Probing cosmological singularities with quantum fields: Open and closed FLRW universes

It was recently pointed out that linear quantum fields φ(x) can be meaningfully propagated across the big bang (and the big crunch) singularities of spatially flat Friedmann, Lemaître, Robertson, Walker (FLRW) universes [A. Ashtekar, Adv. Theor. Math. Phys. 25, 1651 (2021)1095-076110.4310/ATMP.2021.v25.n7.a1]. Recall that φ(x), as well as renormalized observables φ2(x)ren and Tab(x)ren, are distribution valued already in Minkowskian quantum field theories. It was shown that they can be extended as well-defined distributions even when these spacetimes are enlarged to include the big bang (or the big crunch). We generalize these results to spatially closed and open FLRW models, showing that this "tameness"of cosmological singularities is not an artifact of the technical simplifications due to spatial flatness. Our analysis also provides explicit expressions of φ(x)φ(x′),φ2(x)ren and Tab(x)ren in closed and open universes for minimally coupled massless scalar fields and discuss the ambiguities in the definition of Tab(x)ren at the big bang. While the technical expressions are more complicated than in the spatially flat case, there is also an unexpected conceptual simplification: the infrared divergence [L. H. Ford and L. Parker, Phys. Rev. D 16, 245 (1977)PRVDAQ0556-282110.1103/PhysRevD.16.245] is now absent because, in effect, the spatial curvature provides a natural cutoff. Finally, we further clarify the sense in which quantum field theory can continue to be well defined even though the extended spacetime is not globally hyperbolic because of the singularity, and suggest directions for further work.