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Late-Time Correlators and Complex Geodesics in de Sitter Space
by Lars Aalsma, Mir Mehedi Faruk, Jan Pieter van der Schaar, Manus Visser, Job de Witte
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|Authors (as registered SciPost users):||Lars Aalsma|
|Preprint Link:||scipost_202212_00065v2 (pdf)|
|Date submitted:||2023-03-30 20:07|
|Submitted by:||Aalsma, Lars|
|Submitted to:||SciPost Physics|
We study two-point correlation functions of a massive free scalar field in de Sitter space using the heat kernel formalism. Focusing on two operators in conjugate static patches we derive a geodesic approximation to the two-point correlator valid for large mass and at late times. This expression involves a sum over two complex conjugate geodesics that correctly reproduces the large-mass, late-time limit of the exact two-point function in the Bunch-Davies vacuum. The exponential decay of the late-time correlator is associated to the timelike part of the complex geodesics. We emphasize that the late-time exponential decay is in tension with the finite maximal entropy of empty de Sitter space, and we briefly discuss how non-perturbative corrections might resolve this paradox.
Published as SciPost Phys. 15, 031 (2023)
List of changes
We made the following changes to address the referee's comments:
1. We modified the referencing to the paper of Fischetti, Marolf and Wall to make it clear that this was the first paper to point out the role of complex geodesics in correlation functions in de Sitter. We added the sentence ``It was ... higher dimensions.'' near the end of p2.
2. We modified this sentence to ``In addition ... in cosmology.'' to stress that the status of islands in de Sitter space is quite different than for AdS black holes.
3. We added additional explanation around (2.31) to clarify the relation between the integral presented in (2.32), which we used to define the geodesic distance, and the standard expression for the geodesic distance given in (2.31).
4. We improved the explanation the small s-expansion. See the modified paragraph on p10: ``Now $F(x,y;s)$ ... condition (2.27).'' We also added footnote 4 to make it clear that higher powers of s in the heat kernel lead to terms that are suppressed by inverse powers of mass in the propagator.
5. In light of the improved explanation of the small s-expansion we removed this sentence.
6. We added an explanation of this fact in footnote 4.
7. We added footnote 6 to make clear that we are focusing on the Bunch-Davies vacuum and do not consider alpha-vacua.
8. We agree with the referee. In the previous version this statement referred to the wrong equation. We changed the sentence ``It is ... over geodesics'' to refer to (3.1), which does not correspond to a large proper time expansion.
9. We modified Fig. 6 and 7 and distinguished between the Euclidean time coordinate t_E used in Fig. 6 and the Euclidean angle in Fig. 7. The relationship between these two (a priori different) quantities is given around (3.22).
10. We significantly changed the discussion section: ``Assuming the ... firmer footing''. We clarified the difference between the results for the AdS eternal black hole and de Sitter, adding some references to that effect. In addition we clarified our (admittedly) speculative remarks regarding potential non-perturbative corrections that could affect the late-time correlator in de Sitter, including the Euclidean de Sitter black holes instantons. To that effect we also added an additional reference and emphasize that most of the current intuition is based on results in two (or at most three) dimensions.
1. We agree with the referee that knowing the exact answer was useful to understand which geodesics to include. Nonetheless, the geodesics that need to be included can be determined by a saddle point approximation. We did not show this explicitly in our manuscript, but have added a reference to the paper  where this computation was performed explicitly. We added this explanation in the paragraph: ``We would ... is unknown.'' on p16.
2. To add more support to our claim that the geodesic approximation gives a good approximation of the exact result we added Appendix B. Here, we give additional numerical evidence that the difference between the approximation and exact result goes to zero in the limit when the approximation becomes better, as suggested by the referee. In addition, we'd like to stress that our main result - complex geodesics reproducing the exact correlator in the limit of large mass and proper time - is an analytical result. By taking a limit of the exact Wightman function we showed analytically that the correlator involves a sum over complex geodesics. To clarify this point, we added the sentence: ``On top ... Appendix B.'' on p16.
Submission & Refereeing History
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