Quantum Phase Space Description of a Cosmological Minimal Massive Bigravity Model
Julio César Vera-Hernández
备注：
This is a post-peer-review, pre-copyedit version of an article published in General Relativity and Gravitation. The final authenticated version is available online at: https://doi.org/10.1007/s10714-021-02822-2 25 pages, 10 figures
Bimetric gravity theories describes gravitational interactions in the presence of an extra spin-2 field. The Hassan-Rosen (HR) nonlinear massive minimal bigravity theory is a ghost-free bimetric theory formulated with respect a flat, dynamical reference metric. In this work the deformation quantization formalism is applied to a HR cosmological model in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in the minisuperspace. The Friedmann-Lemaître-Robertson-Walker (FLRW) model with flat metrics is worked out and the computation of the Wigner functions for theHartle-Hawking, Vilenkin and Linde wavefunctions are done numerically and, in the Hartle-Hawking case, also analytically. From the stability analysis in the quantum minisuper phase space it is found an interpretation of the mass of graviton as an emergent cosmological constant and as a measure of the deviation of classical behavior of the Wigner functions. Also, from the Hartle-Hawking case, an interesting relation between the curvature and the mass of graviton in a cusp catastrophe surface is discussed
We generalize the scalar tensor bigravity models to the non-minimal kinetic coupling scalar tensor bigravity models with two scalar fields whose kinetic terms are non-minimally coupled to two Einstein tensors constructed by two metrics. We show that a broad class of expanding universes can be explained by some solutions of this model. Then, we study the stability issue of the solutions by means of imposing homogeneous perturbation on the equations of motion and extract the stable solutions.
We study the future cosmological singularities in the framework of massive gravity and minimal massive bigravity theory. In this regards, we consider the possible classes of finite-time future singularities such as sudden, big rip, big freeze and big brake singularities in the massive universe. In dRGT model with an open expanding universe we obtain the sudden singularity in the future at a finite-time which generally without taking account of any particular realistic equation of state, is not avoidable and except the fluid density, all dynamical physical quantities such as pressure approach t…
We construct the complete set of metric-configuration solutions of the ghost-free massive bigravity for the scenario in which the g-metric is the Friedmann-Lemaitre-Robertson-Walker (FLRW) one, and the interaction Lagrangian between the two metrics contributes an effective ideal fluid energy-momentum tensor to the g-metric equations. This set corresponds to the exact background cosmological solution space of the theory.
In this article the phenomenon of degravitation of the cosmological constant is studied in the framework of bigravity. It is demonstrated that despite a sizable value of the cosmological constant its gravitational effect can be only mild. The bigravity framework is chosen for this demonstration as it leads to a consistent, ghost-free theory of massive gravity. We show that degravitation takes place in the limit where the physical graviton is dominantly a gauge invariant metric combination. We present and discuss several phenomenological consequences expected in this regime.
Cosmological perturbations of FRW solutions in ghost free massive bigravity, including also a second matter sector, are studied in detail. At early time, we find that sub horizon exponential instabilities are unavoidable and they lead to a premature departure from the perturbative regime of cosmological perturbations.
In a Friedmann-Robertson-Walker (FRW) space-time background we study the classical cosmological models in the context of recently proposed theory of nonlinear minimal massive bigravity. We show that in the presence of perfect fluid the classical field equations acquire contribution from the massive graviton as a cosmological term which is positive or negative depending on the dynamical competition between two scale factors of bigravity metrics. We obtain the classical field equations for flat and open universes in the ordinary and Schutz representation of perfect fluid. Focusing on the Schutz …
We study the static cosmological solutions and their stability at background level in the framework of massive bigravity theory with Friedmann-Robertson-Walker (FRW) metrics. By the modification proposed in the cosmological equations subject to a perfect fluid we obtain new solutions interpreted as the Einstein static universe. It turns out that the non-vanishing size of initial scale factor of Einstein static universe depends on the non-vanishing three-dimensional spatial curvature of FRW metrics and also the graviton's mass. By dynamical system approach and numerical analysis, we find that t…
We present a comprehensive analysis of classical scalar, vector and tensor cosmological perturbations in ghost-free massive bigravity. In particular, we find the full evolution equations and analytical solutions in a wide range of regimes. We show that there are viable cosmological backgrounds but, as has been found in the literature, these models generally have exponential instabilities in linear perturbation theory. However, it is possible to find stable scalar cosmological perturbations for a very particular choice of parameters. For this stable subclass of models we find that vector and te…
We give a simple derivation of a cosmological bound on the graviton mass for spatially flat FRW solutions in massive gravity with an FRW reference metric and for bigravity theories. This bound comes from the requirement that the kinetic term of the helicity zero mode of the graviton is positive definite. The bound is dependent only on the parameters in the massive gravity potential and the Hubble expansion rate for the two metrics. We derive the decoupling limit of bigravity and FRW massive gravity, and use this to give an independent derivation of the cosmological bound. We recover our previo…
We investigate static and spherically symmetric solutions in the Minimal Theory of Bigravity (MTBG). First, we show that a pair of Schwarzschild-de Sitter spacetimes with different cosmological constants and black hole masses written in the spatially-flat Gullstrand-Painlevé (GP) coordinates is a solution in the self-accelerating branch of MTBG, while it cannot be a solution in the normal branch. We then illustrate how Schwarzschild-de Sitter solutions can become compatible with the normal branch when using different coordinates. We also confirm that the self-accelerating branch of MTBG admits…
A complete classification of exact solutions of ghost-free, massive bigravity is derived which enables the dynamical decoupling of the background, and the foreground metrics. The general decoupling solution space of the two metrics is constructed. Within this branch of the solution space the foreground metric theory becomes general relativity (GR) with an additional effective cosmological constant, and the background metric dynamics is governed by plain GR.
The recent discovery of a ghost-free, non-linear extension of the Fierz-Pauli theory of massive gravity, and its bigravity formulation, introduced new possibilities of interpreting cosmological observations, in particular, the apparent late-time accelerated expansion of the Universe. Here we discuss such possibilities by studying the background cosmology of the model and comparing its predictions to different cosmological measurements. We place constraints on the model parameters through an extensive statistical analysis of the model, and compare its viability to that of the standard model of …
Within the theory of the ghost-free bigravity, we present the most general cosmological solution for which the physical metric is homogeneous and isotropic, while the second metric is inhomogeneous. The solution includes a matter source and exists for any values of the theory parameters. The physical metric describes a universe with the late time acceleration due to the effective cosmological term mimicked by the graviton mass. When perturbed, this universe should rest approximately homogeneous and isotropic in space regions small compared to the graviton Compton length. In the limit where the…
We study higher dimensional scenarios of massive bigravity, which is a very interesting extension of nonlinear massive gravity since its reference metric is assumed to be full dynamical. In particular, the Einstein field equations along with the following constraint equations for both physical and reference metrics of a five-dimensional massive bigravity will be addressed. Then, we study some well-known cosmological spacetimes such as the Friedmann-Lemaitre-Robertson-Walker, Bianchi type I, and Schwarzschild-Tangherlini metrics for the five-dimensional massive bigravity. As a result, we find t…
In de Rham-Gabadadze-Tolley (dRGT) massive gravity and bigravity, a non-minimal matter coupling involving both metrics generically re-introduces the Boulware--Deser (BD) ghost. A non-minimal matter coupling via a simple, yet specific composite metric has been proposed, which eliminates the BD ghost below the strong coupling scale. Working explicitly in the metric formulation and for arbitrary spacetime dimensions, we show that this composite metric is the unique consistent non-minimal matter coupling below the strong coupling scale, which emerges out of two diagnostics, namely, absence of Ostr…
We explore the cosmological solutions of classes of non-linear bigravity theories. These theories are defined by effective four-dimensional Lagrangians describing the coupled dynamics of two metric tensors, and containing, in the linearized limit, both a massless graviton and an ultralight one. We focus on two paradigmatic cases: the case where the coupling between the two metrics is given by a Pauli-Fierz-type mass potential, and the case where this coupling derives from five-dimensional brane constructions. We find that cosmological evolutions in bigravity theories can be described in terms …
In this paper, in the framework of massive bigravity, we study all possible cosmic evolutions by using a method in which the modified Friedmann equation is written in a form where the scale factor evolves like the motion of a particle under a "potential". Massive bigravity provides this potential with the most general mass interaction term which can create new circumstances to find different kinds of cosmological evolutions in the early universe. We classify all possible cosmic evolutions according to the classifications of the energy density as dust, radiation and dust with phantom. Oscillati…
Massive bigravity, a theoretically consistent modification of general relativity with an additional dynamical rank two tensor, successfully describes the observed accelerated expansion of the Universe without a cosmological constant. Recent analyses of perturbations around a cosmological background have revealed power law instabilities in both scalar and tensor perturbations, motivating an analysis of the initial conditions, evolution, and cosmological observables to determine the viability of these theories. In this paper we focus on the tensor sector, and study a primordial stochastic gravit…
We present a survey of the known cosmological and black hole solutions in ghost-free bigravity and massive gravity theories. These can be divided into three classes. First, there are solutions with proportional metrics, which are the same as in General Relativity with a cosmological term, which can be positive, negative or zero. Secondly, for spherically symmetric systems, there are solutions with non-bidiagonal metrics. The g-metric fulfills Einstein equations with a positive cosmological term and a matter source, while the f-metric is anti-de Sitter. The third class contains solutions with b…
We study spherically symmetric black hole solutions in a four-parameter Einstein-Cartan-type class of theories, called "torsion bigravity". These theories offer a geometric framework (with a metric and an independent torsionfull connection) for a modification of Einstein's theory that has the same spectrum as bimetric gravity models. In addition to an Einsteinlike massless spin-2 excitation, there is a massive spin-2 one (of range $κ^{-1}$) coming from the torsion sector, rather than from a second metric. We prove the existence of three broad classes of spherically-symmetric black hole solutio…
Following the path of minimalism in alternative theories of gravity, we construct the "Minimal Theory of Bigravity" (MTBG), a theory of two interacting spin-2 fields that propagates only four local degrees of freedom instead of the usual seven ones and that allows for the same homogeneous and isotropic cosmological solutions as in Hassan-Rosen bigravity (HRBG). Starting from a precursor theory that propagates six local degrees of freedom, we carefully choose additional constraints to eliminate two of them to construct the theory. Investigating the cosmology of MTBG, we find that it accommodate…
In this paper, we propose a toy model of the renormalizable theory describing massive spin two field. Although the model is renormalizable, we show that the model contains ghost. The coupling of the theory with gravity can be regarded as a new kind of bimetric gravity or bigravity. We show that the massive spin two field plays the role of the cosmological constant.
We study the accelerating cosmology in massive $F(R)$ bigravity via the reconstruction scheme. The consistent solution of the FRW equations is presented: it includes Big and Little Rip, quintessence, de Sitter and decelerating universes described by the physical $g$ metric while the corresponding solution of the universe described by the reference $f$ metric is also found. It is demonstrated that in general the cosmological singularities of $g$ metric are not always manifested as cosmological singularities of the reference $f$ metric. We give two consistent ways to describe the Big and Little …
We explore trace-anomaly driven inflation in modified gravity. It is explicitly shown that in $T^2$ teleparallel gravity, the de Sitter inflation can occur, although quasi de Sitter inflation happens in $R^2$ gravity. Furthermore, we investigate the influence of the trace anomaly on inflation. It is found that in $f(T)$ gravity, the de Sitter inflation can end because it becomes unstable due to the trace anomaly, whereas also in higher derivative gravity, the de Sitter inflation can be realized and it will be over thanks to the trace anomaly for smaller parameter regions in comparison with tho…
The theory of bigravity offers one of the simplest possibilities to describe a massive graviton while having self-accelerating cosmological solutions without a cosmological constant. However, it has been shown recently that bigravity is affected by early-time fast growing modes on the tensor sector. Here we argue that we can only trust the linear analysis up to when perturbations are in the linear regime and use a cut-off to stop the growing of the metric perturbations. This analysis, although more consistent, still leads to growing tensor modes that are unacceptably large for the theory to be…
Nonlinear, ghost-free massive gravity has two tensor fields; when both are dynamical, the mass of the graviton can lead to cosmic acceleration that agrees with background data, even in the absence of a cosmological constant. Here the question of the stability of linear perturbations in this bimetric theory is examined. Instabilities are presented for several classes of models, and simple criteria for the cosmological stability of massive bigravity are derived. In this way, we identify a particular self-accelerating bigravity model, infinite-branch bigravity (IBB), which exhibits both viable ba…
We compute the one-loop quantum corrections to the interactions between the two metrics of the ghost-free massive bigravity. When considering gravitons running in the loops, we show how the structure of the interactions gets destabilized at the quantum level, exactly in the same way as in its massive gravity limit. A priori one might have expected a better quantum behavior, however the broken diffeomorphism invariance out of the two initial diffeomorphisms in bigravity has similar consequences at the quantum level as the broken diffeomorphism in massive gravity. From lessons of the generated q…
We study a previously largely unexplored branch of homogeneous and isotropic background solutions in ghost-free massive bigravity with consistent double matter coupling. For a certain family of parameters we find `self-inflated' FLRW cosmologies, i.e. solutions with an accelerated early-time period during the radiation-dominated era. In addition, these solutions also display an accelerated late-time period closely mimicking GR with a cosmological constant. Interestingly, within this family, the particular case of $β_1=β_3=0$ gives bouncing cosmologies, where there is an infinite contracting pa…
Massive gravity theories have been developed as viable IR modifications of gravity motivated by dark energy and the problem of the cosmological constant. On the other hand, modified gravity and modified dark matter theories were developed with the aim of solving the problems of standard cold dark matter at galactic scales. Here we propose to adapt the framework of ghost-free massive bigravity theories to reformulate the problem of dark matter at galactic scales. We investigate a promising alternative to dark matter called dipolar dark matter (DDM) in which two different species of dark matter …
We study the cosmic expansion history of massive bigravity with a viable matter coupling which treats both metrics on equal footing. We derive the Friedmann equation for the effective metric through which matter couples to the two metrics, and study its solutions. For certain parameter choices, the background cosmology is identical to that of LCDM. More general parameters yield dynamical dark energy, which can still be in agreement with observations of the expansion history. We study specific parameter choices of interest, including minimal models, maximally-symmetric models, and a candidate p…
This article extends bimetric formulations of massive gravity to make the mass of the graviton to depend on its environment. This minimal extension offers a novel way to reconcile massive gravity with local tests of general relativity without invoking the Vainshtein mechanism. On cosmological scales, it is argued that the model is stable and that it circumvents the Higuchi bound, hence relaxing the constraints on the parameter space. Moreover, with this extension the strong coupling scale is also environmentally dependent in such a way that it is kept sufficiently higher than the expansion rat…
If the amplitude of primordial gravitational waves is measured in the near-future, what could it tell us about bigravity? To address this question, we study massive bigravity theories by focusing on a region in parameter space which is safe from known instabilities. Similarly to investigations on late time constraints, we implicitly assume there is a successful implementation of the Vainshtein mechanism which guarantees that standard cosmological evolution is largely unaffected. We find that viable bigravity models are subject to far less stringent constraints than massive gravity, where there…
The ghost-free theory of massive gravity with two dynamical metrics has been shown to produce viable cosmological expansion, where the late-time acceleration of the Universe is due to the finite range of the gravitational interaction rather than a nonzero cosmological constant. Here the cosmological perturbations are studied in this theory. The full perturbation equations are presented in a general gauge and analyzed, focusing on subhorizon scales in the quasistatic limit during the matter-dominated era. An evolution equation for the matter inhomogeneities and the parameters quantifying the de…
In this work we study the cosmological perturbations in massive bigravity in the presence of non-minimal derivative couplings. For this purpose we consider a specific subclass of Horndeski scalar-tensor interactions that live on the unique composite effective metric. For the viability of the model both metrics have to be dynamical. Nevertheless, the number of allowed kinetic terms is crucial. We adapt to the restriction of having one single kinetic term. After deriving the full set of equations of motion for flat Friedmann-Lemaitre-Robertson-Walker background, we study linear perturbations on …
In the context of massive (bi-)gravity non-minimal matter couplings have been proposed. These couplings are special in the sense that they are free of the Boulware-Deser ghost below the strong coupling scale and can be used consistently as an effective field theory. Furthermore, they enrich the phenomenology of massive gravity. We consider these couplings in the framework of bimetric gravity and study the cosmological implications for background and linear tensor, vector, and scalar perturbations. Previous works have investigated special branch of solutions. Here we perform a complete perturba…
In this paper we study the generation of primordial perturbations in a cosmological setting of bigravity during inflation. We consider a model of bigravity which can reproduce the $Λ$CDM background and large scale structure and a simple model of inflation with a single scalar field and a quadratic potential. Reheating is implemented with a toy-model in which the energy density of the inflaton is entirely dissipated into radiation. We present analytic and numerical results for the evolution of primordial perturbations in this cosmological setting. We find that the amplitude of tensor perturbati…
Massive bigravity models are interesting alternatives to standard cosmology. In most cases, however, these models have been studied for a simplified scenario in which both metrics take homogeneous and isotropic forms [Friedmann-Lemaître-Robertson-Walker (FLRW)] with the same spatial curvatures. The interest to consider more general geometries arises, in particular, in view of the difficulty so far encountered in building stable cosmological solutions with homogeneous and isotropic metrics. Here we consider a number of cases in which the two metrics take more general forms, namely FLRW with dif…
We discuss the causal diagrams of static and spherically symmetric bigravity vacuum solutions, with interacting metrics $f$ and $g$. Such solutions can be classified into type I (or "non-diagonal") and type II (or "diagonal"). The general solution of type I is known, and leads to metrics $f$ and $g$ in the Schwarzschild-(Anti)de Sitter family. The two metrics are not always diagonalizable in the same coordinate system, and the light-cone structure of both metrics can be quite different. In spite of this, we find that causality is preserved, in the sense that closed time-like curves cannot be p…
We study the integrated Sachs-Wolfe (ISW) effect in ghost-free, massive bigravity. We focus on the infinite-branch bigravity (IBB) model which exhibits viable cosmic expansion histories and stable linear perturbations, while the cosmological constant is set to zero and the late-time accelerated expansion of the Universe is due solely to the gravitational interaction terms. The ISW contribution to the CMB auto-correlation power spectrum is predicted, as well as the cross-correlation between the CMB temperature anisotropies and large-scale structure. We use ISW amplitudes as observed by the WMAP…
The possibility to have singular accelerated evolution in the context of $F(R)$ bimetric gravity is investigated. Particularly, we study two singular models of cosmological evolution, one of which is a singular modified version of the Starobinsky $R^2$ inflation model. As we demonstrate, for both models in some cases, the slow-roll parameters become singular at the Type IV singularity, a fact that we interpret as a dynamical instability of the theory under study. This dynamically instability may be an indicator of graceful exit from inflation and we thoroughly discuss this scenario and the int…
The possibility of matter coupling to two metrics at once is considered. This appears natural in the most general ghost-free, bimetric theory of gravity, where it unlocks an additional symmetry with respect to the exchange of the metrics. This double coupling, however, raises the problem of identifying the observables of the theory. It is shown that if the two metrics couple minimally to matter, then there is no physical metric to which all matter would universally couple, and that moreover such an effective metric generically does not exist even for an individual matter species. By studying p…
We study the implications of the recent detection of gravitational waves emitted by a pair of merging neutron stars and their electromagnetic counterpart, events GW170817 and GRB170817A, on the viability of the doubly coupled bimetric models of cosmic evolution, where the two metrics couple directly to matter through a composite, effective metric. We demonstrate that the bounds on the speed of gravitational waves place strong constraints on the doubly coupled models, forcing either the two metrics to be proportional at the background level or the models to become singly coupled. Proportional b…
We consider modified gravity which may describe the early-time inflation and/or late-time cosmic acceleration of the universe. In particular, we discuss the properties of $F(R)$, $F(G)$, string-inspired and scalar-Einstein-Gauss-Bonnet gravities, including their FRW equations and fluid or scalar-tensor description. Simplest accelerating cosmologies are investigated and possibility of unified description of the inflation with dark energy is described. The cosmological reconstruction program which permits to get the requested universe evolution from modified gravity is developed. As some extensi…
We discuss a scale-free model of bigravity, in which the mass parameter of the standard bigravity potential is promoted to a dynamical scalar field. This modification retains the ghost-free bigravity structure, in particular it remains free of the Boulware-Deser ghost. We investigate the theory's interaction structure, focusing on its consistent scaling limits and strong coupling scales. Furthermore we explore the model's quadratic action, both around generic background configurations and paying special attention to cosmological backgrounds and to the associated background evolution. Finally w…
In the former part, we study the gravitational collapse of pressureless dust and find special solutions, where, in both the physical and fiducial sectors, the exterior and interior spacetime geometries are given by the Schwarzschild spacetimes and the Friedmann-Lemaître-Robertson-Walker universes dominated by pressureless dust, respectively, with specific time slicings. In the case where the Lagrange multipliers are trivial and have no jump across the matter interfaces in both the physical and fiducial sectors, the junction conditions across them remain the same as those in general relativity …
We study the background cosmology of the ghost-free, bimetric theory of gravity. We perform an extensive statistical analysis of the model using both frequentist and Bayesian frameworks and employ the constraints on the expansion history of the Universe from the observations of supernovae, the cosmic microwave background and the large scale structure to estimate the model's parameters and test the goodness of the fits. We explore the parameter space of the model with nested sampling to find the best-fit chi-square, obtain the Bayesian evidence, and compute the marginalized posteriors and mean …
As is well known, both massive gravity and bigravity exhibit the linear van Dam-Veltman-Zakharov (vDVZ) discontinuity that is cured classically by the nonlinear Vainshtein mechanism due to certain low scale strongly coupled interactions. Here we show how both the vDVZ and strong coupling problems can be removed by embedding 4D covariant massive gravity into a certain 5D warped geometry. The 4D theory is a nonlinear strongly coupled massive gravity, that is being coupled to a 5D bulk theory that generates a bulk graviton mass via a one loop diagram. This induced mass leads to an additional 4D k…
We implement the method developed in [1] to construct the most general parametrised action for linear cosmological perturbations of bimetric theories of gravity. Specifically, we consider perturbations around a homogeneous and isotropic background, and identify the complete form of the action invariant under diffeomorphism transformations, as well as the number of free parameters characterising this cosmological class of theories. We discuss, in detail, the case without derivative interactions, and compare our results with those found in massive bigravity.
We analyse the late time cosmology and the gravitational properties of doubly coupled bigravity in the constrained vielbein formalism (equivalent to the metric formalism) when the mass of the massive graviton is of the order of the present Hubble rate. We focus on one of the two branches of background cosmology where the ratio between the scale factors of the two metrics is algebraically determined. The Universe evolves from a matter dominated epoch to a dark energy dominated era where the equation of state of dark energy can always be made close to -1 now by appropriately tuning the graviton …
We find the general conditions for viable cosmological solution at the background level in bigravity models. Furthermore, we constrain the parameters by comparing to the Union 2.1 supernovae catalog and identify, in some cases analytically, the best fit parameter or the degeneracy curve among pairs of parameters. We point out that a bimetric model with a single free parameter predicts a simple relation between the equation of state and the density parameter, fits well the supernovae data and is a valid and testable alternative to $Λ$CDM. Additionally, we identify the conditions for a phantom b…
We investigate the models of cosmological inflation in generalized scalar-tensor gravity, which we consider as a source of deviation from de Sitter dynamics in the case of GR. Within the framework of the proposed approach, the exact equations of cosmological dynamics and parameters of cosmological perturbations are obtained.
We show that the combined minimal and non minimal interaction with the gravitational field may produce the generation of a cosmological constant without self-interaction of the scalar field. In the same vein we analyze the existence of states of a scalar field that by a combined interaction of minimal and non minimal coupling with the gravitational field can exhibit an unexpected property, to wit, they are acted on by the gravitational field but do not generate gravitational field. In other words, states that seems to violate the action-reaction principle. We present explicit examples of this …
We present a novel mechanism for generating a Cosmological Constant and suitably sequestering the vacuum contribution to it, so that the eponymous Cosmological Constant problem is avoided. We do so by resorting to a model endowed with a non-minimal coupling between curvature and matter in an appropriately defined relaxed regime, and show that this shares features with both Unimodular gravity as well as a recent proposal to sequester the vacuum contribution through the use of an external term to the action functional.
A homogeneous and isotropic Universe in the framework of nonlinear sigma model with non-minimal coupling to the target space is considered. A two-component model of such a sort is preliminary investigated. Some solutions for this model are given. Perspectives and directions of development of such a sort of models are designated.
With the help of our own software package DifEqTools, numerical modeling of the cosmological evolution of a system consisting of an asymmetric scalar doublet of nonlinear, minimally interacting scalar fields, a classical field and a phantom field, has been performed. Peculiarities of the behavior of the model near zeroenergy hypersurfaces have been revealed.
A qualitative analysis of a cosmological model based on the asymmetric scalar doublet classical + phantom scalar field with minimal interaction is performed. It is shown that depending on the parameters of the model, the corresponding dynamical system can have 1, 3, or 9 stationary points corresponding to attractive or repulsive centers (1--5) and saddle points (0--4). A physical analysis of the model is performed.
I conjecture non-zero photon and gluon bare masses derived from a cosmological scale. That entails the existence of photon and gluon Bose-Einstein (BE) condensates in a comoving Friedmann-Lemaitre-Robertson-Walker geometry. I derive tantalizing results that set the inter-particle distance in these BE condensates almost at the nucleon scale and a corresponding critical temperature of condensation almost at the end of the quark epoch or the beginning of quark-gluon confinement. My estimates for particle mass and number density in these BE condensates suggest remarkable relations among fundamenta…
