Redshifting the Cosmological Constant in Unimodular Gravity via Nonlinear Quantum Mechanics

TL;DR

This paper tackles the cosmological constant problem by combining unimodular gravity with nonlinear quantum mechanics to de-gravitate vacuum energy. It introduces state-dependent Hamiltonian terms built from metric expectation values, notably an averaged Ricci scalar, which convert the shadow energy associated with unconstrained initial conditions into a decaying component that redshifts away as the universe expands. Consequently, late-time dynamics are governed by matter and radiation, with consistency checks showing compatibility with local gravity tests and cosmological observations; the framework also discusses potential tests via cosmological measurements of the gravitational constant and possible inflationary behavior. The authors also argue that this infrared modification can accommodate inflation and quintessence while remaining radiatively stable, though certain simple inflation models may be unstable in multi-world scenarios, motivating further model-building and observational exploration.

Abstract

The cosmological constant problem represents a profound conflict between quantum field theory and general relativity. Unimodular gravity offers a compelling starting point by de-gravitating the vacuum energy of the Standard Model, but this framework traditionally trades the problem of vacuum energy for a fine-tuning of initial conditions, which manifest as a ``shadow" cosmological constant. In this paper, we resolve this initial conditions problem by proposing a novel modification to gravity based on nonlinear quantum mechanics. We introduce specific state-dependent terms to the Hamiltonian, constructed from expectation values of the metric such as the average Ricci scalar. These terms alter the dynamical equations of gravity such that the shadow energy density associated with unconstrained initial conditions redshifts away with cosmic expansion, rendering it negligible at late times. The resulting cosmology is naturally dominated by matter and radiation without fine-tuning. We demonstrate that this significant infrared modification of gravity is consistent with local and cosmological tests of gravity. We comment on the possibility of testing this solution in cosmological measurements of Newton's constant.

Figures (9)

• Figure 1: The analytic solution for $w = 0$ with $\kappa = -2$, $\bar{\kappa} = -0.025$. The solutions are expanded around $a_0 \approxeq 0.84$ and $b_0 \approxeq 0.0025$ with $\alpha = 1/3.$
• Figure 2: The numerical solution for $w = 0$ with $\alpha = 1/3$, $\kappa = -2$, $\bar{\kappa} = -0.025$ plotted as the ratio of the scale factors $a/b$. The initial conditions for the metric and matter are perturbed around 10 percent. Instead of unimodular time, we show the redshift $z$ that the solution captures. Note that the ratio a/b that the solution asymptotes to is the ratio of $a_0/b_0$ around which the analytic solution is expanded in figure \ref{['fig:analyticw0']}.
• Figure 3: The analytic solution for $w = 1/3$ with $\kappa = -0.5$, $\bar{\kappa} = -0.025$. The solutions are expanded around $a_0 \approxeq 2.33$ and $b_0 \approxeq 0.001$ with $\alpha = 1/10.$
• Figure 4: The numerical solution for $w = 1/3$ with $\alpha = 1/10$, $\kappa = -1/2$, $\bar{\kappa} = -0.025$ plotted as the ratio of the scale factors $a/b$. The initial conditions for the metric and matter are perturbed around 10 percent. Instead of unimodular time, we show the redshift $z$ that the solution captures. The ratio $a/b$ slowly asymptotes to the stable ratio around which the analytic solution in figure \ref{['fig:analyticwr']} is expanded around.
• Figure 5: The analytic solution for $w = -0.9$ with $\kappa = -2$, $\bar{\kappa} = -0.025$. The solutions are expanded around $a_0 \approxeq 0.053$ and $b_0 \approxeq 0.001$ with $\alpha = 1/3.$