  The "Shapiro delay" experienced by an astronomical messenger traveling
through a gravitational field has been used to place constraints on possible
deviations from the equivalence principle. The standard Shapiro delay used to
obtain these constraints is not itself an observable in general relativity, but
is rather obtained by comparing with a fiducial Euclidean distance. There is
not a mapping between the constraints obtained in this manner and alternative
theories that exhibit equivalence principle violations. However, even assuming
that the comparison with the fiducial Euclidean distance is carried out in a
way that is useful for some class of alternative theories, we show that the
standard calculation of these constraints cannot be applied on cosmological
scales, as is often done. Specifically, we find that the Shapiro delay computed
in the standard way (taking the Newtonian potential to vanish at infinity)
diverges as one includes many remote sources. We use an infinite homogeneous
lattice model to illustrate this divergence, and also show how the divergence
can be cured by using Fermi coordinates associated with an observer. With this
correction, one finds that the Shapiro delay is no longer monotonic with the
number of sources. Thus, one cannot compute a conservative lower bound on the
Shapiro delay using a subset of the sources of the gravitational field without
further assumptions and/or observational input. As an illustration, we compute
the Shapiro delay by applying the Fermi coordinate expression to two catalogs
of galaxy clusters, illustrating the dependence of the result on the
completeness of the catalogue and the mass estimates.
