A testable description of space-time foam as a fundamental stochastic gravity-wave background

I develop a phenomenological approach to the description of the noise levels that the space-time foam of quantum gravity could induce in modern gravity-wave detectors. Various possibilities are considered, including white noise and random-walk noise. In particular, I find that the sensitivity level expected for the planned LIGO and VIRGO interferometers and for the next upgrade of the NAUTILUS resonantbar detector corresponds to a white-noise level which can be naturally associated with the Planck length. One of the most natural expectations for quantum gravity, as the theory describing the interplay between gravity and quantum mechanics, is that space-time, when resolved at very short distances, would appear to be “foamy” in the sense of Refs. [1, 2]. The fact that there is still a rather wide collection of intuitions for this fascinating new picture of space-time (see, e.g., the recent Refs. [3, 4], which also provide a good starting point for a literature search backward in time) is due to the technical and conceptual difficulties encountered in the development of theoretical approaches to the quantum-gravity problem. Most quantum-gravity theories have not yet passed even the most basic tests of consistency. The two approaches that have survived at least a few non-trivial consistency tests, the one based on “critical superstrings” [5, 6] and the one based on “canonical/loop quantum gravity” [7, 8, 9], do not have any direct confirmation from experimental data and even theoretical studies of the nature of their physical implications are only at a preliminary stage. While waiting for the emergence of a “full-grown” quantum gravity, possibly through the maturation of one of the mentioned approaches, it is becoming increasingly clear that, by exploiting recent progress in experimental technologies and ideas, we can follow an alternative path [10, 11, 12] toward the exploration of space-time foam. It has been shown [10, 11, 12] that certain types of experiments have become so refined that their sensitivities can be naturally expressed as proportional to the Planck length Lp ∼ 1.6 · 10 m (whose smallness we expect to penalize all quantum-gravity effects). In order to profit from these new experimental possibilities one can set up phenomenological models providing estimates (typically depending on a few unknown parameters 1 Marie Curie Fellow of the European Union (address from February 2000: Dipartimento di Fisica, Universitá di Roma “La Sapienza”, Piazzale Moro 2, Roma, Italy) encoding our ignorance of the correct quantum gravity) of candidate quantum-gravity effects. The hope is that these phenomenological estimates may guide experimentalists toward the discovery of some quantum-gravity phenomena, which in turn would provide much needed hints for the rigorous mathematical work searching for the correct quantum-gravity formalism. [More detailed considerations on the impact that this phenomenological approach might have on the development of quantum gravity can be found in Refs. [7, 8, 9, 13, 14].] In one of these phenomenological proposals I observed [12] that the quantum fluctuations affecting distances in conventional pictures of space-time foam would manifest themselves in the operation of modern gravity-wave detectors in a way that mimics a stochastic gravity-wave background. Just like a stochastic gravity-wave background these quantum-gravity effects would induce stochastic fluctuations in the magnitude of distances, and just like a stochastic gravity-wave background these quantum-gravity effects would be felt in a sensitive gravity-wave detector as an additional source of noise. I also observed that, as done for ordinary stochastic gravity-wave backgrounds, the power spectrum of the strain noise [15] that would be induced in gravity-wave detectors is the most convenient way to characterize models of foam-induced distance fluctuations. This predicted strain noise power spectrum can be compared to the strain noise power spectrum actually found in a given detector, thereby obtaining bounds on the parameters of phenomenological descriptions of the foam-induced distance fluctuations. In the present Letter I use these observations as motivation for a phenomenological approach to the study of space-time foam in which some properties of foam are modeled as a fundamental/intrinsic level of strain noise power spectrum. It may seem hard to develop a phenomenology directly at the level of the foam, without an underlying theory of quantum gravity, but I shall show that the assumption that an appropriate characterization be given by a strain noise power spectrum (which is a function of a single variable) together with the assumption that this fundamental level of strain noise should be a universal property of physics (in a sense that will become clearer below) provide rather strong constraints for the construction of candidate power spectra. Let me start by discussing the possibility that this foam-induced strain noise power spectrum be due to underlying quantum-gravity space-time fluctuations that are of random-walk type. This is a rather simple hypothesis which also fits well the intuition emerging from certain approaches to the more formal analysis of space-time foam (see, e.g., Refs. [3, 4] and references therein). From this simple hypothesis it already follows that the functional form of the strain noise power spectrum is [ρh(f)]random walk ∼ 1