Cosmological evolution in vector-tensor theories of gravity

We present a detailed study of the cosmological evolution in general vector-tensor theories of gravity without potential terms. We consider the evolution of the vector field throughout the expansion history of the Universe and carry out a classification of models according to the behavior of the vector field in each cosmological epoch. We also analyze the case in which the Universe is dominated by the vector field, performing a complete analysis of the system phase map and identifying those attracting solutions which give rise to accelerated expansion. Moreover, we consider the evolution in a universe filled with a pressureless fluid in addition to the vector field and study the existence of attractors in which we can have a transition from matter domination to vector domination with accelerated expansion so that the vector field may play the role of dark energy. We find that the existence of solutions with late-time accelerated expansion is a generic prediction of vector-tensor theories and that such solutions typically lead to the presence of future singularities. Finally, limits from local gravity tests are used to get constraints on the value of the vector field at small (Solar System) scales.

Figure 1

Behavior of the temporal component of the vector field in the different phases of the Universe’s history according to the values of the parameters of the model. The labels Decaying and Growing refer to the behavior with respect to the dominant component, whereas Scaling means that it evolves in the same way as the background.

Figure 2

Behavior of the spatial component of the vector field in the different phases of the Universe’s history according to the values of the parameters of the model. The labels Decaying and Growing refer to the behavior with respect to the dominant component, whereas Scaling means that it evolves in the same way as the background.

Figure 3

This plot shows the different cases explained in the main text in the parameter space. The shaded (green) region corresponds to models in which both $x_{\pm }$ exist (case I) whereas in the white region neither of them is present (case II). The solid line identifies the models for which $x_{+}=x_{-}$ corresponding to case III. The dashed line (whose equation is $2{\omega }_{\lambda }+2{\sigma }_{\lambda }+3=0$) represents those models that have either $x_{-}$ or $x_{+}$ at the origin (case Ib) and the models whose parameters lie on the dotted line (with equation $2{\omega }_{\lambda }+{\sigma }_{\lambda }+1=0$) have the separatrix $x_s$ at the origin. Thus, in the region above (below) that line $x_s$ is negative (positive). Case Ic corresponds to the piece of the dotted line inside the white region whereas the piece of the dotted line inside the gray region corresponds to case IIb. Finally, the (orange) dot with parameters ${\omega }_{\lambda }=1/2$, ${\sigma }_{\lambda }=-2$ gives the case IIIb in which $x_{\pm }=x_s=0$.

Figure 4

These plots show two examples of phase maps for the autonomous system describing a universe dominated by the vector field. The left panel corresponds to a model lying in the gray region in Fig. 3 in which the two critical points $P_{\pm }$ are present whereas the right panel shows the phase map for a model in which these two critical points do not exist. We can see that all the critical points as well as the separatrix are vertical tangents and the $x$ axis is a horizontal separatrix, as explained in the main text.

Figure 5

In this plot we show the different regions obtained in the parameter space according to the features of the critical points. We have shaded (orange) the regions where the eigenvalue of $x$ is negative so that the trajectories in the phase map approach the corresponding point. We have also indicated the sign of each eigenvalue in the form $({\mu }_x,{\mu }_H)$. For the separatrix we indicate the regions where the trajectories approach the separatrix. Finally, the two plots on the right show the 13 regions explained in the text (and summarized in Table ). The bottom-right plot corresponds to a zoom of the shaded (blue) region in the top-right panel above.

Figure 6

In this plot we show the regions in which we get accelerated solutions in a universe dominated by the vector field for each critical point. The darkest shaded regions are those in which the accelerated solutions are attractors. We can see that neither for $P_{+}$ nor for $P_{-}$ can we obtain attracting accelerated solutions.

Figure 7

In these two plots we show two examples of phase maps corresponding to the cases when the two critical points are present (left) and when they are not (right) or, equivalently, when the separatrix is open from above or from below.

Figure 8

In this plot we show the regions where we have the vertex of the separatrix attracting trajectories with accelerated expansion ([green] shaded region) and where there are not solutions attracted by the vertex in which the expansion is accelerated (white region). We see that, in most of the parameter space, we have that the vertex attracts some trajectories in which the equation of state of the vector field diverges evolving toward $-\infty$. The curve plotted in the graph separates those models in which the separatrix is open from above (outer region) and those in which the separatrix is open from below (inner region). Notice that these regions coincide with those in which the critical points $P_{\pm }$ exist (separatrix open from above) and they do not (separatrix open from below), as explained in the main text.