Proper Size of the Visible Universe in FRW Metrics with Constant Spacetime Curvature

TL;DR

This paper investigates the proper size of the visible universe in static FRW spacetimes with constant curvature, establishing it as half the gravitational horizon at the current age, with implications for cosmological measurements.

Contribution

It proves that in certain static FRW cosmologies, the visible universe's size equals half the gravitational horizon, clarifying the interpretation of cosmological distances.

Findings

The proper size of the visible universe is R_h(t_0/2).

This result holds for FRW models with constant spacetime curvature, excluding de Sitter and Lanczos.

Numerical integration confirms the theoretical findings across various cosmologies.

Abstract

In this paper, we continue to examine the fundamental basis for the Friedmann-Robertson-Walker (FRW) metric and its application to cosmology, specifically addressing the question: What is the proper size of the visible universe? There are several ways of answering the question of size, though often with an incomplete understanding of how far light has actually traveled in reaching us today from the most remote sources. The difficulty usually arises from an inconsistent use of the coordinates, or an over-interpretation of the physical meaning of quantities such as the so-called proper distance R(t)=a(t)r, written in terms of the (unchanging) co-moving radius r and the universal expansion factor a(t). In this paper, we use the five non-trivial FRW metrics with constant spacetime curvature (i.e., the static FRW metrics, but excluding Minkowski) to prove that in static FRW spacetimes in which expansion began from an initial signularity, the visible universe today has a proper size equal to R_h(t_0/2), i.e., the gravitational horizon at half its current age. The exceptions are de Sitter and Lanczos, whose contents had pre-existing positions away from the origin. In so doing, we confirm earlier results showing the same phenomenon in a broad range of cosmologies, including LCDM, based on the numerical integration of null geodesic equations through an FRW metric.