Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit

Abstract

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 an arbitrary potential of the form V (|ϕ| 2 ). 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. Nonrelativistic hydrodynamic equations based on the Schrödinger-Poisson equations or on the Gross-Pitaevskii-Poisson equations are recovered in the limit c → +∞.was considered by Matos et al. [11]. They found that real SFs display fast oscillations but that, on the mean, they reproduce the cosmological predictions of the standard ΛCDM model. The study of perturbations was considered by Suárez and Matos [12] for a self-interacting real SF described by the Klein-Gordon-Poisson (KGP) equations and by Magaña et al. [13] for a noninteracting real SF described by the KGE equations. These studies show that the perturbations can grow in the linear regime, leading, in the nonlinear regime, to the formation of structures corresponding to DM halos. This is in agreement with the early work of Khlopov et al. [14] who studied the Jeans instability of a relativistic SF in a static background. The case of a complex self-interacting SF representing BECDM was considered by Chavanis [15] in the context of Newtonian cosmology. His study is based on the GPP equations. Harko [16] and Chavanis [15] developed an approximate relativistic BEC cosmology and found that the perturbations grow faster in BECDM as compared to ΛCDM. Recently, Li et al. [17] developed an exact relativistic cosmology for a complex self-interacting SF/BEC based on the KGE equations. They studied the evolution of the homogeneous background and showed that the Universe undergoes three successive phases: a stiff matter era, followed by a radiation era due to the SF (that exists only for self-interacting SFs), and finally a matter era similar to CDM.Instead of working directly in terms of a SF, we can adopt a fluid approach and work with hydrodynamic equations. In the case of the Schrödinger equation, this hydrodynamic approach was introduced by Madelung [18]. He showed that the Schrödinger equation is equivalent to the Euler equations for an irrotational fluid with an additional quantum potential arising from the finite value ofh. This hydrodynamic representation has been used by Böhmer and Harko [19] and by Chavanis [15,20,21] among others in the case of BECDM and in the case of BEC stars [22]. This hydrodynamic approach has been generalized by Suárez and Matos [12,23] in the context of the KG equation. They used it to study the formation of structures in the Universe, assuming that DM is in the form of a fundamental SF with a λϕ 4 potential.In the works [12,23], the SF is taken to be real and the gravitational potential is introduced by hand in the KG equation, and assumed to be determined by the classical Pois...

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“…The Madelung equations resemble the continuity and Euler equations of classical fluid dynamics, with the addition of a 'quantum pressure' term accounting for resistance against gravitational collapse. The Madelung formalism is discussed in detail in [43,[46][47][48]. Because this hydrodynamical formulation defines the fluid velocity as the gradient of the phase of the field ψ, problems arise when ψ = 0, where the phase is not well defined.…”

Hydrodynamic representation of a cosmic scalar field

“…1 In a recent work, 2,3 we have developed a hydrodynamic representation of the Klein-Gordon-Einstein (KGE) equations in the weak field limit. In these proceedings, for conciseness, we present the nonrelativistic limit of these fluid equations that was obtained in our earlier works, 4,5 and we study the growth of perturbations in the linear regime.…”

Cosmological evolution of a complex scalar field with repulsive or attractive self-interaction

“…In the relativistic case, de Broglie [72][73][74] in his so-called pilot wave theory, showed that the KG equations are equivalent to hydrodynamic equations including a covariant quantum potential. This approach has been generalized to the Klein-Gordon-Poisson (KGP) and KGE equations in the context of DM halos by [75][76][77][78][79]. 2 In this hydrodynamic representation, DM halos result from the balance between the gravitational attraction and the quantum pressure arising from the Heisenberg uncertainty principle or from the self-interaction of the bosons.…”