Ultra-light dark matter with non-canonical kinetics reopening the mass window

Fuzzy dark matter (FDM) with mass around $10^{-22}$ eV is viewed as a promising paradigm in understanding the structure formation of the local universe at small scales. Recent observations, however, begin to challenge FDM in return. We focus on the arguments between the solution to CDM small-scale curiosities and recent observations on matter power spectrum, and find its implication on an earlier formation of small-scale structure. In this article, we propose a scheme of k-ULDM scalar field with a differently-evolving sound speed, thanks to the non-canonical kinetics. With the help of the Dirac-Born-Infeld (DBI) theory, we illustrate to change the behavior of the quantum pressure term countering collapse, therefore change the history of structure growth. We find that it can truly reopen the ULDM mass window closed by the Lyman-$\alpha$ problem. We will discuss such examples in this paper, while more possibilities remain to be explored.