  We present a large suite of cosmological simulations, the FORGE (F-of-R
Gravity Emulator) simulation suite, which is designed to build accurate
emulators for cosmological observables in galaxy clustering, weak gravitational
lensing and galaxy clusters, for the $f(R)$ gravity model. The total of 200
simulations explore the cosmological parameter space around the Planck(2018)
cosmology with a Latin hypercube, for 50 combinations of $\bar{f}_{R0}$,
$\Omega_m$, $\sigma_8$ and $h$ with all other parameters fixed. For each
parameter combination, or node, we ran four independent simulations, one pair
using $1024^3$ particles in $500 h^{-1} Mpc$ simulation boxes to cover small
scales, and another pair using $512^3$ simulation particles in $1500 h^{-1}
Mpc$ boxes for larger scales. Each pair of initial conditions are selected such
that sample variance on large scales is minimised on average. In this work we
present an accurate emulator for the matter power spectrum in $f(R)$ gravity
trained on FORGE. We have verified, using the cross-validation technique, that
the emulator accuracy is better than $2.5\%$ for the majority of nodes,
particularly around the center of the explored parameter space, up to scales of
$k = 10 h Mpc^{-1}$. We have also checked the power spectrum emulator against
simulations which are not part of our training set and found excellent
agreement. Due to its high accuracy on small scales, the FORGE matter power
spectrum emulator is well suited for weak lensing analysis and can play a key
tool in constraining $f(R)$ gravity using current and future observational
data.
