Hydrodynamic representation of a cosmic scalar field

2018

Phys. Rev. D

We study the gravitational instability of a general relativistic complex scalar field with a quartic self-interaction in an infinite homogeneous static background. This quantum relativistic Jeans problem provides a simplified framework to study the formation of the large-scale structures of the Universe in the case where dark matter is made of a scalar field. The scalar field may represent the wave function of a relativistic self-gravitating Bose-Einstein condensate. Exact analytical expressions for the dispersion relation and Jeans length λJ are obtained from a hydrodynamical representation of the Klein-Gordon-Einstein equations [Suárez and Chavanis, Phys. Rev. D 92, 023510 (2015)]. We compare our results with those previously obtained with a simplified relativistic model [Khlopov, Malomed and Y. B. Zeldovich, Mon. Not. R. astr. Soc. 215, 575 (1985)]. When relativistic effects are fully accounted for, we find that the perturbations are stabilized at very large scales of the order of the Hubble length λH . Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with a repulsive self-interaction, and for bosons with an attractive self-interaction (QCD axions and ultralight axions). We show that the Jeans instability is inhibited in the ultrarelativistic regime (early Universe and radiationlike era) because the Jeans length is of the order of the Hubble length (λJ ∼ λH ), except when the self-interaction of the bosons is attractive. By contrast, structure formation can take place in the nonrelativistic regime (matterlike era) for λJ ≤ λ ≤ λH . Since the scalar field has a nonzero Jeans length due to its quantum nature (Heisenberg uncertainty principle or quantum pressure due to the self-interaction of the bosons), it appears that the wave properties of the scalar field can stabilize gravitational collapse at small scales, providing halo cores and suppressing small-scale linear power. This may solve the CDM crisis such as the cusp problem and the missing satellite problem. We also compare our results with those obtained in the case where dark matter is made of fermions instead of bosons. PACS numbers: 95.35.+d, 98.80.Jk, 95.30.Sf

Section: Discussionsupporting

confidence: 58%

Jeans-type instability of a complex self-interacting scalar field in general relativity

Suárez

2018

Phys. Rev. D

“…( 146). However, it can be shown that they coincide for a spatially homogeneous SF [103,112] in certain approximations. The hydrodynamic equations ( 142)-( 145) have a clear physical interpretation.…”

Section: H Hydrodynamic Representation Of the Gross-pitaevskii Equationmentioning

confidence: 99%

Covariant theory of Bose-Einstein condensates in curved spacetimes with electromagnetic interactions: The hydrodynamic approach

Matos

2017

Eur. Phys. J. Plus

We develop a hydrodynamic representation of the Klein-Gordon-Maxwell-Einstein equations. These equations combine quantum mechanics, electromagnetism, and general relativity. We consider the case of an arbitrary curved spacetime, the case of weak gravitational fields in a static or expanding background, and the nonrelativistic (Newtonian) limit. The Klein-Gordon-Maxwell-Einstein equations govern the evolution of a complex scalar field, possibly describing self-gravitating Bose-Einstein condensates, coupled to an electromagnetic field. They may find applications in the context of dark matter, boson stars, and neutron stars with a superfluid core.1 As reported by Bhaumik [4], "Langevin, who was de Broglie's thesis advisor, was very skeptical about de Broglie's theory and contacted Einstein to have his opinion. Einstein strongly supported de Broglie's work and suggested to physicists to look for an evidence of the matter wave. A proof was furnished soon by the accidental discovery of electron waves by Davisson and Germer [5] in observing a diffraction pattern in a nickel crystal. Later, Einstein mentioned to Rabi that he had thought about the equation for matter waves before de Broglie but did not publish it because there was no evidence for it at that time." 2 Here and in the following, the dates correspond to the dates of submission. 3 In this brief review, we stick to what was originally called the "wave (or undulatory) mechanics" initiated by de Broglie [2] and developed by Schrödinger [6][7][8][9]. We shall not discuss in detail what was originally called the "quantum mechanics" (a term introduced by Born) based on the matrix theory of Heisenberg [12] further developed by Born and Jordan [13,14]. In Ref. [15] (18 March 1926), Schrödinger showed the equivalence between his wave mechanics and the Heisenberg-Born-Jordan quantum mechanics.[8], Schrödinger applied his theory to the perturbation of the hydrogen atom caused by an external homogeneous electric field (Stark effect). In his fourth paper (21 June 1926) [9], he obtained for the first time the time-dependent equation (F23) that now bears his name. From this equation, he introduced a (charge) density and a current (of charge) [see Eqs. (174) and (176)] and derived a local conservation equation for ρ = |ψ| 2 [see Eq. (175)]. This gives an interpretation to the wave function in the sense that |ψ| 2 (r, t) characterizes the "presence" of the particle at some point. Schrödinger thought that the wave function represents a particle that is spread out, most of the particle being where the modulus of the wave function |ψ| 2 is large. For example, according to Schrödinger's view, the charge of the electron is not concentrated in a point, but is spread out through the whole space, proportional to the quantity |ψ| 2 . On the other hand, Born (21 July 1926, 16 October 1926 developed a probabilistic interpretation of the wave function. 4 Born proposed that the magnitude of the wavefunction ψ(r, t) does not tell us how much of the particle is at position r at time t, but...

“…The complete study of these relativistic hydrodynamic equations is of considerable interest but it is, of course, of great complexity. In our research papers [24,25], we have started their study in simple cases. We have checked that the hydrodynamic equations of the SFDM model reproduce the evolution of the homogeneous background obtained previously by Li et al [17] directly from the KGE equations: a stiff matter era, followed by a radiation era (for a self-interacting SF), and a matter era.…”

Section: Discussionmentioning

confidence: 99%

“…( 38). 3 However, they coincide for a homogeneous SF in the regime where the SF oscillations are faster than the Hubble expansion [24,25].…”

Section: The Gross-pitaevskii-einstein Equationsmentioning

confidence: 99%

See 1 more Smart Citation

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

Suárez

2015

J. Phys.: Conf. Ser.

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...