Correspondence between strings in the Hagedorn phase and asymptotically de Sitter space

A correspondence between closed strings in their high-temperature Hagedorn phase and asymptotically de Sitter (dS) space is established. We identify a thermal, conformal field theory (CFT) whose partition function is, on the one hand, equal to the partition function of closed, interacting, fundamental strings in their Hagedorn phase yet is, on the other hand, also equal to the Hartle-Hawking (HH) wave function of an asymptotically dS universe. The Lagrangian of the CFT is a functional of a single scalar field, the condensate of a thermal scalar, which is proportional to the entropy density of the strings. The correspondence has some aspects in common with the anti–de Sitter/CFT correspondence, as well as with some of its proposed analytic continuations to a dS/CFT correspondence, but it also has some important conceptual and technical differences. The equilibrium state of the CFT is one of maximal pressure and entropy, and it is at a temperature that is above but parametrically close to the Hagedorn temperature. The CFT is valid beyond the regime of semiclassical gravity and thus defines the initial quantum state of the dS universe in a way that replaces and supersedes the HH wave function. Two-point correlation functions of the CFT scalar field are used to calculate the spectra of the corresponding metric perturbations in the asymptotically dS universe and, hence, cosmological observables in the postinflationary epoch. Similarly, higher-point correlation functions in the CFT should lead to more complicated cosmological observables.