Bose-Einstein Condensate and Liquid Helium He${}^4$: Implications of GUP and Modified Gravity Correspondence

Utilizing the recently established connection between Palatini-like gravity and linear Generalized Uncertainty Principle (GUP) models, we have formulated an approach that facilitates the examination of Bose gases. Our primary focus is on the ideal Bose-Einstein condensate and liquid helium, chosen as illustrative examples to underscore the feasibility of tabletop experiments in assessing gravity models. The non-interacting Bose-Einstein condensate imposes constraints on linear GUP and Palatini $f(R)$ gravity (Eddington-inspired Born-Infeld gravity) within the ranges of $-{10}^{12}\lesssim \sigma \lesssim 3\times {10}^{24}\text{ s}\mathrm{/}\text{kg m}$ and $-{10}^{-1}\lesssim \bar{\beta }\lesssim {10}^{11}{\text{ m}}^2$ ($-4\times {10}^{-1}\lesssim \epsilon \lesssim 4\times {10}^{11}{\text{ m}}^2$), respectively. In contrast, the properties of liquid helium suggest more realistic bounds, specifically $-{10}^{23}\lesssim \sigma \lesssim {10}^{23}\text{ s}\mathrm{/}\text{kg m}$ and $-{10}^9\lesssim \bar{\beta }\lesssim {10}^9{\text{ m}}^2$. Additionally, we argue that the newly developed method employing Earth seismic waves provides improved constraints for quantum and modified gravity by approximately one order of magnitude.