Periodic Reporting for period 1 - InflaBoot (Bootstrapping Inflationary Cosmology)
Berichtszeitraum: 2018-10-01 bis 2020-09-30
"Cosmology is the source, but also the adjudicator, of some of the biggest questions in physics. There is growing evidence that the very early universe underwent a period of rapid, approximately de Sitter (dS), inflationary expansion -- providing an elegant explanation for the primordial density fluctuations inferred from observations of the cosmic microwave
background and the large-scale structure of the universe. Since the energy scale of inflation could be as high as 10^14 Gev, far beyond the reach of our terrestrial experiments, this would make inflation the highest energy observable natural process. We are therefore presented with a remarkable opportunity to probe the laws of physics at these scales, which thus far have remained rather elusive.
In anticipation of increasingly precise observations it is imperative that we understand how to extract information about the laws of physics at such scales. While in recent years there have been significant theoretical advances in our understanding of cosmological correlation functions, we still find it challenging to isolate fundamental physics from them. In particular, we don’t yet have a firm grasp of the rules that govern cosmological correlators, so that, naively, it would appear that anything goes.
At the same time, the last few decades has seen significant advances in our understanding of gravitational observables. In particular, the striking notion that Quantum Gravity is “holographic” in nature -- in studying Quantum Gravity, we are led to consider holographic theories for the boundary observables at infinity. The most precise formulation of such a duality is the AdS/CFT correspondence, where boundary observables of Quantum Gravity in asymptotically anti-de Sitter (AdS) space are given by correlation functions of a Conformal Field Theory (CFT) in flat space-time. In this setting, the rules that govern gravitational observables in anti-de Sitter space are given by those of consistent CFTs, for which various powerful techniques have been developed in last decade within the framework of the “Conformal Bootstrap”.
By extending this understanding of observables in AdS space to dS, which are related by analytic continuation, this would provide key insights into understanding the ""rules of the game"" for cosmological correlators. This is the overall objective of the project. In particular, to extend Conformal Bootstrap techniques and results that so far have been tailored to physics in anti-de Sitter to physics in de Sitter space."
background and the large-scale structure of the universe. Since the energy scale of inflation could be as high as 10^14 Gev, far beyond the reach of our terrestrial experiments, this would make inflation the highest energy observable natural process. We are therefore presented with a remarkable opportunity to probe the laws of physics at these scales, which thus far have remained rather elusive.
In anticipation of increasingly precise observations it is imperative that we understand how to extract information about the laws of physics at such scales. While in recent years there have been significant theoretical advances in our understanding of cosmological correlation functions, we still find it challenging to isolate fundamental physics from them. In particular, we don’t yet have a firm grasp of the rules that govern cosmological correlators, so that, naively, it would appear that anything goes.
At the same time, the last few decades has seen significant advances in our understanding of gravitational observables. In particular, the striking notion that Quantum Gravity is “holographic” in nature -- in studying Quantum Gravity, we are led to consider holographic theories for the boundary observables at infinity. The most precise formulation of such a duality is the AdS/CFT correspondence, where boundary observables of Quantum Gravity in asymptotically anti-de Sitter (AdS) space are given by correlation functions of a Conformal Field Theory (CFT) in flat space-time. In this setting, the rules that govern gravitational observables in anti-de Sitter space are given by those of consistent CFTs, for which various powerful techniques have been developed in last decade within the framework of the “Conformal Bootstrap”.
By extending this understanding of observables in AdS space to dS, which are related by analytic continuation, this would provide key insights into understanding the ""rules of the game"" for cosmological correlators. This is the overall objective of the project. In particular, to extend Conformal Bootstrap techniques and results that so far have been tailored to physics in anti-de Sitter to physics in de Sitter space."
Work performed at the beginning of the project focused the analytic continuation of perturbative Feynman diagram techniques in AdS to dS. To this end a formalism was developed based on a Mellin-Barnes integral representation of conformal correlation functions in momentum space, that placed propagators and, in turn, correlators, in AdS and dS on an equal footing. With this result a dictionary that translates between boundary correlators in AdS and dS could be straightforwardly established, which we applied to compute 4pt exchange diagrams in de Sitter from their AdS counterparts. Although the expressions for the de Sitter exchanges were obtained through perturbative Feynman diagram techniques, the result provided significant guidance for how these results could have been obtained using symmetry and consistency considerations alone (i.e. a Bootstrap approach). In particular, the Mellin-Barnes representation made manifest how the result is fixed by a combination of factorisation, symmetries and boundary conditions. With this guidance, it was shown how exchanges could be obtained simply by solving the Conformal Casimir equation with certain boundary conditions, which was trivialised by the Mellin-Barnes representation.
Using this result, an identity that expresses any exchange in dS as a sum of exchanges in AdS could was written down. This is significant because AdS exchange diagrams have emerged as important building blocks in the Conformal Bootstrap programme and such an identity allows such techniques in the Conformal Bootstrap to be directly imported, for the first time, to dS. It also sheds light on the analytic structure of dS correlators and constraints from unitarity, which are implied from our (comparably firm) understanding of the analogous problem in AdS. Using this result, so far, we have defined Mellin amplitudes and the crossing decomposition of exchanges in dS from their AdS counterparts.
Regarding dissemination, these results have been invited to be presented at numerous major international conferences, as well as invited seminars at institutions in the United States, Europe and Asia.
Using this result, an identity that expresses any exchange in dS as a sum of exchanges in AdS could was written down. This is significant because AdS exchange diagrams have emerged as important building blocks in the Conformal Bootstrap programme and such an identity allows such techniques in the Conformal Bootstrap to be directly imported, for the first time, to dS. It also sheds light on the analytic structure of dS correlators and constraints from unitarity, which are implied from our (comparably firm) understanding of the analogous problem in AdS. Using this result, so far, we have defined Mellin amplitudes and the crossing decomposition of exchanges in dS from their AdS counterparts.
Regarding dissemination, these results have been invited to be presented at numerous major international conferences, as well as invited seminars at institutions in the United States, Europe and Asia.
This project provides the first concrete connection between the conformal bootstrap and cosmological bootstrap programmes. It is the first application of Conformal Bootstrap techniques to de Sitter correlators, obtaining new, completely general results for exchange diagrams in de Sitter space -- which were only available previously in specific cases -- together with new insights on their structure. It also developed a new formalism, a Mellin-Barnes integral representation, for conformal correlation functions in momentum space.