1. Introduction
In order to understand properly the role of symmetries in Cosmology, we have to make a short detour into General Relativity and the relativistic models in general. General Relativity associates the gravitational field with the geometry of spacetime as this is specified by the metric of the Riemannian structure. Concerning the matter, this is described in terms of various dynamical fields which are related to the mater via Einstein equations
Einstein equations are not equations, in the sense that they equate known quantities in terms of unknown ones, that is there is no point to look for a “solution” of them in this form. These equations are rather generators of equations which result after one introduces certain assumptions according to the model required. These assumptions are of two kinds: (a) geometric assumptions; and (b) other non-geometric assumptions among the physical fields which we call equations of state. The first specifies the metric to a certain degree and are called collineations [
1] (or “symmetries” which is in common use and is possible to create confusion in our discussion of symmetries in Cosmology). The collineations are the familiar Killing vectors (KV), the conformal Killing vectors (CKV), etc. The collineations affect the Einstein tensor, which is expressed in terms of the metric. For example, for a KV
X, one has
.
Through Einstein field equations collineations pass over to
and restrict its possible forms, therefore the types of matter that can be described in the given geometry-model. For example, if one considers the symmetries of the Friedmann–Robertson–Walker (FRW) model, which we shall discussed in the following, then the collineations restrict the
to be of the form
where
,
p are two dynamical scalar fields the density and the isotropic pressure,
are the comoving observers
(
and
is the spatial projection operator. An equation of state is a relation between the dynamical variables
Once one specifies the metric by the considered collineations (and perhaps some additional requirements of geometric nature) of the model and consequently the dynamical fields in the energy momentum tensor, then Einstein equations provide a set of differential equations which describe the defined relativistic model. What remains is the solution of these equations and the consequent determination of the physics of the model. At this point, one introduces the equations of state which simplify further the resulting field equations.
When one has the final form of the field equations considers a second use of the concept of ”symmetry” which is the main objective of the current work. Let us refer briefly some history.
In the late 19th century, Sophus Lie, in a series of works [
2,
3,
4] with the title “Theory of transformation groups” introduced a new method for the solution for differential equations via the concept of ”symmetry”. In particular, Lie defined the concept of symmetry of a differential equation by the requirement of the point transformation to leave invariant the set of solution curves of the equation, that is, under the action of the transformation, a point form one solution curve is mapped to a point on another solution curve. Subsequently, Lie introduced a simple algebraic algorithm for the determination of these types of symmetries and consequently on the solution of differential equations. Since then, symmetries of differential equations is one of the main methods which is used for the determination of solutions for differential equations. Some important works which established the importance of symmetries in the scientific society are those of Ovsiannikov [
5], Bluman and Kumei [
6], Ibragimov [
7], Olver [
8], Crampin [
9], Kalotas [
10] and many others; for instance, see [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20].
As it is well known, an important fact in the solution of a differential equation are the first integrals. Inspired by the work of Lie in the early years of the 20th century, Emmy Noether required another definition of symmetry, which is known as Noether symmetry. This symmetry concerns Lagrangian dynamical systems and it is defined by the requirement that the action integral under the action of the point transformation changes up to a total derivative so that the Lagrange equation(s) remain the same [
21]. She established a connection between a Noether symmetry and the existence of a first integral which she expressed by a simple mathematical formula. In addition, Noether’s work except for its simplicity had a second novelty by alloying the continuous transformation to depend also on the derivatives of the dependent function, which was the first generalization of the context of symmetry from point transformations to higher-order transformations. Since then, symmetries play an important role in various theories of physics, from analytical mechanics [
22], to particle physics [
23,
24], and gravitational physics [
1,
25].
In the following, we shall present briefly the approach of Lie and Noether symmetries and will show how the symmetries of differential equations are related to the collineations of the metric in the mini superspace. We shall develop an algorithm which indicates how one should work in order to get the analytical solution of a cosmological model. The various examples will demonstrate the application of this algorithm.
In conclusion, by symmetries in Cosmology, we mean the work of Lie and Noether applied to the solution of the field equations of a given cosmological model scenario.
3. Symmetries of Differential Equations
Consider an
order system of differential equations defined on
of the form
We say that the point transformation (
2) in
generated by the vector field
is a symmetry of the system of equations if it leaves the
set of solutions of the system the same. Equivalently, if we consider the function
on
then a symmetry of the differential equation is a vector field leaving
G invariant.
The main reason for studying the symmetries of a system of differential equations is to find first integrals and/or invariant solutions. Both of these items facilitate the solution and the geometric/physical interpretation of the system of equations.
In the following, we shall be interested in systems of second order differential equations (SODE) of the form
therefore we shall work on the jet space
which is essentially the space
. In this case, the infinitesimal point transformation (
2) is written
and it is generated by the vector field
where
are some general (smooth) functions and
is the prolonged vector field:
where
If then is defined on the base manifold M and is called the first prolongation of X in
6. Generalized Killing Equations
We decompose the Noether condition along the vector
and normal to it. In order to do that, we expand the overdot terms and assume that
are independent variables. The right hand side (rhs) is:
while the rhs gives:
Therefore, we obtain the following equivalent system of equations:
These equations have been called the generalized Killing equations (see Equations (17) and (18) of Djukic in [
32]).
We note that the generalized Killing equations have
unknowns (the
and are only
equations. Therefore, there is not a unique solution
and we are free to fix
variables in order to get a solution. However, this is not a problem because all these solutions admit the same first integral
I of the dynamical equations (because all satisfy the Noether condition (
48)).
6.1. How to Solve the Generalized Killing Equations
Suppose that, by some method, we have determined a quadratic first integral
I of the dynamical equations. Our purpose is to determine a gauged Noether symmetry which will admit the given quadratic first integral. Assume the
gauge conditions
Suppose
is the Lagrangian part of the dynamical equations ( Actually, we only need the kinetic energy which will define the kinetic metric). Let
be the vector field generating a Noether symmetry which admits the first integral
where
is the Noether gauge function. We compute
where
is the kinetic metric determined by the Lagrangian
Then, the second Equation (50) gives that
and we find eventually the expression
The vector we have determined is not the only one possible. For example, one may specify the gauge function f and assume a form for (while maintain the gauge and then use Equation (50) to determine the solution (see example below). However, in all cases, the first integral is the same.
The first integral of a Noether point symmetry
is in addition an invariant of the Noether generator, that is,
The result (
52) means that a Noether point symmetry results in a twofold reduction of the order of Lagrange equations. This is done as follows. The first integral
can be used to replace one of the second order equations by the first order ODE
where
is a constant fixed by the initial (or boundary) conditions. Property (
52) says that this new equation admits the Lie symmetry
(because
); therefore, it can be used to integrate the equation once more, according to well-known methods.
Noether symmetries are mainly applied to construct first integrals which are important to determine the solution of a given dynamical system. It is possible that there exist different (i.e., not differing by a perfect differential) Lagrangians describing the same dynamical equations. These Lagrangians have different Noether symmetries (see, for example, [
33,
34,
35]). Therefore, it is clear that, when we refer to a Noether symmetry of a given dynamical system, we should always mention the Lagrangian function assumed.
Let us discuss the above scenario in a practical case.
Consider the Emden–Fauler equation
where
are arbitrary constants. This equation defines a conservative holonomic dynamical system with Lagrangian
Assume now that the function
where
A is some constant and assume further that
. Equation (
50) gives
where
F is an arbitrary function of its arguments. Replacing this in (
49), we find
which provides the following system
The solution of the latter system is
Still, we do not know the parameters
In order to compute them, we turn to the first integral
Replacing this, we have:
Having computed I, we compute from the relation hence Then, Comparing this with what we have already found, we get We note that, for these values of the as it is correct because the relation is valid only under the assumption
8. Symmetries of SODEs in Flat Space
Obviously, an area where symmetries of ODEs play an important role is the cases in which the kinetic metric (not necessarily the spacetime metric) is flat. These cases cover significant part of Newtonian Physics where the kinetic energy is a positive definite metric with constant coefficients; therefore, there is always a coordinate transformation in the configuration space that brings the metric to the Euclidian metric. Similar remarks apply to Special Relativity and—as we shall see—to Cosmology.
The basic result in these cases is that the maximum number of Lie point symmetries that a SODE can have is
and the maximum number of Noether point symmetries
Moreover, the number of point symmetries which a SODE can possess is exactly one of 0, 1, 2, 3, or 8 [
36]. Similar results exist for higher-order differential equations [
37].
Lie has shown [
2] the important result that
“for all the second order ordinary differential equations which are invariant under the elements of the sl (3, R), there exists a transformation of variables that brings the equation to the form and vice versa”.
In current cosmological models of importance, of interest is the case therefore we shall restrict our attention to two cases
i. The case the of systems which admit the maximum number of Lie point symmetries which for is eight and span the algebra .
ii. The case that the Lie point symmetries span the algebra
8.1. The Case of sl(3, R) Algebra
The prototype dynamical system which admits the
algebra of eight Lie point symmetries is the Newtonian free particle moving in one dimension whose dynamical equation is
where
and a dot means differentiation with respect to the time parameter
t.
Let
be the generator of a Lie point symmetry of (
54). The Lie point symmetries of (
54) are given by the special projective vectors of
(see condition (
35))
To show the validity of the aforementioned Lie’s result, we consider the harmonic oscillator
which also admits the eight Lie point symmetries [
38]
which form another basis of the
Lie algebra. The transformation which relates the two different representations of the
algebra is
It easy to show that under this transformation Equation (
55) becomes
, which is Equation (
54).
In order to calculate the Noether point symmetries of (
54), we have to define a Lagrangian. We recall that the Noether symmetries depend on the particular Lagrangian we consider. Let us assume the classical Lagrangian
Replacing in the Noether condition, we find the associated Noether conditions
whose solution shows that the Noether point symmetries of (
54) for the Lagrangian
are the vector fields
and
, with corresponding non-constant Noether functions the
and
.
Furthermore, from the second theorem of Noether, the corresponding first integrals are calculated easily. The vector
provides the conservation law of energy,
the conservation law of momentum, while
gives the Galilean invariance [
39]. Finally, the vector fields
and
are also important because they can be used to construct higher-order conservation laws.
A more intriguing example is the slowly lengthening pendulum whose equation of motion in the linear approximation is (The time dependence in the ‘spring constant’ is due to the length of the pendulum’s string increasing slowly [
40].),
which also admits eight Lie point symmetries. According to Lie’s result, there is a transformation which brings (
59) to the form (
54). In order to find this transformation, one considers the Noether point symmetries and shows that (
59) admits the quadratic first integral [
41]
where
, is a solution of the second-order differential equation
The first integral (
60) is known as the Lewis invariant.
On the other hand, Equation (
61) is the well-known Ermakov–Pinney Equation [
42] whose solution has been given by Pinney [
43] and it is
where
are constants of which only two are independent, and the functions
are two linearly independent solutions of (
59).
Finally, the transformation which connects the time dependent linear Equation (
59) with (
54) is the following
where
is given by (
62).
8.2. The Case of the sl (2, R) Algebra
The prototype system in this case is the Ermakov system defined by Equations (
59) and (
60) whose Lie point symmetries span the
algebra for arbitrary function
, while any solution for a specific
can be transformed to a solution for another
by a coordinate transformation. The Ermakov system has numerous applications in diverting areas of Physics (see, for instance [
44,
45,
46]).
Let us restrict our considerations to the autonomous case, with
a constant. Equation (
61) becomes
while the elements of the admitted
Lie algebra are
and
Concerning the Noether point symmetries, we consider the Lagrangian
and find that the Lie symmetries
satisfy the Noether condition, hence they are also Noether point symmetries, and lead to the quadratic first integral of energy
and the time dependent first integrals
or
While the first integrals
and
are time-dependent, we can easily construct the time-independent Lewis invariant [
47]. For instance,
is a time-independent first integral.
The two-dimensional system with Lagrangian
describes the simplest generalization of the Ermakov–Pinney system in two-dimensions. It can be shown that the Lie point symmetries
are Noether point symmetries of (
71) with the same first integrals. Again with the use of the time dependent Noether integrals
and
, we are able to construct the autonomous conservation laws [
48]
and
As we shall see below, the Ermakov–Pinney system and its generalizations are used in the dark energy models [
47].
11. Symmetries in Cosmology
The nature of the source which drives the late-acceleration phase of the universe is an important problem of modern cosmology. Currently, the late-acceleration phase of the universe is attributed to a perfect fluid with a negative equation of state parameter, which has been named the dark energy. The simplest dark energy candidate is the cosmological constant model leading to the
CDM cosmology. In this model, the gravitational field equations can be linearized and one is able to write the analytic solution in closed-form. However, in spite of its simplicity, the
CDM cosmological model suffers from two major problems: the fine-tuning problem and the coincidence problem [
54,
55,
56]. In order to overcome these problems, cosmologists introduced dynamical evolving dark energy models. In these models, the dark energy fluid can be an exotic matter source like the Chaplygin gas, quintessence,
essence, tachyons or it can be of a geometric origin provided by a modification of Einstein’s General Relativity [
57,
58,
59,
60,
61,
62,
63,
64,
65,
66,
67,
68,
69,
70]. The dark energy components introduce new terms in the gravitational field equations which are nonlinear or increase the degrees of freedom. Thereafter, the linearization process applied in the case of the
CDM model fails and other mathematical methods must be applied in the study of integrability of the field equations and the construction of analytic solutions.
Two different groups, de Ritis et al. [
71] and Rosquist et al. [
72], applied independently the symmetries of differential equations in order to construct first integrals in scalar field cosmology. In particular, they determined the forms of the scalar field potential, which drives the dynamics of the dark energy, in order for the field equations to admit Noether point symmetries. The classification scheme is based on an idea proposed by Ovsiannikov [
5]. Since then, the classification scheme has been applied to various dark energy models and modified theories of gravity. Some of these classifications are complete while some others lack mathematical completeness leading to incorrect results. The purpose of the current review is to present the application of symmetries of differential equations in modern cosmology.
A cosmological model is a relativistic model and therefore requires two assumptions:
a. A specification of the metric, which is achieved mainly by the collineations for the comoving observers we discussed above and
b. Equations of state which specify the matter of the model universe and are mathematically compatible with the assumed collineations defining the metric. This is done by the introduction of a potential function in the action integral from which the field equations follow.
One important class of cosmological models are the ones in which spacetime brakes in 1 + 3 parts, that is, the cosmic time and the spatial universe, respectively. The latter is realized geometrically by three-dimensional spacelike hypersurfaces which are generated by the orbits of the KVs of a three-dimensional Lie algebra. In 1898, Luigi Bianchi [
73] classified all possible real three-dimensional real Lie algebras in nine types. Each Lie algebra leads to a (hypersurface orthogonal) cosmological model called a Bianchi Spatially homogeneous cosmological model. These nine models have been studied extensively in the literature over the years and have resulted in many important cosmological solutions.
The principal advantage of Bianchi cosmological models is that, due to the geometric structure of spacetime, the physical variables depend only on the time, thus reducing the Einstein and the other governing equations to ordinary differential equations [
74]. However, the gravitational field equations in General Relativity for the Bianchi cosmologies are ordinary second-order differential equations, due to the existence of nonlinear terms, exact solutions have been determined only for a few of them [
75,
76,
77,
78,
79,
80], while there was a debate a few years ago on the integrability or not of the Mixmaster universe (Bianchi type IX model) [
80,
81,
82,
83,
84].
In order to get detailed information on these alternative models, one has to find an analytical solution of the field equations. This can be a formidable task depending on the form of the potential function and the free parameters that it has. The standard method to find an analytical solution is to use Noether point symmetries and compute first integrals of the field equations. Indeed, the application of symmetries of differential equations in the dark energy models started with the use of the first Noether theorem in [
71] and with the consideration of the second Noether theorem in [
72]. Both approaches are equivalent. Since then, Noether point symmetries have been applied to a plethora of models for the determination of first integrals, and consequently analytical solutions. We refer the reader to some of them [
85,
86,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116]. It is important to note that some of the published results are mathematically incorrect. For instance, in [
117], the authors used the Noether conditions in order to solve the dynamical equations of the model, and a posteriori, they determined the symmetries of the field equations. This is not correct because Noether symmetries are imposed by the requirement that they transform the Action Integral in a certain way and not as extra conditions on the Euler–Lagrange equations.
The difficulty with the above approach is that one has to work in spacetime where the geometry is not simple and the field equations are rather complex. To bypass that difficulty, a new scenario has been developed in which one transforms the problem to a minisuperspace defined by the dynamical variables through a Lagrangian which produces the field equations in that space [
118]. Then, one considers the Lagrangian in two parts: the kinematic part which defines the kinetic metric and the remaining part which defines the effective potential. If one knows the homothetic algebra of the kinetic metric [
27], then the application of the results of
Section 10 provide the Noether point symmetries and the corresponding Noether first integrals of the field equations in mini superspace. Therefore, the solution of the field equations is made possible, and, by the inverse transformation, one finds the solution of the original field equations in the original dynamic variables in spacetime. This approach brought new results in various dark energy models and modified theories of gravity [
119,
120,
121,
122]. Some of these results are discussed in the following.
Before we enter into detailed discussion, it is useful to state the scenario of this method of work in order to provide a working tool to new cosmologists not experienced in this field.
11.0.1. Method of Work—Scenario
1. Consider the Action Integral of the model in spacetime and produce the field equations.
2. Change variables and give a new set of field equations in the minisuperspace of dynamical variables in a convenient form.
3. Define a Lagrangian for the field equations in the mini superspace.
4. Read from the Lagrangian the kinetic metric and the effective potential. The new variables must be such that the kinetic metric will be flat or at least one for which one knows already the homothetic algebra. This defines the phrase "convenient form " stated in step 2 above.
5. Apply the results of
Section 3 to get a classification of Noether point symmetries of the field equations and compute the corresponding first integrals in the mini superspace.
6. Using the first integrals, solve the field equations for the various cases of the effective potential and other possible parameters.
7. Apply the inverse transformation and get the solution of the original field equations in terms of the original dynamical variables in spacetime.
In the sections which follow, we apply this scenario to the major cosmological models proposed so far and give the detailed results in each case.
11.1. FRW Spacetime and the CDM Cosmological Model
The FRW spacetime is a decomposable 1 + 3 spacetime in which the three-dimensional hypersurfaces are maximally symmetric spacelike hypersurfaces of constant curvature, which are normal to the time coordinate. The metric of a FRW spacetime is specified modulo a function of time, the scale factor
In comoving coordinates
it has the form
In the Bianchi classification, it is a Type IX spacetime. This spacetime in classical General Relativity for comoving observers (non-comoving observers can support all types of matter)
(
can support matter that is a perfect fluid, that is, the energy momentum tensor, is
where
are the energy density and the isotropic pressure of matter as measured by the comoving observers.
is the tensor projecting normal to the vector
This spacetime has been used in the early steps of relativistic cosmology.
The first cosmological model using this spacetime was the
CDM cosmology, which was a vacuum spatially flat FRW spacetime with matter generated by a non-vanishing cosmological constant
. For this model, Einstein field equations are
and
whose solution is the well-known de Sitter solution
that is a maximally symmetric spacetime (not only the maximally symmetric 3d hypersurfaces). According to earlier comments, there exists a coordinate transformation that brings the system to the linear Equation (
110) [
123]. Indeed, if we introduce the new variable
the field Equation (
109) becomes [
124]
which is the one-dimensional hyperbolic linear oscillator, which admits eight Lie point symmetries.
Let us demonstrate the geometric scenario mentioned above in this simple case. For this, we need to find the maximum number of Noether point symmetries admitted by the field Equation (
109). We choose the variables
and have a two-dimensional mini superspace. A Lagrangian for Equation (
109) in the minisuperspace
is
from where we read the kinetic metric
and the effective potential
The Noether condition (
48) for the Lagrangian (
111) gives that it admits five Noether point symmetries, which is the maximum number of Noether symmetries for a two-dimensional space. These Noether symmetries are
Having the Noether symmetries, we continue with the first integrals and finally get the solution we have already found. Note that not all first integrals are independent. This simple application shows how the geometry of the kinetic metric can be used to recognize the equivalence of well-known systems of classical mechanics with dark energy models.
11.2. Scalar-Field Cosmology
In the case of classical General Relativity with a minimally coupled scalar field (quintessence or phantom), the Action Integral in spacetime is considered to be
Assuming a spatially flat FLRW background and comoving observers, the field equations are
We consider the mini superspace defined by the dynamic variables
A point-like Lagrangian in the mini superspace of the for field Equations (
114) and (
115) is
To bring the Lagrangian in the “convenient form”, we consider the coordinate transformation
to
and again
to
where
and the new variables have to satisfy the following inequality
. In the coordinates
the scale factor becomes
Under the coordinate transformation
the point-like Lagrangian takes the simpler form
in which the metric in the coordinates
is the Lorentzian 2d metric
which is the metric of a flat space while the effective potential is
Application of the previous analysis gives the following classification of Noether point symmetries of the model for various forms of the effective potential [
118]:
To find the solution in the original dynamical variables
we apply the inverse transformation. The result is
11.3. Brans–Dicke Cosmology
The Brans–Dicke action is [
126]
where
is related to the Brans–Dicke parameter.
In the case of the spatially flat FRW background and comoving observers, the point-like Lagrangian in the mini superspace defined by the variables
, which describes the gravitational field equations is
If one performs the coordinate transformation
to
by the equations
Lagrangian (
127) becomes
from which we have that the kinetic metric of the minisuperspace is the conformally flat Lorentzian 2d metric
whose symmetry algebra depends on the values
while the effective potential is
We consider cases.
11.3.1. Case .
For
, the homothetic algebra of the minisuperspace consists of the gradient KVs
the non-gradient KV
and the gradient HV
The symmetry classification provides the following results [
121]:
For arbitrary potential , the dynamical system admits the Noether point symmetry .
For
there are two additional Noether point symmetries
with first integrals
For
there are two additional Noether point symmetries
, with corresponding first integrals
For
, we have the extra Noether point symmetries
with Noether first integrals
For
, we have the extra Noether symmetries
constant, with first integrals
For the potential
the additional symmetry is the vector field
with first integrals
For
the extra Noether point symmetries are
with first integrals
We note that in this case the system is the Ermakov–Pinney dynamical system (because it admits the Noether point symmetry algebra the hence the Lie symmetry algebra is at least .
For
,
we have the Noether point symmetries
,
constant with corresponding first integral
For this potential, the Noether point symmetries form the Lie algebra, i.e., the dynamical system is the two-dimensional Kepler–Ermakov system.
The case corresponds the two-dimensional free particle in flat space and the dynamical system admits seven additional Noether point symmetries.
11.3.2. Case
We have to consider two cases i.e., and It is enough to consider the case , because the results for are obtained directly from those for if we make the substitution
For
the homothetic algebra of the minisuperspaceis given by the vector fields
of (
130,
131) and the vector field
Hence, the symmetry classification provides the following cases:
For arbitrary potential , the dynamical system admits the Noether point symmetry .
All the rest of cases admit additional symmetries.
If
we have the extra Noether point symmetries
with first integrals (
134) with
.
If
we have the Noether point symmetries
with first integrals the (
135) with
.
Noether point symmetries generated by the KV .
If
then we have the Noether point symmetries
with first integrals
If
then we have the Noether point symmetries
with first integrals
If then the system becomes the free particle and admits seven extra Noether point symmetries.
The exact solutions of the models and their physical properties can be found in [
121]. The results from the classification analysis are presented in
Table 3 and
Table 4. For the notation of the admitted Lie algebra, we follow the Mubarakzyanov Classification Scheme [
127,
128,
129].
11.4. f (R)-Gravity
f (
R)-Gravity (in the metric formalism) is a fourth-order theory where the Action Integral in spacetime is assumed to be
In the case of FRW background and comoving observers, the resulting field equations follow from the Lagrangian [
129]
where
is the spatial curvature of the FRW spacetime, and a prime denotes derivative with respect to the dynamical parameter
R, that is
. Since
theory can be written as a special case of Brans–Dicke theory, the so-called O’Hanlon gravity [
130], the results will be similar to that of the previous analysis. However, for completeness, we present them below.
The classification scheme provides the following cases [
119]:
In
Table 5, we collect the results of the classification scheme for
-gravity.
11.5. Two-Scalar Field Cosmology
We consider now a two-scalar field cosmological model in General Relativity with Action Integral
where
describes the coupling between the two scalar fields
in the kinematic part. Moreover, we assume the metric tensor
to be a maximally symmetric metric of constant curvature [
122]. In such a scenario, it is not possible to define new fields in order to remove the coupling in the kinematic part.
Assuming again a spatially flat FRW spacetime and comoving observers the field equations are
where
are the connection coefficients for the metric
In the mini superspace defined by the variables
we introduce the new variables
by the requirements
and the field equations become
These follow from the point-like Lagrangian
which defines the 3d flat Lorentzian kinetic metric
and the effective potential
The kinetic metric admits a seven-dimensional homothetic algebra consisting of the three gradient KVs (translations)
with corresponding gradient functions
given by
the three non-gradient KVs (rotations) which span the
algebra
and the gradient proper HV
The classification of the Noether symmetries for the various potentials
is as follows [
122]:
For arbitrary potential , the field equations admit the Noether point symmetry which provides the constraint equation of General Relativity.
For , the dynamical system is maximally symmetric and admits in total twelve Noether point symmetries.
For
, there exists the additional Noether point symmetry, the vector field
, with conservation law the angular momentum on the two-dimensional sphere, that is,
For
, the system admits six additional Noether point symmetries given by the vector fields
where
and
. The corresponding Noether integrals are expressed as follows:
However, when two constants
are equal, for instance,
, then the dynamical system admits an extra Noether symmetry. That is, it admits the rotation normal to the plane defined by the axes
given by the vector
where
if
and
if
.
For the potential being
, the dynamical system admits the extra Noether symmetries
where the functions
and
are given by the linear second-order differential equations
and
. Finally, the corresponding Noether integrals are as follows:
In both of the last cases, from the admitted algebras of Lie symmetries, it is easy to recognize that the gravitational field equations can be linearized. Indeed, for the potential
under the coordinate transformation
the field equations become
which is the three-dimensional “unharmonic-oscillator”.
On the other hand, for the potential
, we perform the additional transformation
and the field equations are linearized as follows:
11.6. Galilean Cosmology
The cubic Galilean cosmological model in a spatially flat FRW spacetime with comoving observers is defined by the Lagrangian [
131]
From the symmetry condition, we should determine two functions,
and
Indeed, we find that, when [
104]
Lagrangian (
179) admits the Noether point symmetries
which form the
Lie algebra.
The Noether point symmetry
provides as first integral the constraint equation, while
gives the first integral
Furthermore, the same first integral exists in the limit in which . In addition, we remark that when the universe is dominated by the potential of the scalar field, then , and the model reduces to that of a minimally coupled scalar field.
12. Higher-Order Symmetries in Cosmology
In the previous section, we presented classification of cosmological models based on point symmetries. However, these are not the only cases where first integrals are used. Indeed, it is possible for one to extend the classification scheme by applying non-point symmetries, such as the contact symmetries.
In particular, for Lagrangians of the form (
93), it has been found that the vector field
is a contact symmetry for the Action Integral iff the following conditions are satisfied [
10]
Conditions (
183)–(
185) follow directly from the application of the weak Noether condition. From the symmetry condition (
184), it follows that
and
. Furthermore, the second-theorem of Noether provides the first integral
Condition (
183) means that the second rank tensor
is a Killing tensor of order 2 of the metric
. Condition (
185) is a constraint relating the potential with the Killing tensor
and the Noether function
f. Application of contact symmetries in cosmological studies can be found in [
72,
132,
133,
134,
135]. In the following, we present the results for the classification of contact symmetries in scalar-field cosmology and
-gravity.
12.1. Scalar-Field Cosmology from Contact Symmetries
In the polar coordinates (
117), the Lagrangian of the field equations in scalar field cosmology become
while the Killing tensors of rank two for a two-dimensional flat space in Cartesian coordinates
are
Thus, condition (
185) provides the following cases [
132]:
Case A: For the hyperbolic Potential
the field equations admit the contact symmetry
with corresponding Noether Integral
In the special case where
, the field equations admit the second contact symmetry
with corresponding Noether Integral
Case B: For the potential
there exists the contact symmetry
with Noether Integral
Case C: For potential
the admitted contact symmetry is
with corresponding Noether Integral
This potential is equivalent to case B under the transformation
Case D: For
the admitted contact symmetry is
with corresponding Noether Integral
Moreover, when
, the dynamical system admits the additional contact symmetry
with corresponding Noether Integral
Case E: Finally, for the potential
the field equations admit the first integral
generated by the contact symmetry (
199). It is important to note that in all cases the results remain the same under the transformation
.
12.2. -Gravity from Contact Symmetries
Without loss of generality, we define
, where now
-gravity can be written in its equivalent form as a Brans–Dicke scalar field cosmological model. Specifically, the Lagrangian of the field equations is written equivalently as
where
The classification in terms of the contact symmetries provides the following five cases for the potential and the corresponding first integrals.
Case A: For
, the field equations admit the quadratic first integral
Case B: For
the field equations admit the quadratic first integral
Case C: For
the field equations admit the quadratic first integral
Case D: For
the field equations admit the quadratic first integral
Case E: For
the field equations admit the quadratic first integral
However, in order to derive the function
one has to solve the Clairaut Equation (
204). For the above cases, Clairaut equation has a closed-form solution only for some particular forms of
. The analytic forms of
functions that admit contact symmetries are presented in
Table 6.