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Anomalous decay rate of quasinormal modes in Schwarzschild-dS and Schwarzschild-AdS black holes

Recently an anomalous decay rate of the quasinormal modes of a massive scalar field in Schwarzschild black holes backgrounds was reported in which the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, while that beyond this value the behaviour is inverted. In this work, we extend the study to other asymptotic geometries, such as, Schwarzschild-de Sitter and Schwarzschild-AdS black holes. Mainly, we found that such behaviour and the critical mass are present in the Schwarzschild-de Sitter background. Also, we found that the value of the critical mass increases when the cosmological constant increases and also when the overtone number is increasing. On the other hand, despite the critical mass is not present in Schwarzschild-AdS black holes backgrounds, the decay rate of the quasinormal modes always exhibits an anomalous behaviour.