A general framework to test gravity using galaxy clusters III:
  Observable-mass scaling relations in $f(R)$ gravity
  We test two methods, including one that is newly proposed in this work, for
correcting for the effects of chameleon $f(R)$ gravity on the scaling relations
between the galaxy cluster mass and four observable proxies. Using the first
suite of cosmological simulations that simultaneously incorporate both full
physics of galaxy formation and Hu-Sawicki $f(R)$ gravity, we find that these
rescaling methods work with a very high accuracy for the gas temperature, the
Compton $Y$-parameter of the Sunyaev-Zel'dovich (SZ) effect and the X-ray
analogue of the $Y$-parameter. This allows the scaling relations in $f(R)$
gravity to be mapped to their $\Lambda$CDM counterparts to within a few
percent. We confirm that a simple analytical tanh formula for the ratio between
the dynamical and true masses of haloes in chameleon $f(R)$ gravity, proposed
and calibrated using dark-matter-only simulations in a previous work, works
equally well for haloes identified in simulations with two very different --
full-physics and non-radiative -- baryonic models. The mappings of scaling
relations can be computed using this tanh formula, which depends on the halo
mass, redshift and size of the background scalar field, also at a very good
accuracy. Our results can be used for accurate determination of the cluster
mass using SZ and X-ray observables, and will form part of a general framework
for unbiased and self-consistent tests of gravity using data from present and
upcoming galaxy cluster surveys. We also propose an alternative test of
gravity, using the $Y_{\rm X}$-temperature relation, which does not involve
mass calibration.
