Statistical physics on Euclidean Snyder space: connections with the GUP and cosmological implications

TL;DR

This work develops a realization-independent statistical-mechanics framework for Euclidean Snyder space, where noncommutativity is encoded as momentum-space curvature on a 3-sphere of radius $\mathcal{P}$. By reformulating kinetic energy as a geodesic distance and using momentum-space invariants, the authors derive modified partition functions for Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics in both non-relativistic and ultrarelativistic regimes, revealing a universal suppression of energy, entropy, and energy density at high temperatures. When applied to early-Universe cosmology, these corrections lead to a negative deviation in the Hubble rate $H(T)$ compared to standard cosmology, and Big Bang Nucleosynthesis bounds translate into a robust constraint on the Snyder scale, $\mathcal{P}$, which via a GUP mapping yields $\beta = m_P^2/\mathcal{P}^2 \lesssim 10^{60}$. The results show that momentum-space curvature effects are potentially testable with high-precision cosmological data and offer a strong example of how Planck-scale geometry can leave imprints on early-Universe dynamics, distinct from area-quantization (GUP) scenarios. The work thereby strengthens the link between noncommutative geometry and observational cosmology, suggesting targeted probes in high-energy regimes beyond terrestrial experiments.

Abstract

We develop a systematic formulation of statistical mechanics on Euclidean Snyder space, where noncommutativity is geometrically encoded in the curvature of momentum space. Adopting a realization independent approach based on momentum-space invariants, we derive modified partition functions and thermodynamic quantities for systems obeying Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics in both non-relativistic and ultrarelativistic regimes. We show that momentum-space curvature induces temperature-dependent corrections that suppress the energy, entropy and energy density with respect to their standard counterparts. We apply these results to early-Universe cosmology, deriving the corresponding corrections to the Friedmann equations driven by the modified energy density of radiation. Using Big Bang Nucleosynthesis as a precision probe, we derive bounds on the Snyder deformation parameter and, via a phenomenological mapping, on the Generalized Uncerainty Principle (GUP) parameter, providing one of the most stringent cosmological and astrophysical constraints currently available. Our analysis demonstrates that high-energy cosmological processes provide a predictive arena for testing momentum-space curvature and noncommutative geometry effects.

Figures (1)

• Figure 1: Relative variation $\delta T_f/T_f$ as a function of $\mathcal{P}$. The solid blue line shows the theoretical prediction from Eq. \ref{['dtovT']}, while the red dashed line corresponds to the experimental bound from Eq. \ref{['expb']}. We set $T_{f,0}=0.6\,\mathrm{MeV}$ and $Q=1.293\,\mathrm{MeV}$.