Since its original ideas put forward, inflation has been one of the mainstream research topics in early universe cosmology for several decades due to theoretical elegance as well as predictions compatible with cosmological observations such as the Planck result. A simple way to have the inflationary phase, namely, the accelerated expansion of the universe is to assume the universe filled with a scalar field (called inflaton field) which has the negative pressure.
Resolving the big bang puzzles requires that inflation persisted long enough, which could be realized by the inflaton field with a sufficiently flat potential so that the inflaton slowly rolls down the potential during inflation. Numerous inflation models with such a property constructed so far predict a nearly scale-invariant spectrum consistent with observations. Those models (often called slow-roll models), however, have one potential drawback that they cannot predict non-Gaussianities in the primordial density perturbations which could be appeared in future observations.
Non-Gaussianities in the perturbations can be generated by involving many scalar fields or giving up the slow-roll condition. For the last decade, the latter way has been adopted in many works, where the slow-roll condition was replaced with the constant-roll condition in which the second slow-roll parameter is constant but not too small. It was shown that constant-roll models could produce a nearly scale-invariant spectrum in agreement with observations. However, one might need a "fine-tuned" potential to satisfy the exact constant-roll condition.
We have discussed a generalization of the constant-roll inflation by applying the b-function formalism for inflation. The generalized constant-roll condition was proposed in such a way that it approaches the exact constant-roll condition in deep inflationary phase. We have found several analytic solutions to the generalized constant-roll condition. Among them, there were unstable models with the potential not bounded from below. One benefit of generalizing the constant-roll condition is to enlarge the physically viable inflation models. For instance, it was possible to terminate inflation naturally by generalizing the model with constant β-function that was known inappropriate to describe inflation with a natural end. Given that the horizon crossing occurred in deep inflationary phase where the generalized constant-roll condition approaches the exact one, we expect that the predictions for the CMB observables of the models proposed here could be compatible with observations.
This result, has been published in "Indian Journal of Physics" under the title of "Notes on some analytically solvable generalized constant-roll inflation models" (https://doi.org/10.1007/s12648-024-03339-5).
