Power law inflation with electromagnetism

We generalize Ringström's global future causal stability results (Ringström 2009) [11] for certain expanding cosmological solutions of the Einstein-scalar field equations to solutions of the Einstein-Maxwell-scalar field system. In particular, after noting that the power law inflationary spacetimes (Mn+1,gˆ,ϕˆ) considered by Ringström (2009) in [11] are solutions of the Einstein-Maxwell-scalar field system (with exponential potential) as well as of the Einstein-scalar field system (with the same exponential potential), we consider (nonlinear) perturbations of initial data sets of these spacetimes which include electromagnetic perturbations as well as gravitational and scalar perturbations. We show that if (as in Ringström (2009) [11]) we focus on pairs of relatively scaled open sets UR0⊂U4R0 on an initial slice of (Mn+1,gˆ), and if we choose a set of perturbed data which on U4R0 is sufficiently close to that of (Mn+1,gˆ,ϕˆ,Aˆ=0), then in the maximal globally hyperbolic spacetime development (Mn+1,g,ϕ,A) of this data via the Einstein-Maxwell-scalar field equations, all causal geodesics emanating from UR0 are future complete (just as in (Mn+1,gˆ)). We also verify that, in a certain sense, the future asymptotic behavior of the fields in the spacetime developments of the perturbed data sets does not differ significantly from the future asymptotic behavior of (Mn+1,gˆ,ϕˆ,Aˆ=0).