Kourosh Nozari, Milad Hajebrahimi, Sara Saghafi, G. Mustafa, Emmanuel N. Saridakis

The Event Horizon Telescope (EHT) image of M87* provides a direct test of strong-field gravity, measuring an angular shadow diameter $θ_d = 42 \pm 3~μ\mathrm{as}$ and a circularity deviation $ΔC \leq 0.1$. Such observations allow quantitative tests of the Kerr paradigm and of possible deviations from the no-hair theorem. In scalar-tensor extensions of gravity, black holes may possess primary scalar hair, introducing an additional independent parameter beyond mass and spin. In this work, we construct rotating black hole solutions with primary scalar hair in beyond Horndeski gravity and analyze their photon regions and shadow formation. We show that the scalar hair parameter $Q$ induces characteristic modifications of the shadow, and in particular negative $Q$ enlarges the shadow and reduces its oblateness, while positive $Q$ shrinks and enhances its distortion. Modeling M87* within this framework and imposing the EHT bounds on $θ_d$ and $ΔC$, we determine the viable $(a,Q)$ parameter space. We find that current observations do not exclude rotating black holes with primary scalar hair, although the allowed region is significantly restricted for $Q>0$. Finally, the scalar-hair-induced deviations are of order $\mathcal{O}(μ\mathrm{as})$, placing them near the sensitivity threshold of present instruments and within reach of next-generation horizon-scale imaging.