Curvature and Flatness in a Brans-Dicke Universe

TL;DR

This paper analyzes the evolution of a Brans-Dicke universe with curvature, exploring its dynamics, flatness problem, and how curvature influences different cosmological epochs, including inflation and the Planck era.

Contribution

It provides explicit solutions for a curved Brans-Dicke universe during radiation dominance and discusses the flatness problem within this framework, extending standard cosmological insights.

Findings

Curvature divides the Brans-Dicke universe into three distinct classes.

The flatness problem persists in Brans-Dicke gravity, similar to the standard model.

A residual flatness problem exists at the Planck epoch in models addressing the horizon problem.

Abstract

The evolution of a universe with Brans-Dicke gravity and nonzero curvature is investigated here. We present the equations of motion and their solutions during the radiation dominated era. In a Friedman-Robertson-Walker cosmology we show explicitly that the three possible values of curvature $\kappa=+1,0,-1$ divide the evolution of the Brans-Dicke universe into dynamically distinct classes just as for the standard model. Subsequently we discuss the flatness problem which exists in Brans-Dicke gravity as it does in the standard model. In addition, we demonstrate a flatness problem in MAD Brans-Dicke gravity. In general, in any model that addresses the horizon problem, including inflation, there are two components to the flatness issue: i) at the Planck epoch curvature gains importance, and ii) during accelerated expansion curvature becomes less important and the universe flattens. In many cases the universe must be very flat at the Planck scale in order for the accelerated epoch to be reached, thus there can be a residual flatness problem.