Planck Scale Induced Speed of Sound in a Trapped Bose-Einstein Condensate

In the present work, we analyze the corrections caused by an anomalous dispersion relation, suggested in several quantum gravity models, upon the speed of sound in a weakly interacting Bose--Einstein Condensate, trapped in a potential of the form $V(r)\sim r^{2}$. We show that the corresponding ground state energy and consequently, the associated speed of sound, present corrections respect to the usual case, which may be used to explore the sensitivity to Planck--scale effects on these relevant properties associated with the condensate. Indeed, we stress that this type of macroscopic bodies may be more sensitive, under certain conditions, to Planck--scale manifestations than its constituents. In addition, we prove that the inclusion of a trapping potential, together with many--body contributions, improves the sensitivity to Planck--scale signals, compared to the homogeneous system.