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In this work we study the cosmological perturbations inBahamonde-Dialektopoulos-Levi Said (BDLS) theory, i.e. the teleparallel analogof Horndeski gravity. In order to understand the evolution of structure in acosmological model, it is necessary to study its cosmology not only in thebackground but also perturbatively. Both Horndeski and its teleparallel analoghave been analyzed a lot in the literature, but in order to study themquantitatively, we need to know their cosmological perturbations. That is why,we study here the scalar-vector-tensor decomposition of the theory and we alsoexpress the so-called alpha parameters in terms of the arbitrary functions ofthe theory, that designate the deviation from the {\Lambda}CDM model. We haveexplored tensor, vector and scalar perturbation of the action up to secondorder, which drastically opens up new possibilities on searches in theparameter space of scalar-tensor theories in the context of observations.

Cosmological Perturbations in the Teleparallel analog of Horndeski gravity