In this work we focus on the evolution of the linear perturbations in the novel hybrid metric-Palatini theory achieved by adding a f ( \mathcal { R } ) function to the gravitational action .
Working in the Jordan frame , we derive the full set of linearized evolution equations for the perturbed potentials and present them in the Newtonian and synchronous gauges .
We also derive the Poisson equation , and perform the evolution of the lensing potential , \Phi _ { + } , for a model with a background evolution indistinguishable from \Lambda CDM .
In order to do so , we introduce a designer approach that allows to retrieve a family of functions f ( \mathcal { R } ) for which the effective equation of state is exactly w _ { \textrm { eff } } = -1 .
We conclude , for this particular model , that the main deviations from standard General Relativity and the Cosmological Constant model arise in the distant past , with an oscillatory signature in the ratio between the Newtonian potentials , \Phi and \Psi .