We consider a time-dependent deformation of anti–de Sitter (AdS) space-time which contains a spacelike “singularity”—a spacelike region of high curvature. Making use of the AdS/CFT correspondence we can map the bulk dynamics onto the boundary. The boundary theory has a time dependent coupling constant which becomes small at times when the bulk space-time is highly curved. We investigate the propagation of small fluctuations of a test scalar field from early times before the bulk singularity to late times after the singularity. Under the assumption that the AdS/CFT correspondence extends to deformed AdS space-times, we can map the bulk evolution of the scalar field onto the evolution of the boundary gauge field. The time evolution of linearized fluctuations is well-defined in the boundary theory as long as the coupling remains finite, so that we can extend the boundary perturbations to late times after the singularity. Assuming that the spacetime in the future of the singularity has a weakly coupled region near the boundary, we reconstruct the bulk fluctuations after the singularity crossing making use of generic properties of boundary-to-bulk propagators. Finally, we extract the spectrum of the fluctuations at late times given some initial spectrum. We find that the spectral index is unchanged, but the amplitude increases due to the squeezing of the fluctuations during the course of the evolution. This investigation can teach us important lessons on how the spectrum of cosmological perturbations passes through a bounce which is singular from the bulk point of view but which is resolved using an ultraviolet complete theory of quantum gravity.
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During the course of the project we benefited from support by grants from FQXi and from the Mathematical Physics Laboratory of the CRM in Montreal. We gratefully acknowledge support from the CRM during a recent workshop during which this project was completed. The work at McGill has been supported by an NSERC Discovery Grant, and by funds from the Canada Research Chair program. S. D. acknowledges support from the US National Science Foundation under Grants No. NSF-PHY-1214341 and No. NSF-PHY-1521045. E. F. would like to thank CNPq (Science without Borders) for financial support. Y. F. C. is supported in part by the National Youth Thousand Talents Program and by the USTC start-up funding under Grant No. KY2030000049. Y. W. is supported by Grant No. HKUST4/CRF/13G issued by the Research Grants Council (RGC) of Hong Kong. R. B. was supported by a Simons Foundation fellowship and by a Senior Fellowship at the Institute for Theoretical Studies of the ETH Zurich, with support provided by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zurich Foundation.
Brandenberger, Robert H.; Ferreira, Elisa G. M.; Morrison, Ian A.; Cai, Yi-Fu; Das, Sumit R.; and Wang, Yi, "Fluctuations in a Cosmology with a Spacelike Singularity and Their Gauge Theory Dual Description" (2016). Physics and Astronomy Faculty Publications. 427.