Novel triple barrier potential for axial gravitational perturbations of a family of Lorentzian wormholes

We study the behavior of a specific Lorentzian wormhole family under gravitational perturbations. In earlier work [EPJC 80, 850 (2020)], we have proved the stability of a test scalar field in the background of the wormhole family, where the effective potential was that of a double barrier. Continuing with the stability analysis, here we focus on the more physically relevant scenario, that of axial gravitational perturbations. Interestingly, we find that the effective potential is a triple barrier for lower angular momentum modes. This raises important questions on the ringdown of the corresponding wormhole geometry as well as the gravitational wave echo profile that we try to answer through our work. We study in detail how the geometry of each member wormhole affects the quasinormal modes, the time evolution of the signal as well as echoes which are, in general, very feeble in comparison to the main signal. Different `cleaning' techniques have been used to obtain the echo profile in the time evolution of the signal. Lastly, we dwell on the possibility of our wormhole family as a candidate black hole mimicker, as long as its stability is proven under all kinds of perturbations. We briefly present a comparison of the ringdown characteristics of these wormholes with that of a black hole, in support of this speculation.