  Observations of the continuum spectrum emitted by accretion disks around
black holes allows us to infer their properties, including possibly whether
black holes are described by the Kerr metric. Some modified gravity theories do
not admit the Kerr metric as a solution, and thus, continuum spectrum
observations could be used to constrain these theories. We here investigate
whether current and next generation X-Ray observations of the black hole
continuum spectrum can constrain such deviations from Einstein's theory,
focusing on two well-motivated modified quadratic gravity theories: dynamical
Chern-Simons gravity and Einstein-dilaton-Gauss-Bonnet gravity. We do so by
determining whether the non-Kerr deviations in the continuum spectrum
introduced by these theories are larger than the observational error intrinsic
to the observations. We find that dynamical Chern-Simons gravity cannot be
constrained better than current bounds with current or next generation
continuum spectrum observations. Einstein-dilaton-Gauss-Bonnet gravity,
however, may be constrained better than current bounds with next generation
telescopes, as long as the systematic error inherent in the accretion disk
modeling is decreased below the predicted observational error.
