  We study the effects of two popular modified gravity theories, which
incorporate very different screening mechanisms, on the angular power spectra
of the thermal (tSZ) and kinematic (kSZ) components of the Sunyaev-Zel'dovich
effect. Using the first cosmological simulations that simultaneously
incorporate both screened modified gravity and a complete galaxy formation
model, we find that the tSZ and kSZ power spectra are significantly enhanced by
the strengthened gravitational forces in Hu-Sawicki $f(R)$ gravity and the
normal-branch Dvali-Gabadadze-Porrati model. Employing a combination of
non-radiative and full-physics simulations, we find that the extra baryonic
physics present in the latter acts to suppress the tSZ power on angular scales
$l\gtrsim3000$ and the kSZ power on all tested scales, and this is found to
have a substantial effect on the model differences. Our results indicate that
the tSZ and kSZ power can be used as powerful probes of gravity on large
scales, using data from current and upcoming surveys, provided sufficient work
is conducted to understand the sensitivity of the constraints to baryonic
processes that are currently not fully understood.
