Effects of f(R) Gravity on the Gravitational Bound States of Ultra-cold Neutrons: A Laboratory Scale Detection Technique for Dark Energy?

Abstract

f(R) theories are the simplest generalisations of General Relativity, Einstein's theory of gravity. f(R) theories, and modified gravity theories more broadly, have been invoked to explain the accelerated expansion of the universe. A number of “viable” models have been proposed which satisfy cosmological tests and get around present day laboratory and solar system constraints by invoking the chameleon mechanism. It is important to determine whether such alternative models can be distinguished by precision experiments at the laboratory scale. This thesis focuses on recently proposed f(R) models in the context of the upcoming neutron quantum bouncer experiment GRANIT, which has the potential to directly observe violations of Einstein gravity. This experiment works by precisely measuring the energy levels of ultra-cold neutrons trapped in the Earth's gravitational field above a mirror. Neutrons in these states have energies of the order of 1 0 -12 eV (peV) and wavefunctions with spatial extents of ∼ 10 μ m . The size of the neutron wavefunctions gives gravitational spectroscopy sensitivity to any change of the Newtonian potential above the mirror surface on a scale of tens of microns. These experiments have a unique opportunity to study gravity and quantum mechanics in an unprecedented regime. GRANIT claims an energy sensitivity of 0.01 peV with an improvement of one or two orders of magnitude expected over the next several years. The theoretical limit to the sensitivity is 1 0 -7 peV . Previous work by Brax and Pignol has shown that scalar-tensor theories, which are closely related to f(R) theories, can give rise to detectable shifts in the energy levels from their Newtonian values.

Here the energy levels shifts are examined for three viable f(R) models: the exponential gravity model of Linder, the model of Hu and Sawicki and the “new” exponential gravity model of Xu and Chen. It is found that exponential gravity predicts energy level shifts of the order exp ( -1 0 19 ) m N c 2 , where m N is the neutron mass. This is unobservable in any experiment lasting less than the age of the universe. The Hu-Sawicki model predicts much larger shifts which depend on the model parameters n and c 2 , where n, c 2 >0 . It is found that the shifts are below the theoretical sensitivity limit for n ≥ 1 2 , with shifts at n= 1 2 of the order of 1 0 -19 peV . It is unlikely, though not yet ruled out, that n < 1 models could be observed at neutron bouncer experiments operating under extreme conditions. The computation of the shifts for n < 1 models is hampered by convergence issues. This may be immaterial, however, because Hu-Sawicki models with n<2 are not compatible with cosmological inflation in the early universe. The Xu-Chen model is shown to be indistinguishable from an n>2 Hu-Sawicki model at laboratory scales, hence it is unobservable at a neutron bouncer experiment.

The reason for the difference between these results with those of Brax and Pignol is shown to be due to the special form of the scalar potential and matter coupling, which are free in scalar-tensor theory and can be chosen to give a large effect, but are constrained in f(R) gravity. Because of these special features the neutron bouncer experiment can distinguish between f(R) and non-f(R) modified gravity, but not between the f(R) models examined in this thesis.