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The possibility of testing gravity theories with the help of gravitationalwave detections has become an interesting arena of recent research. In thispaper, we follow this direction by investigating the quasinormal modes (QNMs)of the axial perturbations for charged black holes in the Palatini-typetheories of gravity, specifically ($i$) the Palatini $f(R)$ gravity coupledwith Born-Infeld nonlinear electrodynamics and ($ii$) theEddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwellelectromagnetic fields. The coupled master equations describing perturbationsof charged black holes in these theories are obtained with the tetradformalism. By using the Wentzel-Kramers- Brillouin (WKB) method up to 6thorder, we calculate the QNM frequencies of the EiBI charged black holes, theEinstein-Born-Infeld black holes, and the Born-Infeld charged black holeswithin the Palatini $R+\alpha R^2$ gravity. The QNM spectra of these blackholes would deviate from those of the Reissner-Nordstr\"om black hole. Inaddition, we study the QNMs in the eikonal limit and find that for the axialperturbations of the EiBI charged black holes, the link between the eikonalQNMs and the unstable null circular orbit around the black hole is violated.

Probing Palatini-type gravity theories through gravitational wave detections via quasinormal modes