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Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant

Raphael Bousso, Joseph Polchinski

TL;DR

The work tackles the cosmological-constant problem by showing that four-form fluxes in quantum gravity are quantized and that a single flux yields too coarse a λ, but a large set of incommensurate fluxes can produce a dense discretuum of λ values. Through membrane nucleation (Brown-Teitelboim mechanism) in a multi-flux landscape, the dynamics can populate regions with λ in the observational window, while inflationary dynamics and Brownian inflaton motion address the potential empty-universe problem. Key contributions include explicit quantization relations (c = en/Z, F_0 = en/Z), a density-estimating framework for the discretuum, a M-theory compactification setup with hundreds of fluxes, and plausible cosmological scenarios (inflation, kicking, trapping) that yield entropy and density perturbations without fine-tuned microphysics. The findings suggest that a small but nonzero λ can arise dynamically in string/M-theory settings, potentially aided by anthropic selection, though moduli stabilization and detailed microphysics remain important open issues with practical implications for the early-universe cosmology.

Abstract

A four-form gauge flux makes a variable contribution to the cosmological constant. This has often been assumed to take continuous values, but we argue that it has a generalized Dirac quantization condition. For a single flux the steps are much larger than the observational limit, but we show that with multiple fluxes the allowed values can form a sufficiently dense `discretuum'. Multiple fluxes generally arise in M theory compactifications on manifolds with non-trivial three-cycles. In theories with large extra dimensions a few four-forms suffice; otherwise of order 100 are needed. Starting from generic initial conditions, the repeated nucleation of membranes dynamically generates regions with a cosmological constant in the observational range. Entropy and density perturbations can be produced.

Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant

TL;DR

The work tackles the cosmological-constant problem by showing that four-form fluxes in quantum gravity are quantized and that a single flux yields too coarse a λ, but a large set of incommensurate fluxes can produce a dense discretuum of λ values. Through membrane nucleation (Brown-Teitelboim mechanism) in a multi-flux landscape, the dynamics can populate regions with λ in the observational window, while inflationary dynamics and Brownian inflaton motion address the potential empty-universe problem. Key contributions include explicit quantization relations (c = en/Z, F_0 = en/Z), a density-estimating framework for the discretuum, a M-theory compactification setup with hundreds of fluxes, and plausible cosmological scenarios (inflation, kicking, trapping) that yield entropy and density perturbations without fine-tuned microphysics. The findings suggest that a small but nonzero λ can arise dynamically in string/M-theory settings, potentially aided by anthropic selection, though moduli stabilization and detailed microphysics remain important open issues with practical implications for the early-universe cosmology.

Abstract

A four-form gauge flux makes a variable contribution to the cosmological constant. This has often been assumed to take continuous values, but we argue that it has a generalized Dirac quantization condition. For a single flux the steps are much larger than the observational limit, but we show that with multiple fluxes the allowed values can form a sufficiently dense `discretuum'. Multiple fluxes generally arise in M theory compactifications on manifolds with non-trivial three-cycles. In theories with large extra dimensions a few four-forms suffice; otherwise of order 100 are needed. Starting from generic initial conditions, the repeated nucleation of membranes dynamically generates regions with a cosmological constant in the observational range. Entropy and density perturbations can be produced.

Paper Structure

This paper contains 17 sections, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Four-form quantization
  3. Four-form energetics
  4. Four-form quantization
  5. Discussion
  6. Multiple four-forms
  7. M Theory Compactification
  8. Small charges from large dimensions
  9. Cosmology
  10. The Brown-Teitelboim mechanism
  11. The empty universe problem
  12. Inflation and reheating
  13. Kicking the inflaton
  14. Trapping the inflaton
  15. Vacuum selection
  16. ...and 2 more sections

Figures (1)

  • Figure 1: The allowed values of the four-form energy density are given by the radius-squared of points in the grid, whose dimension is the number of four-forms $J$. The spacing in direction $i$ is $q_i$. The negative of the bare cosmological constant corresponds to a $(J-1)$-dimensional sphere, and cancellation is possible if there is at least one grid point sufficiently close to the sphere.