  We numerically investigate the correlation function, the response and the
breakdown of the Fluctuation-Dissipation Theorem (FDT) in active particles
close to the motility-induced critical point. We find a strong FDT violation in
the short time and wavelength regime, where the response function has a larger
amplitude than the fluctuation spectrum. Conversely, at larger spatiotemporal
scales, the FDT is restored and the critical slowing-down is compatible with
the Ising universality class. Building on these results, we develop a novel
field-theoretical description employing a space-time correlated noise which
qualitatively captures the numerical results already at the Gaussian level. By
performing a one-loop renormalization group analysis we show that the
correlated noise does not change the critical exponents with respect to the
equilibrium. Our results demonstrate that a correlated noise field is a
fundamental ingredient to capture the features of critical active matter at the
coarse-grained level.
