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Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

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Abstract

Milne-like spacetimes are a class of \(k = -1\) FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper Ling (Found. of Phys. 50:385–428, 2020), it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong spatially isotropic assumption associated with them. We show how our results yield a “quasi de Sitter” expansion for the early universe which could have applications to inflationary scenarios.

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Notes

  1. These extensions have been noted previously in the physics literature, see e.g. [9].

  2. Using similar arguments as in [15, Appendix B], one can show that Milne-like spacetimes actually admit Lipschitz spacetime extensions through the big bang. This should be compared with the results in [37].

  3. See [14, Thm. 2.6] for a proof. The proof generalizes to the class of timelike curves considered in this paper since it only uses the openness of \(I^\pm \) which follows from [29, Thm. 2.12]. Moreover, the “topological hypersurface” part of the conclusion follows from [29, Thm. A.6].

  4. Our convention for the mean curvature H, which includes the 1/3 factor in front of \(\text {tr}(K)\), coincides with the Hubble parameter, \(a'/a\), which is also denoted by H in the physics literature.

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Acknowledgements

The author gratefully acknowledges being supported by the Harold H. Martin Postdoctoral Fellowship at Rutgers University. He thanks Greg Galloway for many helpful comments and pointing out references [10, 11]. He thanks two anonymous reviewers who greatly improved the quality of the paper. Lastly, he thanks the organizers of Singularity theorems, causality, and all that; a tribute to Roger Penrose for putting together a stimulating conference.

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  1. Fields Institute University of Toronto, Toronto, Canada

    Eric Ling

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Ling, E. Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes. Gen Relativ Gravit 54, 68 (2022). https://doi.org/10.1007/s10714-022-02955-y

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