Construction of slow-roll approximations in the Jordan frame

Models with nonminimally coupled scalar fields and F(R) gravity models are some of the most popular modified gravity models of cosmological inflation. Using the conformal transformation of the metric, one can transform the original modified gravity models to the General Relativity models with minimally coupled scalar fields; in other words, using the Einstein frame. The standard way to construct such modified gravity models is to consider them in the Einstein frame using slow-roll approximation. For the simplest inflationary F(R) gravity models, including the Starobinsky R+R² inflationary model, this method gives precise results, but F(R) gravity models with more complicated functions F(R) usually correspond to the inflaton potential given in the parametric form. Such F(R) functions usually appear in the models that describe not only inflation, but also primordial black hole formation.

The slow-roll approximation is a method used to analyse the dynamical properties of the early universe during inflation. We propose new versions of the slow-roll approximation for inflationary models with nonminimally coupled scalar fields and F(R) gravity models. We investigate the inflationary dynamics in the initial (Jordan) frame without the use of the conformal transformation and rewrite evolution equations in the Jordan frame in a form that is similar to the Friedmann equations in the Einstein frame. The proposed slow-roll approximations are described in the papers by E.O. Pozdeeva, M.A. Skugoreva, A.V. Toporensky, and S.Yu. Vernov, JCAP 05 (2025) 081 and S.V. Ketov, E.O. Pozdeeva, and S.Yu. Vernov, JCAP 12 (2025) 40. This approximation allows us to construct a new inflationary model with the fifth-order polynomial F(R) function. This model is in good agreement with the recent observations of the cosmic microwave background radiation obtained by the Atacama Cosmology Telescope. The presentation is based on these papers and recent results.