Cosmography and the redshift drift in Palatini $f(\mathcal{R})$ theories

We present an application to cosmological models in $f(\mathcal{R})$ theories within the Palatini formalism of a method that combines cosmography and the explicit form of the field equations in the calculation of the redshift drift. The method yields a sequence of constraint equations which lead to limits on the parameter space of a given $f(\mathcal{R})$-model. Two particular families of $f(\mathcal{R})$-cosmologies capable of describing the current dynamics of the universe are explored here: (i) power law theories of the type $f({\cal R})={\cal R}-\beta /{\cal R}^n$,and(ii)theoriesoftheform$f({\cal R})={\cal R}+\alpha \ln{{\cal R}} -\beta$\mathrm{.}Theconstraintson$(n,\beta)$and$(\alpha,\beta)$, respectively, limit the values to intervals that are narrower than the ones previously obtained. As a byproduct, we show that when applied to General Relativity, the method yields values of the kinematic parameters with much smaller errors that those obtained directly from observations.