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) similar to vertical bar phi vertical bar(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) + (t(s)- t)(q) where a(s), b, c, d are constants which are obtained from the dynamical equations and t(s) is the time of the singularity. In the case of quintessence we find q = n + 2 (i.e. 2 < q < 3), while for the case of scalar-tensor quintessence we have a stronger singularity with q = n + 1 (1 < q < 2). We verify these analytical results numerically and extend them to the case where a perfect fluid, with a constant equation of state w = p/rho, is present. We find that the strength of the singularity (value of q) remains unaffected by the presence of a perfect fluid. 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 <(H) over dot> = 3/2 Omega(0m)(1 + z(s))(3)-3H(2). These terms should be taken into account when searching for future or past time such singularities, in cosmological data.