The strength and tensor structure of the Higgs boson's interactions are investigated using an effective Lagrangian, which introduces additional CP-even and CP-odd interactions that lead to changes in the kinematic properties of the Higgs boson and associated jet spectra with respect to the Standard Model. The parameters of the effective Lagrangian are probed using a fit to five differential cross sections previously measured by the ATLAS experiment in the H ->gamma gamma decay channel with an integrated luminosity of 20.3fb(-1) at root s= 8TeV. Inorder to perform a simultaneous fit to the five distributions, the statistical correlations between them are determined by re-analysing the H ->gamma gamma candidate events in the proton-proton collision data. No significant deviations from the Standard Model predictions are observed and limits on the effective Lagrangian parameters are derived. The statistical correlations are made publicly available to allow for future analysis of theories with non-Standard Model interactions. (C) 2015 CERN for the benefit of the ATLAS Collaboration. Published by Elsevier B.V