Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

Abstract We study the small perturbations of the 1 + 3-dimensional Milne model for the Einstein–Klein–Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the 1 + 3 splitting of the Bianchi–Klein–Gordon equations. Moreover, we treat the Bianchi–Klein–Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi–Klein–Gordon equations with spatially covariant derivatives while normal derivative is not allowed. We propose some refined estimates for lapse and the hierarchies of energy estimates to close the energy argument.