Rotating Black Holes with Primary Scalar Hair: Shadow Signatures in Beyond Horndeski Gravity

The Event Horizon Telescope (EHT) image of M87* provides a direct test of strong-field gravity, measuring an angular shadow diameter $\theta_{d}=42\pm 3~\mu\mathrm{as}$ and a circularity deviation $\Delta C\leq 0.1$. 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 a rotating configuration inspired by 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. Adopting M87* as a representative case within this framework and imposing the EHT bounds on $\theta_{d}$ and $\Delta C$, we identify 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 $\mathcal{O}(\mu\mathrm{as})$, placing them near the sensitivity threshold of present instruments and within reach of next-generation horizon-scale imaging.