Cosmological evolution of a scalar-charged degenerate cosmological plasma with Higgs scalar fields
CCosmological evolution of a scalar-charged degenerate cosmologicalplasma with Higgs scalar fields Yu.G. Ignat’ev
Institute of Physics, Kazan Federal University, Kremlyovskaya str., 18, Kazan, 420008, Russia
A mathematical model of the cosmological evolution of statistical systems of scalarly charged particleswith Higgs scalar interaction is formulated and investigated. Examples are given of numerical modeling ofsuch systems, revealing their very remarkable properties, in particular, the formation of paired bursts ofcosmological acceleration.
Intoduction
In [1], a complete mathematical model of the cosmo-logical evolution of the classical Higgs scalar vac-uum field was formulated and studied, both bymethods of qualitative analysis and numerical sim-ulation. In such models, transitions of cosmologicalevolution from the stage of expansion to the stageof compression (and, conversely, for the phantomfield) become possible . Earlier, a comprehen-sive study of incomplete cosmological models wascarried out under the assumption that the Hubbleconstant is non-negative for the cases of the clas-sical Higgs vacuum field [2], the Higgs phantomfield [3], [4] and the asymmetric scalar doublet [5]– [6]. If we discard a number of incorrect results ofthese works, which are just related to the assump-tion that the Hubble constant is non-negative, thenone of the results of these works can be summarizedas follows: at the late stages of evolution, the cos-mological model always goes to inflation. The sameresult was confirmed by studies of the complete [1]model, in which the assumption that the Hubbleconstant was nonnegative was removed. Thus, itcan be argued that cosmological models based onvacuum scalar fields contradict the observationaldata on invariant cosmological acceleration in thelate Universe w < We do not pose in this article the task of compiling anyreview on a huge layer of articles devoted to cosmologicalmodels based on scalar fields. Such a brief review is con-tained in the cited works of the Author, in particular, [6]. responding cosmological models was shown, amongwhich were models with an intermediate ultra-relativistic stage and a final non-relativistic [17]– [18]. However, these studies were based, firstly,on an incomplete mathematical model, secondly,on the quadratic potential of scalar fields and,thirdly, on a scalar singlet. In this connection, theproblem arises of formulating a complete mathe-matical model of cosmological systems of scalarcharged particles with Higgs scalar fields, includ-ing an asymmetric scalar doublet. Note that thephantom scalar field due to the negativity of itskinetic energy should be considered only as partof the usual components of matter. However, aswill be seen from what follows, it is the presenceof a phantom field in the system that ensures thecorrect behavior of the cosmological model. In thisarticle, we formulate a mathematical model of acosmological statistical system of scalar chargedparticles with Higgs scalar fields, examine its basicproperties and show examples of numerical mod-elling.
The foundations of the general relativistic kineticand statistical theory were laid in the 60s in theworks of E. Tauber - J.W. Weinberg [7], N. A.Chernikov (see, for example, [8]), A. A. Vlasov [9]and others. Scalar fields in general relativisticstatistics and kinetics were introduced at the be-ginning of the 80s in the works of the Author [10]– [13]. Further, in [14–16], a mathematical modelof the statistical system of scalarly charged par- a r X i v : . [ phy s i c s . g e n - ph ] S e p ticles was formulated, based on the microscopicdescription and the subsequent procedure for thetransition to kinetic and hydrodynamic models.Here we will refer to the paper cite Ignat15, whichcontains a correct generalization of the relativistictheory both to the case of phantom scalar fieldsand to the sector of negative dynamic masses ofscalar charged particles m ∗ = m a + N (cid:88) r =1 q ra Φ r , (1)where m a is some bare particle mass of rest of theparticle, which it may be zero, Φ r is a scalar fieldof type r , q r is the scalar charge of a particle withrespect to this field ( r = 1 , N ).Strict macroscopic consequences of the kinetictheory are the transport equations, including theconservation law of a certain vector current corres-ponding to the microscopic conservation law in re-actions of some fundamental charge G (if there issuch a conservation law) – ∇ i (cid:88) a g a n ia = 0 , (2)as well as the laws of conservation of the energy -momentum of the statistical systems: ∇ k T ikp − (cid:88) r σ r ∇ i Φ r = 0 , (3)where n ia is a numerical vector, T ikp is the energy- momentum tensor (MET) of particles; σ r is thedensity of scalar charges with respect to the fieldΦ r [16], so that T ikp = (cid:88) a T ika ; σ r = (cid:88) a σ ra . (4)Under conditions of local thermodynamic equili-brium (LTE), the statistical system is isotropicand is described by locally equilibrium distributionfunctions: f a = 1e ( − µ a +( u,p )) /θ ± , (5)where µ a is the chemical potential, θ is the localtemperature, u i is the unit time-like vector of thedynamic velocity of the statistical system, the sign“+ ” corresponds to fermions, “ − ” – to the bosons.Further, the kinematic momentum of the particle p i lies on the effective mass surface:( p, p ) = m ∗ ⇒ ˜ p = (cid:112) m ∗ + ˜ p , (6) where ˜ p ( i ) are the reference projections of the mo-mentum vector, p is the square of the physicalmomentum. In this case, the macroscopic momentstake the form of the corresponding moments of theideal fluid for each of the components [14]: n ia = n a u i , (7) T ika = ( ε a + p a ) u i u k − p a g ik , (8)while( u, u ) = 1 . (9)The normalization relation (9) implies the well-known identity: u k,i u k ≡ . (10)Therefore, the conservation laws (3) can be reducedto the form: ( ε p + p p ) u i,k u k =( g ik − u i u k ) (cid:18) p p,k + (cid:88) r σ r Φ r,k (cid:19) ; (11) ∇ k [( ε p + p p ) u k ] = u k (cid:18) p p,k + (cid:88) r σ r Φ r,k (cid:19) , (12)and the law of conservation of the fundamentalcharge G (2) becomes: ∇ k n r u k = 0 , n r ≡ (cid:88) a q ra n a . (13)Macroscopic scalars under LTE conditions havethe form [16] : n a = 2 S + 12 π m ∗ ∞ (cid:90) sh x ch xdxe − γ a + λ ∗ ch x ± ε p = (cid:88) a S + 12 π m ∗ ∞ (cid:90) sh x ch xdxe − γ a + λ ∗ ch x ± p p = (cid:88) a S + 16 π m ∗ ∞ (cid:90) sh xdxe − γ a + λ ∗ ch x ± T p = (cid:88) a S + 12 π m ∗ ∞ (cid:90) sh xdxe − γ a + λ ∗ ch x ± σ r = (cid:88) a S + 12 π q ra m ∗ ∞ (cid:90) sh xdxe − γ a + λ ∗ ch z ± , (18) To reduce the record, we omit the particle sort index insome places where T p is the trace of MET particles, ε p = (cid:80) ε a , p p = (cid:80) p a , σ r = (cid:80) σ ra , λ ∗ = | m ∗ | /θ , γ a = µ a /θ and S is the spin of particles.Thus, under LTE conditions formally on 5 + N macroscopic scalar functions ε p , p p , n r and 3independent components of the velocity vector u i macroscopic conservation laws give 4 + N inde-pendent equations (11 ) – (13) . However, not allindicated macroscopic scalars are functionally inde-pendent, since all of them are determined by locallyequilibrium distribution functions (5). At solved aseries of conditions of chemical equilibrium, whenonly one chemical potential remains independent,the solved mass equation of the surface and givenscalar potentials and the scale factor the 4 + 2 r ofmacroscopic scalar ε p , p p , n r , σ r are determined bytwo scalars — some chemical potential µ and localtemperature θ . Thus, the system of equations (11)– (13) turns out to be completely defined. In contrast to the works [14] - [17] in this paper wewill consider Higgs scalar fields with the Lagrangefunction:L rs = 18 π (cid:18) e r g ik Φ ( r ) ,i Φ ( r ) ,k − V r (Φ r ) (cid:19) , (19)where the indicator e r = +1 for the classical scalarfield and e r = − V r (Φ r ) is the potential energy of the scalar field( V = (cid:80) r V r ): V r (Φ r ) = − α r (cid:18) Φ r − m r α r (cid:19) , (20) α r is the self-action constant, m r is the mass ofscalar bosons, L s = (cid:80) r L rs .Further T ikr = 18 π (cid:18) e r Φ ,ir Φ ,kr − e r g ik Φ r,j Φ ,jr + g ik V r (Φ r ) (cid:19) (21)is the energy - momentum tensor of the r -th scalarfield, T iks = (cid:80) r T ikr . Next, we omit the constantterm in the Higgs potential (20), since it leads to asimple redefinition of the cosmological constant λ . one of the equations (11) is dependent on the restdue tothe identity (10) See details in [14, 16].
The scalar fields Φ r are determined by theequations for charged scalar fields with the source[15] :e r (cid:3) Φ r + V (cid:48) Φ r = − πσ r , (22)where (cid:3) ψ is the d’Alembert operator on the metric g ik . It can be shown that, due to (3) and (22), theconservation law for the complete MET system “plasma + charged scalar fields ” is identical: ∇ i T ik = ∇ i (cid:0) T ikp + T iks (cid:1) ≡ . (23) As a background, we consider the spatially flatFriedmann metric ds = dt − a ( t )( dx + dy + dz ) , (24)and as a background solution, we consider a homo-geneous isotropic distribution of matter, when allthermodynamic functions and scalar fields dependonly on time. It is easy to verify that u i = δ i con-verts equations (11) into identities, and the systemof equations (12) – (13) reduces to 1+ N equations:˙ ε p + 3 ˙ aa ( ε p + p p ) = (cid:88) r σ r ˙Φ r ; (25)˙ n r + 3 ˙ aa n r = 0 . (26)Thus, there remain 2 differential equations for thetwo thermodynamic functions µ and θ . Whenpassing to the limit µ → θ → In connection with the normalization of the Lagrangefunction of the scalar field, different from the normalizationof the article [15], the scalar source function on the right sideis multiplied by 2
We first consider a one-component statisticalsystem of scalarly charged fermions under condi-tions of complete degeneracy: θ → . (27)when the locally equilibrium distribution functionof fermions (5) takes the form of a step function[16]: f ( x, P ) = χ + ( µ − (cid:112) m ∗ + p ) , (28)where χ + ( z ) is the Heaviside step function. In thiscase, however, we will admit the presence of severalscalar fields with respect to which the same particlecan have different scalar charges q r .The result of integrating macroscopic densities(14) – (18) with respect to the distribution (28)expressed in elementary functions [16]: n = 1 π p F ; (29) ε p = m ∗ π F ( ψ ); (30) p p = m ∗ π ( F ( ψ ) − F ( ψ )) (31) σ r = q r · m ∗ π F ( ψ ) , (32)where the dimensionless function is introduced ψψ = p F /m ∗ , (33)equal to the ratio of the Fermi momentum p F tothe effective mass of the fermion, and to reducewriting, the functions F ( psi ) and F ( psi ) wereintroduced: F ( ψ ) = ψ (cid:112) ψ − ln( ψ + (cid:112) ψ ); (34) F ( ψ ) = ψ (cid:112) ψ (1 + 2 ψ ) − ln( ψ + (cid:112) ψ ) . (35)The functions F ( x ) and F ( x ), firstly, are odd: F ( − x ) = − F ( x ); F ( − x ) = − F ( x ) , (36) and, secondly, they have the following asymptotics: F ( x ) | x → (cid:39) x ; F ( x ) | x → (cid:39) x ;( F ( x ) − F ( x )) | x → (cid:39) x ; (37) F ( x ) | x →∞ (cid:39) x | x | ; F ( x ) | x →∞ (cid:39) x | x | . (38)It is easy to verify the validity of the identity: ε p + p p ≡ m ∗ π ψ (cid:112) ψ . (39)Note that in the extended theory [15], the effectivemass ( ?? ) can also be a negative quantity. The re-quirement of symmetry between particles and an-tiparticles ( q → − q ) leads to the condition thatthe seed mass is equal to zero in the formula ( ?? ).Therefore, for the effective particle mass we have: m ∗ = q Φ . (40)Moreover, we do not exclude the possibility of anegative effective mass of fermions, in particular,the effective masses of particles and antiparticlesin this case will differ in sign, since q = − q . Notethat the effective mass of particles is not a heavymass, which, in contrast to the effective mass, is de-termined by the total energy, i.e., m = (cid:112) m ∗ + p ,so m ( p = 0) = | m ∗ | (see [15]). So, due to ψ ( − Φ) = − ψ (Φ) and the oddness properties (36) of the for-mula (30) – (32), the following transformation laws ε p ( − Φ) = − ε p ( − Φ); p p ( − Φ) = − p p (Φ); σ ( −| Φ) = σ (Φ) . (41)We also note that in the case of a seemingly morestandard version of m ∗ = | q Φ | (cid:62)
0, compatibilityproblems for the basic equations arise.Further, the MET of the scalar field in the un-perturbed state also takes the form MET of an idealisotropic fluid: T iks = ( ε s + p s ) u i u k − p s g ik , (42)moreover: ε s = 18 π (cid:88) r (cid:18) e r r + V r (Φ r ) (cid:19) ; (43) p s = 18 π (cid:88) r (cid:18) e r r − V r (Φ r ) (cid:19) , (44)so that: ε s + p s = 18 π (cid:88) r e r ˙Φ r . (45)The equation of scalar fields in the Friedmann met-ric takes the form:e r (cid:18) ¨Φ r +3 ˙ aa ˙Φ r (cid:19) + m r Φ r − α r Φ r = − πσ r ( t ) , (46)where ( r = 1 ,N ). We consider the standard Einstein equations withthe Λ - term: G ik ≡ R ik − Rδ ik = 8 πT ik + Λ δ ik . (47)We write Einstein’s independent background equa-tions for the Friedmann metric (24):2 ¨ aa + ˙ a a + (cid:88) r (cid:18) e r ˙Φ r − m r Φ r α r Φ r (cid:19) +8 πp p = Λ;(48)3 ˙ a a − (cid:88) r (cid:18) e r ˙Φ r m r Φ r − α r Φ r (cid:19) − πε p = Λ . (49)Due to the energy - momentum conservation law(23), (25) of the field equations (46) one of theEinstein equations (48) - (49) is differentially algeb-raic consequence of the remaining equations. In [1]shows that to study a dynamical system it is moreconvenient to consider their difference instead ofthese Einstein equations, taking into account theidentity for the Hubble constant H = ˙ a/a , –˙ H ≡ ¨ aa − ˙ a a . Thus, we obtain the necessary equation:˙ H + 4 π ( ε + p ) = 0 , (50)where ε = ε p + ε s and p = p p + p s .Further, according to [1], we introduce the totalenergy, E , of cosmological matter: E = 18 π (3 H − Λ) − ε, (51)with the help of which the Einstein equation (49)can be given a simple form: E = 0 , (52) reflecting the fact that the total energy of the spa-tially flat Friedman universe is zero.Differentiating in time the total energy (51) tak-ing into account the field equations (46) and therelations (25), (43), (45) and (50), we obtain theenergy conservation law ddt E = 0 ⇒ E = E . (53)Thus, the consequence of the considered system ofdynamic equations (25), (46) and (50) is the law ofconservation of the total energy of the cosmologicalsystem (53) E = E . the Einstein equation (49)is a particular integral of this system E = 0. Asimilar situation arises for the vacuum scalar fields[1]. This means that the first integral (52) can beconsidered as the initial condition in the Cauchyproblem for the cosmological model.In particular, for a degenerate Fermi system,taking into account (39) we obtain from (50) theequation:˙ H + (cid:88) r e r ˙Φ r π m ∗ ψ (cid:112) ψ = 0 . (54)The system of equations (25), (46) and (54) to-gether with the definitions (30) – (32) describesa closed mathematical model of the cosmologicalevolution of a completely degenerate Fermi systemwith scalar interaction .Differentiating the energy density of the Fermisystem (30) taking into account the identity (39)we bring the energy conservation law for the Fermisystem (25) to the form of the equation: ddt ln m ∗ ψa = 0 . (55)From this, taking into account the definition of thefunction ψ (33) we get: ap F = Const . (56)From here, taking into account (7) we obtain thelaw of conservation of the number of fermions (see[16]): a n = Const . (57)Thus, despite the apparent complexity of the equa-tion (25), its solution is easy to find - from the lawof conservation of energy of the Fermi system, thelaw of conservation of the number of particles is ob-tained. We can say that the law of conservation ofthe scalar charge in the form (13) is, at least in ourcase, redundant. note that, unlike the law of con-servation of electric charge, this law does not followfrom anywhere, but, nevertheless, is fulfilled.Following [1] we also introduce a nonnegativeeffective energy of the cosmological system E eff ac-cording to (51) and (52) E eff = ε + Λ8 π (cid:62) ⇒ e2 ˙Φ + m Φ − α Φ m ∗ π F ( ψ ) + Λ (cid:62) . (58) In this article, we will consider a cosmologicalmodel based on a one-component degenerate Fermisystem and a single scalar field Φ. An importantcircumstance is that the energy conservation law ofthe statistical system (25) for a degenerate Fermisystem is completely equivalent to the equation(55). Instead of choosing the non-negative dynamicvariable a ( t ) (cid:62)
0, choosing ξ ( t ) ξ = ln a, ξ ∈ ( −∞ , + ∞ ); (59)˙ ξ = H, (60)we use the integral (56) of the energy conservationlaw (55) to find the function ψ ( t ) (33) ψ = p f e − ξ m ∗ . (61)Thus, taking into account (40) we obtain the ex-pression for the function ψ ( t ): ψ = p f e − ξ q Φ ≡ β Φ e − ξ , (62)where β = p f q ; p f = p f ( ξ = 0) . (63)Further, for the scalar charge density (32) we ob-tain the expression σ = q Φ π F ( ψ ) . (64) Assuming further˙Φ = Z, (65)we write the field equation in these notations (46)˙ Z = − HZ − em Φ + e Φ (cid:18) α − q π F ( ψ ) (cid:19) . (66)Further, the equation (54) takes the form:˙ H = − eZ π q Φ ψ (cid:112) ψ . (67)Moreover, the first integral of the system of equa-tions (52) takes the form:Σ E : 3 H − Λ − q Φ π F ( ψ ) − eZ − m Φ α Φ . (68)The equation (68) is an algebraic equation for thedynamic variables Φ , ξ, Z, H and describes someahypersurface in the arithmetic space R = { Φ , ξ,Z, H } , which we will call [1] Einstein hypersurface .All phase trajectories of the dynamical system (65),(60), (66) and (71), as well as the starting points,must lie on Einstein hypersurface. Since (68) is thefirst integral of a dynamical system, for to solve theCauchy problem, it suffices to require that the ini-tial point of the dynamic trajectory of the Einsteinhypersurface belong to.Further, the points of the phase space R , atwhich the effective energy (58) is negative, are notavailable for the dynamical system.These points lieon the hypersurface of the phase space S E ⊂ R ,which is a cylinder with the axis OH : S E : Λ + q Φ π F ( ψ ) + eZ m Φ − α Φ , (69)moreover, the hypersurface of zero effective energy(69) touches the Einstein hypersurface (68) in thehyperplane H = 0:Σ E ∩ S E = H = 0 . (70)Further, as can be seen from the equation (67)in the case of a scalar neutral statistical system( q ≡ e : for a classical scalar field ( e = +1) ˙ H < e = −
1) always˙
H >
0. The play of these factors during cosmo-logical evolution can also fine-tune the model pa-rameters to ensure the desired behavior. In thepresence of charged matter, its contribution to thisgame, as can be seen from (67), is determined bythe sign of the scalar potential: for Φ > < Z from (68) in (67):˙ H = − H + Λ + eq Φ π F ( ψ ) + − m Φ α Φ π q Φ ψ (cid:112) ψ . (71) The singular points of a dynamical system repre-sented by a normal autonomous system of differ-ential equations are determined by algebraic equa-tions obtained by equating to zero the derivativesof all dynamic variables. Thus, from (60), (65),(66) and (71) we obtain the system of algebraicequations for finding the coordinates of the singu-lar points: Z = 0; (72) H = 0; (73)Φ (cid:18) α − q π F ( ψ ) (cid:19) − m Φ = 0; (74)43 π q Φ ψ (cid:112) ψ = 0 . (75)In addition, we must take into account the inte-gral of the total energy (68), – the coordinates ofthe singular point must satisfy this equation, whichgiven (72), takes the form:Λ + eq Φ π F ( ψ ) − m Φ α Φ . (76)From (75) it follows 1. ψ = 0 or 2. Φ = 0. Wefirst investigate the first possibility ψ = 0. Since F (0)0, according to (74) we get the equation on Φ α Φ − m Φ = 0 , (77) where do we get the roots from:Φ = 0; Φ ± = ± (cid:114) m α . (78)In this case, the remaining equation (76) gives therelationship between the fundamental constantsΛ = Λ ≡ m α , (79)at which there exist a singular point M Φ M ± Φ : (cid:18) ± (cid:114) m α , + ∞ , , (cid:19) , (Λ = m α > . (80)This case corresponds to the singular points of thevacuum scalar field without charged fermions [1],and the cosmological constant Λ is fully generatedby the Higgs field.We are now investigating the second possibilityΦ = 0. In this case, the equation (74) becomes anidentity, and the equation (76) gives Λ = 0. Whatabout ψ → ±∞ and the dynamic variable ξ cantake any values: M ± ξ : (cid:18) ξ , , , (cid:19) , ( ∀ ξ , Λ = 0) . (81)Calculating the basic matrix of the dynamical sys-tem (65), (60), (66) and (71) A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂X i ∂x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , it is easy to show that in both cases - (80) and(81) this matrix is degenerate, therefore a qualita-tive theory differential equations for our dynamicsystem, in contrast to a dynamic system with vac-uum scalar fields, does not give anything. Let ustherefore proceed to numerical modeling. First, we note that the topology of the Einstein hy-persurface (68) can be quite complex, and at thesame time, the phase trajectories of the systemcan be complex, since they lie on this hypersurface.Bearing in mind a wide variety of dynamic systembehavior models depending on the fundamental pa-rameters of the model P = [ e, α, β, m, q, Λ] , in thisarticle we restrict ourselves to the study of somespecial cases, relating full study and general con-clusions to a more detailed article, which we hopeto present in the near future. In the future, forbrevity, we will describe the initial conditions forour model with the ordered set I = [Φ , ξ , Z , (cid:15) ] , where the indicator (cid:15) = ± H >
0, and thevalue −
1, if the initial state of the dynamic systemcorresponds to the compression phase H <
0. Re-call that the initial value of the Hubble constant isdetermined from the equation (68).
In Fig. 1 – 2 shows an example of a phase trajectoryof a cosmological system based on a classical scalarfield, with parameters P = [1 , . , . , , . , . I = [0 . , , . , ± { Φ , Z } ; H > H >
H <
0. Therefore, thephase trajectories in the upper half-plane
H > H min >
0, andthe trajectories in the lower half-plane, on the con-trary, leave the maximum point H max < { Φ , Z, H } . In theupper part of Einstein’s surface, the trajectoriesroll down to the lower point, and in the lowerpart roll down from the top.Figure 3: The evolution of the Hubble constant H ( t ) in the system under study.In Fig. 3 - 6 shows the behavior of the basicphysical parameters of the model under study withthe classical Higgs scalar field: Hubble constant H ,effective energy E eff (58), invariant cosmologicalacceleration ww = 1 + ˙ HH (82)and invariant curvature σ = (cid:113) R ijkl R ijkl = H (cid:112) w ) . Figure 4: Evolution of the effective energy E eff ( t ) in the system under study.Figure 5: The evolution of invariant cosmo-logical acceleration w in the system understudy. In Fig. 7 – 12 shows the results of numericalsimulations for a cosmological statistical systemwith phantom interaction with parameters P =[ − , . , . , , . , .
01] and initial conditions I = [0 . , , . , ± σ ( t ) in the system under study.In Fig. 8 – 9 shows the phase trajectories of themodel with a phantom scalar field on the Einsteinsurface in a three - dimensional section { Φ , Z, H } ,and the first case corresponds to (cid:15) = +1 and thesecond - (cid:15) = −
1. We see how in the first case thetrajectory rises from the lower part of the left cavityof the Einstein surface to its upper point, startingfrom the points of the neck, and in the second case,the trajectory rises from the lower part of the rightcavity of the Einstein surface to the upper point ofthe left plane, slipping through the neck.Figure 7: Phase trajectory of a model witha phantom scalar field in the plane { Φ , Z } ; H > H → σ →
0. Nevertheless, thesepossible bursts are of certain interest for observa-tional cosmology, so they can give evidence of achange in the compression mode to the expansionmode in the early Universe.Figure 8: Phase trajectory of the phantomfield, (cid:15) = +1.We will especially dwell on the behavior of the in-variant cosmological acceleration w for a statisticalsystem with a phantom field (see Fig. 11 and Fig.12). Here we observe two giant bursts of accelera-tion for a system that started from the expansionphase (Fig. 11) and the compression phase (Fig.12), and the graphs in these figures are, in fact, amirror image of each other.In Fig. 11 the giant surge of acceleration w ∼ precedes the supergiant w → + ∞ , and in Fig.12 the burst sequence changes. First, we note thatovergig acceleration bursts w → + ∞ are associ-ated with the passage of the point H = 0 (see theformula (82)) and are not observed in models witha quadratic interaction potential. Secondly, the gi-ant bursts of w ∼ are characteristic of a theorywith the quadratic potential [17] – [18]. Figure 9: Phase trajectory of the phantomfield, (cid:15) = − E eff ( t ) in the system under study.In the near future, we intend to publish moredetailed studies of the models of cosmological evo-lution of scalarly charged statistical systems withHiggs scalar fields. Funding
This work was funded by the subsidy allocated toKazan Federal University for the state assignmentin the sphere of scientific activities.1Figure 11: The evolution of invariant cosmo-logical acceleration w in the system understudy (cid:15) = +1.Figure 12: The evolution of invariant cosmo-logical acceleration w in the system understudy (cid:15) = − References [1] Yu.G. Ignatyev (Ignat’ev) and D.Yu. Ignat’ev,Grav. and Cosmol., , 29 (2020).[2] Yu. G. Ignat’ev, Grav. and Cosmol., , 131(2017).[3] Yu. G. Ignatyev (Ignat’ev), Russ. Phys. J., ,2074 (2017).[4] Yu. G. Ignat’ev and A. A. Agathonov, Grav.and Cosmol., , 230 (2017). [5] Yu. G. Ignat’ev and I. A. Kokh, Russ. Phys.J.,
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