We present first-order post-Newtonian (1PN) approximations of a general imperfect fluid and of an axion as a coherently oscillating massive scalar field, both in the cosmological context. For the axion, using the Klein transformation and Madelung transformation we derive the Schrödinger and Madelung hydrodynamic formulations, respectively, in an exact covariant way and to 1PN order. Complete sets of equations for the 1PN formulations are derived without fixing the temporal gauge condition. We study the linear instability in cosmology and a static limit for both fluid and axion; these are presented independently of the gauge condition to 1PN order, thus are naturally gauge invariant.

• Received 29 September 2022
• Accepted 27 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.083508

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Published by the American Physical Society