Noether symmetries and stability of ideal gas solutions in Galileon cosmology

A class of generalized Galileon cosmological models, which can be described by a pointlike Lagrangian, is considered in order to utilize Noether’s theorem to determine conservation laws for the field equations. In the Friedmann-Lemaître-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Because of the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

Figure 1

In the figures are presented some orbits of (34) on the Poincarè plane for $p=1/6$, $p=1/3$, for radiation ($p=1/2$) and matter ($p=2/3$) dominated universes, for zero acceleration solution ($p=1$), and for accelerated solution ($p=3$).

Figure 2

Plot of the solutions in the plane $({\mathcal{E}}_r,{\mathcal{V}}_r)=(r\sin \theta ,r\cos \theta )$ for the system (37).

Figure 3

Some integral curves for the full system (40). It can be shown that the (0, 0) solution is stable as $\tau \to \infty$ by choosing $c_2$ small enough and for $\epsilon >0$.

Figure 4

Poincarè projection of the system (42).