Unveiling the inert Triplet desert region with a pNGB Dark Matter and its Gravitational Wave signatures

In this work, we extend the scalar sector of the conventional hyperchargeless inert triplet model (ITM) to include a second dark matter (DM) candidate, which appears to be a pseudo-Nambu-Goldstone boson (pNGB). The usual ITM with an extended scalar sector offers a DM candidate along with novel signatures at different experiments, e.g., colliders, gravitational wave detectors, etc. Nevertheless, hitherto unseen experimental detections have placed stringent constraints on the ITM parameter space. Moreover, triplet masses lighter than $1.9$ TeV, consistent with the existing or upcoming collider sensitivity reach, are already excluded from the DM observable, as they yield an underabundant relic density due to a strong $SU(2)_L$ gauge annihilation. Inclusion of a pNGB DM, via a complex $SU(2)_L$ scalar singlet and through the soft-breaking of a $U(1)$ symmetry, helps to revive the sub-TeV regime of the triplet DM. This resurgence relies on a proficient conversion between the two DM species. Using this inter-conversion, with the triplet DM as the lighter one between the two, we show that it is possible to push the triplet DM contribution to $50\% - 60\%$ of the total relic density. This offers a significant improvement over the traditional ITM with a single DM candidate, where the same can at most reach $10\% - 20\%$. Besides, the concerned bipartite DM framework also offers the possibility of a first-order phase transition along various constituent field directions. Among these, the one along the real $SU(2)_L$ singlet direction can be a strong one which subsequently yields detectable gravitational wave signals at the upcoming space-based gravitational wave detectors such as LISA, BBO, DECIGO, etc., alongside distinctive and complementary signatures at the various DM and collider quests.