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We analyze the mode stability of odd-parity perturbations of black holes withlinearly time-dependent scalar hair in shift-symmetric Horndeski theories. Weshow that a large class of black hole solutions in these theories suffer fromghost or gradient instability, while there are some classes of solutions thatare stable under linear odd-parity perturbations in the context of modeanalysis.

Linear perturbation analysis of hairy black holes in shift-symmetric Horndeski theories: Odd-parity perturbations