# Dynamical analysis of generalized Galileon cosmology

###### Abstract

We perform a detailed dynamical analysis of generalized Galileon cosmology, incorporating also the requirements of ghost and instabilities absence. We find that there are not any new stable late-time solutions apart from those of standard quintessence. Furthermore, depending on the model parameters the Galileons may survive at late times or they may completely disappear by the dynamics, however the corresponding observables are always independent of the Galileon terms, determined only by the usual action terms. Thus, although the Galileons can play an important role at inflationary or at recent times, in the future, when the universe will asymptotically reach its stable state, they will not have any effect on its evolution.

a,b]Genly Leon c,d]and Emmanuel N. Saridakis \affiliation[a]Departamento de Matemática, Universidad Central de Las Villas, Santa Clara CP 54830, Cuba \affiliation[b]Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso, Chile \affiliation[c]Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece \affiliation[d]CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA \emailAdd \emailAddEmmanuelS \keywordsGalileon cosmology, dark energy, dynamical analysis \arxivnumber

## 1 Introduction

There are two approaches one can follow in order to describe the observed universe acceleration. The first is to introduce the concept of dark energy, usually adding extra scalar fields (see [1] and references therein) in the right-hand-side of the field equations of General Relativity, while the second is to modify the left-hand-side of the general relativistic field equations, that is to modify the gravitational theory itself (see [2, 3] and references therein).

A recently re-discovered, very general class of scalar-field theories is based on the introduction of higher derivatives in the action, with the requirement of maintaining the equations of motion second order. Although the most general second-order theories avoiding the Ostrogradsky instabilities [4] were already derived in [5], a particular class, dubbed Galileon, was constructed in [6] for the Minkowski metric and in [7, 8] for dynamical geometries, while in [9] the results of [5] were re-discovered in the context of (extended) Galileon framework. In this formalism the Lagrangian is suitably constructed in order for the field equations to be invariant under the Galilean symmetry , in the limit of Minkowski spacetime, with constants. The four-dimensional Lagrangian that preserves these symmetries contains five unique terms consisting of scalar combinations of , and (since General Relativity does not accept the satisfaction of Galilean symmetry in curved spacetime, higher derivatives are necessary in the action [10, 11, 12]), and the corresponding couplings are not renormalized by loop corrections [13]. Furthermore, a significant advantage is that the derivative self-couplings of the scalar screen the deviations from General Relativity at high gradient regions (small scales or high densities) through the Vainshtein mechanism [14], thus satisfying solar system and early universe constraints [16, 15, 17].

Application of the above construction in cosmological frameworks gives rise to the Galileon cosmology, which proves to be very interesting and has been investigated in detail in the literature. In particular, one can investigate the universe evolution and late-time acceleration [19, 23, 20, 24, 18, 21, 25, 26, 22, 27], inflation [28, 29, 30, 31, 32, 33, 34, 35] and non-Gaussianities [36, 37, 38, 39, 40], the reheating of the post-inflationary universe [41], the growth history [42, 43, 44], the cosmological bounce [45, 46, 47], the cosmological perturbations [48, 49, 51, 52, 50], the spherical solutions [57, 58, 53, 54, 55, 56], the stability issues [59, 60, 61, 62, 63, 64], and also he can use observational data in order to constrain the parameters of the theory [66, 65, 67, 73, 69, 70, 72, 68, 71]. Moreover, one can study and extend the properties of Galileon theory itself [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94], and examine the relation of Galileons with other frameworks [95, 97, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106].

Since Galileon cosmology exhibits interesting phenomenological features, in the present work we perform a phase-space and stability analysis of such a scenario, investigating in a systematic way the possible cosmological behaviors, focusing on the late-time stable solutions. Such an approach allows us to bypass the high non-linearities of the cosmological equations, which prevent any complete analytical treatment, obtaining a qualitative description of the global dynamics of these models, which is independent of the initial conditions and the specific evolution of the universe. Moreover, in these asymptotic solutions we calculate various observable quantities, such as the dark energy density and equation-of-state parameters and the deceleration parameter, and in order to ensure that these solutions are free of ghosts and Laplacian instabilities we additionally calculate the relevant perturbations quantities. Interestingly enough, our analysis shows that Galileon cosmology does not exhibit any new stable late-time solutions apart from those of standard quintessence, and moreover the corresponding observables are always independent of the Galileon terms. Thus, although the Galileons can play an important role at inflationary or at recent times, in the future, when the universe will asymptotically reach its stable state, they will not have any effect on its evolution.

The plan of the work is the following: In section 2 we briefly review the Galileon cosmological paradigm and in section 3 we perform a detailed phase-space analysis. In section 4 we discuss the cosmological implications and the physical behavior of the scenario. Finally, in section 5 we summarize the obtained results.

## 2 Generalized Galileon cosmology

In this section we briefly review Galileon cosmology for the most generalized scenario, presenting the background cosmological equations and the conditions for the absence of instabilities [78, 9, 18]. As it is known, in order to avoid the Ostrogradski instability [4] it is desirable to keep the equations of motion at second order in derivatives, and thus the most general 4-dimensional scalar-tensor theories having second-order field equations are described by the Lagrangian [18]

(1) |

where

(2) | |||

(3) | |||

(4) | |||

(5) |

The functions and () depend on the scalar field and its kinetic energy , while is the Ricci scalar, and is the Einstein tensor. and () respectively correspond to the partial derivatives of with respect to and , namely and . We mention that the above Lagrangian was first discovered by Horndeski [5] in a different but equivalent form [32].

Apart from the above scalar-tensor sectors, in a realistic cosmological scenario one needs to take into account the matter content of the universe, described by the Lagrangian , corresponding a perfect fluid with energy density and pressure . Then the total action is given by

(6) |

where is the determinant of the metric .

Let us make here an important comment on the above total action, in which
we will focus on the present work. In particular, we do not include a
possible coupling between the scalar field and the matter sector. This
is usual in many cosmological works
[19, 20, 21, 22, 42, 43, 44, 48, 49, 50, 67, 68, 69, 70, 71]. However, we mention that in
its original incarnation the Galileon arises non-minimally coupled to
matter [6, 7, 11]. Therefore,
strictly speaking, in the present work we view the Galileon theory as a
scalar-field, dark-energy, construction, and not as a modified gravity.
However, we have in mind that a possible coupling between the Galileon and
the matter sector, which allows for the realization of the Vainshtein
mechanism, could lead to significantly different cosmological
behavior.^{1}^{1}1We thank the referee for this comment.

In the following we impose a flat Friedmann-Robertson-Walker (FRW) background metric of the form , where is the cosmic time, are the comoving spatial coordinates, is the lapse function, and is the scale factor. Varying the action (6) with respect to and respectively, and setting , we obtain

(7) |

(8) |

where dots denote derivatives with respect to , and we also defined the Hubble parameter . Variation of (6) with respect to provides its evolution equation

(9) |

with

(10) | |||||

(11) | |||||

Finally, the evolution equation for matter takes the standard form

(12) |

We mention here that the four equations (7), (8), (9) and (12), are not independent due to the Bianchi identities. In particular, the scalar equation (9) can be acquired from the other three equations [18].

In order to be able to perform the dynamical analysis of Galileon cosmology we need to focus on some more specific models. One class of Galileon scenarios has the above Lagrangian with the ansatzes:

(13) |

corresponding to the action

(14) |

Such an action is able to capture the basic, quite general, and more interesting Galileon terms (one could straightforwardly include ansatzes with higher powers of , such as the covariant Galileon model [7], however for simplicity we remain to the above simple but non-trivial Galileon action). In this case, the gravitational field equations (7) and (8) become

(15) | |||

(16) |

where we have defined the effective dark energy sector with energy density and pressure respectively:

(17) | |||

(18) |

The scalar field equation (9) becomes

(19) |

and we can immediately see that using (17),(18) it can be rewritten to the standard form

(20) |

Furthermore, we can define the dark energy equation-of-state parameter as

(21) |

One can clearly see that in this scenario, according to the form of , can be quintessence-like, phantom-like, or experience the phantom divide crossing during the evolution, which is a great advantage of Galileon cosmology.

Without loss of generality in the following we restrict the analysis to the dust matter case, that is we assume that . In this case it is convenient to introduce two additional quantities with great physical significance, namely the “total” equation-of-state parameter

(22) |

and the deceleration parameter

(23) |

We close this section by mentioning that in order for the above scenario to be free of ghosts and Laplacian instabilities, and thus cosmologically viable, two conditions must be satisfied [20, 18, 62]. In particular, using the ansatzes (13) and units where , these write as [18]

(24) |

for the avoidance of Laplacian instabilities associated with the scalar field propagation speed, and

(25) |

for the absence of ghosts, where in our case

(26) | |||

(27) |

Finally, we stress that according to (21) and (24),(25) the phantom phase can be free of instabilities and thus cosmologically viable, as it was already shown for Galileon cosmology [18].

## 3 Phase space analysis

In this section we perform a detailed phase-space and stability analysis of generalized Galileon cosmology. As usual, we first transform the above dynamical system into its autonomous form [107, 108, 109, 110, 111], where X is the column vector constituted by suitably chosen auxiliary variables, f(X) the corresponding column vector of the autonomous equations, with primes corresponding to derivatives with respect to . Next we extract its critical points demanding , and in order to determine their stability properties we expand around as , with U the column vector of the perturbations. Therefore, for each critical point we expand the perturbation equations up to first order as , where the matrix contains the coefficients of the perturbation equations. Lastly, the eigenvalues of calculated at each critical point determine its type and stability.

In the scenario at hand we introduce the auxiliary variables:

(28) |

Using these variables the Friedmann equation (15) becomes

(29) |

Moreover, using (29) and (17) we can write the matter and dark energy density parameters as:

(30) |

Note that in the limit the above quantities are well-defined, and they coincide with the usual quintessence ones [107].

In order to proceed forward we need to consider a specific scalar-field potential and a specific coupling function with the Galileon term. Concerning the usual assumption in dynamical investigations in the literature is to assume an exponential potential of the form

(31) |

since exponential potentials are known to be significant in various cosmological models [112, 113, 107, 108, 109, 110, 111] (equivalently, but more generally, we could use potentials satisfying 114, 115, 116]). Concerning , and in order to remain general, we will consider two ansatzes, namely the exponential one , which is the case for arbitrary but nearly flat potentials [

(32) |

and the power-law one

(33) |

The corresponding analysis will be performed separately in the following two subsections.

### 3.1 Scenario 1: Exponential potential and exponential coupling function

In this subsection we consider the exponential potential (31) and the exponential coupling function (32). In this case, using the auxiliary variables (28), the equations (15), (16) and (19) can be transformed to the autonomous form

(34) |

(35) |

(36) |

defined in the (non-compact) phase space Note that in this case the variable is not needed.

Using (17) and the Friedmann equation (29), we can write the density parameters as:

(37) |

while for the dark-energy equation-of-state parameter (21) we obtain:

(38) |

As we mentioned above, according to the variable , that is according to the coupling function , in this scenario can be quintessence-like, phantom-like, or experience the phantom divide crossing during the evolution. Furthermore, the total equation-of-state parameter (22) becomes

(39) |

and the deceleration parameter (23) reads

(40) | |||||

Finally, the two instability-related quantities (24) and (25) are respectively written as

(41) |

and

(42) |

#### 3.1.1 Finite phase-space analysis

Let us now proceed to the phase-space analysis. The real and physically meaningful critical points of the autonomous system (34)-(36) (that is corresponding to an expanding universe, and thus possessing , with ), are obtained by setting the left hand sides of the equations to zero, and they are presented in Table 1. In the same table we provide their existence conditions. For each critical point of Table 1 we calculate the matrix of the linearized perturbation equations of the system (34)-(36), and in order to determine the type and stability of the point we examine the sign of the real part of the eigenvalues of . The details of the analysis and the various eigenvalues are presented in Appendix A.1, and in Table 1 we summarize the stability results. Moreover, for each critical point we calculate the values of , , and given by (37)-(40), as well as the instability-related quantities and given in (3.1),(42), and we summarize the results in Table 2.

Cr. P. | Exist for | Stability | |||

always | unstable for , | ||||

saddle point otherwise | |||||

always | unstable for , | ||||

saddle point otherwise | |||||

unstable for , | |||||

saddle point otherwise | |||||

unstable for , | |||||

saddle point otherwise | |||||

stable node for , | |||||

stable node for , | |||||

saddle point otherwise | |||||

stable node | |||||

stable node for , | |||||

0 | stable node for , | ||||

stable spiral for , | |||||

stable spiral for , | |||||

0 | 0 | 0 | always | saddle |

Cr. P. | q | |||||
---|---|---|---|---|---|---|

arbitrary | ||||||

0 | 0 | |||||

fz | 0 | arbitrary | 0 |

#### 3.1.2 Phase-space analysis at infinity

Since the dynamical system (34)-(36) is non-compact, there could be features in the asymptotic regime which are non-trivial for the global dynamics. Therefore, in order to complete the phase-space analysis we have to extend the investigation using the Poincaré central projection method [117].

We consider the Poincaré variables

(43) |

with and angle to this range since the physical region is given by ) [117, 118, 119, 120]. Therefore, the points at “infinity” () are those with Moreover, the physical phase space is now given by , such that (we restrict the

(44) |

and

Inverting relations (43) and substituting into (37),(38), we obtain the dark energy density and equation-of-state parameters as a function of the Poincaré variables, namely:

(45) |

(46) |

with , and similarly substituting into (3.1), (40) we obtain the corresponding expressions for the total equation-of-state and deceleration parameters:

(47) |