Eq. (36) for which $u$ is the cyclic
variable. So one has

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{u}}=6\dot{u}=\mbox{Conserved}$ $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{v}}=-\frac{8}{3}\dot{v}$ (56)

Hence, the Hamiltonian of the system takes the form

$\mathcal{H}=\frac{1}{12}p_{u}^{2}-\frac{3}{16}p_{v}^{2}-4kc_{p}^{2}v^{2}$
(57)

Thus the WD equation takes the following form:

$\bigg{[}-\frac{1}{12}\frac{\partial^{2}}{\partial
u^{2}}+\frac{3}{16}\frac{\partial^{2}}{\partial
v^{2}}-4kc_{p}^{2}v^{2}\bigg{]}\chi(u,v)=0$ (58)

The operator version of the conserved momentum in Eq. (56) can be written as
follows:

$i\frac{\partial\chi(u,v)}{\partial u}=\Sigma_{0}~{}\chi(u,v)$ (59)

Now writing $\chi(u,v)=A(u)B(v)$, one has

$\displaystyle i\frac{dA}{du}$ $\displaystyle=$ $\displaystyle\Sigma_{0}~{}A$
$\displaystyle\mbox{i.e.,}~{}A(u)$ $\displaystyle=$ $\displaystyle
A_{0}\exp(-i\Sigma_{0}~{}u)$ (60)

with $A_{0}$ being the constant of integration. Using Eq. (60) the WD equation
(58) becomes a differential equation in $B$ as follows:

$\displaystyle\frac{3}{16}\frac{d^{2}B}{dv^{2}}-4kc_{p}^{2}v^{2}B+\frac{\Sigma_{0}^{2}}{12}B$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\mbox{i.e.,}~{}\frac{d^{2}B}{dv^{2}}-(\lambda v^{2}-\mu)B$
$\displaystyle=$ $\displaystyle 0$ (61)

with $\lambda=\frac{64}{3}kc_{p}^{2},~{}\mu=\frac{4}{9}\Sigma_{0}^{2}$.

Case-II: $c_{p}\neq c_{q}$

The Lagrangian of the system (given by Eq. (37)) shows the variable ‘$u$’ to
be cyclic and the conserved momentum has the following expression:

$p_{u}=\frac{\partial
L}{\partial\dot{u}}=24c_{p}c_{q}\dot{u}=\Lambda_{0},~{}\mbox{a constant}$ (62)

while the momentum conjugate to the variable ‘$v$’ is given by

$p_{2}=\frac{\partial L}{\partial\dot{v}}=-\frac{2}{3c_{p}c_{q}}\dot{v}$ (63)

Hence, the Hamiltonian of the system is expressed as follows:

$\mathcal{H}=\frac{1}{48c_{p}c_{q}}p_{u}^{2}-\frac{3c_{p}c_{q}}{4}p_{v}^{2}-kv^{2}$
(64)

and consequently the WD equation takes the following form:

$\bigg{[}\frac{1}{48c_{p}c_{q}}\frac{\partial^{2}}{\partial
u^{2}}+\frac{3c_{p}c_{q}}{4}\frac{\partial^{2}}{\partial
v^{2}}-kv^{2}\bigg{]}\xi(u,v)=0$ (65)

The operator version of the conserved momentum as before shows

$\displaystyle\xi(u,v)$ $\displaystyle=$ $\displaystyle C(u)D(v)$
$\displaystyle\mbox{with}~{}C(u)$ $\displaystyle=$ $\displaystyle
C_{0}\exp(-i\Lambda_{0}u)$ (66)

Thus from the above WD equation (65) and using the separation of variables,
the differential equation for $D$ reduces to

$\frac{d^{2}D}{dv^{2}}-(lv^{2}-s)D=0$ (67)

with
$l=\frac{4k}{3c_{p}c_{q}},~{}s=\frac{\Lambda_{0}^{2}}{36c_{p}^{2}c_{q}^{2}}$.
The solution of this second-order differential equation takes the following
form:

$\displaystyle{D(v)}$ $\displaystyle=$
$\displaystyle{C_{1}\sqrt{v}J_{\frac{1}{4}}\bigg{(}\frac{1}{2}\sqrt{-l}v^{2}\bigg{)}+C_{2}\sqrt{v}Y_{\frac{1}{4}}\bigg{(}\frac{1}{2}\sqrt{-l}v^{2}\bigg{)}~{}~{}~{}~{}\mbox{when}(s=0)}$
(68) $\displaystyle=$
$\displaystyle{\frac{C_{1}M_{\frac{1}{4}\frac{s}{\sqrt{l}},\frac{1}{4}}\bigg{(}\sqrt{l}v^{2}\bigg{)}}{\sqrt{v}}+\frac{C_{2}W_{\frac{1}{4}\frac{s}{\sqrt{l}},\frac{1}{4}}\bigg{(}\sqrt{l}v^{2}\bigg{)}}{\sqrt{v}}~{}~{}~{}~{}\mbox{when}(s\neq
0)},$

where $J$ and $Y$ are usual Bessel functions and $M$ and $W$ are known as
Whittaker functions. We have represented the wave function graphically both
for zero and non-zero $s$ in Figs.3 and 4, respectively. From the figures, we
see that at $u=0$, $v=0$ (i.e, $l=0$), wave function has finite nonzero value.
Therefore in the present model it is possible to avoid the Big Bang
singularity using quantum cosmology near the initial singularity.

Figure 3: The wave function when $s=0$
Figure 4: Graphical representation of the wave function when $s\neq 0$

##  VI Conclusion

This work is an example where symmetry analysis particularly Noether symmetry
has been extensively used both in classical and quantum cosmology. Here,
chameleon field DE model has been considered in the background of homogeneous
and isotropic flat FLRW space–time. Although the full quantum theory is
described on the infinite-dimensional superspace, but here we shall confine to
minisuperspace which is a 2D Lorentzian manifold.

Although the Einstein field equations are nonlinear coupled differential
equation, but using a transformation in the augmented space and introducing
geometric inner product it is possible to identify the cyclic variable(s) so
that the field equations simplified to a great extent and consequently
classical cosmological solutions are evaluated. There are two sets of
solutions for two different choices of the arbitrary constants involved. Both
the solutions show an expanding model of the universe with accelerating and
decelerating phases (depending on the choices of the arbitrary constants
involved). In particular, the present model describes the decelerating phase
only for the choice $c_{p}=c_{q}$ (Fig. 1), while the model makes a transition
from decelerating phase to accelerating phase and then again it goes to
decelerating phase for the choice $c_{p}\neq c_{q}$ (Fig. 2).

On the other hand, the application of Noether symmetry to the minisuperspace
shows the path for solving WD equation. The conserved momentum due to Noether
symmetry, after converting to quantum version, shows an oscillatory solution
to the WD equation and consequently it gives the semi-classical limit of
quantum cosmology. Furthermore, the nonoscillatory part of the WD equation is
an ordinary differential equation having solution in the form of Bessel
function or Whittaker function. The graphical presentation of this part of the
solution has been shown in Figs. (3 and4), which clearly shows that the
present quantum cosmological model can overcome the big bang singularity i.e.,
the present model may describe the early era of evolution without any
singularity. Finally, one may conclude that Noether symmetry analysis is very
useful in describing quantum cosmology in minisuperspace model and also leads
to possible solution of the WD equation.

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