  The dispersion measure (DM) of fast radio bursts (FRBs) provides a unique way
to probe ionised baryons in the intergalactic medium (IGM). Cosmological models
with different parameters lead to different DM-redshift ($\mathrm{DM}-z$)
relations. Additionally, the over/under-dense regions in the IGM and the
circumgalactic medium of intervening galaxies lead to scatter around the mean
$\mathrm{DM}-z$ relations. We have used the Evolution and Assembly of GaLaxies
and their Environments (EAGLE) simulations to measure the mean $\mathrm{DM}-z$
relation and the scatter around it using over one billion lines-of-sight
between redshifts $0<z<3$. We investigated two techniques to estimate
line-of-sight DM: `pixel scrambling' and `box transformations'. We find that
using box transformations (a technique from the literature) causes strong
correlations due to repeated replication of structure. Comparing a linear and
non-linear model, we find that the non-linear model with cosmological
parameters, provides a better fit to the $\mathrm{DM}-z$ relation. The
differences between these models are the most significant at low redshifts
($z<0.5$). The scatter around the $\mathrm{DM}-z$ relation is highly
asymmetric, especially at low redshift $\left(z<0.5\right)$, and becomes more
Gaussian as redshift approaches $z\sim3$, the limit of this study. The increase
in Gaussianity with redshift is indicative of the large scale structures that
is better probed with longer lines-of-sight. The minimum simulation size
suitable for investigations into the scatter around the $\mathrm{DM}-z$
relation is 100~comoving~Mpc. The $\mathrm{DM}-z$ relation measured in EAGLE is
available with an easy-to-use python interface in the open-source FRB redshift
estimation package FRUITBAT.
