Cosmology of the very early universe

TL;DR

This paper explores early universe cosmological solutions near the Planck scale, revealing conditions for inflation, the impact of curvature and cosmological constant, and identifying critical thresholds for different evolutionary outcomes.

Contribution

It provides a comprehensive analysis of early universe models with radiation, curvature, and cosmological constant, highlighting the role of initial conditions and critical values for inflation.

Findings

Positive curvature leads to a Big Crunch without inflation.

A critical cosmological constant determines inflation viability.

Loitering solutions can evolve into inflation or collapse.

Abstract

We study cosmological solutions for the very early universe beginning at the Planck scale for a universe containing radiation, curvature and, as a simplification of a possible scalar field potential, a cosmological constant term. The solutions are the natural counterpart of the well known results for a post-inflationary universe of non-relativistic matter, curvature and a cosmological constant. Contrary to the common belief that inflation arises independently of the initial curvature we show that in the positive curvature case the universe collapses again into a Big Crunch without allowing the cosmological term to dominate and to produce inflation. There is a critical value for the cosmological constant which divide the regions where inflation is allowed from those where inflation cannot occur. One can also have loitering solutions where the scale factor remains almost constant growing to produce inflation (or decreasing to a Big Crunch) after a time which depends on the amount of energy above (or below) the critical energy. At the critical energy the solution approaches asymptotically a particular value for the scale factor (Einstein's static pre-inflationary universe). The cases where the cosmological term vanishes or becomes negative are also studied providing a complete discussion of Friedmann models.