The method of generating functions in exact scalar field inflationary cosmology

Abstract

The construction of exact solutions in scalar field inflationary cosmology is of growing interest. In this work, we review the results which have been obtained with the help of one of the most effective methods, viz., the method of generating functions for the construction of exact solutions in scalar field cosmology. We also include in the debate the superpotential method, which may be considered as the bridge to the slow roll approximation equations. Based on the review, we suggest a classification for the generating functions, and find a connection for all of them with the superpotential. PACS numbers: 98.80. Bp, 98.80.Cq, 98.80.Es, 98.80.Jk I. EARLY INFLATION AND THE IMPLEMENTED SCALAR FIELD

Superpotential method for chiral cosmological models connected with modified gravity

“…where superpotential W is a differentiable function and C W is a constant part of the Hubble function that plays a special role (see, for example, [33]). It is convenient to write C W separately.…”

“…for all A. This is rather tight restriction which is equivalent to the decomposition method used in the works [33,36,39].…”

“…Under condition h 12 (χ, ψ) = 0 one has the same solutions for the trivial case of a constant diagonal tensor h AB . Similarly, one can generalize any exact solutions in single-field models (see, for example, [33,87]) on this special class of chiral cosmological models with two components.…”

“…In the case of single scalar cosmology with a generic potential, the integration procedure is reduced to solving the Ivanov-Salopek-Bond equation [7,30]. There are many methods of obtaining physically relevant solutions, including that with the Higgs potential [30][31][32][33]. The reduction of a CCM to a single scalar field model was proposed in [34].…”

“…We do not seek solutions for a given potential but construct the potential of the scalar field such that the resulting model has exact solutions with important physical properties. Such a method is similar to the Hamilton-Jacobi method (also known as the superpotential method or the first-order formalism) and is applied to cosmological models with minimally [7,30,33,[40][41][42][43][44][45][46][47][48][49] and nonminimally [50] coupled scalar fields. This method has been used, in particular, to find exact solutions in single scalar field inflationary models 2 [8,[59][60][61][62].…”