[Paper Review] Constraining Curvature Parameters via Topology

Boudewijn F. Roukema, Jean‐Pierre Luminet|arXiv (Cornell University)|Mar 30, 1999

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper proposes a geometric method to constrain cosmological curvature parameters Ω₀ and λ₀ using topological images of astrophysical objects or cosmic microwave background (CMB) features observed via multiple light paths in a multiply connected universe. By analyzing the redshifts and angular separations—especially tangential separations—of these topological images, the method achieves precision of ~1% for Ω₀ and ~10% for λ₀ without relying on dynamical assumptions or H₀, offering a robust, geometry-based constraint on cosmic curvature.

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\^ıtre 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 $Ω_0$ and $λ_0$? The way that the constraints on $Ω_0$ and $λ_0$ 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 $Ω_0$ and $λ_0$ 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, $H_0.$

Motivation & Objective

• To investigate how topological imaging in a multiply connected universe can constrain the curvature parameters Ω₀ and λ₀.
• To determine the precision with which these parameters can be measured using observed topological images of astrophysical objects or CMB features.
• To develop a method independent of dynamical assumptions (e.g., virial theorem) and the Hubble constant H₀.
• To quantify the dependence of parameter constraints on redshifts and angular separations of topological images.

Proposed method

• Model the universe as a multiply connected Friedmann-Lemaître space with non-trivial topology on sub-horizon scales.
• Identify topological images of a single object or CMB feature formed by photons traversing different spatial paths.
• Use observed redshifts and angular separations (radial and tangential) of these images to infer geometric constraints on Ω₀ and λ₀.
• Apply geometric relations derived from the curvature of space to relate image separations to cosmological parameters.
• Focus on tangential separations as the dominant source of constraint precision due to their sensitivity to curvature.
• Assume no dynamical modeling is required—constraints are purely geometric and independent of H₀.

Experimental results

Research questions

• RQ1 How precisely can curvature parameters Ω₀ and λ₀ be constrained using topological images of astrophysical objects?
• RQ2 What is the role of tangential versus radial angular separations in constraining cosmological parameters?
• RQ3 To what extent can this method achieve precision independent of the Hubble constant H₀ or dynamical assumptions?
• RQ4 How do redshifts of topological images affect the accuracy of curvature parameter estimation?
• RQ5 Can CMB 'spots' identified with lower-redshift objects via missions like Planck or MAP provide comparable constraints?

Key findings

• Tangential separations of topological images provide tighter constraints on Ω₀ and λ₀ than radial separations.
• For astrophysical objects at redshifts z ≲ 3, the method can constrain Ω₀ to within ~1% and λ₀ to within ~10%.
• CMB 'spots' identified with lower-redshift objects by Planck or MAP would yield similar precision in parameter estimation.
• The method is purely geometric and does not require assumptions about dynamics, such as the virial theorem.
• Constraints are independent of the Hubble constant H₀, enhancing robustness against calibration uncertainties.
• The approach is viable for both astrophysical objects and CMB features, offering a new, independent route to cosmological parameter estimation.