Some models of quantum gravity predict that the very structure of spacetime is ‘ frothy ’ due to quantum fluctuations .
Although the effect is expected to be tiny , if these spacetime 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 spacetime 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 spacetime 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 \simeq 2.34 , corresponding to a comoving distance of \simeq 5.8 Gpc .
Using a heuristic model where the energy fluctuations grow as \sigma _ { E } / E = ( E / E _ { P } ) ^ { \alpha } , where E _ { P } \simeq 1.22 \times 10 ^ { 28 } ~ { } { eV } is the Planck energy , we rule out models with \alpha \leq 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 spacetime steps is drawn from a continuous distribution .
We conclude , if photons take discrete steps through spacetime and accumulate Planck-scale uncertainties at each step , then our ESPRESSO observations require that the step size must be at least \gtrsim 10 ^ { 13.2 } l _ { P } , where l _ { P } is the Planck length .