Tunneling/WKB and anomaly methods for Rindler and de Sitter space–times

We consider the WKB/tunneling method explicitly in the context of its application to the Rindler and de Sitter space–times. We also present two gravitational anomaly methods (consistent and covariant). The anomaly methods applied to the Rindler space–time do not reproduce the expected Unruh radiation, because the Rindler space–time does not have an anomaly. The consistent and the covariant anomaly methods give different results for the de Sitter space–time. In contrast to them, the WKB/tunneling method is a semiclassical calculation in which the radiation is regarded as a tunneling of quantum fields across the horizon. The tunneling method is applicable in all the indicated cases. But to recover the correct Gibbons–Hawking temperature, a previously overlooked temporal piece that contributes to the total action must be taken into account.