Analytic Singular Slow-roll Inflation

We study a class of minimally coupled scalar field theories which leads to analytic solutions for the Hubble rate and the scalar field, where the scalar field obeys a generalized tracking law $\dot{\phi}^2\sim H^{-m}$. The inflationary phenomenology for this class of models can be studied fully analytically. The resulting phenomenology is compatible with the ACT data and for limiting cases, the spectral index is bluer than the ACT constraints and tends to the value $n_{\mathcal{S}}=0.98$, while in the limiting case, the tensor-to-scalar ratio takes very small values, nearly zero. In addition, we prove analytically that the phenomenology is a one-parameter model, and the inflationary observables encode the scaling exponent $m$ of the generalized kinetic attractor $\dot{\phi}^2\sim H^{-m}$. Furthermore, the tensor-to-scalar ratio and the spectral index have a simple linear and $m$-dependent relation. More importantly, the resulting cosmology describes a Universe that has a finite scale factor at $t=0$, thus non-singular, evolves and expands realizing a slow-roll inflationary era and after that it reaches classically a pressure singularity. Classically, the Universe can pass through this singularity, and a turnaround cosmology is realized with the Universe contracting after the turnaround point. However, before the singularity is realized classically, the quantum phenomena dominate the evolution, avoiding the singularity. Specifically we consider the Nojiri-Odintsov conformal anomaly mechanism and we qualitatively show that the conformal anomaly erases the classical singular evolution and at the same time it enhances particle creation, which eventually reheats the Universe. Thus in this model the scalar field oscillations and the numerous couplings of the inflaton to the Standard Model particles are not required for reheating.