Constraining alternatives to the Kerr black hole

The recent observation of the shadow of the supermassive compact object M87$^*$ by the Event Horizon Telescope (EHT) collaboration has opened up a new window to probe the strong gravity regime. In this paper, we study shadows cast by two viable alternatives to the Kerr black hole, and compare them with the shadow of M87$^{*}$. The first alternative is a horizonless compact object (HCO) having radius $r_0$ and exterior Kerr geometry. The second one is a rotating generalisation of the recently obtained one parameter ($r_0$) static metric by Simpson and Visser. This latter metric, constructed by using the Newman-Janis algorithm, is a special case of a parametrised rotating non-Kerr geometry obtained by Johannsen. Here, we constrain the parameter $r_0$ of these alternatives using the results from M87$^{*}$ observation. We find that, for the mass, inclination angle and the angular diameter of the shadow of M87$^{*}$ reported by the EHT collaboration, the maximum value of the parameter $r_0$ must be in the range $2.54r_{+}\leq r_{0,max}\leq 3.51r_{+}$ for the dimensionless spin range $0.5\leq a_{*}\leq 0.94$, with $r_{+}$ being the outer horizon radius of the Kerr black hole at the corresponding spin value. We conclude that these black hole alternatives having $r_0$ below this maximum range (i.e. $r_0\leq r_{0,max}$) is consistent with the size and deviation from circularity of the observed shadow of M87$^{*}$.