One-loop $\lambda \phi^4$ theory in Robertson-Walker spacetimes: adiabatic regularization and analytic approximation

摘要

The renormalization of a scalar field theory with a quartic self-coupling (a$\lambda \phi^4$ theory) via adiabatic regularization in a generalRobertson-Walker spacetime is discussed. The adiabatic counterterms arepresented in a way that is most conducive to numerical computations. Avariation of the adiabatic regularization method is presented which leads toanalytic approximations for the energy-momentum tensor of the field and thequantum contribution to the effective mass of the mean field. Conservation ofthe energy-momentum tensor for the field is discussed and it is shown that thepart of the energy-momentum tensor which depends only on the mean field is notconserved but the full renormalized energy-momentum tensor is conserved asexpected and required by the semiclassical Einstein's equation. It is alsoshown that if the analytic approximations are used then the resultingapproximate energy-momentum tensor is conserved. This allows a self-consistentbackreaction calculation to be performed using the analytic approximations. Theusefulness of the approximations is discussed.