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Big Bang Nucleosynthesis constraints on space-time noncommutativity

Teodora Maria Matei, Cristian Croitoru, Tiberiu Harko

TL;DR

This study tests Planck-scale spacetime noncommutativity by embedding three deformed photon dispersion models into the early-Universe radiation fluid during Big Bang Nucleosynthesis. By deriving low-temperature corrections to $ ho$ and $P$ for each model and integrating these into a modified Friedmann evolution via the PRyMordial framework, the authors predict shifts in $T_f$, $N_{ m eff}$, and light-element abundances. They constrain the deformation parameters with an MCMC analysis using helium-4 and deuterium data, finding $\lambda \approx 2.91\times10^{-3}\,\mathrm{MeV}^{-1}$, $\beta_0 \approx 3.53\times10^{-4}\,\mathrm{MeV}^{-2}$, and $\alpha_0 \approx 8.55\times10^{-4}\,\mathrm{MeV}^{-1}$ (Model III often being marginally favored by information criteria). The results show all three models yield abundances within observational bounds and demonstrate that BBN provides meaningful constraints on noncommutative-spacetime scenarios, with implications for quantum gravity phenomenology and future cosmological probes.

Abstract

We consider the implications of the modified dispersion relations, due to the noncommutativity of the spacetime, for a photon gas filling the early Universe in the framework of the Big Bang Nucleosynthesis (BBN) processes, during the period of light elements formation. We consider three types of deformations present in the dispersion relations for the radiation gas, from which we obtain the low temperature corrections to the energy density and pressure. The cosmological implications of the modified equations of state in the BBN era are explored in detail for all radiation models. The effects induced on the nucleosynthesis process by spacetime noncommutativity are investigated by evaluating the abundances of relic nuclei (Hydrogen, Deuterium, Helium-3, Helium-4, and Lithium-7). The primordial mass fraction estimates and their deviations due to changes in the freezing temperature impose an upper limit on the energy density of the deformed photon gas, which follows from the modified Friedmann equations. The deviations from the standard energy density of the radiative plasma are therefore constrained by the abundances of the Helium-4 nuclei. Upper limits on the free parameters of the spacetime noncommutativity are obtained via a numerical analysis performed using the \texttt{PRyMordial} software package. The primordial abundances of the light elements are obtained by evaluating the thermonuclear reaction rates for the considered noncommutative spacetime models. An MCMC (Markov Chain Monte Carlo) analysis allows to obtain restrictions on the free parameters of the modified dispersion relations. The numerical and statistical approach is implemented in the python code \texttt{PRyNCe}, available on GitHub.

Big Bang Nucleosynthesis constraints on space-time noncommutativity

TL;DR

This study tests Planck-scale spacetime noncommutativity by embedding three deformed photon dispersion models into the early-Universe radiation fluid during Big Bang Nucleosynthesis. By deriving low-temperature corrections to and for each model and integrating these into a modified Friedmann evolution via the PRyMordial framework, the authors predict shifts in , , and light-element abundances. They constrain the deformation parameters with an MCMC analysis using helium-4 and deuterium data, finding , , and (Model III often being marginally favored by information criteria). The results show all three models yield abundances within observational bounds and demonstrate that BBN provides meaningful constraints on noncommutative-spacetime scenarios, with implications for quantum gravity phenomenology and future cosmological probes.

Abstract

We consider the implications of the modified dispersion relations, due to the noncommutativity of the spacetime, for a photon gas filling the early Universe in the framework of the Big Bang Nucleosynthesis (BBN) processes, during the period of light elements formation. We consider three types of deformations present in the dispersion relations for the radiation gas, from which we obtain the low temperature corrections to the energy density and pressure. The cosmological implications of the modified equations of state in the BBN era are explored in detail for all radiation models. The effects induced on the nucleosynthesis process by spacetime noncommutativity are investigated by evaluating the abundances of relic nuclei (Hydrogen, Deuterium, Helium-3, Helium-4, and Lithium-7). The primordial mass fraction estimates and their deviations due to changes in the freezing temperature impose an upper limit on the energy density of the deformed photon gas, which follows from the modified Friedmann equations. The deviations from the standard energy density of the radiative plasma are therefore constrained by the abundances of the Helium-4 nuclei. Upper limits on the free parameters of the spacetime noncommutativity are obtained via a numerical analysis performed using the \texttt{PRyMordial} software package. The primordial abundances of the light elements are obtained by evaluating the thermonuclear reaction rates for the considered noncommutative spacetime models. An MCMC (Markov Chain Monte Carlo) analysis allows to obtain restrictions on the free parameters of the modified dispersion relations. The numerical and statistical approach is implemented in the python code \texttt{PRyNCe}, available on GitHub.

Paper Structure

This paper contains 22 sections, 97 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dispersion relations in noncommutative spacetimes
  3. General statistical framework
  4. Model I: $\left[\hat{x}^i, \hat{t}\right] = i\lambda \hat{x}^i$, $\left[\hat{x}^i, \hat{x}^j\right] = 0$
  5. Model II: $\left[\hat{x}^i, \hat{p}^j\right] = i\hbar \delta^{ij} (1 + \beta p^2)$, $\left[\hat{p}^i, \hat{p}^j\right] = 0$
  6. Model III: $\left[\hat{x}^i, \hat{p}^j\right] = i\hbar \left(\delta^{ij} - \alpha_0(p \delta^{ij} + \frac{p^i p^j}{p}) + \alpha_0^2(p^2 \delta^{ij} + 3p^i p^j) \right)$, $\left[\hat{p}^i, \hat{p}^j\right] = 0$
  7. Big Bang Nucleosynthesis in noncommutative Spacetimes
  8. Overview of the BBN process and light element formation
  9. Nucleosynthesis in the presence of a deformed photon gas
  10. Model I: $\omega(k) = kc(1 + \lambda \hbar \omega)$
  11. Model II: $\omega(k) = kc \sqrt{1 - 2\beta_0 \hbar^2 \omega^2}$)
  12. Model III: $\omega(k) = kc \sqrt{1 - 2\alpha_0 \hbar \omega}$
  13. Numerical integration with PRyMordial
  14. Directory layout.
  15. Physical setup and ODE system.
  16. ...and 7 more sections

Figures (6)

  • Figure 1: Trace plot showing chain convergence for $\alpha_0$ dispersion parameter.
  • Figure 2: Prior and posterior distribution overlay for the $\alpha_0$ parameter in Model III.
  • Figure 3: The posterior distributions or the deformed photon gas parameter models $\theta = \{\lambda, \beta_0, \alpha_0\}$, estimated with the MCMC algorithm. The plot density $\rho(\theta)$ is visualized against the deformed parameters $\theta$.
  • Figure 4: The resulted nuclei abundances considering the three models of deformed photon gas (first model - upper left, second model - upper right, third model - bottom). The model parameters have the mean and standard deviations from Table. \ref{['table_results']}.
  • Figure 5: The temperature dependent Hubble function evolution for the modified dispersion models from spacetime noncommutativity and the Standard Model.
  • ...and 1 more figures