Quantum Backreaction in Effective Brans-Dicke Bianchi I Cosmology

Abstract: We investigate the effective quantum evolution of the Bianchi type I cosmological model within the Brans-Dicke framework using an effective Hamiltonian approach that includes expectation values, quantum dispersions and cross-correlation terms between different degrees of freedom. For the case $ω< -3/2$, where energy conditions are violated and bouncing solutions exist classically, we demonstrate that quantum backreaction effects significantly smooth the bounce of directional scale factors, with the bounce occurring at scales set by the quantum state width. For the conformally invariant case $ω= -3/2$, quantum corrections cause scale factors to enter accelerated expansion phases more rapidly than in the classical limit. Most significantly, we show that cross-correlation terms between canonical variables are essential for obtaining physically consistent effective dynamics: neglecting these terms leads to spurious divergences and unphysical behavior. When correlations are included, small-amplitude oscillations appear shortly after the bounce, rapidly damping to classical trajectories. We interpret these oscillations as quantum remnant effects encoding information about correlations between gravitational and scalar field degrees of freedom. Our results demonstrate that cross-correlations carry crucial quantum information that substantially influences cosmological dynamics, with implications for quantum gravity phenomenology near singularities. We compare our findings with existing loop quantum cosmology results.