Hydrodynamics of ultralight complex scalar field dark matter and its impact on the growth of structure

The mass window of ultralight axion dark matter motivated by suppressing the growth of structure on subgalactic scales, $m\sim 10^{-22}\,\mathrm{eV}$, is now severely constrained by various observation data (e.g. Lyman-$\alpha$ forest). As an attempt to reopen this mass window, we investigate an alternative ultralight dark matter candidate, the complex scalar field dark matter (SFDM). We derive the relativistic hydrodynamics of the complex SFDM in the framework of cosmological perturbation theory. Our formalism contains two novel ingredients uniquely associated with the complex SFDM model: the Eckart frame defined by the conserved Noether current, and the stiff gauge condition, $c_s^2\equiv (\delta P/\delta\rho)|_s=1$. In the Eckart frame, the complex SFDM is effectively an imperfect fluid with a dissipative energy flux, distinguishing itself from axion dark matter. The energy flux can affect the growth of density fluctuations dynamically. Meanwhile, we apply the stiff gauge condition to find new constitutive equations for the complex SFDM. We revisit the homogeneous evolution of the complex SFDM and present illustrative early-stage solutions for perturbations of the complex SFDM in a simplified setting. We demonstrate the effects of varying the model parameters on the evolution of the perturbation variables.