Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: I. General formalism and perturbations analysis

Using a generalization of the Madelung transformation, we derive the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. We consider a complex self-interacting scalar field with a $\lambda|\varphi|^4$ potential. We study the evolution of the homogeneous background in the fluid representation and derive the linearized equations describing the evolution of small perturbations in a static and in an expanding universe. We compare the results with simplified models in which the gravitational potential is introduced by hand in the Klein-Gordon equation, and assumed to satisfy a (generalized) Poisson equation. We study the evolution of the perturbations in the matter era using the nonrelativistic limit of our formalism. Perturbations whose wavelength is below the Jeans length oscillate in time while pertubations whose wavelength is above the Jeans length grow linearly with the scale factor as in the cold dark matter model. The growth of perturbations in the scalar field model is substantially faster than in the cold dark matter model. When the wavelength of the pertubations approaches the cosmological horizon (Hubble length), a relativistic treatment is mandatory. In that case, we find that relativistic effects attenuate or even prevent the growth of pertubations. This paper exposes the general formalism and provides illustrations in simple cases. Other applications of our formalism will be considered in companion papers.