Exact Geodesic Distances in FLRW Spacetimes
  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.
