Quantum Backreaction in Effective Brans-Dicke Bianchi I Cosmology

We investigate the effective quantum evolution of the Bianchi type I cosmological model within the Brans-Dicke framework, using an effective Hamiltonian approach including expectation values, quantum dispersions, and cross-correlation terms between different degrees of freedom. We show that cross-correlation terms are essential for a physically consistent effective dynamics: neglecting them leads to spurious divergences, violation of the Heisenberg uncertainty relations, and unphysical behavior. For w < -3/2, where bouncing solutions exist already classically, quantum backreaction smooths the bounce, suppresses shear anisotropy, and the energy density peak is slightly enhanced and remains well-behaved, in contrast to the unphysical divergences that arise when cross-correlations are neglected. For the conformally invariant case w = -3/2, quantum corrections accelerate the approach to de Sitter expansion while enhancing shear anisotropy. When correlations are included, small-amplitude damped oscillations appear shortly after the bounce, which we interpret as quantum remnant effects encoding Planck scale correlation information between gravitational and scalar field degrees of freedom. The effective energy density remains finite for both regimes. Our results demonstrate that cross-correlations carry crucial quantum information influencing cosmological dynamics, with implications for quantum gravity phenomenology, inflationary model building, and primordial observational signatures.