Monodromies and functional determinants in the conformal field theory driven quantum cosmology
Year: 2012
Type: article
Abstract: We apply the monodromy method for the calculation of the functional determinant of a special second-order differential operator $\mathbit{F}=\ensuremath{-}{d}^{2}/d{\ensuremath{\tau}}^{2}+\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{g}/g$, $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{g}\ensuremath{\equiv}{d}^{2}g/d{\ensuremath{\tau}}^{2}$, subject to periodic boundary conditions with a periodic zero mode $g=g(\ensuremath{\tau})$. This operator arises in applications of the early Universe theory and, in particular, determines the one-loop statistical sum for the microcanonical ensemble in cosmology generated by a conformal field theory (CFT). This ensemble realizes the concept of cosmological initial conditions by generalizing the notion of the no-boundary wave function of the Universe to the level of a special quasithermal state which is dominated by instantons with an oscillating scale factor of their Euclidean Friedmann-Robertson-Walker metric. These oscillations result in the multinode nature of the zero mode $g(\ensuremath{\tau})$ of $\mathbit{F}$, which is gauged out from its reduced functional determinant by the method of the Faddeev-Popov gauge-fixing procedure. The calculation is done for a general case of multiple nodes (roots) within the period range of the Euclidean time $\ensuremath{\tau}$, thus generalizing the previously known result for the single-node case of one oscillation of the cosmological scale factor. The functional determinant of $\mathbit{F}$ expresses in terms of the monodromy of its basis function, which is obtained in quadratures as a sum of contributions of time segments connecting neighboring pairs of the zero mode roots within the period range.
Authors: A. O. Barvinsky, D. V. Nesterov
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W136686555