Exact geodesic distances in FLRW spacetimes

William J. Cunningham, David Rideout, James Halverson, and Dmitri Krioukov
Physical Review D
96, 103538
November 27, 2017


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.

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