Cosmological perturbations for ultralight axionlike particles in a state of Bose-Einstein condensate

For ultralight scalar particles like axions, dark matter can form a state of the Bose-Einstein condensate (BEC) with a coherent classical wave whose wavelength is of order galactic scales. In the context of an oscillating scalar field with mass m, this BEC description amounts to integrating out the field oscillations over the Hubble timescale H-1 in the regime m≫H. We provide a gauge-invariant general relativistic framework for studying cosmological perturbations in the presence of a self-interacting BEC associated with a complex scalar field. In particular, we explicitly show the difference of BECs from perfect fluids by taking into account cold dark matter, baryons, and radiation as a Schutz-Sorkin description of perfect fluids. We also scrutinize the accuracy of commonly used Newtonian treatment based on a quasistatic approximation for perturbations deep inside the Hubble radius. For a scalar field starting to oscillate after matter-radiation equality, we show that, after the BEC formation, a negative self-coupling hardly leads to a Laplacian instability of the BEC density contrast. This is attributed to the fact that the Laplacian instability does not overwhelm the gravitational instability for self-interactions within the validity of the nonrelativistic BEC description. Our analysis does not accommodate the regime of parametric resonance, which can potentially occur for a large field alignment during the transient epoch prior to the BEC formation.