#  Reconstruction of $\Lambda$CDM Universe from Noether symmetries in
Chameleon gravity

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

We apply the Noether symmetries to constrain the unknown functions of
chameleon gravity in the cosmological scenario of a spatially flat
Friedmann–Lemaître–Robertson–Walker space-time with an ideal gas. For this
gravitational model the field equations admit a point-like Lagrangian with as
unknown functions the scalar field potential and the coupling function which
is responsible for the chameleon mechanism. Noether’s first theorem provides
us with four sets of closed-form functional forms for which variational
symmetries exist. We construct the corresponding conservation laws and we use
them in order to determine new analytic solutions in chameleon gravity. From
the analysis of the physical properties of the new solution it follows that in
the late universe they can reproduce the $\Lambda$CDM model without having to
assume the presence of a pressureless fluid in the cosmological fluid.

Chameleon gravity; Scalar field; Cosmology; Noether symmetries.

##  I Introduction

The physical origin of the dark energy mechanism which is responsible for the
late-time acceleration phase of the universe puzzles the scientific community
q1 ; q2 . Cosmologists have proposed various theoretical models to explain
dark energy q3 ; q4 . Gravitational theories with a scalar field have special
relevance because they provide a very simple mechanism for the description of
the cosmological observations at the late universe q5 ; q6 ; q7 . Within,
scalar fields have been used for the description of inflation through the
inflaton mechanism q8 ; q9 . The novelty of scalar fields is that they provide
a dynamical dark energy scenario which can surpass the major problems of
$\Lambda$-cosmology q10 . The degrees of freedom introduced in the field
equations from the scalar fields can provide the dynamical variables related
to the modification of the Einstein-Hilbert Action Integral q11 ; q12 ; q13 .

Quintessence is a very simple dark energy candidate described by a minimally
coupled canonical scalar field with a nonzero potential function q14 . In
quintessence cosmology acceleration occurs when the scalar field varies at a
slow rate. There is a plethora of quintessence models which have been proposed
in the literature for different potential functions which give a different
cosmological history and evolution q15 ; q16 ; q17 . The quintessence model
can describe the Chaplygin gas (-like) fluids q18 ; q19 ; q20 , as to provide
matter components in the cosmological fluid q21 . There are various extensions
of the quintessence model in the literature. The phantom scalar field has a
similar Lagrangian to the quintessence model, but it has a negative kinetic
energy term. The latter leads to cosmological behaviour which a faster
expansion of the universe is provided by the exponential expansion of the de
Sitter solution, as a result of which this fast expansion is related to the
Big Rip singularity q22 ; q23 . For other scalar field models we refer the
reader to q24 ; q25 ; q26 ; q27 ; q28 ; q29 ; q30  and references therein.

In ch1 ; ch2  an extension has been proposed of the quintessence scalar field
model with a chameleon mechanism for the mass of the scalar field.
Specifically, in the cosmological scenario of General Relativity with a scalar
field and an ideal gas, a coupling function which depends upon the scalar
field as been introduced at the Lagrangian of the ideal gas. As a consequence
the mass of the scalar field depends upon an effective potential function with
the contribution of the coupling function and the energy density of the ideal.
This mechanism gives to the scalar field a small mass in open space with low
energy density and a large mass on compact objects such on the surface of the
Earth. Because of that property, the magnitude for the violation of the
equivalence principle and of the fifth force components are different in the
open space and in the solar system which provides a “hide” mechanism to the
scalar field. That is why it is called “chameleon”.

The coupling function responsible for the chameleon mechanism in ch1  has been
considered to be of the exponential form. However, this chameleon model ch1
has its origin on a more geometrical background. Specifically, the chameleon
model with the exponential coupling is equivalent to the Weyl integrable
spacetime (WIS) salim96 . In WIS the scalar field is introduced in the
gravitational Action Integral with a geometric construction method. The scalar
field in WIS plays the role of the conformal factor between two conformally
related metrics. The gravitational Lagrangian is constructed by the Levi-
Civita connection of the conformal metric which provides the scalar field
degrees of freedom in the field equations. In the case that there is no matter
source the WIS reduces to General Relativity with a scalar field. There exists
a difference between the chameleon mechanism ch1  and the WIS salim96 . In the
chameleon mechanism the scalar field is a quintessence with positive kinetic
density, on the other hand, in WIS the scalar field can have negative kinetic
energy, because a new parameter has been introduced in the kinetic term of the
scalar field Lagrangian.

The chameleon mechanism has been applied in cosmology as a dark energy
alternative. The mechanism provides an interaction between the elements of the
dark sector of the universe. Coupled dark energy models have drawn the
attention of cosmologists because they can gives a theoretical description on
the analysis of the cosmological observations such as the cosmic coincidence
problem ame1 ; ame2 ; pavon ; delcampo , gravitational waves observations gwo
or the $H_{0}$ tension hot ; hot2 ; hot3 ; hot4 . In Wetterich-ide1  it was
found that the interaction is a mechanism which can lead to a dynamical
cosmological constant term. There is a plethora of studies in the literature
which deal with the WIS or the chameleon scalar field on cosmological scales.
In w1  it has been shown that the chameleon scalar field can play the role of
the inflaton field related to the inflationary era. This study shows that
inflation has a geometric origin related to the conformally related metric.
The conditions for the late-time acceleration in WIS were determined w2 ;
nevertheless in w3  it has been derived that the scalar field in chameleon
theory can have a phantom behaviour, which lead to a Big Rip singularity w4 .
The chameleon mechanism provides the limit of hyperbolic inflation when the
matter source is described by a stiff fluid, that is, by a massless scalar
field hy . In this limit the chameleon mechanism is reduced to the Chiral
model hy1 ; hy2 . The coupling function between the scalar field and the
matter source can take other functional forms beyond the exponential coupling
w5 ; w6 ; w7 . Over and above the chameleon mechanism it has been introduced
in other scalar field theories such as in k-essence, in tachyonic fields and
others, see for instance w8 ; w9 ; w10  and references therein. Before, we
mentioned that the chameleon mechanism contributes to the mass of the scalar
field in compact objects; however, in cosmological scales when the dark matter
(or the radiation) dominates to the universe during the chameleon mechanism
has a non-trivial contribution, which means that it can affect the dynamics in
large scales. A direct observation of the chameleon mechanism is presented in
dch1

Although the field equations for chameleon gravity in a
Friedmann–Lemaître–Robertson–Walker universe are of second-order, there are
very few known exact and analytic solutions in the literature. Recently in ns1
analytic solutions were determined for the case of WIS with the use of the
variational symmetries, while the method of variational symmetries has been
applied also and in the limit of the Chiral model ns2 . Specifically the
variational symmetry conditions are applied in order to constrain the unknown
functions and parameters of the gravitational Lagrangian. Such a method has
been widely used in cosmology with many interesting results ns2 ; ns3 ; ns4 ;
ns5 ; ns6 ; ns7 ; ns8 . For a detailed review of the application of the
variational symmetries in gravitational physics we refer the reader in ns9 .

In this work we apply the variational symmetries, aka Noether symmetry
analysis, for the geometric constraint of the scalar field potential and of
the coupling function in chameleon cosmology. We perform a classification
problem as established by Ovsiannikov in ovs . A recent attempt to answer this
classification problem is presented in ns10 . However, the authors considered
the ideal gas important for the chameleon mechanism to be the cosmological
constant. In this case the chameleon mechanism is cancelled and the theory is
equivalent to the quintessence model ns11 . The field equations in chameleon
cosmology are of second-order and they form an autonomous system which admits
a point-like Lagrangian function. According to Noether’s second theorem, for
every variational symmetry there exists a corresponding conservation law. The
latter can be used to investigate the integrability properties of the field
equations and to determine similarity transformations which are applied to
reduce the order of the dynamical system. The variational symmetries in this
study lead to new analytic and exact solutions in chameleon cosmology. The
derivation of analytic solutions in closed-form expression is important
because we can study the behaviour of the given gravitational theory by using
analytic techniques. The structure of the paper follows.

In Section II we present the gravitational model of our consideration which is
that of a minimally coupled scalar field cosmology in a spatially flat
Friedmann–Lemaître–Robertson–Walker (FLRW) geometry with a chameleon
mechanism. The scalar field is coupled to the Lagrangian function of the
matter source. For the latter we assume that it describes an ideal gas. For
this model the field equations are of second-order and they are described by a
point-like Lagrangian with two unknown functions, the scalar field potential
and the coupling function which defines the chameleon mechanism. In Section
III we apply the theory of variational symmetries, i.e. Noether symmetries, as
a selection rule to constrain the unknown functions of the theory. We found
that there exist four sets of functions for which the field equations admit
nontrivial Noether symmetries. For these cases we use Noether’s second theorem
to write the admitted conservation laws. For the new cosmological models
derived by the symmetry analysis in Section IV we derive the analytic solution
for the field equations. We investigate the evolution of the cosmological
parameters and we show that these new integrable models can explain the
present acceleration phase of the universe, while these chameleon models mimic
the $\Lambda$CDM cosmology either if we have considered the ideal gas to be
different from that of a pressureless fluid related to the cold dark matter.
Finally, in Section V we summarize our results.

##  II Chameleon cosmology

Consider now a four-dimensional Riemannian manifold $V^{n}$ with metric tensor
$g_{\mu\nu}$ and Ricci scalar $R=R\left(g\right)$. Furthermore, we assume the
existence of a scalar field $\phi\left(x^{\mu}\right)$ and of a matter source
described by the Lagrangian function
$L_{m}\left(\psi\left(x^{\mu}\right)\right)$.

For the above, in the context of General Relativity we assume the
gravitational Action Integral w5

$S=\int\sqrt{-g}d^{4}x\left(\frac{R}{2}-\frac{1}{2}g^{\mu\kappa}\nabla_{\mu}\phi\left(x^{\nu}\right)\nabla_{\kappa}\phi\left(x^{\nu}\right)-V\left(\phi\right)-F\left(\phi\right)L_{m}\left(x^{\nu}\right)\right),$
(1)

where $F\left(\phi\right)$ is the coupling function between the scalar field
and the matter source and describes the chameleon mechanism.

Variation with respect to the metric tensor provides the gravitational field
equations of General Relativity

$G_{\mu\nu}=T_{\mu\nu}^{eff},$ (2)

where $T_{\mu\nu}^{eff}$ is the effective energy-momentum tensor which
describe the degrees of freedom from the scalar field and the matter source.

Moreover, the equation of motion for the scalar field is

$\Delta\phi=V_{,\phi}+f_{,\phi}T_{\mu}^{\mu}$ (3)

in which $T^{\mu\nu}=\frac{\partial L_{m}}{\partial g_{\mu\nu}},$ and $\Delta$
is the Laplace operator for the background metic $g_{\mu\nu}$, defined as
$\Delta=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{g}g^{\mu\nu}\frac{\partial}{\partial x^{\nu}}\right)$,
that is, $\Delta\phi=g^{\mu\nu}\phi_{,\mu\nu}-\Gamma^{\mu}\phi_{\mu}$. We
remark that, when $T_{\mu}^{\mu}$ corresponds to the cosmological constant,
then the latter equation of motion reduces to the usual Klein-Gordon equation
for a modified potential and that the chameleon mechanism is eliminated.

On the other hand, when $T^{\mu\nu}$ describes a pressureless ideal gas with
energy density $\rho_{m}$ the modified Klein-Gordon equation becomes w5

$\Delta\phi=V_{,\phi}+f_{,\phi}\rho_{m}.$ (4)

Last but not least, the matter source satisfies the equation of motion
$\nabla_{\nu}T^{eff~{}~{}\mu\nu}=0$, where $\nabla_{\nu}$ is the covariant
derivative defined by the Levi-Civita connection with respect to the metric
tensor $g_{\mu\nu}$. Consequently, from (4) it follows that when $\rho_{m}$ is
large enough the chameleon term has a non-zero contribution to the mass of the
scalar field.

For a spatially flat FLRW background geometry with scale factor
$a\left(t\right)$ and line element

$ds^{2}=-dt^{2}+a^{2}\left(t\right)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (5)

the gravitational field equations are

$3H^{2}=\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)-\rho_{m}f\left(\phi\right),$
(6)
$2\dot{H}+3H^{2}=-\left(\frac{1}{2}\dot{\phi}^{2}-V\left(\phi\right)-f\left(\phi\right)p_{m}\right),$
(7)

where $H=\frac{\dot{a}}{a}$ is the Hubble function; $\rho_{m}$ and $p_{m}$ are
the energy density and pressure components for a perfect fluid described by
the Lagrangian function $L_{m}\left(\psi\left(x^{\mu}\right)\right)$. We have
assumed the scalar field and the matter source to inherit the symmetries of
the background geometry which means that $\rho_{m}=\rho_{m}\left(t\right)$ and
$\phi=\phi\left(t\right)$.

The modified Klein-Gordon equation (3) is expressed as

$\ddot{\phi}+3H\dot{\phi}+V_{,\phi}+f_{,\phi}\rho_{m}=0,$ (8)

while that for the perfect fluid the equation of motion is

$\dot{\rho}_{m}+3H\left(\rho_{m}+p_{m}\right)-\dot{\phi}\left(\ln
f\right)_{,\phi}\rho_{m}=0$ (9)

Let the perfect fluid to be an ideal gas with constant equation of state
parameter $w_{m}$, that is, $p_{m}=w_{m}\rho_{m}$. From equation (9) we find
that lan1 ,

$\rho_{m}=\rho_{m0}a^{-3\left(w_{m}+1\right)}f\left(\phi\right)$ (10)

with $\rho_{m0}$ an integration constant.

By replacing in (1) we end with the point-like Lagrangian for the field
equations

$\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\dot{\phi}^{2}-a^{3}V\left(\phi\right)-a^{-3w_{m}}F\left(\phi\right),~{}F\left(\phi\right)=f\left(\phi\right)^{2}\,.$
(11)

Indeed, the field equations follow from the variation of
$\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi}\right)$ with respect to the
dynamical variables $a$ and $\phi$, while equation (6) can be seen as the
Hamiltonian constraint for the dynamical system. Indeed, Lagrangian (11) is
time independent and it admits as Noether symmetry the vector field
$\partial_{t}$, with conservation law the Hamiltonian function, where from (6)
it follows that the “energy” has a fixed value.

We continue our analysis with the investigation of the existence of functional
forms for the potential $V\left(\phi\right)$ and for the coupling
$F\left(\phi\right)$ where extra Noether symmetries exist. In particular, we
apply the theory of variational symmetries, that is, Noether’s first theorem,
and we require the variation of the Action Integral (1) to be invariant under
one-parameter point transformations.

In the following we consider that $w_{m}\in[0,1)$. The limit $w_{m}=1$
corresponds to the Chiral model which has been studied before in ns2 .

##  III Variational symmetries

In this Section we solve the classification problem for the admitted Noether
symmetries of the field equations with Lagrangian function (11).

###  III.1 Symmetries of differential equations

We review briefly the basic elements for the variational symmetries of
differential equations.

We assume the dynamical system of second-order differential equations

$\mathbf{\ddot{x}}=\mathbf{\Omega}\left(t,\mathbf{x},\mathbf{\dot{x}}\right)$
(12)

which follows from the variation of the Lagrangian function

$\mathcal{L}=\mathcal{L}\left(t,\mathbf{x},\mathbf{\dot{x}}\right).$ (13)

In the augmented space $\\{t,\mathbf{x}\\}$ we consider the one-parameter
point transformation with generator the vector field

$X=\xi\left(t,\mathbf{x}\right)\partial_{t}+\mathbf{\eta}\left(t,\mathbf{x}\right)\partial_{\mathbf{x}}.$
(14)

Then, according to Noether’s first theorem, if there exist a boundary function
$g\left(t,\mathbf{x},\mathbf{\dot{x}}\right)$ such that the following
condition be true

$X^{\left[1\right]}\mathcal{L}+\mathcal{L}\dot{\xi}=\dot{g},$ (15)

then the variation of the Action Integral with Lagrangian function
$\mathcal{L}\left(t,\mathbf{x},\mathbf{\dot{x}}\right)$ remains invariant
under the application of the one-parameter point transformation with generator
$X$.

When (15) is true, the vector field $X$ is a variational symmetry, or a
Noether symmetry, for the dynamical system (12). $X^{\left[1\right]}$ is the
first prolongation of $X\,\ $in the jet space
$\left\\{t,\mathbf{x},\mathbf{\dot{x}}\right\\}$ defined as

$X^{\left[1\right]}=X+\left(\mathbf{\dot{\eta}}-\mathbf{\dot{x}}\dot{\xi}\right)\partial_{\mathbf{\dot{x}}}.$
(16)

Noether’s second theorem states that for every variational symmetry vector
there exists a corresponding conservation law. Thus, if $X$ is a Noether
symmetry vector field, the function $I\left(X\right)$ defined as

$I\left(X\right)=\xi\left(\frac{\partial\mathcal{L}}{\partial\mathbf{\dot{x}}}\dot{x}-\mathcal{L}\right)-\frac{\partial\mathcal{L}}{\partial\mathbf{\dot{x}}}\mathbf{\eta}+g,$
(17)

is a conservation law for the dynamical system (12).

###  III.2 Variational symmetries in chameleon cosmology

For the Lagrangian model (11) with the chameleon mechanics, the augmented
space has dimension three and is $\left\\{t,a,\phi\right\\}$. We assume the
generator $X$ of a one-parameter point transformation in the base manifold,

$X=\xi\left(t,a,\phi\right)\partial_{t}+\eta_{a}\left(t,a,\phi\right)\partial_{a}+\eta_{\phi}\left(t,a,\phi\right)\partial_{\phi},$
(18)

with first prolongation

$X^{\left[1\right]}=X+\left(\dot{\eta}_{a}-\dot{a}\dot{\xi}\right)\partial_{\dot{a}}+\left(\dot{\eta}_{\phi}-\dot{\phi}\dot{\xi}\right)\partial_{\dot{\phi}}.$
(19)

From the symmetry condition (15) and for Lagrangian function (11) we determine
a linear system of first-order partial differential equations with respect the
coefficient functions $\xi,~{}\eta_{a}$ and $\eta_{\phi}$ of the symmetry
vector $X$. This system is presented in Appendix A. The solution of the
symmetry condition (15) depends upon the functional form of the potential
function $V\left(\phi\right)$ and of the coupling scalar $F\left(\phi\right)$.
As we discussed above, for arbitrary functions $V\left(\phi\right)$ and
$F\left(\phi\right)$ the variational symmetry $X^{1}=\partial_{t}$ exists
always with conservation law the Hamiltonian
$H=\frac{\partial\mathcal{L}}{\partial\mathbf{\dot{x}}}\dot{x}-L$. However,
from (6) it follows $H=0$.

We omit the presentation of the calculations. We remark that in the following
we consider $F\left(\phi\right)$ is a non-constant function.

####  III.2.1 Case A

For the potential function

$V_{A}\left(\phi\right)=\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}-V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)^{2}$

and the coupling function

$F_{A}\left(\phi\right)=F_{0}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}-V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)^{-2w_{m}},$
(20)

i.e.
$F_{A}\left(\phi\right)=F_{0}\left(V_{A}\left(\phi\right)\right)^{-w_{m}}$,
there exist the variational symmetries

$X^{2}=\frac{1}{\sqrt{a}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)\partial_{a}-\frac{\sqrt{6}}{a^{\frac{3}{2}}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)\partial_{\phi}$
(21)

and

$X^{3}=t\left(\frac{1}{\sqrt{a}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)\partial_{a}-\frac{\sqrt{6}}{a^{\frac{3}{2}}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)\partial_{\phi}\right).$
(22)

The corresponding boundary functions $g\left(t,a,\phi\right)$ are
$g\left(X^{2}\right)=const$ and
$g\left(X^{3}\right)=-4a^{\frac{3}{2}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)$.

Consequently, from Noether’s second theorem and expression (17) the
conservation laws are

$I\left(X^{2}\right)=-6\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)a^{\frac{1}{2}}\dot{a}-\sqrt{6}a^{\frac{3}{2}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)\dot{\phi}$
(23)

and

$I\left(X^{3}\right)=tI\left(X^{2}\right)+4a^{\frac{3}{2}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right).$
(24)

We remark that for $V_{1}V_{2}=0$ it follows that
$V_{A}\left(\phi\right)\simeq e^{\pm\frac{\sqrt{6}}{2}\phi}$ and
$F_{A}\left(\phi\right)\simeq e^{\mp\frac{\sqrt{6}}{2}w_{m}\phi}$.
Furthermore, for $V_{2}=V_{1}$, it follows that
$V_{A}\left(\phi\right)\simeq\sinh^{2}\left(\frac{\sqrt{6}}{4}\phi\right)$,
$F_{A}\left(\phi\right)\simeq\sinh^{-2w_{m}}\left(\frac{\sqrt{6}}{4}\phi\right)$.
On the other hand for $V_{2}=-V_{1}$ we find that
$V_{A}\left(\phi\right)\simeq\cosh^{2}\left(\frac{\sqrt{6}}{4}\phi\right)$,
$F_{A}\left(\phi\right)\simeq\sinh^{-2w_{m}}\left(\frac{\sqrt{6}}{4}\phi\right)$.

####  III.2.2 Case B

The dynamical system with potential function

$V_{B}\left(\phi\right)=\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}-V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)^{2}+\frac{4}{3}\Lambda$
(25)

and the coupling function $F_{B}\left(\phi\right)=F_{A}\left(\phi\right)$ the
resulting Noether symmetries and boundary values are

$Y^{1}=e^{\sqrt{\Lambda}t}X^{2}~{},~{}g\left(Y^{1}\right)=-4\sqrt{\Lambda}e^{\sqrt{\Lambda}t}a^{\frac{3}{2}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right),$
(26)
$Y^{1}=e^{-\frac{2\sqrt{3\Lambda}}{3}t}X^{2}~{},~{}g\left(Y^{2}\right)=4\sqrt{\Lambda}e^{-\sqrt{\Lambda}t}a^{\frac{3}{2}}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right).$
(27)

The corresponding conservation laws are

$I\left(Y^{1}\right)=e^{\sqrt{\Lambda}t}\left(I\left(X^{2}\right)-\sqrt{\Lambda}~{}g\left(Y^{1}\right)\right)$
(28)

and

$I\left(Y^{2}\right)=e^{-\sqrt{\Lambda}t}\left(I\left(X^{2}\right)+\sqrt{\Lambda}g\left(Y^{2}\right)\right).$
(29)

Indeed, from these two time-dependent conservation laws we can construct the
time-independent function
$I\left(Y^{1}Y^{2}\right)=I\left(Y^{1}\right)I\left(Y^{2}\right)$, that is,

$I\left(Y^{1}Y^{2}\right)=\left(I\left(X^{2}\right)\right)^{2}-16\Lambda
a^{2}\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}+V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)^{2}.$
(30)

For specific values of the free parameters $V_{1}$ and $V_{2}$ the scalar
field potential is expressed by specific functions. Indeed, for
$V_{1}V_{2}=0\,$, we find the exponential potential
$V_{B}\left(\phi\right)\simeq
e^{\pm\frac{\sqrt{6}}{2}\phi}+\frac{4}{3}\Lambda$. For $V_{2}=V_{1}$, the
potential function is
$V_{B}\left(\phi\right)\simeq\sinh^{2}\left(\frac{\sqrt{6}}{4}\phi\right)+\frac{4}{3}\Lambda$
while for $V_{2}=-V_{1}$. It follows that
$V_{B}\left(\phi\right)\simeq\cosh^{2}\left(\frac{\sqrt{6}}{4}\phi\right)+\frac{4}{3}\Lambda$.

####  III.2.3 Case C

In the third case provide by the symmetry analysis, the scalar field potential
is expressed as

$V_{C}\left(\phi\right)=V_{0}e^{\lambda\phi}$ (31)

with coupling function

$F_{C}\left(\phi\right)=F_{0}e^{\frac{\phi}{2}\lambda\left(w_{m}-1\right)}.$
(32)

This case correspond to the WIS and it has been widely studied before in ns1 .

The nontrivial variational symmetry is

$Z^{1}=2t+\frac{2}{3}a\partial_{t}+\frac{4}{\lambda}\partial_{\phi}$ (33)

with corresponding conservation law

$I\left(Z^{1}\right)=-4a^{2}\dot{a}+\frac{4}{\lambda}a^{3}\dot{\phi}\text{.}$
(34)

####  III.2.4 Case D

In the last case of the classification scheme the scalar field potential
vanishes, that is, $V_{D}\left(\phi\right)=0$ and
$F\left(\phi\right)=F_{0}e^{-\frac{\sqrt{6}}{2}w_{m}\phi}$.

The Noether symmetry vectors are

$Z^{2}=2t+\frac{2}{3}a\partial_{a}+\frac{2\sqrt{6}\left(1-w_{m}\right)}{w_{m}}\partial_{\phi},$
(35)
$Z^{3}=e^{\frac{\sqrt{6}}{4}\phi}\left(\frac{1}{\sqrt{a}}\partial_{a}-\frac{\sqrt{6}}{a^{\frac{3}{2}}}\partial_{\phi}\right)$
(36)

and

$Z^{4}=te^{\frac{\sqrt{6}}{4}\phi}\left(\frac{1}{\sqrt{a}}\partial_{a}-\frac{\sqrt{6}}{a^{\frac{3}{2}}}\partial_{\phi}\right).~{}g\left(Z^{4}\right)=-4a^{\frac{3}{2}}e^{\frac{\sqrt{6}}{4}\phi}.$
(37)

The corresponding conservation laws are derived to be

$I\left(Z^{2}\right)=-4a^{2}\dot{a}+\frac{2\sqrt{6}\left(1-w_{m}\right)}{w_{m}}a^{3}\dot{\phi},$
(38)
$I\left(Z^{3}\right)=e^{\frac{\sqrt{6}}{4}\phi}\left(-6a^{\frac{1}{2}}\dot{a}-\sqrt{6}a^{\frac{3}{2}}\dot{\phi}\right)$
(39)

and

$I\left(Z^{4}\right)=tI\left(Z^{3}\right)-g\left(Z^{4}\right)\,.$ (40)

This integrable model is a special case of case $C$, for $V_{0}$ and
$\lambda=-\sqrt{6}\frac{w_{m}}{w_{m}-1}$.

We conclude that for these four sets of scalar fields and coupling functions
the field equations admit two conservation laws which are independent and in
involution. The number of independent conservation laws is the same as the
number of dynamical variables. Consequently for the above four sets of the
free functions the cosmological field equations form a Liouville integrable
dynamical system.

##  IV Analytic solutions

In this Section we continue our analysis with the derivation of new analytic
solutions for the cosmological models determined by the Noether symmetry
analysis presented above. The cosmological models of cases C and D have been
investigated in detail in ns1 . Hence, in this work we focus upon the
derivation of the solutions for the hyperbolic potential functions of cases A
and B. Furthermore, the evolution of physical parameters and the asymptotic
solutions of the cosmological solutions are discussed.

###  IV.1 Cases A & B

Cases A and B share the same coupling function $F\left(\phi\right)$ while the
scalar field potential functions differ in a constant parameter. Consider now
the scalar field potential of case $B$, that is,

$V_{B}\left(\phi\right)=\left(V_{1}e^{\frac{\sqrt{6}}{4}\phi}-V_{2}e^{-\frac{\sqrt{6}}{4}\phi}\right)^{2}+\frac{4}{3}\Lambda$
(41)

and the new new set of dependent variables
$\left(a,\phi\right)\rightarrow\left(x,y\right)$ with transformation rule

$a=\left(x^{2}-y^{2}\right)^{\frac{1}{3}}~{},~{}\phi=\frac{2\sqrt{6}}{3}\arctan
h\left(\frac{y}{x}\right).$

In the new variables the cosmological point-like Lagrangian (11) becomes

$\mathcal{L}_{A}\left(x,\dot{x},y,\dot{y}\right)=\frac{4}{3}\left(\dot{x}^{2}-\dot{y}^{2}\right)+\left(V_{1-2}x+V_{1+2}y\right)^{2}+\frac{4}{3}\Lambda\left(x^{2}-y^{2}\right)+F_{0}\rho_{m0}\left(V_{1-2}x+V_{1+2}y\right)^{-2w_{m}},$
(42)

where in the new variables the field equations are

$\frac{4}{3}\left(\dot{x}^{2}-\dot{y}^{2}\right)-\left(V_{1-2}x+V_{1+2}y\right)^{2}-\frac{4}{3}\Lambda\left(x^{2}-y^{2}\right)-F_{0}\rho_{m0}\left(V_{1-2}x+V_{1+2}y\right)^{-2w_{m}}=0,$
(43)
$\ddot{x}-\frac{3}{4}V_{1-2}\left(\left(V_{1-2}x+V_{1+2}y\right)+w_{m}F_{0}\rho_{m0}\left(V_{1-2}x+V_{1+2}y\right)^{-2w_{m}-1}\right)-\Lambda
x=0,$ (44)

and

$\ddot{y}+\frac{3}{4}V_{1+2}\left(\left(V_{1-2}x+V_{1+2}y\right)+w_{m}F_{0}\rho_{m0}\left(V_{1-2}x+V_{1+2}y\right)^{-2w_{m}-1}\right)-\Lambda
x=0$ (45)

with $V_{1-2}=V_{1}-V_{2}$ and $V_{1+2}=V_{1}+V_{2}$.

We proceed by considering some specific values of the free parameters $V_{1}$,
$V_{2}$. In particular we investigate the cases for which (a) $V_{1}+V_{2}=0$,
(b) $V_{1}-V_{2}=0$ and (c) $V_{1}$, $V_{2}$ are arbitrary.

####  IV.1.1 Solution for $V_{1}+V_{2}=0$

For $V_{1}+V_{2}=0$ the field equations are simplified to the following
expressions

$\frac{4}{3}\left(\dot{x}^{2}-\dot{y}^{2}\right)-V_{1-2}x^{2}-\frac{4}{3}\Lambda\left(x^{2}-y^{2}\right)-F_{0}\rho_{m0}\left(V_{1-2}x\right)^{-2w_{m}}-=0,$
(46)
$\ddot{x}-\frac{3}{4}\left(V_{1-2}\right)\left(V_{1-2}x+w_{m}F_{0}\rho_{m0}\left(V_{1-2}x\right)^{-2w_{m}-1}\right)-\Lambda
x=0$ (47)

and

$\ddot{y}-\Lambda x=0.$ (48)

For $\Lambda=0$ we observe that the Noetherian conservation laws
$I\left(X^{2}\right)$ and $I\left(X^{3}\right)$ are
$I\left(X^{2}\right)\simeq\dot{y}$ and $I\left(X^{3}\right)\simeq t\dot{y}-y$.
On the other hand, for $\Lambda\neq 0$, the conservation laws
$I\left(Y^{1}\right)$ and $I\left(Y^{2}\right)$ are
$I\left(Y^{1}\right)=e^{\sqrt{\Lambda}t}\left(\dot{y}-\sqrt{\Lambda}y\right)$
and
$I\left(Y^{2}\right)=e^{-\sqrt{\Lambda}t}\left(\dot{y}+\sqrt{\Lambda}y\right)$
from which we can defined the time-independent conservation law
$I\left(Y^{1}Y^{2}\right)=\dot{y}^{2}-y^{2}$.

The solution of equation (48) is $y\left(t\right)\simeq\left(t-t_{0}\right)$
for $\Lambda=0$ and
$y\left(t\right)\simeq\sqrt{\frac{I\left(Y^{1}Y^{2}\right)}{\Lambda}}\sinh\left(\sqrt{\Lambda}\left(t-t_{0}\right)\right)$
when $\Lambda\neq 0$.

With the use of the conservation laws, the constraint equation (46) becomes

$\frac{dx}{\sqrt{\frac{3}{4}\left(I_{0}\right)^{2}+\frac{3}{4}\left(\left(\left(V_{1-2}\right)^{2}+\frac{16}{9}\Lambda\right)x^{2}+F_{0}\rho_{m0}\left(V_{1-2}x\right)^{-2w_{m}}\right)}}=\pm
dt\text{ }$ (49)

with $I_{0}=I\left(X^{2}\right)$ or $I\left(X^{2}\right)$
$=I\left(Y^{1}Y^{2}\right)$. Thus the analytic solution is expressed in terms
of the elliptic integral (49). We observe that, when the chameleon mechanism
is eliminated, the analytic solution for the unified dark matter potential for
the quintessence field ns11  is recovered.

Indeed, for large values of $x$, because $w_{m}\geq 0$, the surviving terms in
(47) are

$\ddot{x}-\frac{3}{4}\left(\left(V_{1-2}x\right)^{2}+\frac{16}{9}\Lambda\right)x\simeq
0.$ (50)

The chameleon mechanism does not contribute to the field equations and the
model is reduced to that of quintessence theory. The asymptotic solution at
this limit is expressed by the closed-form function

$x\left(t\right)\simeq
x_{1}\sinh\left(\sqrt{\frac{3}{4}\left(\left(V_{1-2}x\right)^{2}-\frac{16}{9}\Lambda\right)}t+x_{2}\right).$
(51)

Consider now that
$\frac{3}{4}\left(\left(V_{1-2}x\right)^{2}-\frac{16}{9}\Lambda\right)>0$,
then the scale factor for $x>>1$, that is, for $t>>0$, it is asymptotically
described by the hyperbolic function

$a\left(t\right)\simeq\left(x_{1}\right)^{\frac{2}{3}}\sinh^{\frac{2}{3}}\left(\sqrt{\frac{3}{4}\left(\left(V_{1-2}x\right)^{2}-\frac{16}{9}\Lambda\right)}t+x_{2}\right).$
(52)

The latter describes the $\Lambda$CDM universe ns11 . Recall that for the
$\Lambda$CDM model the scale factor is
$a\left(t\right)=\left(\frac{\Omega_{m}}{\Omega_{\Lambda}}\right)^{\frac{1}{3}}\sinh^{\frac{2}{3}}\left(\frac{3}{2}H_{0}\sqrt{\Omega_{\Lambda}}t\right)$.

In Fig. 1 we present the qualitative evolution for the effective equation of
state parameter $w_{eff}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}$ and for the
scalar field $\phi$ for various values of the free parameters. We observe that
in the late universe; that is, for small values of the redshift, the solution
is very close to that of $\Lambda$CDM cosmology with the de Sitter solution as
a future attractor.

Figure 1: Qualitative evolution of the effective equation of state parameter
$w_{eff}\left(z\right)$ (left column) and of the scalar field
$\phi\left(z\right)$ (right column) for the scalar field potential of cases A
and B for $V_{1}+V_{2}=0$. The plots at the first row are for $\Lambda=0$ and
$y=I_{0}t.$ In the second row the plots are for $\Lambda=0.1$ and
$y\left(t\right)=\sqrt{\frac{I_{0}}{\Lambda}}\sinh\left(\sqrt{\Lambda}t\right).$
For the plots we selected
$\left(V_{1},I_{0},F_{0}\rho_{m0}\right)=\left(1,0.1,0.01\right)$ and initial
condition $x\left(0\right)$ such that $a\left(0\right)\simeq 0$. Solid lines
are for $w_{m}=0.01$, dashed lines are for $w_{m}=0.05$ and dotted lines are
for $w_{m}=0.1$. Dashed-dotted lines are for the $\Lambda$CDM solution with
$a\left(t\right)=a_{0}\sinh^{\frac{2}{3}}\left(\omega t\right)$ and
$w_{eff}=-\tanh^{2}\left(\omega t\right)$. We observe that in the late
universe the model mimics the $\Lambda$CDM theory with de Sitter universe as a
future attractor.

####  IV.1.2 Solution for $V_{1}-V_{2}=0$

We assume now the case for which $V_{1}-V_{2}=0$. In this case the field
equations are

$\frac{4}{3}\left(\dot{x}^{2}-\dot{y}^{2}\right)-\left(V_{1+2}y\right)^{2}-F_{0}\rho_{m0}\left(V_{1+2}y\right)^{-2w_{m}}=0,$
(53) $\ddot{x}-\Lambda y=0$ (54)

and

$\ddot{y}+\frac{3}{4}V_{1+2}\left(V_{1+2}y+w_{m}F_{0}\rho_{m0}\left(V_{1+2}y\right)^{-2w_{m}-1}\right)-\Lambda
y=0,$ (55)

where now the corresponding Noetherian conservation laws are expressed as
$I\left(X^{2}\right)\simeq\dot{x}$ and $I\left(X^{3}\right)\simeq t\dot{x}-x$,
for $\Lambda=0$; or
$I\left(Y^{1}\right)=e^{\sqrt{\Lambda}t}\left(\dot{x}-\sqrt{\Lambda}X\right)$
and
$I\left(Y^{2}\right)=e^{-\sqrt{\Lambda}t}\left(\dot{x}+\sqrt{\Lambda}x\right)$
when $\Lambda\neq 0$

This dynamical system has the same solution process as in the previous case
with $V_{1}+V_{2}=0$. We omit the analysis of this case.

####  IV.1.3 Solution for arbitrary $V_{1},~{}V_{2}$

We define the new variable $\,Z=V_{1+2}y-V_{1-2}x$ and we write the field
equations as

$\displaystyle 0$
$\displaystyle=4\left(2V_{1}\dot{y}-\dot{Z}\right)\left(2V_{2}\dot{y}-\dot{Z}\right)-F_{0}\rho_{m0}Z^{-2w_{m}}+$
$\displaystyle-\frac{1}{V_{1-2}^{2}}\left(16V_{1}V_{2}\Lambda
y^{2}-8\left(V_{1+2}\right)\Lambda
y+\left(3\left(V_{1-2}^{2}+4\Lambda\right)Z^{2}\right)\right)~{},$ (56)
$0=\ddot{Z}+\left(3V_{1}V_{2}-\Lambda\right)Z-V_{1}V_{2}F_{0}w_{m}\rho_{m0}Z^{-1-2w_{m}}$
(57)

and

$0=\ddot{y}-\Lambda
y+\frac{3}{4}\left(V_{1+2}\right)F_{0}w_{m}\rho_{m0}Z^{-1-2w_{m}}-\frac{3}{4}\left(V_{1+2}\right)Z.$
(58)

We remark that equation (57) is of the form of equation (47). Thus the
analytic solution is expressed in terms of elliptic integrals. By replacing
into (58) we find the second-order differential equation of the form
$\ddot{y}-\Lambda y=h\left(t\right)$, which is an integrable equation.

Consider now the simple Subcase with $\Lambda_{1}\Lambda_{2}=0$, say
$\Lambda_{2}=0$. Then the field equations become

$-4\left(2V_{1}\dot{y}-\dot{Z}\right)\dot{Z}-\frac{1}{V_{1}^{2}}\left(-8\left(V_{1}\right)\Lambda
y+\left(3\left(V_{1}^{2}+4\Lambda\right)Z^{2}\right)\right)-F_{0}\rho_{m0}Z^{-2w_{m}}=0,$
(59) $\ddot{Z}-\Lambda Z=0$ (60)

and

$\ddot{y}-\Lambda
y+\frac{3}{4}\left(V_{1}\right)F_{0}w_{m}\rho_{m0}Z^{-1-2w_{m}}-\frac{3}{4}\left(V_{1}\right)Z=0.$
(61)

For $\Lambda=0$ the analytic solution of this system is

$Z=Z_{1}t+Z_{2},$ (62)
$y\left(t\right)=\frac{1}{8}\left(\frac{3F_{0}\rho_{m0}}{Z_{1}^{2}\left(2w_{m}-1\right)}\left(Z_{1}t+Z_{2}\right)^{1-2w_{m}}-\left(Z_{1}t+Z_{2}\right)V_{1}t\right)+y_{1}t+y_{2},$
(63)

with constraint equation
$Z_{2}^{2}-\frac{4}{3V_{1}^{2}}Z_{1}\left(Z_{1}-2V_{1}y_{1}\right)=0$. For
$\Lambda\neq 0$ the closed-form solution is expressed in terms of the
hyperbolic function. Thus we omit it. However, it is easy that for
$\Lambda\neq 0$ the de Sitter universe is the late-time attractor of this
cosmological model.

In Fig. 2 we present the qualitative evolution for the equation of state
parameter and the scalar field for this cosmological model for various values
of the free parameters.

Figure 2: Qualitative evolution of the effective equation of state parameter
$w_{eff}\left(z\right)$ (left column) and of the scalar field
$\phi\left(z\right)$ (right column) for the scalar field potential of cases A
and B for $V_{2}=0$. The plots in the first row are for $\Lambda=0$,
$\left(Z_{1},Z_{2},V_{1},F_{0}\rho_{m0}\right)=\left(1,0,1,0.1\right)$. In the
second row the plots are for $\Lambda=4$ and for
$Z\left(t\right)=Z_{1}e^{\sqrt{\Lambda}t}$. Solid lines are for $w_{m}=0.01$,
dashed lines are for $w_{m}=0.1$ and dotted lines are for $w_{m}=\frac{1}{3}$.
We observe that in the late universe the late time attractor gives
$w_{eff}<-\frac{1}{3}$ which corresponds to an accelerated universe. For
$\Lambda\neq 0$ the future attractor is the de Sitter universe.

##  V Conclusions

Noether symmetry analysis is a powerful approach for the investigation of
integrable models in gravitational theories. Specifically, Noether’s first
theorem is applied to constrain the unknown functions and parameters of the
gravitational Lagrangian such that nontrivial variational symmetries exist.
The latter are directly related with the existence of conservation laws for
the equations of motion. The exact relation between the variational symmetries
and the conservation laws is given by Noether’s second theorem. In
gravitational physics the field equations are nonlinear and with free
functions and parameters. In ns11  it has been shown that the symmetry
analysis is a geometric selection rule for the determination of the unknown
functions and parameters. Indeed, the variational symmetries of the
gravitational Lagrangian are constructed by isometries and the homothecy of
the minisuperspace. This describes the geometry in the manifold on which the
dynamical variables of the gravitational theory lie.

In this study we considered the Noether symmetry analysis in chameleon
cosmology. We determined the scalar field potential and the coupling function
of the chameleon mechanism by the requirement that the gravitational field
equations admit Noether point symmetries. We derived four families of scalar
field potentials and coupling functions for which the field equations admit
conservation laws constructed from the second theorem of Noether. The equation
of state parameter for the ideal gas plays an important role in these
functions. Two of the derived models belong to the WIS family of models and
they have been studied before. However, for the two new models the
conservation laws indicate that the models are Liouville integrable and we
determined the analytic solutions in closed-form expressions.

From the analysis of the cosmological parameters and the asymptotic solution
(52) we infer that for small values of the redshift; i.e. in the late
universe, these models can reproduce the $\Lambda$CDM with the de Sitter
universe to be a future attractor. That is a characteristic of special
interest for this theory because we have that the $\Lambda$CDM behaviour is
recovered without us to have required the existence of a pressureless fluid in
the cosmological model.

We conclude that the application of Noether symmetries in this model provided
us not only with analytic solutions which can be used as toy models, but also
as unified dark energy model. In future work we plan to investigate further if
this model can explain the cosmological observations.

###### Acknowledgements.

This work was partially financially supported in part by the National Research
Foundation of South Africa (Grant Numbers 131604). The author thanks the
support of Vicerrectoría de Investigación y Desarrollo Tecnológico (Vridt) at
Universidad Católica del Norte through Núcleo de Investigación Geometría
Diferencial y Aplicaciones, Resolución Vridt No - 096/2022.

##  Appendix A Noether symmetry conditions

We present the determining system provided by the Noether symmetry condition
(15) for the Lagrangian function (11).

$\xi_{,a}=0~{},~{}\xi_{,\phi}=0~{},$ (64)
$6a\eta_{,t}^{a}+g_{,a}=0~{},~{}a^{3}\eta_{,t}^{\phi}-g_{,\phi}=0,$ (65)
$2a\eta_{,a}^{a}+6\eta^{a}-a\xi_{,t}=0,$ (66)
$a^{2}\eta_{,a}^{\phi}-6\eta_{,\phi}^{a}=0,$ (67)
$3\eta^{a}+2a\eta_{,\phi}^{\phi}-\xi_{,a}=0,$ (68)
$3a^{2}\eta^{a}\left(V+w_{m}\rho_{m0}a^{3w_{m}-3}F\right)+a^{3}\eta^{\phi}\left(V_{,\phi}-\rho_{m0}a^{3w_{m}-3}F_{,\phi}\right)+\left(a^{3}V\left(\phi\right)-\rho_{m0}a^{-3w_{m}}F\right)\xi_{,t}=-g_{,t}.$
(69)

The Noether symmetry conditions gives the coefficient components of the
Noether vector field $X$. The solution of the Noether conditions it depends on
the functional forms of the free functions $V\left(\phi\right)$ and
$F\left(\phi\right)$. Hence for various values of these functions as presented
in Section III the admitted Noether symmetries differ.

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