Poincar\'e Gauge Theory With Coupled Even And Odd Parity Dynamic Spin-0 Modes: Dynamic Equations For Isotropic Bianchi Cosmologies

We are investigating the dynamics of a new Poincar'e gauge theory of gravity model, which has cross coupling between the spin-0${}^{+}$ and spin-0${}^{-}$ modes. To this end we here consider a very appropriate situation—homogeneous-isotropic cosmologies—which is relatively simple, and yet all the modes have non-trivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies; for this case we find an effective Lagrangian and Hamiltonian for the dynamical system. The Lagrange equations for these models lead to a set of first order equations that are compatible with those found for the FLRW models and provide a foundation for further investigations. Typical numerical evolution of these equations shows the expected effects of the cross parity coupling.