# Exact Geodesic Distances in FLRW Spacetimes

## TL;DR

This paper develops methods to compute exact geodesic distances in flat FLRW spacetimes, providing closed-form solutions for simple cases and a fast numerical approach for complex models relevant to cosmology.

## Contribution

It introduces a general framework for calculating geodesic distances in flat FLRW spacetimes, including closed-form solutions and a numerical method for complex cases.

## Key findings

- Closed-form solutions for geodesic distances in flat spacetimes with dark energy or matter.
- A fast numerical method for geodesics in mixed dark energy and matter spacetimes.
- A general approach to determine geodesic connectedness based on the scale factor.

## Abstract

Geodesics are used in a wide array of applications in cosmology and astrophysics. However, it is not a trivial task to efficiently calculate exact geodesic distances in an arbitrary spacetime. We show that in spatially flat (3+1)-dimensional Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes, it is possible to integrate the second-order geodesic differential equations, and derive a general method for finding both timelike and spacelike distances given initial-value or boundary-value constraints. In flat spacetimes with either dark energy or matter, whether dust, radiation, or a stiff fluid, we find an exact closed-form solution for geodesic distances. In spacetimes with a mixture of dark energy and matter, including spacetimes used to model our physical universe, there exists no closed-form solution, but we provide a fast numerical method to compute geodesics. A general method is also described for determining the geodesic connectedness of an FLRW manifold, provided only its scale factor.