Sudden Future Singularities in Quintessence and Scalar-Tensor Quintessence Models

We demonstrate analytically and numerically the existence of geodesically complete singularities in quintessence and scalar tensor quintessence models with scalar field potential of the form $V(\phi)\sim \vert \phi\vert^n$ with $0<n<1$. In the case of quintessence, the singularity which occurs at $\phi=0$, involves divergence of the third time derivative of the scale factor (Generalized Sudden Future Singularity (GSFS)), and of the second derivative of the scalar field. In the case of scalar-tensor quintessence with the same potential and with a linear minimal coupling ($F(\phi)=1-\lambda \phi$), the singularity is stronger and involves divergence of the second derivative of the scale factor (Sudden Future Singularity (SFS)). We show that the scale factor close to the singularity is of the form $a(t)=a_s+b(t_{s}-t) + c(t_{s}-t)^2 +d(t_{s}-t)^q$ where $a_s,b,c,d$ are constants obtained from the dynamical equations and $t_s$ is the time of the singularity. In the case of quintessence we find $q=n+2$ (ie $2<q<3$), while for the case of scalar-tensor quintessence $q=n+1$ ($1<q<2$). We verify these analytical results numerically and extend them to the case where a perfect fluid is present. The linear and quadratic terms in $(t_{s}-t)$ that appear in the expansion of the scale factor around $t_s$ are subdominant for the diverging derivatives close to the singularity, but can play an important role in the estimation of the Hubble parameter. Using the analytically derived relations between these terms, we derive relations involving the Hubble parameter close to the singularity, which may be used as observational signatures of such singularities in this class of models. For quintessence with matter fluid, we find that close to the singularity $\dot H=\frac{3}{2}\Omega_{0m} (1+z_{s})^{3}-3H^{2}$.