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We present the extension of the effective field theory framework to themildly non-linear scales. The effective field theory approach has beensuccessfully applied to the late time cosmic acceleration phenomenon and it hasbeen shown to be a powerful method to obtain predictions about cosmologicalobservables on linear scales. However, mildly non-linear scales need to beconsistently considered when testing gravity theories because a large part ofthe data comes from those scales. Thus, non-linear corrections to predictionson observables coming from the linear analysis can help in discriminating amongdifferent gravity theories. We proceed firstly by identifying the necessaryoperators which need to be included in the effective field theory Lagrangian inorder to go beyond the linear order in perturbations and then we construct thecorresponding non-linear action. Moreover, we present the complete recipe tomap any single field dark energy and modified gravity models into thenon-linear effective field theory framework by considering a general action inthe Arnowitt-Deser-Misner formalism. In order to illustrate this recipe weproceed to map the beyond-Horndeski theory and low-energy Horava gravity intothe effective field theory formalism. As a final step we derived the 4th orderaction in term of the curvature perturbation. This allowed us to identify thenon-linear contributions coming from the linear order perturbations which atthe next order act like source terms. Moreover, we confirm that the stabilityrequirements, ensuring the positivity of the kinetic term and the speed ofpropagation for scalar mode, are automatically satisfied once the viability ofthe theory is demanded at linear level. The approach we present here will allowto construct, in a model independent way, all the relevant predictions onobservables at mildly non-linear scales.

Tackling non-linearities with the effective field theory of dark energy and modified gravity