Anisotropic singularities in modified gravity models

We show that the common singularities present in generic modified gravity models governed by actions of the type $S=\int d^{4}x\sqrt{-g}f(R,\varphi ,X)$, with $X=-\frac{1}{2}{g}^{ab}{\partial }_a\varphi {\partial }_b\varphi$, are essentially the same anisotropic instabilities associated to the hypersurface $F(\varphi )=0$ in the case of a nonminimal coupling of the type $F(\varphi )R$, enlightening the physical origin of such singularities that typically arise in rather complex and cumbersome inhomogeneous perturbation analyses. We show, moreover, that such anisotropic instabilities typically give rise to dynamically unavoidable singularities, precluding completely the possibility of having physically viable models for which the hypersurface $\frac{\partial f}{\partial R}=0$ is attained. Some examples are explicitly discussed.

Figure 1

Typical trajectories for the model (29) projected on the $(h,\dot{h})$ plane, with $\alpha =2$ (the same value adopted in the analysis of 16). The (attractive de Sitter) fixed point in this case corresponds to $h\approx 0.7157$. The $\rho$-constant parabolas are shown. The solid one, in particular, correspond to $\rho =\alpha -1$ (the singular $\mathcal{F}$ surface, defined by $F(R)=0)$. The regions corresponding to $F(R)>0$ and $F(R)<0$ are, respectively, the region above and the region below the parabola $\rho =\alpha -1$. As one can see, homogeneous and isotropic solutions can cross without problems the singular surface.