Constraints on Equivalence Principle Violation from Gamma Ray Bursts
  Theories of gravity that obey the Weak Equivalence Principle have the same
Parametrised Post-Newtonian parameter $\gamma$ for all particles at all
energies. The large Shapiro time delays of extragalactic sources allow us to
put tight constraints on differences in $\gamma$ between photons of different
frequencies from spectral lag data, since a non-zero $\Delta \gamma$ would
result in a frequency-dependent arrival time. The majority of previous
constraints have assumed that the Shapiro time delay is dominated by a few
local massive objects, although this is a poor approximation for distant
sources. In this work we consider the cosmological context of these sources by
developing a source-by-source, Monte Carlo-based forward model for the Shapiro
time delays by combining constrained realisations of the local density field
using the Bayesian origin reconstruction from galaxies algorithm with
unconstrained large-scale modes. Propagating uncertainties in the density field
reconstruction and marginalising over an empirical model describing other
contributions to the time delay, we use spectral lag data of Gamma Ray Bursts
from the BATSE satellite to constrain $\Delta \gamma < 2.1 \times 10^{-15}$ at
$1 \sigma$ confidence between photon energies of $25 {\rm \, keV}$ and $325
{\rm \, keV}$.
