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Modified gravity models with scale-dependent linear growth typically exhibitan enhancement in the power spectrum beyond a certain scale. The conventionalmethods for extracting cosmological information usually involve inferringmodified gravity effects via Redshift Space Distortions (RSD), particularlythrough the time evolution of $f\sigma_8$. However, classical galaxy RSDclustering analyses encounter difficulties in accurately capturing thespectrum's enhanced power, which is better obtained from the broad-band powerspectrum. In this sense, full-shape analyses aim to consider survey data usingcomprehensive and precise models of the whole power spectrum. Yet, a majorchallenge in this approach is the slow computation of non-linear loop integralsfor scale-dependent modified gravity, precluding the estimation of cosmologicalparameters using Markov Chain Monte Carlo methods. Based on recent studies, inthis work we develop a perturbation theory tailored for Modified Gravity, oranalogous scenarios introducing additional scales, such as in the presence ofmassive neutrinos. Our approach only needs the calculation of thescale-dependent growth rate $f(k,t)$ and the limit of the perturbative kernelsat large scales. We called this approximate technique as fk-Perturbation Theoryand implemented it into the code fkpt, capable of computing the redshift spacegalaxy power spectrum in a fraction of a second. We validate our modeling andcode with the $f(R)$ theory MG-GLAM and General Relativity NSeries sets ofsimulations. The code is available at https://github.com/alejandroaviles/fkpt

fkPT: Constraining scale-dependent modified gravity with the full-shape galaxy power spectrum