  Recent analyses of the cosmic microwave background (CMB) and the Lyman-alpha
forest indicate a mild preference for a deviation from a power law primordial
matter power spectrum (a so-called negative `running'). We use an extension to
the BAHAMAS suite of cosmological hydrodynamic simulations to explore the
effects that a running scalar spectral index has on large-scale structure
(LSS), using Planck CMB constraints to initialize the simulations. We focus on
5 key statistics: i) the non-linear matter power spectrum ii) the halo mass
function; iii) the halo two-point auto correlation function; iv) total mass
halo density profiles; and v) the halo concentration-mass relation. In terms of
the matter power spectrum, we find that a running scalar spectral index affects
all k-scales examined in this study, with a negative (positive) running leading
to an amplification (suppression) of power. These effects should be easily
detectable with upcoming surveys such as LSST and Euclid. In the mass range
sampled, a positive running leads to an increase in the mass of galaxy groups
and clusters, with the favoured negative running leading to a decrease in mass
of lower-mass (M <~ 10^13 M_solar) halos, but an increase for the most massive
(M >~ 10^13 M_solar) halos. Changes in the mass are generally confined to 5-10%
which, while not insignificant, cannot by itself reconcile the claimed tension
between the primary CMB and cluster number counts. We find that running does
not significantly affect the shapes of density profiles of matched halos,
changing only their amplitude. Finally, we demonstrate that the observed
effects on LSS due to a running scalar spectral index are separable from those
of baryonic effects to typically a few percent precision.
