Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

Abstract

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-Lemaître-Robertson-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.

Constraints on quintessence scalar field models using cosmological observations

“…We note that the canonical scalar field models are more informative compared to the non-canonical scalar field models. The physics of canonical scalar field models, which is mostly contained in the potential, V (φ(t)) where φ(t) is the underlying field, have gained a considerable interest in the cosmological community due to explaining various stages of the universe evolution, see [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] (also see [58,59,60,61,62]). The selection of quintessence scalar field models should not be much arbitrary, according to recently introduced swampland [63,64] and refined swampland [65] criteria.…”

Noether symmetries as a geometric criterion to select theories of gravity

“…The Noether Symmetry Approach is outlined in [3] and in [2]. Applications to scalar-tensor cosmologies are reported in [4,5,6,7]. The most general scalar-tensor theory, giving second order field equations, the so-called Horndeski gravity, is discussed in [8].…”

Noether symmetries and stability of ideal gas solutions in Galileon cosmology

“…In the following we assume that g (φ) is not constant or zero. It is important to mentioned that the analysis we perform here it is different from the one which presented in [50]. The main reason is that in this work we see the Lagrangian as a regular system in contrary to [50] which we saw it as a singular system.…”

“…It is important to mentioned that the analysis we perform here it is different from the one which presented in [50]. The main reason is that in this work we see the Lagrangian as a regular system in contrary to [50] which we saw it as a singular system. The two different approaches are complementary, for details see the discussion in [60,61].…”

“…The class of transformations we consider to leave invariant the action and, hence, the field equations are the so-called point transformations. The latter have been applied in various cosmological models and modified theories of gravity for the determination of new cosmological solutions, see for instance [41][42][43][44][45][46][47][48][49][50] and references therein. It is important to mention that except for the point transformations, other classes of mappings or different methods for the study of integrability have been applied in gravitational theories, some of these are presented in [51][52][53][54][55][56][57][58].…”