Constraining the Cardoso-Pani-Rico metric with future observations of SgrA$^*$

SgrA$^*$, the supermassive black hole candidate at the center of our Galaxy, seems to be one of the most promising object to test the Kerr black hole hypothesis with near future observations. In a few years, it will hopefully be possible to measure a number of relativistic effects around this body, and the combination of different observations can be used to constrain possible deviations from the Kerr solution. In this paper, I discuss the combination of three promising techniques in the framework of the Cardoso-Pani-Rico parametrization: the observation of blobs of plasma orbiting near the innermost stable circular orbit, the detection of the black hole shadow, and timing observations of a radio pulsar in a compact orbit. The observations of blobs of plasma and of the shadow can probe the strong gravitational field around SgrA$^*$, while the radio pulsar would be sensitive to the weak field region at larger radii. In the case of a fast-rotating object, the combination of the three measurements could provide strong constraints on the actual nature of SgrA$^*$. For a non-rotating or slow-rotating object, the bounds would be weak.