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 $\delta E \simeq \xi_1 mcp/2M_p$ produces a shift in the condensation temperature $T_c$ of about $\Delta T_{c}/T_{c}^{0} \simeq 10^{-6} \xi_1 $ for typical laboratory conditions. Such a shift allows to bound the deformation parameter up to $|\xi_1| \lesssim 10^4$. Moreover we show that it is possible to enlarge $\Delta T_{c}/T_{c}^{0}$ 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.