Transition from Inflation to Dark Energy in Superfluid Vacuum Theory

Transition from Inflation to Dark Energy in Superfluid Vacuum Theory

Konstantin G. Zloshchastiev, Institute of Systems Science, Durban University of Technology, Durban, South Africa.

Quantum Reports 2025, 7, 7. Received 8 January 2025; revised 31 January 2025; accepted 6 February 2025; published 8 February 2025. Academic editor: Orlando Luongo.

Abstract

The laminar constant-velocity superflow of a physical vacuum modelled by logarithmic quantum Bose liquid is considered. We demonstrate that this three-dimensional non-relativistic quantum flow generates a four-dimensional relativistic quinton system, which comprises the dilaton and quintom (a combination of the quintessence and tachyonic phantom fields); all three fields are thus shown to be projections of the dynamical evolution of superfluid vacuum density and its fluctuations onto the measuring apparatus of a relativistic observer. The unified model describes the transition from the inflationary period in the early universe to the contemporary accelerating expansion of the universe, commonly referred to as the "dark energy" period. The quintessence and tachyonic scalar components of the derived model turn out to be non-minimally coupled, which is a hitherto unexplored generalization of cosmological phantom models.

Keywords: quantum gravity; cosmology; superfluid vacuum; inflation; dark energy.

1. Introduction

The period of cosmological inflation, which occurred in the early universe, was characterized by the expansion of space at an exponential rate. Most popular theories of this inflation are constructed by non-minimal coupling of the scalar field, called dilaton (inflaton, in cosmological terminology), to Einstein gravity. It was believed that the accelerated expansion of the universe ended a long time ago and was replaced by the non-accelerated expansion, commonly referred to as the Big Bang, during which all known elementary particles, including baryons and cosmic microwave background photons, were formed.

However, it was discovered relatively recently that after the inflationary period has ended, and all known elementary particles have been formed, the universe is still continuing to expand with acceleration, only at a slower rate. This re-acceleration period is commonly referred to as the dark energy (DE) epoch. Following this discovery, various theories were proposed to solve the dark energy and dark matter (DM) problems, the Lambda-CDM model being probably the most popular among them.

A number of low-redshift observations later revealed that there are discrepancies between the values of the Hubble parameter at the present time from observations of Cepheids in the Large Magellanic Cloud, the gravitational lensing of quasar measurement, and the value predicted by the Lambda-CDM model using Planck CMB data. This phenomenon, commonly referred to as Hubble tension, posed additional challenges for the Lambda-CDM model, and some of them have not yet been resolved, to the best of our knowledge.

This left the question of a complete cosmological model open once again, not to mention that the Lambda-CDM model alone does not explain the nature and origin of dark matter per se but serves as a phenomenological approach and a curve-fitting summary of astronomical data. Another currently open question is how to explain the transition from inflation to the DE period, because they are usually described by very different theories: scalar–tensor gravities on the inflation side and a plethora of models on the DE side. If the dilaton/inflaton field did exist in the early universe, then what happened to it at later times? If it was "used up" to produce the conventional matter, then how did the fields, which generate the "dark energy" and "dark matter" phenomena we currently observe, appear? Is it possible to construct a cosmological model which would not only be a unified model of inflationary and dark epochs but also originated from a theory of quantum gravity itself?

In this paper, we propose to answer these questions by using the superfluid vacuum theory (SVT), which is the theory of physical vacuum, and a theory of quantum gravity at the same time.

This paper is organized as follows. In the next section, we give a brief review of the superfluid vacuum theory based on the logarithmic liquid model. In Section 3, we consider a cosmological model which arises from the logarithmic superfluid vacuum theory, assuming a simple superflow (laminar and constant velocity) of the physical vacuum. In Section 4, we demonstrate a transition from the dilaton-driven inflation to the cosmological expansion driven by one of the candidates for the dark energy, quintom. We show that even the simplest laminar superflow generates the quinton system, which unifies the inflaton and quintom (quintessence field coupled to tachyonic phantom) models. In Section 5, we propose the generalized quinton model of dark energy and study its basic properties. Conclusions are drawn in Section 6.

2. Superfluid Vacuum Theory

According to the SVT paradigm, a physical vacuum is a quantum liquid with suppressed dissipative fluctuations (superfluid) "living" in three-dimensional Euclidean space, whereas four-dimensional curved spacetime and Lorentz symmetry are induced phenomena, occurring through so-called superfluid–spacetime correspondence. This theory is, in fact, a framework for constructing models of superfluid vacuum by assuming one or the other structure and dynamics thereof.

Logarithmic nonlinearity naturally occurs in the theory of laboratory quantum liquids, such as Bose–Einstein condensates of alkali atoms and helium superfluid, where it provides a more accurate fitting of experimental curves and even resolves certain puzzles. This motivates us to use this nonlinearity to describe the background superfluid as well.

Let us introduce the state vector and the wavefunction in a position representation, which obeys the normalization condition (Equation 1: the volume integral of the squared modulus of the wavefunction, that is of the density, equals the total mass M, itself the constituent particle mass m times the particle number N, and is positive), where M and V are the total mass and volume of the system, and m and N are the mass and number of constituent particles; here and in what follows, angle brackets indicate Dirac's bra–ket notation.

We assume that the liquid is described by the state vector whose dynamics obey the logarithmic Schrödinger equation (Equation 2: the time derivative of the wavefunction is driven by the usual kinetic term, an external potential, and a logarithmic term proportional to the logarithm of the density divided by a critical density), where b, the critical density, and the ratio of the reduced Planck constant to the particle mass are real-valued parameters, and the external potential is a function of position and time. For brevity, we assume the external potential is zero.

One can show that the inviscid flow of the logarithmic liquid is observed as four-dimensional curved spacetime by an observer who can operate only with the small-amplitude low-momentum fluctuations in this fluid (in what follows, referred to as small fluctuations). In other words, Lorentz symmetry is not an exact symmetry in superfluid vacuum theory, but an induced effect and approximation. The mapping which relates these two pictures, quantum three-dimensional Euclidean and classical four-dimensional relativistic, is the superfluid–spacetime correspondence mentioned at the beginning of this section.

These small fluctuations are observed by the above-mentioned observer as relativistic particles, transforming according to the irreducible representations of the Poincaré group, for which reason this observer is referred to as the R(elativistic)-observer. The approach thus has two types of observers: the F(ull)-observer who can "see" the vacuum as three-dimensional quantum fluid, and the R-observer who is unable to observe any underlying Euclidean objects or processes, but observes four-dimensional relativistic phenomena instead.

In particular, the matter observed by the R-observer is defined, up to an overall factor, by the induced stress–energy tensor (Equation 3: the induced stress–energy tensor is proportional to the Einstein tensor built from the induced metric, that is the Ricci tensor minus half the metric times the scalar curvature), where the Ricci tensor and scalar curvature are derived from the induced metric, whereas the latter comes about as a result of superfluid dynamics via the superfluid–spacetime correspondence. The right-hand side of the definition depends on a theory of gravity which one assumes; here, we adhere to the canonical GR-type one, without higher-order Riemannian terms, topological invariants, torsion, et cetera.

Apart from explaining the occurrence of the Lorentz symmetry and relativistic phenomena in nature, superfluid vacuum theory is a well-defined quantum theory (with respect to the F-observer); therefore, it can be regarded as a theory of quantum gravity observed by the R-observer. The consistent workflow is not to regard Lorentz-covariant gravity as an effective theory for macroscopic measurements by the R-observer, but to quantize the underlying Euclidean superfluid, and then use the above-mentioned superfluid–spacetime correspondence as a "dictionary" to translate the outcomes into the R-observer's language.

In most cases, however, one does not know the wavefunction of the vacuum but does know the energy–momentum tensor; therefore, one must reverse-engineer it from the energy–momentum tensor. In this reverse workflow, the inverse superfluid–spacetime correspondence acts as a gravity quantization procedure, because it delivers a transition from the (classical) metric to the (quantum-mechanical) wavefunction.

One of the predictions of the logarithmic SVT is the formula for the speed of light (Equation 4: the squared speed of light in the limit of small fluctuations goes as the reduced Planck constant divided by twice the particle mass, multiplied by the difference between the eigenfrequency of the vacuum state and the logarithmic coupling b), where the eigenfrequency belongs to a given quantum state of the physical vacuum. Logarithmic nonlinearity plays a crucial role here, because if one starts not with the logarithmic equation but with any other nonlinear Schrödinger equation of the general class, then the speed of light would no longer have a constant (independent of density) limit, because the function that occurs in the derivation of the speed of light is constant if and only if the nonlinearity is a logarithm. Such a limit is necessary for the compatibility of SVT with relativity postulates in the small-fluctuations regime, and also for defining the fundamental constant of the speed of light in vacuum.

3. SVT Cosmology

Within the framework of superfluid vacuum theory, let us consider the global flow of superfluid vacuum absent of any distortions, and its observational consequences for R-observers.

In the simplest possible case, the phase of a superfluid wavefunction is a linear function with respect to spatial coordinates and time. This corresponds to a laminar constant-velocity flow in Euclidean space along one of directions, if viewed by F-observers. The R-observers, however, would see a totally different picture, according to superfluid–spacetime correspondence. They will discover themselves, by measuring the trajectories of probe particles, as "living" in a conformally flat four-dimensional spacetime. In the leading-order approximation, their metric can be written as the Minkowski metric multiplied by a conformal factor equal to the superfluid density, that is to the squared modulus of the wavefunction (Equation 5). For the R-observer, this metric is defined up to an overall factor, which determines the choice of physical frame, but here we assume for simplicity that this factor is one. The logarithmic nonlinearity is crucial for this conformal flatness — basically, due to a constant value of the speed of light.

According to the Petrov classification, the class of conformally flat spacetimes includes all universes with acceleration, where they differ from each other by their conformal factors; there is an approach to the study of cosmology in these coordinates alone. For example, de Sitter spacetime can be written in this conformal form with a density that falls off as the inverse square of conformal time measured from a reference instant (Equation 6), where conformal time in this case coincides with the Euclidean time of the F-observer.

In this case, R-observers find themselves inside a four-dimensional de Sitter spacetime which expands exponentially, whereas the F-observer "sees" the homogeneous three-dimensional superfluid with quadratically decreasing density as time passes. Note that singularity exists for the R-observer, when the metric's conformal factor vanishes or diverges, but not for the F-observer, because the infinite value of superfluid density is disallowed by the normalization condition, whereas the zero value is regular and asymptotic. This illustrates our earlier remarks about superfluid vacuum theory being well defined as a theory of gravity: spacetime singularities menace its small-fluctuation (relativistic) limit, but not the full underlying theory.

From the metric, using the definition of the induced stress–energy tensor and the conformal-transformation formulae of the appendix in one of the intermediate steps, one can reverse-engineer the basic Lorentz-invariant action functional describing the gravitational interaction experienced by R-observers. One then obtains the dilaton action (Equation 7, written in Planck units), in which the Ricci scalar is multiplied by an exponential of the dilaton field, a kinetic term for the dilaton appears with a fixed coefficient, and a constant topological term is added. Here the dilaton field is the logarithm of the ratio of the superfluid background density to the critical density, the constant term is a reference value for counting the energy of a scalar field, and the dimension parameter equals two in four dimensions.

One can see that the background superfluid induces, in the R-observer's picture, not only spacetime but also the scalar field.

4. The Transition

The dilaton model can be directly applied to early-universe cosmology because it can describe exponential expansion during the inflationary epoch, for instance of the de Sitter type; it also explains the origin of the dilaton field from superfluid vacuum density.

Let us assume that the R-observer is in the middle of the inflation era, say, described by the de Sitter universe. Then the F-observer observes during the same period of time that the background density of the superfluid vacuum decreases as time passes.

This means that at some stage of evolution, fluctuations in density (i.e., the wavefunction's amplitude), which are always present in a quantum realm, become no longer negligible, although still small if compared to the background value. From the viewpoint of the R-observer, it means that the Lagrangian acquires small corrections, which can break the original symmetry.

We thus assume the perturbation (Equation 8: the density is the background density plus a fluctuation whose magnitude is much smaller than the background, while the derivatives of the fluctuation and of the background are of the same order of magnitude). In this approximation, in the leading order, the perturbation of the model yields an action (Equation 9) which, once the two combinations are named, is written in terms of a new pair of relativistic scalar fields: the quintessence field, proportional to the logarithm of the ratio of background density to critical density, and a second field proportional to the full density including its fluctuation (Equation 10).

Furthermore, to extract more physical information from this action, let us rewrite it in the Einstein frame. Under a conformal transformation of the metric by the squared ratio of background density to critical density (Equation 11), and using the appendix formula for the scalar curvature, the action transforms into a form (Equation 12) in which the Ricci scalar appears alone, the quintessence field has an ordinary kinetic term, and the second field has a kinetic term multiplied by an exponential of the quintessence field, together with an exponential potential.

In this form, induced gravity action reveals an important feature of the model: the kinetic couplings for the two scalars have opposite signs, indicating that if one of them is bradyonic then the other one must be tachyonic.

The tachyon occurrence cannot be explained within the framework of the "orthodox" theory of relativity, because it would require drastic changes to its postulates and mathematical structure. Within the SVT framework, the explanation is rather natural: at a certain stage of evolution, when superfluid vacuum fluctuations became sufficiently large, the original scalar field decayed into the quintessence and phantom fields, with the latter being tachyonic.

Whereas the original dilaton field is instrumental in the inflationary models of the early universe, the combination of the quintessence and phantom fields, often referred to as the quintom, can be used to describe the accelerated expansion occurring nowadays, and is regarded as a form of "dark energy". Quintom cosmology is also instrumental in explaining the Hubble tension mentioned in the Introduction.

One can notice the crucial difference between the conventional quintom cosmology, where kinetic couplings of the scalars are postulated to be constant, and the action derived here, where the kinetic coupling of the phantom field is a function of the quintessence field. This coupling ensures that at large positive values of quintessence (in the F-observer's picture, it corresponds to the background density becoming much larger than the critical value), the phantom decouples from the system, thus making the latter purely quintessential. At small values of the quintessence field, one recovers the plain quintom cosmological model.

Another effect is the transition between the conformal and Einstein frames during the dilaton–quintom transition. In scalar–tensor theories of gravity, the question of which physical frame is physical is always a good one. In this model, the Einstein-frame metric is more instrumental because it explicitly takes into account the tachyonic nature of the phantom field, whereas the conformal-frame metric is more suitable when dealing with the dilaton-driven inflationary period. Both frames momentarily coincide in the instant point where the dilaton and quintessence fields vanish and induced spacetime becomes empty in the R-observer picture; in the F-observer picture, this corresponds to the non-perturbed superfluid density reaching its critical value.

5. Quinton Model of Dark Energy

Using the Einstein-frame action as a starting point, let us formulate a general model of a current-epoch cosmology motivated by superfluid vacuum theory. Restoring the Einstein gravitational constant, we write the quinton action (Equation 13): the Einstein–Hilbert term, an ordinary kinetic term for the quintessence field, a kinetic term for the phantom field multiplied by an exponential of the quintessence field with a free rate constant, an exponential potential, an ad hoc potential perturbation depending on both fields, and a matter action.

The matter action is added to account for the other matter and radiation content of the universe which was generated during the inflaton–quinton transition. In this action, we added the scalar potential perturbation, which can be chosen ad hoc, as is common in cosmological models involving scalar fields. We also relaxed the non-minimal coupling's rate constant to a free scale parameter of the model. These generalizations are expected to account for self- and mutual interactions of scalar fields and spacetime geometry with quantum matter and radiation, which inevitably occur. For the same reason, we also assume that the matter Lagrangian density can depend, in general, not only on the matter's fields but also on the quinton fields.

Note that the quintessence and phantom fields are now non-minimally coupled, for which reason their effects cannot be separated from each other as clearly as before. In the limit where the quintessence field goes to zero, one obtains the conventional quintom cosmology action, but otherwise the system's dynamics become more complicated. To begin with, the scalar field equations turn out to be significantly entangled (Equations 14 and 15), and the non-minimal coupling couples the quintessence field component to the non-quinton matter if the latter interacts with the tachyonic component.

Furthermore, by varying the action with respect to the metric, we obtain the Einstein equations of motion sourced by the sum of two stress–energy tensors (Equation 17): that of the non-quinton matter and radiation, and that of the quinton (Equation 18), which carries the ordinary quintessence contribution minus the exponentially coupled phantom contribution, together with the potential.

Let us resort now to a special case of the spatially flat Friedmann–Lemaître–Robertson–Walker geometry with time-dependent fields (Equation 19), for which the equations of motion reduce to a system of ordinary differential equations (Equations 20 to 23): the Friedmann equation for the squared Hubble parameter in terms of the matter and quinton densities, its time derivative in terms of densities and pressures, and two second-order equations for the quintessence and phantom fields with Hubble friction, the non-minimal coupling term and the potential gradients.

Here the dot denotes a derivative with respect to time, and densities and pressures are defined via diagonal components of their respective stress–energy tensors, so that the quinton density is the quintessence kinetic term minus the exponentially coupled phantom kinetic term plus the potential, and the quinton pressure is the same kinetic combination minus the potential (Equations 24 and 25). The Hubble parameter is, as usual, the time derivative of the scale factor divided by the scale factor.

In this model, dark energy is attributed to the quinton — its density and pressure are the quinton density and pressure (Equation 26) — and its equation of state is the ratio of that pressure to that density (Equation 27) in the perfect-fluid approximation; an exact form of the equation of state can be obtained by substituting the found solutions into these formulae and eliminating the time variable from the resulting equations.

Further study of this model depends on specifying properties of the non-quinton matter and its interaction with the quinton, which is an extensive topic on its own.

6. Discussion and Conclusions

In this report, we considered the laminar flow with constant velocity of the physical vacuum modelled by logarithmic superfluid. We demonstrated that this three-dimensional non-relativistic quantum flow generates a four-dimensional Lorentz-symmetric "quinton" system, which consists of the dilaton and the quintom, a combination of the quintessence and tachyonic phantom fields, and explains a transition between them.

All three fields were shown to be projections of the Euclidean dynamical evolution of superfluid vacuum density and its fluctuations onto the measuring apparatus of a relativistic observer; their four-dimensional action functionals were not postulated but derived from a single quantum mechanical theory. This unified cosmological model describes the transition from the inflationary period in the early universe to the contemporary accelerating expansion of the universe, commonly referred to as the "dark energy" period.

It should be emphasized that the model, and even its "perturbed" generalization, is obviously the simplest possible one, because it neglects any large distortions in the laminar flow of the background superfluid (in the F-observer picture), which would otherwise induce and introduce additional fields and terms in a Lagrangian. Nevertheless, even such a simple kind of flow is already capable of resolving, in a unified way, at least three important problems in the modern theory of gravity and cosmology: the emergence of spacetimes with large-scale accelerated expansion leading to the occurrence of the inflationary period in the early universe, a generation mechanism for long-range scalar fields (which are not otherwise predicted by the Standard Model of particle physics), and the transition from the inflationary era to the current "dark energy" epoch.

One can also recall that the logarithmic model in the weak-gravity limit with inhomogeneous and rotationally symmetric superfluid density (which is another limit of SVT different from a homogeneous laminar flow) quantitatively explains the non-Keplerian behaviour of rotating curves in galaxies: the fittings closely correspond with observational data, even for those galaxies whose rotation velocity profiles do not have flat asymptotics. These effects are usually attributed to the phenomenon known as "dark matter".

To conclude, we showed that the dilaton-driven inflation and the effects attributed to "dark energy" can be viewed as different manifestations of the same object and a kind of matter, superfluid vacuum.

Funding and acknowledgements

This research was funded by the Department of Higher Education and Training of South Africa and in part by the National Research Foundation of South Africa (Grant Nos. 95965 and 132202). No new data were created in this study. The author is grateful to L. Tannukij, and thanks P. Stannard for proofreading; the author acknowledges an invitation and a full waiver on article processing charges by MDPI. The author declares no conflicts of interest.

(Appendix A, which collects the conformal-transformation formulae for the Christoffel symbols, Ricci tensor, scalar curvature and Einstein tensor, and the reference list are omitted for length; the complete text is at the source.)