Quantum geometric formulation of Brans-Dicke theory for Bianchi I spacetime

TL;DR

The paper formulates quantum-corrected effective dynamics for Brans-Dicke theory in a Bianchi I spacetime within loop quantum gravity, using two quantization schemes (muA and muB) in the Jordan frame. By polymerizing the connection in the BD Hamiltonian, it shows that both schemes replace the classical singularity with a quantum bounce, and that, unlike GR-based LQC, all three directional scale factors can grow after the bounce and eventually resemble their pre-bounce values depending on initial BD data. The analysis demonstrates robustness of singularity resolution beyond GR and reveals scheme-dependent differences in post-bounce isotropization and bounce timing, with a bounded effective energy density in the muB scheme and lower bounds on triads in muA. The work sets the stage for perturbation studies and potential observational signatures in anisotropic BD-LQC cosmologies, including possible inflationary dynamics following the bounce.

Abstract

This paper investigates Bianchi I spacetimes within the Jordan frame of Brans-Dicke theory, incorporating the framework of effective loop quantum gravity. After developing general formulas, we analyze the robustness of classical singularity resolution due to quantum geometric effects using two common quantization schemes. We then compare the resulting physical properties. We find that both schemes replace classical singularities with regular quantum bounces. Notably, in contrast to similar studies based on general relativity, we find that all three directional scale factors of the Bianchi I spacetimes increase and after the quantum bounce they reach values similar to their initial values, leading to a merging with classical spacetimes in both schemes.

Figures (4)

• Figure 1: The evolution of the directional scale factors $a_I \; (I=1,2,3)$ of the Bianchi-I spacetimes, where (a) is for LQC and (b) - (d) are for BDT. The initial conditions for LQC are $c_1=-0.13$, $c_2=-0.12$, $p_1=10^3$, $p_2=2\times10^3$, $p_3=3\times10^3$. For BDT, we adopt the same initial conditions as those of LQC with the addition of $(\phi,\dot{\phi})=(1.5,0)$ for (b), $(\phi,\dot{\phi})=(1.0,5.5\times10^{-5})$ for Panel (c), and $(\phi,\dot{\phi})=(1.0,-5.5\times10^{-5})$ for (d), respectively, and set $\omega=2\times10^5$Zhang:2017sym for the BD parameter in all these three last. All plots correspond to the $\bar{\mu}_A$ scheme.
• Figure 2: The same as in Fig. \ref{['a_I_A']}, but for the $\bar{\mu}_B$ scheme.
• Figure 3: The average scale factor $a$ of the Bianchi I spacetime for the $\bar{\mu}_A$ [(a)] and $\bar{\mu}_B$ [(b)] schemes, where the LQC model is presented by the black curve and the BDT models are presented by the blue, yellow and red curves, respectively for the choices, $(\phi,\dot{\phi})=(1.5,0)$, $(\phi,\dot{\phi})=(1.0,5.5\times10^{-5})$, and $(\phi,\dot{\phi})=(1.0,-5.5\times10^{-5})$. The other initial conditions are the same as those given in Fig. \ref{['a_I_A']} for the $\mu_A$ scheme, and those given in Fig. \ref{['a_I_B']} for the $\mu_B$ scheme.
• Figure 4: The average Hubble parameters $H$ of the Bianchi I spacetime for the $\bar{\mu}_A$ scheme [(a)] and the $\bar{\mu}_B$ scheme [(b)], where the LQC model is presented by the black curve and the BDT models are presented by the blue, yellow and red curves, respectively, for the choices, $(\phi,\dot{\phi})=(1.5,0)$, $(\phi,\dot{\phi})=(1.0,5.5\times10^{-5})$, and $(\phi,\dot{\phi})=(1.0,-5.5\times10^{-5})$. The other initial conditions are the same as those given in Fig. \ref{['a_I_A']} for the $\mu_A$ scheme and Fig. \ref{['a_I_B']} for the $\mu_B$ scheme.