Inflation and integrable one‐field cosmologies embedded in rheonomic supergravity

Abstract

In this paper we show that the new approach to the embedding of the inflationary potentials into supergravity, presented in a quite recent paper [11] of Ferrara, Kallosh, Linde and Porrati can be formulated within the framework of standard matter coupled supergravity, without the use of the new minimal auxiliary set and of conformal compensators. The only condition is the existence of a translational Peccei Quinn isometry of the scalar Kähler manifold. We suggest that this embedding strategy based on a nilpotent gauging amounts to a profound Copernican Revolution. The properties of the inflaton potential are encoded in the geometry of some homogeneous one-dimensional Kähler manifolds that now should be regarded as the primary object, possibly providing a link with microscopic physics. We present a simple and elegant formula for the curvature of the Kähler manifold in terms of the potential. Most relevant consequence of the new strategy is that all the integrable potentials quite recently classified in a paper [7] that we have coauthored, are automatically embedded into supergravity and their associated Kahler manifolds demand urgent study. In particular one integrable potential that provides the best fit to PLANCK data seems to have inspiring geometrical properties deserving further study.

Integrable cosmological potentials

“…Differentiating (30) by q 2 and eliminating q 2 -derivatives of A(q 1 , q 2 ) and B(q 1 , q 2 ) with the help of (27), we get several more linear ODEs. Hence A(q 1 , q 2 ) and B(q 1 , q 2 ) have the form (21). Substituting these expressions into (30) and equating coefficients at q…”

“…Differentiating (50) several times by q 2 and eliminating q 2 -derivatives ofC,D, A and B with the help of (49), (51) and (27), we arrive at a system of linear equations with respect to the independent variable q 1 . This implies that A(q 1 , q 2 ) and B(q 1 , q 2 ) are quasi-polynomials of the form (21) and…”

On the topology of the inflaton field in minimal supergravity models

“…Independently from integrability, the geometrical basis of the construction of minimal supergravity models of inflation introduced in [5] was analyzed in another pair of recent papers [15,16]. It was pointed out that the root from any given positive definite potential V (φ) to the corresponding minimal supergravity model is a map, named by two of us the D-map, in whose image there is a two-dimensional Kähler surface Σ admitting at least a one-dimensional group of isometries G. Various aspects of this map were explored, but a fundamental question remained so far unanswered about the global topology both of the surface Σ and of its isometry group.…”

“…Let us recall such a notion, according to the discussion of [35] and [15]. In [35] the construction was applied to N = 2 supergravity so that the composite connections to be gauged were those emerging in Special Kähler Geometry.…”

“…Eq. (4.5) was first derived in [15]. 6 6 The curvature R defined as the component of dω in the basis E 1 ∧ E 2 differs by a factor 1 2 with respect to the curvature defined in standard curved index tensor calculus.…”