  If the universe is multiply connected and small the sky shows multiple images
of cosmic objects, correlated by the covering group of the 3-manifold used to
model it. These correlations were originally thought to manifest as spikes in
pair separation histograms (PSH) built from suitable catalogues. Using
probability theory we derive an expression for the expected pair separation
histogram (EPSH) in a rather general topological-geometrical-observational
setting. As a major consequence we show that the spikes of topological origin
in PSH's are due to translations, whereas other isometries manifest as tiny
deformations of the PSH corresponding to the simply connected case. This result
holds for all Robertson-Walker spacetimes and gives rise to two basic
corollaries: (i) that PSH's of Euclidean manifolds that have the same
translations in their covering groups exhibit identical spike spectra of
topological origin, making clear that even if the universe is flat the
topological spikes alone are not sufficient for determining its topology; and
(ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological
origin. These corollaries ensure that cosmic crystallography, as originally
formulated, is not a conclusive method for unveiling the shape of the universe.
We also present a method that reduces the statistical fluctuations in PSH's
built from simulated catalogues.
