Superfluid Rayleigh–Plesset extension of FLRW cosmology
Introduction
Hydrodynamic phenomena have long served as rich-in-wisdom analogies for astrophysics and cosmology. The Mach shock wave context of the black hole evaporation introduced by Unruh [1] four decades ago is just a famous example which attracted a lot of attention, but clearly it is not the only case. Superfluid hydrodynamics with its embodiments as Bose–Einstein condensates and superlight bosons is another notorious example in strong association with galactic halos and the concept of dark matter [2], [3], [4], whereas various other superfluid analogs of quantum field cosmic phenomena can be found in a review by Volovik [5]. Let us also recall that the idea of a superfluid universe has been central in the works of the late Professor Kerson Huang who summarized the publications with his collaborators along many years in an inspiring book on this topic [6].
On the other hand, in the latter case of cosmology, the fluid and classical mechanics analogies start to abound [7], [8], [9], [10]. Among the fluid analogies, a less known, but equally promising one has been recently introduced by Rousseaux and Mancas [11], that we address here from the superfluid point of view. It is based on the similarities between the Rayleigh-Plesset (RP) equation [12], [13], which is a reshaped form of the Navier–Stokes equations describing the dynamical evolution of bubbles in fluids [14], and the cosmological dynamics as introduced by Friedmann, Lemaître, Robertson, and Walker almost one century ago. The modern compelling components of the universe, such as dark energy and dark matter, may require the addition of supplementary terms to the latter equations. This is what Banerjee et al. [15] have recently done by adding a fourth power term in the inverse of the scale factor as due to dark energy to enhance the late expansion rate of the universe, while Rousseaux and Mancas [11] speculated on the possible addition of more viscosity terms in the cosmological evolution, motivated by the case of the RP equation for which the addition of this kind of terms is a common usage, as revealed by some references cited in [11].
Extensions of the FLRW cosmology are generically motivated by the dark energy era [16], but in this paper, motivated by [11], we investigate in an explicit way the addition of polynomial nonlinearities to the usual FLRW evolution equation. In particular, we are interested not only in the common cubic power of the usual RP equation, but also in the effects of an additional nonlinear power of order six which occurs in the Rayleigh-Plesset framework developed in the study of multielectron bubbles (henceforce MEBs) in superfluid Helium-4 laboratory physics [17], [18], [19]. Such a term takes into account the repulsive Coulomb interaction of the electrons forming a thin layer at the surface of the bubbles. By considering such a phenomenological detail in theoretical cosmology, we place ourselves from the beginning in a particular case of the class of inhomogeneous fluid models of the universe which are widely discussed in the literature, see e.g. [20] and references therein. However, we believe that the detail that we emphasize here may be important not only for the laboratory phenomenology, but also in cosmology and astrophysics.
We also remark that the addition of power nonlinearities is well known in other research areas. It is sufficient to recall that a quintic power is added to the cubic nonlinear Schrödinger equation to handle the rich phenomenology of soliton propagation in nonlinear waveguides, and the case of laboratory BECs, where the cubic–quintic Ginzburg–Landau equation has been for several decades the standard equation for modeling the BEC behavior. Differently from the case of the single cubic nonlinearity which is integrable by the inverse scattering method, the solutions of the cubic quintic cases can develop blow-ups, finite time singularities where the amplitude of solution reaches infinity in a finite time [21]. The FLRW cosmology is not far from these areas in the dynamical nonlinear perspective. The standard FLRW equations with physical content characterized by normal (not ‘super-negative’-dark [22], [23]) forms of equation of state are integrable, see e.g. [24] and the references to textbooks therein, but adding more power nonlinearities can generate future-in-time singularities.
The rest of the paper is focused on obtaining some representative cosmological scaling factors of this extended FLRW model for zero cosmological constant which are obtained as solutions of the Weierstrass elliptic equation. In Section 2, we establish the nonlinear differential equation corresponding to this generalized FLRW model. In the case of the coasting universe, by a change of variable of the Sundman type, this generalized FLRW equation is turned into a Weierstrass elliptic equation whose standard mathematical setup is recalled. In Section 3, we present some sets of scale factors of the universe obtained as parametric solutions of this model in the form of rational expressions of Weierstrass elliptic functions including the special and degenerate cases [25]. In Section 4, we work in the standard conformal time variable in which presumably closed-form results cannot be obtained, and display some numerical results for the vacuum case and the same coasting case. Finally, Section 5 contains some conclusions and further remarks on the results.
Section snippets
The FLRW equation with superfluid Rayleigh-Plesset terms and the elliptic equation
The standard RP equation for the time-evolving radius, , of a bubble in a fluid reads to which a sixth power term in should be added in the case of MEBs. The symbols in the coefficients are: – the liquid density, – the liquid kinematic viscosity, – the surface tension, and – the relative pressure drop at the bubble with respect to the pressure in the liquid at infinity, and the dot denotes the time derivative. For this
The parametric solutions
We are now ready to present a whole wealth of parametric solutions of this extended FLRW model using as parameter. Although all possible cases can be enlisted in a systematic manner, for a superfluid universe or one containing superfluid bubbles one should consider the simultaneous presence of the Laplace pressure and electrostatic terms, i.e., both and should not be zero simultaneously. When both of them are chosen to be zero it corresponds to the standard FLRW cosmology. The case when
The conformal time approach
The rational solutions in terms of Weierstrass elliptic functions, and degenerate cases that we introduced above owe their existence to the usage of the Sundman time variable. They have the peculiarity that they belong only to the case and besides the majority of them occur for the integration constant . This is due to the fact that the integrating factor method works only for the three halves value of the parameter . It is natural to ask what happens for other values of . In this
Discussion and conclusion
We have introduced and analyzed in some detail a nonlinear differential equation for a modified FLRW cosmology for zero cosmological constant, which is based on an analogy with a Rayleigh-Plesset equation encountered in the area of multielectron bubbles in superfluid He-4, a well studied laboratory phenomenon. At the speculative level, one can assume the existence of equivalents of such bubbles even at astrophysical scales, as a kind of microcavities hovering in the galactic halos where they
CRediT authorship contribution statement
Haret C. Rosu: Writing - original draft, Supervision, Formal analysis. Stefan C. Mancas: Calculations, Writing - review & editing. Chun-Chung Hsieh: Formal analysis, Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
We wish to thank the anonymous referee for important remarks that helped us to improve the content of this paper.
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