Dynamical Analysis of an Integrable Cubic Galileon Cosmological Model
Abstract
Recently a cubic Galileon cosmological model was derived by the assumption that the field equations are invariant under the action of point transformations. The cubic Galileon model admits a second conservation law which means that the field equations form an integrable system. The analysis of the critical points for this integrable model is the main subject of this work. To perform the analysis, we work on dimensionless variables different from that of the Hubble normalization. New critical points are derived while the gravitational effects which follow from the cubic term are studied.
pacs:
98.80.k, 95.35.+d, 95.36.+xJuly 9, 2021
I Introduction
A theory which has drawn the attention of the scientific society in the last few years is the Galileon gravity nik ; gal02 . It belongs to the modified theories of gravity in which a noncanonical scalar field is introduced and the field equations are invariant under the Galilean transformation. The Action integral of the Galileon gravity belongs to the Horndeski theories hor , which means that the gravitational theory is of secondorder m10 . The vast applications of study for Galileons in the literature covers all the areas of gravitation physics from neutron stars, black holes, acceleration of the universe, for instance see cha ; char ; chr2 ; chr3 ; bar ; bellini ; bartolo ; babi ; Bhat ; genlyGL ; genlyGL2 ; m07 ; m08 ; m09 ; m10 ; m11 and references therein.
In this work we are interested in the cosmological scenario and specifically in the socalled Galileon cosmology chow ; defelice ; cedric . In cosmology, the Galileon field has been applied in order to explain various phases of the evolution of the universe Teg ; Kowal ; Komatsu ; Ade15 ; planck2015 . Specifically the new terms in the gravitational Action integral can force the dynamics in a way such that the model fits the observations. The mechanics can explain the inflation era inf1 ; inf2 ; inf3 ; inf4 ; inf5 ; inf6 ; inf7 as also the late timeacceleration of the universe late1 ; late2 ; late3 ; late4 ; late5 ; late6 . Last but not least the growth index of matter perturbations have been constrained in matter1 .
As we mentioned in the previous paragraph, Galileon gravity belongs to the Hondenski theories and specifically there is an infinite number of different models which can be constructed from a general Lagrangian. A simple model is the cubic Galileon model bar ; bellini ; bartolo ; babi ; Bhat ; genlyGL ; genlyGL2 where the Action Integral is that of a canonical scalar field plus a new term which has a cubic derivative on the Galileon field. The theory can be seen as a first extension of the scalar field cosmology. Due to this cubic term, the nonlinearity and the complexity of the field equations is increased dramatically. Recently in Dimakis:2017kwx a cubic Galileon model was derived which admits an additional conservation law and where the field equations formed an integrable dynamical system.
Integrability is an important issue in all the areas of physics and mathematical sciences. The reason is that while a dynamical system can be studied numerically, it is unknown if an actual solution which describes the “orbits” exist. The integrable cubic Galileon model admits special solutions which describes an ideal gas universe, that is, powerlaw scale factors. While this is similar to the solution of the canonical scalar field, we found that the power of the powerlaw solution is not strongly constrained by the Galileon field and that is because of the cubic term. On the other hand a special property of that model is that when the potential in the Action integral dominates, then the cubic term disappears which mean that the theory approach is that of a canonical scalar field. However as we shall see from our analysis, the existence of the conservation law provides new dynamics which have not been investigated previously.
The scope of this work is to study the evolution of the integrable cubic Galileon cosmological model. For that we perform an analysis of the critical points. In particular, in Section II we briefly discuss the cubic Galileon cosmology and we review the integrable case that was derived before in Dimakis:2017kwx . Section III includes the main material of our analysis where we rewrite the field equations in dimensionless variables. We define variables different from that of the normalizaton where we find that the dynamical system is not bounded. Because of the latter property the critical points at the finite and the infinite regions are studied. At the finite region we find various critical points which can describe the expansion history of our universe as also the matter dominated era. Appendices A, B, C, D, E and F include important mathematical material which justify our analysis. One important property of the integrable model that we study is that there is a limit in which the terms in the field equations (which follow from the cubic term of the Galileon Lagrangian) vanish and the model is then reduced to that of a canonical scalar field cosmological model. Hence, in order to study the effects of the cubic term in Section IV we perform an asymptotic expansion of the solution when the cubic term dominates the universe. In Section V we extend our analysis to the case where an extra matter term is included in the gravitational Action integral. Finally, we discuss our results and draw our conclusions in Section VI.
Ii Cubic Galileon Cosmology
The cubic Galileon model is defined by the following Action integral
(1) 
which has various cosmological and gravitational applications.
In the cosmological scenario of a homogeneous and isotropic universe with zero spatially curvature the line element of the spacetime is that of the FLRW metric
(2) 
where is the scale factor of the universe.
Indeed for this line element, variation with respect to the metric tensor in (1) provides the gravitational field equations
(3) 
and
(4) 
while variation with respect to the field , provides the (modified) “KleinGordon” equation
(5) 
which describes the evolution of the field, and . Recall that we have assumed that the Galileon field inherits the symmetries of the spacetime, that is if is an isometry of (2), i.e. , then inherits the symmetries of the spacetime if and only if . Therefore is only a function of the “” parameter, i.e. .
An equivalent way to write the field equations is by defining fluid components such as energy density and pressure which corresponds to the Galileon field. Indeed if we consider the energy density
(6) 
and the pressure term
(7) 
the field equations take the form , where is the energy momentum tensor corresponding to the Galileon field and
(8) 
in which is the normalized comoving observer . Last but not least equation (5) is now equivalent to the Bianchi identity , that is,
(9) 
Last but not least the dark energy equationofstate parameter is defined as follows.
(10) 
It can be shown that with a proper election of , the equationofstate parameter can realize the quintessence scenario, the phantom one and cross the phantom divide during the evolution, which is one of the advantages of Galileon cosmology. In general the specific functions of and are unknown and for different functions there will be a different evolution.
ii.1 The extra conservation law
Recently in Dimakis:2017kwx two unknown functions were specified by the requirement that the gravitational field equations form an integrable dynamical system. Specifically the following functions were found
(11) 
There exists a symmetry vector which provides, with the use of Noether’s second theorem, the following conservation law for the field equations
(12) 
Because of the nonlinearity of the field equations the general solution cannot be written in a closedform. However from the symmetry vector invariant curves have been defined and by using the zeroth order invariants some power law (singular solutions) have been derived. In particular the following solutions were obtained:
(13) 
and
(14) 
These solutions are special solutions since they exist for specific initial conditions. In order to study the general evolution of the system we perform an analysis in the phasespace.
A phasespace analysis for this cosmological model has been performed previously in genlyGL , however the integrable case with and given by the expressions (11) were excluded from genlyGL . Moreover there is a special observation in the integrable case in the sense that . The latter means that when dominates, becomes very small and the cubic Galileon model reduces to that of a canonical scalar field which can mimic also the cosmological constant when .
In the following we write the field equations in new dimensionless variables and we perform our analysis.
Iii Evolution of the dynamical system
From equation (3) one immediately sees that the Hubble function can cross the value , from negative to positive values, or viceverca. This means that the standard normalizaton is not useful and new variables have to be defined. We introduce the new variables
(15) 
and the parameter so we obtain the threedimensional dynamical system
(16) 
where and functions are defined by the following expressions
(17a)  
(17b)  
(17c)  
(17d)  
and prime denotes the new derivative Interestingly, the system (16) admits the first integral . 
(18) 
which is the Friedmann’s first equation and constrains the evolution of the solution. Thus, the dynamics are restricted to a surface given by (18). For a fixed value of , the first and last equations in (16) are invariant under the discrete symmetry . Thus, the fixed points related by this discrete symmetry have the opposite dynamical behavior. By definition, .
Let’s compare with the variables introduced in genlyGL defined by
Since we have chosen here and such that we have the relations
Furthermore, some cosmological parameters with great physical significance are the effective equation of state parameter (because we set ) and the ‘deceleration parameter”
(19) 
Some conditions for the cosmological viability of the most general scalartensor theories has to be satisfied by extended Galileon dark energy models; say the model must be free of ghosts and Laplacian instabilities DeFelice:2010pv ; DeFelice:2011bh ; Appleby:2011aa . In the special case of the action (1) (in units where ), we require for the avoidance of Laplacian instabilities associated with the scalar field propagation speed that DeFelice:2011bh
(20)  
(21) 
Meanwhile, for the absence of ghosts it is required that
(22) 
Finally, we have from the Eqs. (10), (21) and (22) that the phantom phase can be free of instabilities and thus cosmologically viable, as it was already shown for Galileon cosmology DeFelice:2011bh .
iii.1 Analysis at the finite region
The fixed points/ fixed lines at the finite region of the system
(16), and a summary of their stability conditions are presented in
table 1. In table 2 we display several
cosmological parameters for the fixed points at the finite region of the
system (16). The discussion about the physical interpretation of
these points and the points at infinity is left for section
III.4
Label: Coordinates  Existence  Eigenvalues  Stability 

Always  Undetermined  Numerical analysis  
Always  stable for  
or ,  
saddle for  
or .  
(see appendix C)  
Always  unstable for  
or ,  
saddle for  
or .  
(see appendix C)  
A.S. (see Appendix D)  
A.U. (see Appendix D)  
Where is a real root of  
such that  Numerical analysis  Numerical analysis  
Where is a real root of  
such that  Numerical analysis  Numerical analysis  
Always  Sink for  
Saddle otherwise  
Always  Source for  
Saddle otherwise  
sink for  
saddle otherwise  
source for  
saddle otherwise  
saddle  
saddle  
A.S. for  
A.U. for  
(see Appendix E).  
A.S. for  
A.U. for  
(see Appendix F).  
A.S. for  
A.U. for  
(see Appendix F). 
Label  Physical interpretation  

.  
or  The Galileon mimics radiation for  
, or  The Galileon mimics matter for .  
Powerlawsolution for  
The Galileon mimics dust for  
.  
.  
The Galileon mimics dust.  
.  
de Sitter solution.  
.  
see section III.4  Accelerated solution for  
, or  
The Galileon mimics dust.  
.  
The Galileon mimics dust.  
for .  
The Galileon mimics dust  
for  
.  
de Sitter solution.  
.  
see section III.4  de Sitter solution.  
We proceed with the determination of critical points at the infinity.
iii.2 Analysis at “infinity”
Because the phase space of the system is unbounded we introduce the Poincarè compactification and a new time derivative
(23) 
The dynamics on the “cylinder at infinity” can be obtained by setting ; the dynamics in the coordinates is governed by the equations
(24a)  
(24b) 
We linearize around a given fixed point on the “cylinder at infinity” by introducing , with . Notice that , so to examine the stability of the fixed points at the cylinder, and from the interior of it, we have to estimate how evolves, not only the stability in the plane . We obtain for the expansion rate
(25) 
Label: Coordinates  Coordinates  Stability  

Nonhyperbolic  
Saddle  
Saddle  
Saddle  
Saddle 
The stability condition of a fixed point along is then . The full stability of the above fixed points is summarized in the table 3.
iii.3 Numerical analysis
Let us complete our analysis by performing some numerical simulations. Specifically we choose the constants of the model to satisfy the conditions
hence,
Moreover we impose the condition , which guarantees the stability of the perturbation of the scaling solution Dimakis:2017kwx . Because we have chosen , this leads to the “allowed” region on the parameter space displayed in figure 1.
In figure 2, it is presented a Poincarè projection of the system (16) on the invariant set . The green dots correspond to the points (that we solved numerically). In the special case we have , thus, the polynomial is quadratic and there are only two roots of . The blue contour is defined by . As shown in the figures this line is singular, and attracts some orbits. The brown solid line corresponds to intersection of the invariant surface and the invariant set . In the top figures, attracts some orbits, but others are attracted by one the of green points associated to . is not the attractor of the whole phase space since . In the bottom figures is the attractor, not just in this invariant set but in the whole phase space since .
iii.4 Discussion
In this section we discuss the stability conditions, cosmological properties and physical meaning of the (lines of) fixed points in both finite and infinite regions.

always exists. To analyze the stability we resort to numerical examination.
In the Appendix B we proved, using normal forms calculations, that the fixed point corresponds to the cosmological solution (64). Moreover, in order to improve the range and accuracy we calculate the diagonal first order Pade approximants
around . This yields the following approximate expressions
(26a) (26b) (26c) while the scale factor is calculated to be (27) Due to the new conservation law (12), the allowed values of the constants are:
It is interesting to note that the powerlaw solution (13) satisfies the condition
(28) Additionally, after the substitution of the functional forms of in (28), and the substitution of 28) is satisfied for all the values of . it follows that the restriction (
There exists a relation between and given by
(29) such that implies . Thus, as , this powerlaw solution approaches the as . For large we can invert to have
(30) Thus, we can take as approximation for large :
Furthermore
