Exact tail function in the full FLRW geometry

Derive the tail function V(x,y) by solving its wave equation directly in the full Friedmann–Lemaître–Robertson–Walker background, rather than using the flat-spacetime approximation near the present epoch.

Background

The retarded Green function is written in Hadamard form, with the tail function V(x,y) determined by a wave equation in the timelike domain. Because the exact FLRW solution is analytically intractable in the treatment presented, the paper approximates V by its flat-spacetime expression near the current epoch.

An exact or quantitatively more accurate FLRW calculation is needed to assess corrections from the evolving cosmological geometry, especially for contributions originating at early times.