Trapped Bose-Einstein condensates with Planck-scale induced deformation of the energy-momentum dispersion relation

We show that harmonically trapped Bose-Einstein condensates can be used to constrain Planck scale physics. In particular we prove that a Planck-scale induced deformation of the Minkowski energy-momentum dispersion relation $\delta E\simeq {\xi }_{1}mcp\mathrm{/}2M_p$ produces a shift in the condensation temperature $T_c$ of about $\mathrm{\Delta }{T}_c\mathrm{/}{T}_c^0\simeq {10}^{-6}{\xi }_1$ for typical laboratory conditions. Such a shift allows to bound the deformation parameter up to $\mathrm{\mid }{\xi }_1\mathrm{\mid }\lesssim {10}^4$. Moreover we show that it is possible to enlarge $\mathrm{\Delta }{T}_c\mathrm{/}{T}_c^0$ and improve the bound on ${\xi }_1$ lowering the frequency of the harmonic trap. Finally we compare the Planck-scale induced shift in $T_c$ with similar effects due to interboson interactions and finite size effects.