The wave equation in Friedmann-Robertson-Walker space-times and asymptotics of the intensity and distance relationship of a localised source

Variations in redshift measurements of Type 1a supernovae and intensity observations from large sky surveys are an indicator of a component of acceleration in the rate of expansion of space-time. A key factor in the measurements is the intensity-distance relation for Maxwell’s equations in Friedmann-Robertson-Walker (FRW) space-times. In view of future measurements of the decay of other fields on astronomical time and spatial scales, we determine the asymptotic behavior of the intensity-distance relationship for the solution of the wave equation in space-times with an FRW metric. This builds on previous work done on initial value problems for the wave equation in FRW space-time [Abbasi, B. and Craig, W., Proc. R. Soc. London, Ser. A 470, 20140361 (2014)]. In this paper, we focus on the precise intensity decay rates of the special cases for curvature k = 0 and k = −1, as well as giving a general derivation of the wave solution for −∞ < k < 0. We choose a Cauchy surface {(t, x) : t = t0 > 0} where t0 represents the time of an initial emission source, relative to the Big Bang singularity at t = 0. The initial data [g(x), h(x)] are assumed to be compactly supported; supp(g, h) ⊆ BR(0) and terms in the expression for the fundamental solution for the wave equation with the slowest decay rate are retained. The intensities calculated for coordinate time {t : t > 0} contain correction terms proportional to the ratio of t0 and the time differences ρ = t − t0. For the case of general curvature k, these expressions for the intensity reduce by scaling to the same form as for k = −1, from which we deduce the general formula. We note that for typical astronomical events such as Type 1a supernovae, the first order correction term for all curvatures −∞ < k < 0 is on the order of 10−4 smaller than the zeroth order term. These correction terms are small but may be significant in applications to alternative observations of cosmological space-time expansion rates.

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