Sudden Future Singularities and their observational signatures in Modified Gravity

We verify the existence of Generalized Sudden Future Singularities (GSFS) in quintessence models with scalar field potential of the form $V(\phi)\sim \vert \phi\vert^n$ where $0<n<1$ and in the presence of a perfect fluid, both numerically and analytically, using a proper generalized expansion ansatz for the scale factor and the scalar field close to the singularity. This generalized ansatz includes linear and quadratic terms, which dominate close to the singularity and cannot be ignored when estimating the Hubble parameter and the scalar field energy density; as a result, they are important for analysing the observational signatures of such singularities. We derive analytical expressions for the power (strength) of the singularity in terms of the power $n$ of the scalar field potential. We then extend the analysis to the case of scalar tensor quintessence models with the same scalar field potential in the presence of a perfect fluid, and show that a Sudden Future Singularity (SFS) occurs in this case. We derive both analytically and numerically the strength of the singularity in terms of the power $n$ of the scalar field potential.