We investigate the cosmological perturbations around all three branches of spatially flat universe with different connections in symmetric teleparallel gravity. The model we consider can cover both the case of f(Q) model and that of the non-minimal coupling between a scalar field and the non-metricity scalar. We focus on analyzing and comparing the propagation behavior and stability of the tensorial and non-tensorial gravitational waves on spatially flat universe with different connections.
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