Emergent Cosmological Intervals from Ultralight Bose-Einstein Condensates in Modified Measure Theory

Abstract

We present a rigorous mathematical synthesis of Modified Measure Theory (MMT) and the cosmology of ultralight Bose-Einstein Condensates (BEC). By identifying the phenomenological "measure field" of standard MMT as the macroscopic wavefunction of a cosmological BEC, we naturally derive a fundamental geometric scale—the Interval ΔLΔL. This interval emerges dynamically from the correlation length (Compton wavelength) of the condensate, resolving the arbitrary nature of the measure field without violating Lorentz invariance. We demonstrate that the conformal mismatch between the dynamical measure and the standard Riemannian volume element sources geometric torsion, effectively acting as a dynamical dark energy component. This synthesis establishes the physical and geometric foundation for the Ramanujan-Bose Interval Cosmology (RBIC), which is essential for the subsequent resolution of the H0H0 and S8S8 cosmological tensions. Furthermore, we show that this dynamical interval safely evades current constraints on Lorentz Invariance Violation (LIV) from advanced gravitational wave (LVK O4b) and high-energy photon (Fermi-LAT) observations.