Constraining Curvature Parameters via Topology

TL;DR

This paper explores how detecting multiple topological images in the universe can precisely constrain curvature parameters, offering a purely geometrical method independent of dynamical assumptions and the Hubble constant.

Contribution

It provides a detailed analysis of how topological image detections can constrain universe curvature parameters with high precision, independent of dynamical models.

Findings

Multiple topological images can constrain  and \u03bb0 to within 1% and 10%.

CMB spots linked to low-redshift objects can yield similar constraints.

Constraints are independent of the Hubble constant, H0.

Abstract

If the assumption that physical space has a trivial topology is dropped, then the Universe may be described by a multiply connected Friedmann-Lema\^{\i}tre model on a sub-horizon scale. Specific candidates for the multiply connected space manifold have already been suggested. How precisely would a significant detection of multiple topological images of a single object, or a region on the cosmic microwave background, (due to photons arriving at the observer by multiple paths which have crossed the Universe in different directions), constrain the values of the curvature parameters $\Omega_0$ and $\lambda_0$? The way that the constraints on $\Omega_0$ and $\lambda_0$ depend on the redshifts of multiple topological images and on their radial and tangential separations is presented and calculated. The tangential separations give the tighter constraints: multiple topological images of known types of astrophysical objects at redshifts $z \ltapprox 3$ would imply values of $\Omega_0$ and $\lambda_0$ preciser than $\sim 1%$ and $\sim 10%$ respectively. Cosmic microwave background `spots' identified with lower redshift objects by the Planck or MAP satellites would provide similar precision. This method is purely geometrical: no dynamical assumptions (such as the virial theorem) are required and the constraints are independent of the Hubble constant, $H_0.$