A limit on Planck-scale froth with ESPRESSO

Some models of quantum gravity predict that the very structure of space-time is 'frothy' due to quantum fluctuations. Although the effect is expected to be tiny, if these space-time fluctuations grow over a large distance, the initial state of a photon, such as its energy, is gradually washed out as the photon propagates. Thus, in these models, even the most monochromatic light source would gradually disperse in energy due to space-time fluctuations over large distances. In this paper, we use science verification observations obtained with ESPRESSO at the Very Large Telescope to place a novel bound on the growth of space-time fluctuations. To achieve this, we directly measure the width of a narrow Fe II absorption line produced by a quiescent gas cloud at redshift z similar or equal to 2.34, corresponding to a comoving distance of similar or equal to 5.8 Gpc. Using a heuristic model where the energy fluctuations grow as sigma(E)/E = (E/E-P)(alpha), where E-P similar or equal to 1.22 x 10(28) eV is the Planck energy, we rule out models with alpha <= 0.634, including models where the quantum fluctuations grow as a random walk process (alpha = 0.5). Finally, we present a new formalism where the uncertainty accrued at discrete space-time steps is drawn from a continuous distribution. We conclude, if photons take discrete steps through space-time and accumulate Planck-scale uncertainties at each step, then our ESPRESSO observations require that the step size must be at least >= 10(13.2) l(P), where l(P) is the Planck length.