Abstract
We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.
Highlights
We find that the net integrated curvature including variations of the warp and conformal factors can be tuned small enough to enable the quantum Casimir energy to compete in the volume stabilization mechanism, illustrating this with back reacted solutions in large patches of the space
We obtained de Sitter and inflation models from hyperbolic compactifications of Mtheory. These have an effective potential (3.4) whose leading term arises from the negative curvature g(7)(−R(7) + . . . ) =
The internal curvature combines with flux, Casimir energy, and warping effects in Ve f f in a comprehensive mechanism to stabilize the internal dimensions, supporting accelerated expansion of the universe
Summary
The observational discovery of the accelerated expansion of the universe demands a much more complete theoretical understanding. We find that the net integrated curvature including variations of the warp and conformal factors can be tuned small enough to enable the quantum Casimir energy to compete in the volume stabilization mechanism, illustrating this with back reacted solutions in large patches of the space. Via a simple comparison of two integrated combinations of the equations of motion, we derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters These results indicate metastable de Sitter solutions, including examples fairly close to our fiducial hyperbolic metric in the bulk of the internal space. This constitutes a set of nonlinear partial differential equations (PDEs) for deformations {A( y), B( y), h( y), C6}, including deviations from the fiducial hyperbolic metric (1.1) This problem is well posed, as many concrete examples of appropriate finite volume hyperbolic spaces are known explicitly [15,25–27]. After explaining our framework in the bulk of the paper, we will comment on other future directions and applications of our results to aspects of observational cosmology and de Sitter quantum gravity, along with connections to mathematics
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