Polymerized spacetime dynamics with multi-field source: unraveling the pre-inflationary Universe

Abstract

We study a multi-field model in Loop Quantum Cosmology for a maximally symmetric spacetime governed by the Einstein--Hilbert action minimally coupled to scalar fields. Using a Legendre transformation, we formulate the Hamiltonian dynamics in canonically equivalent geometrodynamical and Yang--Mills--type representations, incorporating nontrivial couplings through a geometric structure on the multi-field configuration space. Implementing the $\barμ$-scheme polymerization, we obtain the loop-quantum-corrected Friedmann equations. By focusing on the two-field model as an example, we analyze the effective dynamics for specific potentials. The \textit{quantum bouncing, transition, and slow-roll inflationary} phases are investigated numerically, and viability of the models is assessed by evaluating the number of e-folds during the inflationary phase for certain given initial conditions. The global behavior of the background evolution is further examined through linear stability and dynamical-systems analyses.

Figures (26)

• Figure 1: Potential profile of the hybrid potential model Eq. (\ref{['hybridV']}). This profile is evaluated for the parameter values $M=\;1 \;m_{Pl}, \;\lambda=\;0.3,\;g=\;10^{-5},\;m=\;0.01\; m_{Pl}$.
• Figure 2: Potential profile of the string-inspired model Eq. (\ref{['stringV']}). The parameters are assumed to be: $m=10^{-6}{m_{Pl}}$ and $g=3\times10^{-4}$.
• Figure 3: Evolution of (a) the scale factor $a(t)$, and (b) $\dot{ H(t)}$ for the Linde's hybrid potential model \ref{['fig:hybridmodel_profile']} with a canonical kinetic space represented by the metric \ref{['eq:metric_hybridmodel']}, in the LQC framework. The $\chi_B$ value is fixed at $0.01~m_{Pl}$ and its velocity $\dot{\chi_B}= 0.05~m_{Pl}^2$. The initial values of clock field and its velocity are fixed at $(\Phi_B, \dot{\Phi}_B)=(0.5~m_{Pl}, 0.5~m_{Pl}^2)$. The initial values of the field $\phi$ are taken as $1$, $2$, and $3~m_{Pl}$, the model parameters are $m=0.01\; m_{Pl},\; g= 10^{-5},\; M= 1 \;m_{Pl},\;\lambda= 0.3$ which are consistently used throughout this analysis. These plots highlight the modifications introduced by quantum gravitational corrections compared to the classical dynamics.
• Figure 4: Each panel corresponds to the plot of $\ddot{a}(t)$ for different values of $\phi_B$ in the hybrid model \ref{['fig:hybridmodel_profile']} within the LQC framework, with model parameters and initial conditions identical to those in Fig. \ref{['fig:figlm_bounce_phase']}. It can be clearly seen that the acceleration $\ddot{a}(t)$ remains positive throughout the slow-roll phase.
• Figure 5: Evolution of the component fields $\phi(t)$, $\chi(t)$ and $\Phi(t)$ for the hybrid potential model \ref{['fig:hybridmodel_profile']} in the LQC framework. Each panel, corresponding to a distinct value of $\phi_B$, illustrates the dynamics of the fields from bounce onwards. The clock field $\Phi(t)$ remains constant throughout the evolution, unaffected by the dynamical interactions.