K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models

Abstract

We determine the k-essence Lagrangian of a relativistic barotropic fluid. The equation of state of the fluid can be specified in different manners depending on whether the pressure is expressed in terms of the energy density (model I), the rest-mass density (model II), or the pseudo rest-mass density for a complex scalar field in the Thomas-Fermi approximation (model III). In the nonrelativistic limit, these three formulations coincide. In the relativistic regime, they lead to different models that we study exhaustively. We provide general results valid for an arbitrary equation of state and show how the different models are connected to each other. For illustration, we specifically consider polytropic and logotropic dark fluids that have been proposed as unified dark matter and dark energy models. We recover the Born-Infeld action of the Chaplygin gas in models I and III and obtain the explicit expression of the reduced action of the logotropic dark fluid in models II and III. We also derive the two-fluid representation of the Chaplygin and logotropic models. Our general formalism can be applied to many other situations such as Bose-Einstein condensates with a |φ|4 (or more general) self-interaction, dark matter superfluids, and mixed models.

Jeans Instability of Dissipative Self-Gravitating Bose–Einstein Condensates with Repulsive or Attractive Self-Interaction: Application to Dark Matter

“…In general, ρ m = ρ except (i) for a noninteracting SF (V = 0), (ii) when V is constant, corresponding to the ΛFDM model (see Appendix E of [359]), (iii) and in the nonrelativistic limit c → +∞. Remark: More general results valid beyond the TF approximation are given in [87,360].…”

“…We consider a relativistic complex SF ϕ(x µ ) = ϕ(x, y, z, t) which is a continuous function of space and time. It can represent the wavefunction of a relativistic BEC [20,87]. The total action of the system, which is the sum of the Einstein-Hilbert action of general relativity + the action of the SF, can be written as…”

“…Therefore, according to Eqs. (B35) and (B36), the energy density and the pressure derived from the Lagrangian (B28) are given by [87,347] = ρV tot (ρ) + V tot (ρ) = ρc 2 + ρV (ρ) + V (ρ), (B37)…”

“…This relativistic equation of state P ( ) was first obtained by Colpi et al [58] in the context of boson stars. It was studied in detail by Chavanis and Harko [86] in connection to general relativistic BEC stars and by Li et al [364] and Suárez and Chavanis [347] in a BECDM cosmology (see also [87,298,360,365,366] for related studies). This equation of state reduces to that of an n = 1 polytrope [see Eq.…”

“…Nontopological solitons [12-14, 54, 55, 60, 61, 113-137] may be regarded as the nongravitational precursors of 8 See Ref. [87] and Appendix B for a detailed derivation of this equation of state. The hydrodynamic decription of boson stars with a repulsive self-interaction in the TF limit has been studied in detail by Chavanis and Harko [86] in the frawework of the Oppenheimer-Volkoff equations.…”