#### Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

Eur. Phys. J. C
Integrability from point symmetries in a family of cosmological Horndeski Lagrangians
N. Dimakis 1
Alex Giacomini 1
Andronikos Paliathanasis 0 1
0 Institute of Systems Science, Durban University of Technology , PO Box 1334, Durban 4000 , Republic of South Africa
1 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile , Valdivia , Chile
For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann-LemaîtreRobertson-Walker space-time. We are interested in point transformations which leave invariant the field equations. Noether's theorem is applied to determine the conservation laws for a family of models that belong to the same general class. The cosmological scenarios with or without an extra perfect fluid with constant equation of state parameter are the two important cases of our study. The de Sitter universe and ideal gas solutions are derived by using the invariant functions of the symmetry generators as a demonstration of our result. Furthermore, we discuss the connection of the different models under conformal transformations while we show that when the Horndeski theory reduces to a canonical field the same holds for the conformal equivalent theory. Finally, we discuss how singular solutions provides nonsingular universes in a different frame and vice versa.
1 Introduction
The plethora of phenomena which have been discovered
the last few years have led to the consideration of
alternative/modified gravitational theories [1]. These extended
theories of gravity provide additional terms in the field
equations which – in conjunction to those of general relativity –
can explain the various phases of the universe. In physical
science the existence of a fundamental axiom, such as
Hamilton’s principle, is of paramount importance. Among the
theories that are generated by an action integral, those that are of
second-order in respect to their equations of motion possess
a distinguished role; from Newtonian Mechanics to General
Relativity.
A particular family of theories which has drawn attention
are the so-called Horndeski theories. Horndeski, in 1974 [2]
derived the most general action for a scalar field in a
fourdimensional Riemannian space in which the Euler–Lagrange
equations are at most of second-order. All relevant scalar–
tensor theories with this property, such as the Brans–Dicke
[3–5], the Galileon [6,7] and the generalized Galileon [8]
belong to the general family of Horndeski theories. Although
the latter are of second-order, they provide non-canonical
nonlinear field equations which – even for the simplest line
element of the underlying space – might not be able to lead
to solutions in terms of closed-form expressions. Due to the
high level of nonlinearity, numerical methods are applied
in order to approximate the evolution of the system.
However, whether a solution actually exists is not always known.
For that reason, in this work, we are motivated to study the
integrability of the field equations for a class of families of
Hordenski theories (or equivalently those of the generalized
Galileon Lagrangian [9]).
There are various methods to study the integrability of
a system of differential equations. Two of the most famous
are: (a) the existence of invariant transformations, i.e.
symmetries and (b) the singularity analysis; for a recent
discussion and comparison of these two methods see [10]. Both
of these have been applied widely in gravitational
theories [11–17]. In several cases the gravitational field
equations, after a specific ansatz for the line element is adopted,
can be derived by point-like Lagrangians [18]. The
application of Noether’s theorem over the corresponding
minisuperspace action has been utilized for the determination
of conservation laws/analytical solutions in various
models both at the classical as well as in the quantum level,
for instance see [19–30] and references therein. Noether’s
theorem is the main mathematical tool that we use in this
study.
From the various different classes of generators of the
invariant transformations we consider the most simple;
the one corresponding to the so-called point symmetries.
Point symmetries are the generators of the invariant
transformations in the base manifold in which the
dynamical system is defined. The application of Noether’s
theorem for point symmetries leads to conservation laws
linear in the momentum. Well known conservation laws of
this type from classical mechanics are those of the
momentum and the angular momentum. The plan of the paper it
follows.
In Sect. 2, we present our model which is a special
consideration of the Hordenski Lagrangian and can be seen as a first
generalization of the canonical scalar–tensor theories.
Moreover, we consider the cosmological scenario of an isotropic
and homogeneous universe in which an extra fluid exists
with constant equation of state parameter. For that model
the field equations are calculated while the mini-superspace
Lagrangian is discussed. Furthermore our model admits four
unknown functions which define the specific form of the
cosmological model. In Sect. 3, we apply Noether’s
theorem in the mini-superspace Lagrangian in order to specify
the unknown form of the functions which define the model
and derive the corresponding Noetherian conservation laws.
For the completeness of our analysis we consider separately
the cases with or without an ideal gas and the cases with
zero or non-zero spatial curvature for the underlying
spacetime. In order to demonstrate the usefulness of our results
we present some closed-form (special) solutions in Sect. 4;
these are derived from the invariant functions of the admitted
symmetry vectors. Finally, in Sect. 5 we discuss our results
and specifically we discuss the relation between our models
under conformal transformations and we show how a
singular universe is mapped to a nonsingular universe under the
change of the frame.
2 The model
The gravitational action integral that we consider is of the
form
S =
√−g h(φ)R − ω(2φ) φ,μφ,μ − V (φ)
− g(2φ) φ,μφ,μ φ d4x + Sm ,
(
1
)
that falls into the class of a generalized Galileon (or
Horndeski) model [9,31], where Sm indicates any
possible additional matter content. Furthermore, for the
background geometry, we consider a four-dimensional space of
Lorentzian signature, isotropic and homogeneous with
spatially flat three-dimensional geometry; that is, a Friedmann–
Lemaître–Robertson–Walker (FLRW) space-time line
element
ds2 = −N (t )2dt 2 + a(t )2 dx 2 + dy2 + dz2 .
(
2
)
The action integral (
1
) is the simplest generalization of those
scalar–tensor theories where the kinetic energy is not coupled
with functions entailing derivatives of φ. As we can see, when
g (φ) → 0, an action integral of that type, including the
Brans–Dicke case, is recovered.
Our goal is to derive all admissible gravitation models in
(
1
) that possess an integral of motion of the form previously
described. In that respect, note that the existence of ω(φ)
is not trivial since its absorption with a re-parameterization.
φ = f (ϕ) = ω(ϕ)−1/2dϕ leads to the transformation of
the Laplacian of φ as (primes denote differentiation with
respect to the argument)
φ = f (ϕ) ϕ + f (ϕ)ϕ,μϕ,μ,
introducing a new kinetic energy squared term in action (
1
).
This is in contrast to what happens in the case of a scalar field
whose action does not involve a coupling with derivatives of
φ over its kinetic term, i.e. when in our case g (φ) = 0, where
without loss of generality we can always select ω (φ) to be a
constant.
The mini-superspace Lagrangian that we can derive with
the help of the gravitation plus scalar field part of (
1
) and (
2
)
by integrating out the spatial degrees of freedom reads1
L N , a, a˙ , φ, φ˙ =
6ah(φ)a˙ 2
N
+ a3 N V (φ),
(
3
)
(
4
)
(
5
)
where the dot stands for differentiation with respect to time
t .
In what regards the matter part, we start our
investigation by considering the Sm = 2 Lm d4x contribution in
(
1
) to be that of a perfect fluid obeying the barotropic
equation of state P = γρ, with P(t ) and ρ(t ) the pressure and
energy density, respectively. In order to perform the variation
of Lm = √−gρ with respect to the metric we need to a priori
define a continuity equation [32]. We choose the following
well-known relation for an ideal gas in a FLRW space-time:
ρ˙ aa → ρ(t ) = ρ0a−3(γ +1),
ρ = −3(1 + γ ) ˙
where ρ0 is a constant of integration. Thus, the relevant
addition to the mini-superspace Lagrangian is
1 We assume that the field φ inherits the symmetries of the FLRW
space-time (
2
).
E N =
Eq =
∂ Ltot
∂ N
∂ Ltot
∂q
= 0,
d
− dt
∂ Ltot
∂q˙
= 0,
q = (a, φ),
are completely equivalent to the field equations of motion of
(
1
) for the metric
ω 1
h(φ)Gμν = 2 φ,μφ,ν − 2 gμν
ω φ,κ φ,κ + V (φ)
2
1
− 2 gμν G,(κ1)φ,κ + G((1μ)φ,ν)
− 21 G,(1X)φ,μφ,ν φ + Tμ(mν)
and the scalar field
h (φ)R + ω(φ)φ,κ ;κ −
ω (φ)
2
φ,κ φ,κ − V (φ)
for the energy-momentum tensor of the fluid with uμ =
( N1 , 0, 0, 0) the comoving velocity.
3 Symmetries and conservation laws
By considering the mini-superspace action to be form
invariant under point transformations generated by2
2 For details on Noether’s theorem we refer the reader to [33].
(
7
)
(8a)
(8b)
(
9
)
(
10
)
(
11
)
(
12
)
Lm (N , a) = √−gρ(t ) = Nρ0a−3γ .
As a result the total Lagrangian reads
Ltot = L + 2Lm =
where pr(
1
)Y is the first prolongation of Y , i.e. the
extension of the generator in the first jet space spanned by
(t, a, φ, N , a˙ , φ˙ , N˙ ). Clearly, since N˙ does not appear in (
4
),
we can disregard the relevant term and just write
pr(
1
)Y = Y +
∂
1 ∂a˙ +
2
∂
∂φ˙
,
where i = ddξti − q˙i ddχt , q = (a, φ), i = 1, 2.
Application of (
13
) leads to an overdetermined system of
partial differential equations to be solved for χ (t, a, φ, N ),
ξi (t, a, φ, N ), i = 1, . . . 3, and F (t, a, φ, N ). The
former is formed by gathering the coefficients of all terms
involving the derivatives of a, φ and N , to which the
functions entering the generator (
12
) have no dependence. By
gradually integrating the equations, we obtain the
following results (the basic steps to the solution can be found in
Appendix A):
• An infinite-dimensional symmetry group generated by
(
13
)
(
14
)
(
15
)
∂ ∂
Y∞ = χ (t ) ∂t − N χ˙ (t ) ∂ N
,
with χ (t ) an arbitrary function of time. Its appearance
reflects the fact that the mini-superspace action for (
4
) is
form invariant under arbitrary time transformations. Its
existence leads, through Noether’s second theorem, to a
differential identity between the Euler–Lagrange
equations of motion (
8
). The latter implies the well known
fact that not all of them are independent, i.e. Ltot is a
constrained (or singular) Lagrangian. It is well known that
when an infinite-dimensional symmetry group is present
(i.e. Noether’s second theorem is applicable) the system
is necessarily singular; however, the inverse is not true
[34]. In our case this particular symmetry is a remnant
of a more general group, the four-dimensional
diffeomorphism, χ (x )μ ∂x∂μ , under which the full gravitational
action (
1
) is form invariant.
• Additionally to the previous group – which always exists
for time re-parameterization invariant Lagrangians [35] –
we obtain a symmetry generator if h(φ), ω(φ), g(φ) and
V (φ) satisfy certain criteria. In particular, we see that if
γ = 31 , then
a ∂ φ ∂ 3γ ∂
Y = 3γ − 1 ∂a − λ ∂φ + N 3γ − 1 ∂ N
satisfies (
13
) whenever
3(γ −1)λ
h(φ) = φ 3γ −1 , g(φ) = g0φ3(λ−1),
in which λ is a constant. Note that the corresponding
gauge function appearing in (
13
) is trivial in the
calculation, i.e. F (t, a, φ, N ) =const. and this remains that way
for every case that we examine later on in the analysis.
Note also that (
16
) and (
17
) describe a particular solution
of (
13
) in the special case when γ = −1. When the fluid
contribution plays the role of a cosmological constant,
it can be absorbed inside the potential V (φ) (as can be
seen by the form of Ltot) and the general solution of the
γ = −1 case is the same as the one that we get if we
completely omit the fluid.
On the other hand, for the special case where the
perfect fluid describes radiation, i.e. γ = 1/3 the situation
changes and the symmetry generator assumes the general
form
Y = a g˜(2φg˜)(g˜φ)(2φ) ∂∂a + gg˜˜((φφ)) ∂∂φ + N g˜(φ)g˜ (φ) ∂ .
2g˜ (φ)2 ∂ N
(
18
)
while the functions entering the action need to be
h(φ) = g˜ (1φ) , g(φ) = g0 gg˜˜((φφ))33 ,
V0 g˜ (φ)
V (φ) = g˜ (φ)2 , ω(φ) = ω0 g˜(φ)2 −
3g˜ (φ)2
g˜ (φ)3 , (
19
)
where g˜(φ) is an arbitrary non-constant function. As a
result, for γ = 1/3, there exist an infinite family of
models belonging to the general class of actions of the form
(
1
) that possesses an integral of motion of the type we
are investigating.
3.1 Particular case: ρ0 = 0
Let us see how the situation alters if we remove the ideal
gas from our considerations. In other words let us use the
Lagrangian given by (
4
), in the criterion (
13
) instead of the
Lagrangian Ltot. Then, in addition to the diffeomorphism
group characterized by (
15
) – which always exists for time
re-parameterization invariant Lagrangians [35] – we obtain
a symmetry generator
Y = a
(λ + 3)g˜ (φ)2 − 3g˜(φ)g˜ (φ) ∂ g˜(φ) ∂
6g˜ (φ)2 ∂a − g˜ (φ) ∂φ
+N
(λ + 3)g˜ (φ)2 − g˜(φ)g˜ (φ) ∂
2g˜ (φ)2 ∂ N
(
21
)
that satisfies (
13
) when
1
h(φ) = g˜ (φ) , g(φ) = g0
g˜ (φ)3
g˜(φ)λ+6
, V (φ) = V0 g˜g˜(φ()φλ)+23 ,
with λ being a constant and g˜(φ) again an arbitrary
(nonconstant) function of φ. Once more, we have an infinite set
of physically different models that are characterized by a
function g˜(φ). We can see that, in comparison to the γ = 1/3
case, result (
21
) is identical to (
19
) in the special case when
λ = −3.
It is useful to study how the inclusion of spatial curvature
(either positive or negative) may alter the conditions under
which a symmetry generator appears. If we consider the line
element
ds2 = −N 2dt 2 +
dx 2 + dy2 + dz2 ,
(
22
)
a(t )2
k r 2)2
(1 + 4
where r 2 = x 2 + y2 + z2, Lagrangian (
4
) is modified by the
addition of an extra term in the potential and reads
Lk =
6a2h (φ)a˙ φ˙
N
+a3 N V (φ) − 6 k a N h(φ).
(
23
)
Thus, for the same class of point transformations we
considered earlier and with the application of the infinitesimal
symmetry criterion (
13
), it is straightforward to derive the result
given by (
20
) and (
21
) under the condition that λ = −3 or
equivalently result (
18
) and (
19
) corresponding to the
radiation fluid case where γ = 1/3. In other words when k = 0 we
have the same situation as in the γ = 1/3 case. The function
g˜(φ) once more remains arbitrary, under the restriction of
course of not being constant. As we can see, even though the
non vanishing of k imposes a restriction (λ = −3) in
comparison to the k = 0 case, there still exist an infinite number
of models admitting an integral of motion of the type we
consider in this work.
Moreover, for the combined case where we have both a
non-zero spatial curvature and a perfect fluid, in other words
when the Lagrangian is being given by Lk +2Lm , it comes as
no surprise that the existence of an integral of motion implies
that γ = 1/3. In fact the result is once more exactly the same
as what we see in (
18
) and (
19
).
Having derived the previous results, we can use Noether’s
second theorem and derive the integrals of motion
corresponding to each case. It can easily be verified that
(
20
)
d I
dt = −ξ3 E0 + Ai Ei , i = 1, 2
(
24
)
(
25
)
where the conserved charge is
∂ L ∂ L
I = ξ1 ∂a˙ + ξ2 ∂φ˙
while Ai Ei denotes a linear combination of the two spatial
equations of motion. Of course, in place of L there can be
either Ltot of (
7
), L of (
4
) or Lk of (
23
), depending on the
generator that we use and the case that we examine. In every
situation, we have on mass shell the corresponding conserved
quantity (
25
) for each symmetry vector field Y . The fact that
in every case but the generic fluid with γ = 1/3 an arbitrary
function g˜(φ) is involved in the generator as well as in the
action itself, implies that we possess an infinite collection of
models – corresponding to different sets of functions h(φ),
g(φ), V (φ) and ω(φ) – which admit at least one integral
of motion of this type; hence being integrable. Note here
that the fact that we are led to an infinite number of models
through the arbitrariness of g˜(φ) is owed to the use of the
reparameterization invariant Lagrangian (
4
). Had we chosen to
adopt the gauge N = 1 at the Lagrangian level, then only
a particular case would have emerged where g˜(φ) = φκ ,
with the corresponding χ (t ) for this symmetry being χ (t ) ∝
t . Of course, due to the system being autonomous, ∂t also
exists; the latter being the only remnant of (
15
) after fixing
the gauge. The difference appearing here in the arbitrariness
of g˜(φ) lies in the consideration of Lagrangian (
4
), which
naturally generates the constraint equation E0 = ∂∂NL = 0
in (
13
) and (
24
), allowing the derivation of a larger class of
symmetries.
The existence of an integral of motion of the form (
25
)
is of great interest in the search of solutions. The system at
useful if the result is used in a way to distinguish any existing
solutions that are invariant under the action of the generator,
i.e. derive the characteristic of Y for a given model g˜(φ) so
as to deduce a possible relation between a and φ and use the
latter together with the integral of motion and the constraint
equation.
4 Invariant solutions
In what follows, and in order to demonstrate the importance
of our results, we present a few illustrative applications of
invariant solutions that we derive in the gauge N = 1 with
and without a perfect fluid. For a treatment of how the
conserved quantity can be utilized to lead to more general
solutions expressed in an arbitrary gauge we refer the reader to
Appendix B. Note that the solutions which we present here
are derived with a mentality of keeping all terms inside the
action, i.e. we do not express any additional solutions which
may have a vanishing V0, g0 or ω0.
1. Perfect fluid, γ = 1/3 case Let as consider the FLRW
space-time with zero spatial curvature, i.e. k = 0. As is
well known invariant solutions can be constructed with
the help of the symmetry generator. For example, in the
case of a generic perfect fluid with γ = 1/3, generator
(
16
) implies an invariant relation of the form a 1−λ3γ φ =
const. Hence, it is easy to derive the solution
3γ −1
a(t ) = t γ /3, φ (t ) = φ0t 3γ λ
(
26
)
with
hand possesses two degrees of freedom that are bound by
a constraint equation. Thus, the existence of I implies that,
in principle, we need only to solve the constraint equation
together with I = const. in order to fully integrate the
system. As a result, our problem is immediately reduced to one
that involves only first order differential equations. The
usefulness of I = const. also lies in the fact that it is linear in
a˙ , an advantage which the constraint equation does not have
since it is nonlinear in both velocities a˙ and φ˙ . Nevertheless,
even for the simplest of models defined by g˜(φ), the situation
can be highly complicated. In most of cases it may be more
The on mass shell value of the conserved quantity I for
solution (
26
) is I = 6γ (3γγ+−11)ρ0 .
It is interesting to note what occurs when the theory
assumes a Brans–Dicke-like form, i.e. when h(φ) = φ
and ω(φ) = ω0φ−1. This happens when λ = 33(γγ −−11) and
it leads to a particular solution with γ = −1
(cosmological constant contribution in the action), which reads
a(t ) = t −1/3, φ (t ) = φ0t 2
where now ω0 = −5/3, V0 = −2ρ0, while g0 is kept
arbitrary. We can observe that solution (
27
) satisfies the
(
27
)
same condition as the vacuum Brans–Dicke solution (for
a spatially flat universe), namely that a(t )3φ (t ) ∝ t (in
the gauge N = 1) [36]. This is explained by the fact
that for γ = −1 the potential in (
17
) becomes a constant
with the value V0 = −2ρ0 so that it effectively cancels
the cosmological constant contribution of the fluid in (
7
).
Nevertheless, for an arbitrary γ , we can see how the
symmetry that is present in the Brans–Dicke model –
satisfying a6φ=constant – is generalized in the class of models
we are considering here. In particular we encounter the
more general relation a 1−λ3γ φ =constant, implied by the
existing symmetry generator. Of course in our case, (
26
)
is not a general solution of the equations of motion but
a particular one, which however, as a power law, is of
special cosmological interest.
2. In the absence of a fluid, g˜(φ) = e−μφ case. For a
spatially flat FLRW line element, we choose the function
appearing in (
21
) to be g˜(φ) = e−μφ . After an
appropriate redefinition of the constants ω0, g0 and V0, we derive
the corresponding model to be
h(φ) = eμφ ,
ω(φ) = ω0eμφ ,
g(φ) = g0eμ(λ+3)φ , V (φ) = V0e−μ(λ+1)φ .
(
28
)
A special solution in terms of a power law for the scale
factor exists
a(t ) = t σ , φ (t ) = μ(λ2+ 2) ln(φ0t ), λ = −2, (
29
)
when the constants ω0 and V0 are given by
ω0 =
V0 =
2g0φ02((λ + 2)(3σ − 1) − 2)
μ(λ + 2)2
+μ2((λ + 2)(λ + 3)σ + λ),
4g0φ04(3σ −1)+2μ3φ02(λ+2)((λ+2)σ +1)(3(λ+2)σ −λ)
μ3(λ+2)3
,
while the rest parameters remain free. It can be seen that
on this solution the conserved quantity I (the Noetherian
conservation law) becomes zero.
3. Case g˜(φ) = φ−μ, without a perfect fluid Again in
(
21
), we make the choice g˜(φ) = φ−μ and obtain (once
more with an appropriate redefinition of the constants
V0, g0, ω0 in the action and λ → λ/μ) the subsequent
set of functions entering the action
h(φ) = φμ+1, ω(φ) = ω0φμ−1,
g(φ) = g0φ3(μ−1)+λ, V (φ) = V0φ2−μ−λ.
(
30
)
Two solutions for which the conserved quantity is again
zero can be easily derived for this model.
2(μ + 1) (κ(μ+1) − σ )
κ
3
+ 2 g0κ3σ +κ2(μ+1)2 +5κ(μ+1)σ +6σ 2.
4. Case k = 1, g˜(φ) = e−μφ , without a perfect fluid Let
us furthermore consider the case with non-zero spatial
curvature k = 0. Then, from the two particular types of
solutions we examined above in cases 2 and 3, only for
g˜(φ) = e−μφ arises the following configuration:
2
a(t ) = t, φ (t ) = − μ ln (φ0 t )
ω0 = k − 8g0μφ02, V0 = 4μφ02 2g0μφ02 − k , (
33
)
a(t ) = eσ t , φ (t ) = φ0eκt
when the following relations hold:
λ = 1 − 2μ,
ω0 = g0κ (κ(1 − μ)+3σ ) −
The first solution is again a power law,
a(t ) = t σ , φ (t ) = φ0t κ ,
where
2
λ = κ − 2μ + 1,
ω0 = −g0κ2(μ − 1)φ02/κ + g0κ(3σ − 1)φ02/κ
are the constants appearing in (
30
).
Once more we can also distinguish a Brans–Dicke
subcase that appears when μ = 0. It is interesting to observe
that condition a(t )3φ (t ) ∝ t of the vacuum Brans–Dicke
model does not apply here, although it was still relevant
in the corresponding model of case 1. As we can see, the
power σ in the scale factor expression is not connected
to the power of time in the φ (t ). Only if we choose to
turn the action into the pure Brans–Dicke form does (
31
)
with μ = 0 satisfy the aforementioned condition: By
setting g0 = 0 and V0 = 0, which immediately results in
κ = 1 − 3σ or κ = −2σ .
Another solution we can derive for the g˜(φ) = φ−μ
model leads to a space-time of a constant scalar Ricci
curvature, a de Sitter universe that is,
(
31
)
(
32
)
which, as long as the scale factor is concerned, it
corresponds to the k = −1 solution of Einstein’s gravity
in vacuum, that is, the Milne solution. Here, it is given
for any sign of k with the difference being carried in the
coupling (through ω0 and V0) that is necessary for its
existence. The integral of motion – calculated with the
help of (
25
), (
23
) and (
18
) – assumes on mass shell the
2 k .
value I = μφ02
As is obvious, we are able – just by choosing a particular
function g˜(φ) – to find through (
21
) or (
19
) the corresponding
class of integrable models that admit a conserved quantity of
the aforementioned form. As we already stated, it is due to
the existence of the constraint equation (8a) that one can in
principle obtain the general solution by considering only the
∂ L
first order system E0 = ∂ N = 0 and I =const. However,
because of the complexity of the equations this is a highly
nontrivial task.
5 Discussion
We remark that, for the cases without the extra matter fluid
where we have obtained an infinite set of models owed to the
arbitrariness of g˜(φ), there exists an interesting connection
among them. These models can be mapped to each other by
conformal transformations. Take for example two different
models whose action is characterized by two different scalar
field functions g˜1(φ1) and g˜2(φ2). Then the transformation
φ1 = g˜1−1 ◦ g˜2(φ2), gμ(1ν) =
1
(g˜1−1 ◦ g˜2) (φ2) gμ(2ν),
(
34
)
where gμ(1ν) and gμ(2ν) are the two corresponding space-times,
maps the one action to the other. What is more, if we try
to map the relevant form of the action to the Einstein frame
we can see that an interesting “degeneration” occurs3: The
different models that we get for the various g˜(φ) all collapse
to a single action. Take for example the case where the initial
action is characterized by the functions of φ given by (
21
).
It can be seen to be mapped into an action of the form
S¯ =
1 1 ,μ
−g¯ 2 R¯ − 2
,μ − V1eσ
−e−σ
ω1 ,μ
,μ ¯
+ ω2( ,μ
,μ)2
d4x (
35
)
by a conformal transformation in the metric and a
reparametrization of φ ( ). The new and the old variables are
related by the expressions
3 A similar situation can be seen to arise in the result of [21], where again
infinite models in a family of scalar–tensor theories was recovered. By
means of conformal transformations they can be mapped in the Einstein
frame to a case of a minimally coupled scalar field with exponential
potential.
(
37
)
(
38
)
(
39
)
(
40
)
(
36
)
V0g˜(φ)λ+3
4V1
with V1 a constant and σ , ω1 and ω2 in (
35
) being related
to the initial constants g0, V0, ω0 and λ. We can see that the
difference of this action lies in the existence of a term that is
quadratic to the kinetic energy of the scalar field, while in (
1
)
we considered a theory that has at most linear expressions in
X = − 21 φ,μφμ.
However, what is also interesting here is that when
V ( ) eσ dominates then the contribution of the terms
e−σ ω1 ,μ ,μ ¯ + ω2( ,μ ,μ)2 0, which means
that (
35
) reduces to the action integral of a minimally
coupled scalar field. The latter is exactly the limit which relates
a canonical scalar field between the Jordan and the Einstein
frames.
In general, great care needs to be taken when using
conformal transformations. The fact that all the different models
for the various choices of g˜(φ) can be mapped to (
35
) does
not make them equivalent to the latter, or even to each other
(due to (
34
)) for that matter. Two actions in order to be
physically equivalent, they need to be mapped by gauge
transformations of the theories under consideration. In the case of
gravitational actions, those are the four-dimensional
diffeomorphism of space-time. As we know, conformal
transformations cannot always be attributed to coordinate changes.
As a result, the gravitational space-time arising in each
situation is generally different.
In order to see that consider the nonsingular solution (
32
)
corresponding to a de Sitter universe, which came out of a
model characterized by g˜(φ) = φ−μ. If we choose to map
this model to (
35
) then, by virtue of (
36
), we obtain
N¯ (t ) = −
a¯ (t ) = −
μ
2φ (t )μ+1
= c1 eκ(μ+1)t ,
μ
2φ (t )μ+1 eσ t = c1 e[κ(μ+1)+σ ]t ,
where N¯ , a¯ are the new lapse and scale factor, respectively,
while c1 is a constant. It is easy to see that if you go from
this metric
ds2 = −N¯ 2dt 2 + a¯ 2(dx 2 + dy2 + dz2)
to the gauge where N¯ = 1 (so as to compare with what we
have in (
32
)) by performing the time transformation
N¯ (t )dt = dτ ⇒ t =
ln κ(μ+1)τ
c1
κ(μ + 1)
we derive, with the appropriate scalings in x , y and z, the
line element
ds2 = −dτ 2 + τ ψ (dx 2 + dy2 + dz2),
where ψ is a constant. What was an exponential solution
with a constant Ricci scalar in the theory we are
investigating, has now become a power law with R(τ ) ∝ τ −2 and a
space-time with curvature singularity at τ = 0 in the system
described by action (
35
). Hence we can see that, whenever
the conformal transformation does not correspond to a
general coordinate transformation, the gravitational properties
of the system are bound to change and the solutions
represent different geometries. For discussions on the relation
between analytical solutions and physical quantities between
the Jordan and the Einstein frames see [37–39] and references
therein.
Additionally to the previous geometrical considerations,
extra care needs to be taken when a fluid is used as a matter
source. This is owed to the fact that you need to pre-define a
continuity equation in order to derive a rule for the variation
of the energy density ρ with respect to the metric gμν . In the
beginning of our analysis we considered a perfect fluid that is
characterized by continuity equation (
5
). It is a well-known
fact that the latter is not conformally invariant, while its
solution is necessarily utilized at the level of the mini-superspace
Lagrangian. Henceforth, after a conformal transformation is
being made, one needs to take into account a different fluid,
which is now interacting with the scalar field, in order to
make such a correspondence possible. This also results in
a change of the physical behavior of the system, since the
properties of the matter source need to be altered.
In a future work we plan to study the integrability of other
Horndeski theories by including more terms in the action
integral and for other kind of transformations which leaves
the field equations invariants, such as the generalized
symmetries as also to investigate the effects of the conformal
transformations in Hordenski theories.
Acknowledgements This work is financial supported by FONDECYT
grants 3150016 (ND), 1150246 (AG) and 3160121 (AP). AP thanks the
Durban University of Technology for the hospitality provided while part
of this work was performed.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Calculation of the symmetry generator
As indicated in the main text, application of (
13
) leads to
an overdetermined system of partial differential equations.
The latter is formed by gathering and demanding that are
zero the coefficients multiplying terms involving derivatives
of a, φ and N . For Lagrangian Ltot as written in (
7
) we infer
The fourth order coefficients a˙ 2φ˙ 2 and a˙ φ˙ 2 N˙ each imply
∂a ξ2 = 0 and ∂N ξ2 = 0,
which means that ξ2 = ξ2(t, φ). However, with the help of
χ = χ (t ), we can get a further restriction from the coefficient
of a˙ φ˙ 2 that leads to
∂t ξ2 = 0 ⇒ ξ2 = ξ2(φ).
Due to (A1) and (A3), the terms involving φ˙ 3 N˙ and φ˙ 3 bring
about the conditions
∂N ξ1 = 0
∂t ξ1 = 0
⇒ ξ1 = ξ1(a, φ).
Thanks to the above restrictions (A1), (A3) and (A4) we also
get
from the coefficients of a˙ 3, a˙ 2 N˙ and a˙ 2φ˙ that we need to set,
respectively,
(A1)
(A2)
(A3)
(A4)
(A5)
(A6)
(A8)
⎧ ∂N F = 0
⎪
⎨ ∂φ F = 0 ⇒ F = F (t )
⎪⎩ ∂a F = 0
ξ3(t, a, φ, N )
= N
2∂a ξ1(a, φ) +
from N˙ , φ˙ and a˙ , respectively.
The equation emanating from the coefficient of a˙ 2 can be
solved algebraically with respect to ξ3(t, a, φ, N )
ξ1(a, φ)
a
+
ξ2(φ)h (φ)
h(φ)
− χ˙ (t ) ;
ξ0(φ)
+ a1/5 .
the latter being a consequence of the fact that no derivative
of N enters in the Lagrangian. After this step one can see that
the equation extracted from the coefficient of a˙ φ˙ 3 is
−3aξ2(φ) + 3aξ 211(φ)h (φ) = 0
ah(φ)ξ2(φ)g (φ) − g(φ) [h(φ) (5a∂a ξ1(a, φ) + ξ1(a, φ)
(A7)
and it can be immediately integrated to yield
1
ξ1(a, φ) = 6 a
ξ2(φ)g (φ)
g(φ)
−
3ξ2(φ)h (φ)
h(φ)
+ 3ξ2(φ)
With the dependence with respect to a being now
completely specified, the relation produced by the coefficient of
a˙ φ˙ involves only unknown functions of φ. Hence, we can now
start gathering coefficients with respect to powers of a that
appear inside it. With the help of a useful re-parametrization
g(φ) = g1(φ)3 one straightforwardly obtains
ξ2(φ) =
c3
ξ0(φ) = h(φ)3/2 ,
where the ci ’s indicate constants of integration. The only
appearance of time in the remaining coefficient is inside the
zero-th order component (that multiplies no derivative of a˙ ,
φ˙ or N˙ ) and it implies that
F (t ) = const.
Thus, the gauge function becomes trivial.
With the results up to here all dependence of the functions
of the generator with respect to a, t and N is specified. Inside
each of the remaining equations we can gather coefficients
with respect to a, since now the unknown functions involve
only φ. However, due to the existence of γ the way that the
coefficients are to be gathered depends on its value. We can
distinguish two main cases:
• Case γ = 1/3. The equation produced by the zero-th
order coefficient leads to
with general solution g2 = φ +c8c9 . Thus, the final function
needed so that all equations are satisfied is
At this point the system of equations is completely
satisfied. By absorbing the trivial constant c9 inside φ with
a translation and with an appropriate parametrization of
the rest of the constants we obtain result (
16
), (
17
).
Additionally, we can observe that the function χ (t ) remained
arbitrary throughout the calculation and this explains the
existence of (
15
).
• Case γ = 1/3. With this choice of γ the zeroth order
equation implies
(A17)
(A18)
(A19)
(A20)
(A9)
(A10)
(A11)
(A12)
(A13)
(A14)
(A15)
(A16)
c4h(φ)2
V (φ) = (c1 + c2g1(φ))3
3(γ −1)
h(φ) = c5 (c1 + c2g1(φ)) 3γ −1
c3 = 0.
We note that, since we want the most general result, we
avoid any special solution that leads to vanishing of any
of the functions involved in our starting action. The φ˙ 2
component leads to
1+3γ
ω(φ) = c6 (c1 + c2g1(φ)) 1−3γ g12
while the φ˙ 4 gives rise to the third order equation
g1 g1g1 − 2(g1 )2g1 + g1 (g1)2 = 0
which under a transformation g1 = exp
becomes
g2(φ)dφ
g2 g2 − 2(g2)2 = 0
It is a matter of re-parametrizing the constants of
integration and an introduction of a new function g˜(φ) as
sga1m(φe)c=omg˜m(1φe)nttos oabstianinthreespurletv(i1o8u)s, c(1a9se),afnodr
othfecoarubristerathrieness of χ (t ), hold.
For the sake of completeness we have to point out that
apart from the case γ = 1/3 there are also other values
of γ that can lead to a different gathering of terms in the
coefficients of powers of a; namely γ = −1 and γ = −7/5.
The first gives the same result as the case without fluid (see
Eqs. (
20
), (
21
)) with the sole difference of a potential that is
V (φ) = V0 g˜λg+3 − 2ρ0 in place of the one appearing in (
21
).
That is one t˜hat cancels the cosmological constant role of
the fluid. The second case, γ = −7/5, after appropriate
reparametrizations leads to the exact same result as the generic
γ = 1/3 result and that is why we do not make a separate
presentation of it here.
Appendix B: More general solutions
In Sect. 4 we presented some invariant solutions where the
functional dependence between a and φ can be extracted with
the help of the infinitesimal symmetry generator. Although
these are particular solutions which can be derived in a rather
simple manner, they are the most cosmologically
interesting. Even though we have proven the existence of an integral
of motion I , thus reducing the problem to solving two first
order differential equations: the constraint and I =const.
The acquirement of the general solution is still a very
difficult task, especially due to the nonlinearity of the constraint
equation in both derivatives involved a˙ and φ˙ .
Here, we want to exhibit a method with which the integral
of motion I can be used to derive the solution for the spatially
flat case in the absence of a perfect fluid, when I = 0. Some
of the solution we derived in Sect. 4 lead to I = 0 but still they
are not the general solution of this case, but rather particular
solutions of it. Although this method does not lead to the full
solution (where I = const.) of the system, it can be applied
for an arbitrary function g˜(φ). Thus, giving in closed form
the full solution with the property I = 0 for any model of
the integrable type we are considering.
At first let us construct the conserved quantity I of (
25
)
by using the ξ1 and ξ2 of (
20
)
ξ1 = a
(λ + 3)g˜ (φ)2 − 3g˜(φ)g˜ (φ)
6g˜ (φ)2
g˜(φ)
, ξ2 = − g˜ (φ) (B1)
and make the transformation a(t ) = exp
the relation I = 0 reduces to
u 4(λ + 3)N 2g˜(φ)λ+6g˜ (φ) − 6g0g˜(φ)φ˙2g˜ (φ)4
+ φ˙ g˜ (φ)2 2ω0 N 2g˜(φ)λ+5 − g0(λ + 7)φ˙2g˜ (φ)3
+ g˜(φ)g˜ (φ) 3g0φ˙2g˜ (φ)3 − 2(λ + 3)N 2g˜(φ)λ+5
u(t )dt . Then
= 0,
(B2)
which can be algebraically solved with respect to the function
u(t ). Substitution of the latter inside the constraint equation
∂ L
∂N = 0 leads to
8(λ + 3)2V0g˜(φ)4(λ+5) N 8 − 4φ˙ 2 3ω02 − ω0(λ + 3)2
+6g0(λ + 3)V0) g˜(φ)3(λ+5)g˜ (φ)3 N 6
+ 2g0φ˙ 4 (6ω0(λ + 7) + 9g0V0
−2(λ + 6)(λ + 3)2 g˜(φ)2(λ+5)g˜ (φ)6 N 4
+ 3g02(−3ω0 + λ(λ + 2) − 19)
× φ˙ 6g˜(φ)λ+5g˜ (φ (t ))9 N 2 + 9g03φ˙ 8g˜ (φ)12 = 0,
(B3)
which is an eighth order algebraic equation with respect to
N (t ). However, only even powers of N appear. So, by
making a substitution N (t ) = √n(t ) in (B3), the latter reduces to
a fourth order polynomial equation and can be solved
analytically with respect to n(t ). We refrain for giving the
expression here, but general formulas can be found in the
bibliography [40]. As a result we are able, in the case of I = 0, to
derive the solution of the system in a purely algebraic manner
for every admissible function g˜(φ) without even fixing the
gauge, since we have not set N = 1 or any of the rest two
degrees of freedom to be a specific function of t . We only
satisfied dynamical equations, even though it was just for the
special case I = 0. The solution is expressed through the
relations
N (t ) =
n(φ, φ˙ ), a(t ) = e u(φ,φ˙)dt
(B4)
in terms of an arbitrary function φ (t ), i.e. the scalar field
plays effectively the role of time.
1. T. Clifton , P.G. Ferreira , A. Padilla , C. Skordis , Phys. Rep . 513 , 1 ( 2012 )
2. G.W. Horndeski , Int. J. Ther. Phys . 10 , 363 ( 1974 )
3. C. Brans , R.H. Dicke , Phys. Rev . 124 , 195 ( 1961 )
4. G. Kofinas , M. Tsoukalas , EPJC 76 , 686 ( 2016 )
5. A. Cisterna , T. Delsate , M. Rinaldi , Phys. Rev. D 92 , 044050 ( 2015 )
6. A. Nicolis , R. Rattazzi , E. Trincherini, Phys. Rev. D 79 , 064036 ( 2009 )
7. C. Deffayet , G. Esposito-Farese , A. Vikman , Phys. Rev. D 79 , 084003 ( 2009 )
8. M. Trodden , K. Hinterbichler , Class. Quant. Grav. 28 , 204003 ( 2011 )
9. T. Kobayashi , M. Yamaguchi , J. Yokoyama , Prog. Theor. Phys . 126 , 511 ( 2011 )
10. A. Paliathanasis , P.G.L. Leach , Int. J. Geom. Meth. Mod. Phys . 13 , 1630009 ( 2016 )
11. S. Cotsakis, P.G.L. Leach , H. Pantazi , Grav. Cosm. 4 , 314 ( 1998 )
12. S.D. Majaraj , P.G.L. Leach , J. Math. Phys. 37 , 430 ( 1996 )
13. A.M. Msomi , K.S. Govinder , S.D. Maharaj , Int. J. Theor. Phys . 51 , 1290 ( 2012 )
14. S. Cotsakis , J. Demaret , Y. De Rop , L. Querella, Phys. Rev. D 48 , 4595 ( 1993 )
15. S. Cotsakis, P.G.L. Leach , J. Phys . A: Math. Gen. 27 , 1625 ( 1994 )
16. A. Paliathanasis , J.D. Barrow , P.G.L. Leach , Phys. Rev. D 94 , 023525 ( 2016 )
17. A. Paliathanasis , P.G.L. Leach , Phys. Lett. A 380 , 2815 ( 2016 )
18. R.S. Palais, Commun. Math. Phys. 69 , 19 ( 1979 )
19. S. Capozziello , R. De Ritis, C. Rubano , P. Scudellaro , Riv. Nuovo Cim . 19 , 1 ( 1996 )
20. S. Capozziello , E. Piedipalumbo , C. Rubano , P. Scudellaro, Phys. Rev. D. 80 , 104030 ( 2009 )
21. P.A. Terzis , N. Dimakis , T. Christodoulakis, Phys. Rev. D 90 , 123543 ( 2014 )
22. Y. Zhang , Y.-G. Gong , Z.-H. Zhu , Phys. Lett. B 688 , 13 ( 2010 )
23. B. Vakili , Phys. Lett. B 664 , 16 ( 2008 )
24. B. Vakili , Phys. Lett. B 738 , 488 ( 2014 )
25. H.M. Sadjadi , Phys. Lett. B 718 , 270 ( 2012 )
26. A. Paliathanasis , Class. Quantum Gravit . 33 , 075012 ( 2016 )
27. A. Zampeli , T. Pailas, Petros A. Terzis , T. Christodoulakis, JCAP 16 ( 05 ), 066 ( 2016 )
28. M.F. Shamir , M. Ahmad , EPJC 77 , 55 ( 2017 )
29. K. Atazadeh , F. Darabi , EPJC 72 , 2016 ( 2012 )
30. H. Dong , J. Wang , X. Meng , EPJC 73 , 2543 ( 2013 )
31. G. Leon, E.N. Saridakis , JCAP 13 ( 03 ), 025 ( 2013 )
32. B. F. Schutz , Phys. Rev. D 2 ( 12 ), 2762 ( 1970 )
33. G.W. Bluman , S. Kumei , Symmetries of Differential Equations (Springer, New York, 1989 )
34. K. Sundermeyer , Constrained Dynamics, Lect. Notes Phys . Vol. 169 , Springer-Verlag ( 1982 )
35. T. Christodoulakis, N. Dimakis , Petros A. Terzis, Phys . A: Math. Theor. 47 , 095202 ( 2014 )
36. P. Chauvet, Astrophys. Space Sci . 90 , 51 - 58 ( 1983 )
37. V. Faraoni , E. Gunzig, Int. J. Theor. Phys . 38 , 217 ( 1999 )
38. M. Postma , M. Volponi , Phys. Rev. D 90 , 103516 ( 2014 )
39. M. Tsamparlis , A. Paliathanasis , S. Basilakos , S. Capozziello , Gen. Rel. Grav. 45 , 2003 ( 2013 )
40. M. Abramowitz , I.A. Stegun , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972 )