  Observations of the black hole shadow of supermassive black holes, such as
Sagittarius A* at the center of our Milky Way galaxy, allow us to study the
properties of black holes and the nature of strong-field gravity. According to
the Kerr hypothesis, isolated, stationary, and axisymmetric astrophysical black
holes are described by the Kerr metric. The Kerr hypothesis holds in General
Relativity and in some modified gravity theories, but there are others in which
it is violated. In principle, black hole shadow observations can be used to
determine if the Kerr metric is the correct description for black holes, and in
turn, they could be used to place constraints on modified gravity theories that
do not admit the Kerr solution. We here investigate whether black hole shadow
observations can constrain deviations from general relativity, focusing on two
well-motivated modified quadratic gravity theories:
Einstein-dilaton-Gauss-Bonnet gravity and dynamical Chern-Simons gravity. We
show that current constraints on Einstein-dilaton-Gauss-Bonnet gravity are
stronger than any that could be placed with black hole shadow observations of
supermassive black holes. We also show that the same holds for dynamical
Chern-Simons gravity through a systematic bias and a likelihood analysis when
considering slowly-rotating supermassive black holes. However, observations of
more rapidly-rotating black holes, with dimensionless spins
$|\vec{J}|/M^{2}\simeq0.5$, could be used to better constrain dynamical
Chern-Simons gravity.
