Noncommutative Bianchi I and III Cosmology Models: Radiation Era Dynamics and gamma Estimation

TL;DR

This paper investigates noncommutative (NC) deformations in locally rotationally symmetric Bianchi I and Bianchi III cosmologies coupled to a radiation fluid within a Schutz Hamiltonian formalism, introducing a deformation parameter $\gamma$ in the Poisson algebra.NC effects are implemented via Bopp shifts to first order in $\gamma$, yielding a NC Hamiltonian $H_c$ and corresponding equations of motion for the commutative variables $(a_c,\beta_c,T_c)$ that determine the physical evolution through $a_{nc}=a_c+(\gamma/2)T_c$ and $\beta_{nc}=\beta_c$.Numerical solutions in the radiation era show that $\gamma<0$ tends to enhance expansion and promote isotropization, while $\gamma>0$ can slow expansion and retain residual anisotropy, with the magnitude and timing of these effects depending on spatial curvature $\kappa$.An observational estimate of $\gamma$ suggests it was most relevant in the early, high-density universe and becomes negligible today, supporting NC dynamics as a potential early-universe mechanism that might mimic, but differ from, a cosmological constant.

Abstract

Understanding the early evolution of the universe requires models that incorporate possible quantum and anisotropic effects in its dynamics. In this work, we analyze the dynamical evolution of locally rotationally symmetric anisotropic cosmological models of Bianchi type I (flat curvature) and Bianchi type III (open curvature) within a noncommutative phase space framework characterized by a deformation parameter gamma. Using a Hamiltonian formulation based on Schutz formalism for a perfect radiation fluid, we introduce noncommutative Poisson brackets that allow for geometric corrections to commutative dynamics. The resulting equations are solved numerically under various initial conditions, enabling the study of the impact of gamma and the energy density C on the universe expansion and anisotropy evolution. The results show that gamma < 0 enhances expansion and favors isotropization, while gamma > 0 tends to slow expansion and preserve residual anisotropy, especially in the open curvature model. It is estimated that the influence of non-commutativity was significant during the early stages of the universe, decreasing toward the present time, suggesting that this approach could serve as an effective alternative to the cosmological constant in describing the evolution of the early universe.

Figures (14)

• Figure 1: The graphs (a) and (b) show the numerical solutions obtained under initial conditions: $a_c(0) = 1$, $\dot{a}_c(0) = 1$, $\beta_c(0) = 1$, $\dot{\beta}_c(0) = 1$, and $T_c(0) = 0$. For each selected value of $\gamma= \{0,\, -0.2,\, 0.2\}$, the corresponding value of $C = \{ 6.58747, \, 5.43337, \, 7.74158\}$ was calculated using the constraint $H_c = 0$, while keeping the other initial conditions fixed.
• Figure 2: The graphs (a) and (b) show the numerical solutions obtained under initial conditions: $a_c(0) = 1$, $\dot{a}_c(0) = 1$, $\beta_c(0) = 1$, $T_c(0) = 0$, and the NC parameter $\gamma = -0.01$. For each selected value of $C= \{0,\, 1,\, 3\}$, the corresponding value of $\dot{\beta}_c(0) = \{ 5.78536, \, 5.50429, \, 4.8607\}$ was calculated using the constraint $H_c = 0$, while keeping the other initial conditions fixed.
• Figure 3: The graphs (a) and (b) show the numerical solutions obtained under initial conditions: $\dot{a}_c(0) = 1$, $\beta_c(0) = 1$, $\dot{\beta}_c(0) = 1$, $T_c(0) = 0$, and the NC parameter $\gamma = -0.01$. For each selected value of $a_c(0) = \{1,\, 1.5,\, 2\}$, the corresponding value of $C = \{6.5298, \, 15.5508, \, 28.1657\}$ was calculated using the constraint $H_c = 0$, while keeping the other initial conditions fixed.
• Figure 4: The graphs (a) and (b) show the numerical solutions obtained under initial conditions: $a_c(0) = 1$, $\beta_c(0) = 1$, $\dot{\beta}_c(0) = 1$, $T_c(0) = 0$, and the NC parameter $\gamma = -0.01$. For each selected value of $\dot{a}_c(0) = \{1,\, 1.5,\, 2\}$, the corresponding value of $C = \{ 6.5298, \, 14.7829, \, 25.9576\}$ was calculated using the constraint $H_c = 0$, while keeping the other initial conditions fixed.
• Figure 5: The graphs (a) and (b) show the numerical solutions obtained under initial conditions: $a_c(0) = 1$, $\dot{a}_c(0) = 1$, $\dot{\beta}_c(0) = 1$, $T_c(0) = 0$, and the NC parameter $\gamma = -0.05$. For each selected value of $\beta_c(0) = \{1,\, 5,\, 10\}$, the corresponding value of $C = \{6.2990, \, 102.7980, \, 2920.6400\}$ was calculated using the constraint $H_c = 0$, while keeping the other initial conditions fixed.