Abstract
The previously introduced class of two-parametric phenomenological inflationary models in general relativity in which the slow-roll assumption is replaced by the more general, constant-roll condition is generalized to the case of f(R) gravity. A simple constant-roll condition is defined in the original Jordan frame, and exact expressions for a scalaron potential in the Einstein frame, for a function f(R) (in the parametric form) and for inflationary dynamics are obtained. The region of the model parameters permitted by the latest observational constraints on the scalar spectral index and the tensor-to-scalar ratio of primordial metric perturbations generated during inflation is determined.
Highlights
In [3], we showed that the model can satisfy the latest observational constraint on the spectral a e-mail: motohashi@ific.uv.es b e-mail: alstar@landau.ac.ru index of the curvature power spectrum and the tensor-toscalar ratio. This constant-roll construction refers to inflationary models in General Relativity (GR) where gravity is not modified but a new scalar field has to be introduced
In the opposite limit one can construct inflationary models without new scalar fields, by changing the gravity sector only, as typified by the R + R2 model [7] and its f (R) gravity modifications [8,9,10,11,12]. This purely geometrical approach is equivalent to introducing a scalar degree of freedom, which can be explicitly seen by performing a conformal transformation from the Jordan frame to the Einstein frame
In contrast to previous works [1,2,3] where the constant-roll condition was effectively imposed in the Einstein frame, since inflation in GR was considered, we impose a new constant-roll condition in the original Jordan frame where the form of equations is simpler ; see e.g. Eq (11) below
Summary
This constant-roll construction refers to inflationary models in General Relativity (GR) where gravity is not modified but a new scalar field has to be introduced. For f (R) gravity the master first-order equation for HJ in the original Jordan frame considered as a function of the Ricci scalar RJ has even a simpler form, which can be obtained as follows. For a generic f (R) function, substituting the constant-roll condition (12) to (3) and integrating it, we obtain a very simple and elegant relation which has to be satisfied for all models in this class at all times:. The inflaton velocity |dφ/dtE| is always decreasing, it approaches φ = φb spending infinite Einstein-frame time This process develops small-scale inhomogeneity of the Universe. For the case β < −3 and γ = −1, Ricci curvature is always negative, which is another reason why we do not consider this parameter set, in addition to the negative potential mentioned above. Using the relation d NJ dtJ dtE dtJ d NJ dφ (34)
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