The equation of vacuum state and the structure formation in universe
aa r X i v : . [ g r- q c ] J un Equation of a vacuum state and a structure formation in theuniverse
S.L.Cherkas
Institute for Nuclear Problems, Bobruiskaya 11, Minsk 220030, Belarus
V.L. Kalashnikov
La Sapienza Universit´a di Roma, Via Eudossiana 18, 00189 - Roma, RM, ItaliaThe vacuum is considered as some fluid emergent from the zero-pointfluctuations of the quantum fields contributing to the vacuum energy den-sity and pressure. The equation of vacuum state and the speed of vacuumsound-waves are deduced under the assumption of zero vacuum entropy.The evolution of the background space-time metric resembles that of theMilne’s-like universe. In the framework of the five-vector theory of gravita-tion allowing an arbitrary choice of the energy density reference level, thedynamics of the vacuum, pressureless matter, and space-time metrics per-turbations are traced under this background. The obtained results show thevery early formation of the Universe structure without the need for darkmatter. Thus, a vacuum can be considered as some of the dark-energy-matter unification.
1. Introduction
Clarification of the role of vacuum in the formation of the cosmic mi-crowave background anisotropy (CMB) and the matter structures in theevolving Universe remains one of the key issues of modern cosmology de-spite the numerous hypotheses and suggestions [1–7]. At the same time,the application of the standard renormalization procedure [8] to gravita-tion seems not feasible due to the impossibility to define a vacuum statewhich is invariant under the general coordinate transformations [5, 8, 9].Nevertheless, one might intuitively feel that some “pieces” of the vacuumenergy density and pressure have to be omitted, while the others are tobe taken into account as was demonstrated on the example of the Gowdy’smodel [10]. The issue of huge vacuum energy indicates that the most diverg-ing part of the vacuum energy density has no physical meaning and has to (1)
Equation˙of˙a˙vacuum˙state printed on June 23, 2020 be discarded. Formally, this is impossible in the frameworks of the generaltheory of relativity (GR) because of any non-zero energy density contributesto space-time curvature. However, this procedure guaranteeing against theunphysical “piece” of the vacuum energy density can be realized within theso-called five-vector theory of gravitation (FVT) [11], in which the energydensity is defined up to some constant. The remaining part of the vacuumenergy can be treated as corresponding to some“fluid” possessing definiteequation of state (ES).The idea to describe a vacuum as some “fluid” defined by ES seemsvery tempting, starting from the concepts of a quantum “ether” [12, 13]to the models of quintessence, K-essence, and cosmological Chaplygin’sgas [14–16]. Generally speaking, the the situation looks as follows: there isno “ether” in the flat space-time owing to vacuum invariance relatively theLorentz transformations, but in the presence of gravitation, the picture isdifferent because there is no invariant vacuum state relatively the general co-ordinate transformations. As a result, one might conjecture the existence ofsome preferred reference frame indicating the existence of “ether” identifiedwith a quantum vacuum.
2. Violation of gauge invariance in a framework of FVT
In GR, any spatially uniform energy density (including that of zero-pointfluctuations of the quantum fields) causes the expansion of the universe. Us-ing the Planck level of UV-cutoff results in the Planckian vacuum energydensity ρ vac ∼ M p [2], which leads to the universe expanding with thePlanckian rate [17]. In this sense, the vacuum energy problem is an ob-servational fact [18]. One of the possible solutions is to build a theory ofgravity, allowing an arbitrarily reference level of energy density. One suchtheory has long been known. That is the unimodular gravity [19–24], whichadmits an arbitrary cosmological constant. However, under using of thecomoving momentums cutoff, the vacuum energy density scales with timeas radiation [18, 25], but not as the cosmological constant.Recently, another theory has been suggested [11], which also leads to theFriedman equation defined up to some arbitrary constant. This constantcorresponds to the invisible radiation and, thus, can compensate the vacuumenergy.Five-vector theory of gravity (FVT) [11] assumes the gauge invarianceviolation in GR by constraining the class of all possible metrics in varyingthe standard Einstein-Hilbert action. This theory arises if one varies thestandard Einstein-Hilbert action over not all possible space-time metrics quation˙of˙a˙vacuum˙state printed on June 23, 2020 g µν , but over some class of conformally-unimodular metrics [11] ds ≡ g µν dx µ dx ν = a (1 − ∂ m P m ) dη − γ ij ( dx i + N i dη )( dx j + N j dη ) , (1)where x µ = { η, x } , η is conformal time, γ ij is a spatial metric, a = γ / is alocally defined scale factor, and γ = det γ ij . The spatial part of the interval(1) reads as dl ≡ γ ij dx i dx j = a ( η, x )˜ γ ij dx i dx j , (2)where ˜ γ ij = γ ij /a is a matrix with the unit determinant.The interval (1) is similar formally to the ADM one [26], but with thelapse function N changed by the expression 1 − ∂ m P m , where P m is a three-dimensional (relatively rotations) vector, and ∂ m is a conventional particularderivative. Finaly, restrictions ∂ n ( ∂ m N m ) = 0 and ∂ n ( ∂ m P m ) = 0 arisebecause they are the Lagrange multipliers in FVT. Hamiltonian H andmomentum P i constraints in the gauge (1) have the same form [11] as inGR, so in FVT, and obeys the constraint constraint algebra which gives theconstraint evolution equation [11]. Let us write it in the particular gauge N = 1, N i = 0: ∂ η H = ∂ i (cid:16) ˜ γ ij P j (cid:17) , (3) ∂ η P i = 13 ∂ i H . (4)One could notice that the evolution of constraints governed by (3), (4)admit adding some constant to H . Thus the constraint H not necessarilymust be zero, but H = const is also admitted. This fact explains why theFriedmann equation in FVT is satisfied up to some arbitrary constant. Itis also interesting how black holes look in the framework of FVT [27].
3. Vacuum as a fluid
Let us consider a quantum scalar field ˆ φ ( η, x ) against a classical backgroundof the uniform, flat, expanding Universe with a space-time metric: ds ≡ g µν dx µ dx ν = a ( η ) (cid:16) dη − ˜ γ ij dx i dx j (cid:17) , (5)where ˜ γ ij = diag { , , } is a Euclidean 3-metric. At this moment, at leastone fundamental scalar field is known that is the Higgs boson [28]. Besides, In this gauge, a space-time metric is presented as a product of a common multiplierby a 4-dimensional matrix with a determinant equal to -1, including a 3-dimensionalspatial block with unit determinant.
Equation˙of˙a˙vacuum˙state printed on June 23, 2020 as was shown in [29], the gravitational waves contribute to the vacuumenergy density in the same manner as a scalar field.The operators of the energy density and pressure of a scalar field can bewritten as ˆ ρ φ = 1 V Z V ˆ φ ′ a + ( ∇ ˆ φ ) a ! d r , ˆ p φ = 1 V Z V ˆ φ ′ a − ( ∇ ˆ φ ) a ! d r , (6)where V is some normalizing volume which can be equaled unity. Pressureand density of all kinds of matter define the scale factor evolution of theflat Universe by the equations: − M p a ′ + ρa = const, (7) M p a ′′ = ( ρ − p ) a , (8)where the Planck mass M p = q πG . In the frameworks of the FVT [30],the Friedmann equation (7) is satisfied up to some constant allowing toavoid the problem of huge vacuum energy, which diverges as fourth degreeof momentum. A scalar field could be expanded over the plane wave modesˆ φ ( r ) = P k ˆ φ k e i kr , which are expressed through the creation and annihila-tion operators [8]: ˆ φ k = ˆa + − k χ ∗ k ( η ) + ˆa k χ k ( η ) . (9)The complex functions χ k ( η ) satisfy the relations [8] χ ′′ k + k χ k + 2 a ′ a χ ′ k = 0 ,a ( η )( χ k χ ′ k ∗ − χ ∗ k χ ′ k ) = i (10)and can be found in the adiabatic approximation: χ k ( η ) = exp (cid:18) − i R η r k − a ′′ ( τ ) a ( τ ) dτ (cid:19) √ a ( η ) r k − a ′′ ( η ) a ( η ) . (11)Let us calculate the mean vacuum energy density of a scalar field ρ v a = a Z (cid:16) < | ˆ φ ′ | > + < | ( ∇ ˆ φ ) | > (cid:17) d r = quation˙of˙a˙vacuum˙state printed on June 23, 2020 a X k < | ˆ φ ′ k ˆ φ ′− k | > + k < | ˆ φ k ˆ φ − k | > = a X k χ ′ k ∗ χ ′ k + k χ ∗ k χ k ≈
12 4 π (2 π ) k max k max a ′ a + O ( a ′ ) + O ( a ′ a ′′ ) + O ( a ′′′ + ... ) ! , (12)where it is implied that a ′ , a ′′ has the second-order of smallness, a ′ , a ′′ a ′ are the third-order and so on [31]. Two first terms in Eq. (12) divergeas 4th and 2nd momentum degrees, respectively. The first term can beomitted if the Friedmann equation (7) is satisfied up to some constant. Inthe calculation of the second term, one could use the ultra-violet cut-off k max ∼ M p at the Planck mass level [29].One could ask why the k -cutoff of comoving momentums is used insteadof, for instance, a cutoff of physical momentums related to p = k /a ( a is theuniverse scale factor)? The answer could be that it is relatively simple toconstruct a theory with the k -cutoff, but it is challenging to introduce the p -cutoff fundamentally. For instance, merely considering gravity on a latticegives rather fundamental theory with comoving momentums restricted bythe period of a lattice.Let us for definiteness to imply that the FVT model is considered onsome lattice in which the coordinate x takes discrete values. As a result,one finds for the vacuum energy density: ρ v = a ′ a M p S , (13)where S = 12 M p X k k = 1 M p (2 π ) Z d k k = k max π M p . The vacuum energy density calculated is about of the critical density of M p H /
2. However, as a result of an arbitrary constant on the right-handside of Eq. (7), the concept of critical density loses its fundamental roleas a demarcating line between closed and opened Universes. Here, we willconsider a flat universe ad hoc.On the way to the vacuum ES finding, the next quantity has to becalculated: < | ˆ ρ φ − p φ | > = − a Z (cid:16) < | ˆ φ ′ | > − < | ( ∇ ˆ φ ) | > (cid:17) d r = − a X k < | ˆ φ ′ k ˆ φ ′− k | > − k < | ˆ φ k ˆ φ − k | > = − a X k a ( χ ′ k ∗ χ ′ k − k χ ∗ k χ k ) ≈ a (cid:16) aa ′′ − a ′ (cid:17) X k k + O ( a ′ ) + O ( a ′ a ′′ ) + O ( a ′′′ ) + ..., (14) Equation˙of˙a˙vacuum˙state printed on June 23, 2020 - - - z q Fig. 1. Deceleration parameter dependence on the redshift z . Solid black, graysolid and gray dashed curves correspond to the standard ΛCDM model, the vacuumdomination model (20) of the present paper and the mean value of the observationaldata reconstruction [32], respectively. Thin dashed curves point the 1 σ and 2 σ errorchannels of the reconstruction. which does not contain the terms ∼ k max . The omission of items containinghigher-order derivatives in (14) leads to ρ v − p v = 1 a (cid:16) aa ′′ − a ′ (cid:17) M p S . (15)The vacuum pressure from Eqs. (15) and (13) is: p v = M p S a (cid:18) a ′ − a ′′ a (cid:19) . (16)It is easy to check that the vacuum energy density and pressure determinedby Eqs. (13), (16) satisfy ρ ′ v + 3 a ′ a ( ρ v + p v ) = 0 . (17)Eq. (17) is one of the keystones describing the universe evolution. It allowsconsidering a vacuum as some “fluid” or “substance” with the well-defineddynamical ES, which can be expressed explicitly. For this goal, one needsto find a dependence of the scale-factor a on the conformal time. In a moregeneral case, when the universe filled with a cold dust-like matter besides avacuum, Eqs. (7), (8) take the form − M p a ′ + ρ v a + 12 M p Ω m H a = const, (18) M p a ′′ = ( ρ v − p v ) a + 12 M p Ω m H , (19) quation˙of˙a˙vacuum˙state printed on June 23, 2020 where Ω m is a dimensionless constant characterizing the density of matterand H is a value of a of Hubble constant at the present time η = η , whenthe universe scale factor equals unity. ( a ) - - - - - a w ( b ) a c s Fig. 2. Dependencies of the vacuum ES (a) and the velocity of the scalar “soundwaves” (b) on the universe scale factor. Solid line - S = 2 .
3, Ω m = 0 .
3, dashedline - S = 2 .
3, Ω m = 0 . The constant on the right hand side of Eq. (18) equals H M p ( S +Ω m − a ( η ) = 1 and a ′ ( η ) = H . The resulting Hubble constantdependence on scale factor is: H ( a ) = ˙ aa = 1 a dadη = H a s S + Ω m − − Ω m aS a − − , (20)where a dot denotes the differentiation on the cosmic time dt = adη . Finally,ES reduces to w v = p v /ρ v = (cid:18) − a ′′ aa ′ (cid:19) = 2 a Ω m − a ( S + Ω m −
1) + S ( S + Ω m − a − S ) (( a − m − S + 1) , (21)where it is taken into account that a ′ = a H ( a ) and a ′′ = a H ( a ) dda (cid:0) a H ( a ) (cid:1) .Eq. (21) is not singular up to the “Big Rip” (see, e.g., [33]) at a = √ S ,which will come in future if S >
1. Explicit calculation of the vacuumenergy density leads to ρ v = H M p S ( S + Ω m − − a Ω m )2 a ( S − a ) . (22)Let us remind that Eq. (17) leads to ρa w ) = const for simple de-pendencies of p = wρ , where w = const . In this case, it is easy to write theUniverse expansion law a ′′ a − w = const from Eq. (8). Thus, a ∼ η w ,except for w = − / a ∼ exp ( H η ) . (23) Equation˙of˙a˙vacuum˙state printed on June 23, 2020
The last case corresponds to the Milne’s-like Universe [34], i.e., to the ex-ponential expansion in “conformal time” and to the linear one in “cosmictime” dt = adη . It is necessary to remind that Milne’s Universe is spatiallyopen, while we consider a flat Universe ad hoc.For the vacuum ES w = const the evolution becomes nontrivial. Eqs.(13), (16) result in the defined ES, if the expansion law is known, for ex-ample, w vac ∼ / − w . Thus, if w isclose to 1 / w = 1 / H ( a ) ∼ /a at smallscale factor according to Eq. (20), i.e., as it is for the Milne’s Universe.The results of calculation of the deceleration parameter q ( z ) = − ¨ a ˙ a = zH dH ( z ) dz − z ≥
2, where the deceleration parameter is closeto zero, and then comes to an acceleration phase. More general backgroundmodel is discussed in [18].Let us once more explain proximity to the Milne’s law of the Universeexpansion at the simple particular case of Ω m = 0 in which ρ v = H M p S ( S − a ( S − a ) , (24) w v = 13 − a S − a ) . (25)Eq. (8) leads to M p a ′′ a = ρ v (1 − w v ) a = H M p S ( S − S − a ) , (26)and one has a ′′ a ∼ const at small a , i.e., approximately the Milne’s-likeUniverse. It is instructive to compare that with the case of w = − / ρ = H M p a when a ′′ a = const , i.e., exactly the Milne’s expansion law.Validity of Eq. (17) allows describing a vacuum as some absolutelyelastic “fluid” with a “sound-speed”: c s = p ′ ρ ′ = 2 (cid:0) a Ω m − a Ω m S + (7 a S − a − S )(Ω m + S − (cid:1) a − S ) (5 a Ω m − a Ω m S + (4 S − a )(Ω m + S − . (27) quation˙of˙a˙vacuum˙state printed on June 23, 2020 According to Eqs. (12), (14), the waves of the Planck-order frequencygive the main contribution to the vacuum pressure and density. Thesefrequencies exceed the frequencies of “vacuum sound waves”. That is, thelocal compressions/expansions in a vacuum caused by these sound wavescan be considered as the expansion and collapse of some “small universes”.Eq. (27) implies that the birth of particles from a vacuum, which wouldincrease its entropy, is negligible. This means that an adiabatic vacuumis under consideration so that a “fluid” remains a vacuum during all theUniverse evolution in the process of the scalar sound waves propagation.Fig. 2 demonstrates that a dust-like pressureless matter has a little im-pact on the vacuum ES and the corresponding sound wave speed. The lastincreases from 1 / √ a = q S . That is, Eq. (27) demon-strates that the “Big Rip” occurs earlier, than it follows from ES (21).Regarding the speed of light excess, it is difficult to say from the aboveempirical model whether one deals with the physical effect [35] or with aconsequence of neglecting of a vacuum entropy.Unlike the linearly expanding Universe with the ES of w = − /
4. Masses and vacuum
A massless quantum field is considered above. In the case of the massivefields, Pauli’s idea could be actual (see [25] and Refs.). That is a contributionof masses to vacuum energy from bosons and fermions should compensateeach other. As a result, the main part of vacuum energy density is ρ v = 14 π a Z k max k p k + a m dk ≈ π (cid:18) k max a + m k max a + m m a k max !!(cid:19) . (28)Simultaneously, three different principles could explain why the main partof vacuum energy does not contribute to the Universe evolution. The term k max is omitted in the FVT gravity, where the energy reference level isarbitrary. The terms ∼ m are a pure mass contribution to the vacuumdensity. However, the condensates precipitate in the Standard Model ofElectroweak Interactions to generate masses itself. A density of condensates Equation˙of˙a˙vacuum˙state printed on June 23, 2020 has the same order of m . Overall compensation of m − terms includingcondensates should be considered. This problem stills unresolved yet, but itimplies some unknown symmetry mass generating potentials in Lagrangianallowing the compensation with the accuracy at least of the order of ∼ m ν ,where m ν is the neutrino mass. The only informative terms for particlephysics are ∼ k max m , which gives m H + N A m A + 6 m W + 3 m Z = 12 m t , (29)where the top quark mass is m t = 173 . GeV , the Higgs boson mass is m H = 125 GeV , the charged vector boson mass is m W = 80 , GeV , theneutral vector boson mass is m Z = 91 . GeV , and m A is a mass of unknown A − bosons contributing with the N A − weight. Thus, the physics of vacuumbeyond the Milne-like stage of the Universe expansion anticipates unknownbosons: a single boson m A ∼ Gev , or, for instance, four bosons m A ∼ Gev .
5. Formation of matter structures in the universe
The ES and the scalar waves speed in a vacuum found in the previoussection could serve as the basis for the description of the perturbations evo-lution of vacuum, radiation, and matter in the expanding Universe. As wasmentioned above, the Friedmann equation is satisfied up to some constantin the FVT that allows choosing an arbitrary reference level of the vacuumenergy density. Briefly, the FVT theory is based on the standard Einstein-Hilbert action which is varied not over all the possible metrics, but oversome restricted class of them [30]. As a result, the Hamiltonian constraintturns out to be weaker than that in GR. Perturbations of the metric of theexpanding Universe looks as ds = a ( η ) (1 + 2 A ) dη − X m =1 ∂ m F ! δ ij − ∂ i ∂ j F ! dx i dx j ! . (30)The metric (30) belongs to the class of metrics (1) admissible in FVT [30].Perturbations of density, pressure and 4-velocity of every c -fluid areconsidered as ρ c ( η, x ) = ρ c ( η ) + δρ c ( η, x ), p c ( η, x ) = p c ( η ) + δp c ( η, x ), u µc = 1 a ( η ) { (1 − A ) , ∇ v c ( η, x ) } , (31)where v c is a velocity potential. The resulting system of equations wasobtained for the Fourier components of δρ c ( η, x ) = P k δρ c k ( η ) e i kx ... − A ′ k + 6 A k α ′ + k F ′ k + 18 M p e α X c V c k = 0 , (32) quation˙of˙a˙vacuum˙state printed on June 23, 2020 H a L z D k H b L z D k H c L - - z D k H d L z D k H e L z D k H f L z D k Fig. 3. The inhomogeneity growth factors for the different perturbation wave num-bers: a,b- k/h = 0 . − , c,d- k/h = 0 .
001 Mpc − , e,f- k/h = 0 .
01 Mpc − .Solid curves correspond to a matter, dashed curves correspond to a vacuum.Ω m = 0 . m = 0 .
03 for (b,d,f), respectively. − α ′ A ′ k − A k α ′ − k A k + k F k + 18 M p e α X c δρ c k + 4 A k ρ c = 0 , (33) − A k − (cid:0) F ′′ k + 2 α ′ F ′ k (cid:1) + k F k = 0 , (34) − (cid:0) A ′′ k + 2 α ′ A ′ k (cid:1) − A k α ′′ − A k α ′ − k A k + k F k − M p e α X c A k (3 p c − ρ c ) + 3 δp c k − δρ c k = 0 , (35) − α ′ ( δp c k + δρ c k ) − A ′ k ( ρ c + p c ) − δρ ′ c k + k V c k = 0 , (36) Equation˙of˙a˙vacuum˙state printed on June 23, 2020 ( ρ c + p c ) A k + 4 V c k α ′ + δp c k + V ′ c k = 0 , (37)where V c = ( p c + ρ c ) v c corresponds to every kind of a fluid.Let us remind that “gauge invariant” potentials are usually under con-sideration in GR that corresponds to the metric ds = a ( η ) (cid:16) (1 + 2Φ( η, x )) dη − (1 − η, x )) δ ij dx i dx j (cid:17) , (38)as well as the “gauge invariant” density contrasts and the velocity potentials:˜ δ c k ( η ) = δρ c k ( η ) ρ c ( η ) + ρ ′ c ( η )2 ρ c ( η ) F ′ k ( η ) , ˜ v c k = V c k ( η ) ρ c ( η ) + p c ( η ) − F ′ k ( η )2 , Φ k ( η ) = A k ( η ) + a ′ ( η ) F ′ k ( η ) + a ( η ) F ′′ k ( η )2 a ( η ) , Ψ k ( η ) = − a ′ ( η ) F ′ k ( η )2 a ( η ) − A k ( η ) + 16 k F k ( η ) . (39)If the Friedmann equation is satisfied exactly, Eqs. (32) - (37) can berewritten in the terms of “invariant” quantities that results in the knownequations [38, 39]. However, if Friedman equation is satisfied up to onlysome constant, the fundamental system is (32) - (37). In this case, it isimpossible to rewrite this system in the terms of invariant variables becauseof the metric (38) does not belong to a class of metrics regarding which theaction varies in the FVT gravity [10].Here, the authors consider a linear evolution of the inhomogeneities ofpressureless matter and vacuum beyond “the last scattering surface”, whenradiation decouples with the matter, and the Universe structure starts todevelop [40, 41]. As is known, the anisotropy of CMB imprints the degreeof spatial inhomogeneity of a baryon-photon plasma at the last scatteringsurface.After decoupling, the inhomogeneities growth with the Universe evolu-tion results in the formation of structures such as galaxies, clusters, andsuperclusters ( D k ≥ , , and 10 , respectively, [42]. Let’s calculatethe inhomogeneity growth factor (the “density contrast” factor): D k ( z ) = ˜ δ k ( z ) / ˜ δ k (1100) , (40)where z = 1100 is the redshift corresponding approximately to the lastscattering surface. Eq. (40) contains the “invariant” variables, i.e., thecalculation is performed in the reference system (30), but one turns finallyto the expressions (39) which are the reference-frame invariant.As is seen from Fig. 3, a, b , the inhomogeneities at the extra-large scaledecrease for both matter and vacuum. At the intermediate scale (Fig. 3, c, quation˙of˙a˙vacuum˙state printed on June 23, 2020 d ), vacuum decouples with matter in a sense that its perturbations growsslower. The value of the growth factor suggests that the linear theory isstill valid, because the typical value of inhomogeneities at the last scatteringsurface is estimated as 10 − − − . Multiplying these value by the growsfactor results in quantity less than unity. At smaller scales of the orderof galaxy clusters shown in Fig. 3, e, f , the inhomogeneities enter into anonlinear regime. In the standard ΛCDM model this scale is “slightly”-nonlinear ( D k ≤ − ), but it is strongly nonlinear in our model.We conjecture that as an evidence of early and more intensive structureformation demonstrated by the modern observational data [41, 43–45].At smaller scales one might conjecture that such vacuum clusterizationwould be considered as a “dark-matter halo” formation, but such nonlinearregimes are far beyond the scope of the present paper considering onlylinear perturbations evolution. The above calculations are performed fortwo values of the pressureless matter Ω m = 0 .
03, as in the standard model,and Ω m = 0 .
3. The last value is preferable for Milne’s-like Universe becausethe nucleosynthesis in linearly coasting cosmology demands this value ofbaryonic matter to provide necessary amount of helium [46, 47].
6. Conclusion
It is shown that the description of a vacuum as some elastic medium(“fluid”) leads to the ES with the defined speed of scalar “sound waves”.Such a representation can be considered as a basis for the precision cosmol-ogy of the Milne’s-type Universe [36, 37, 48], with expansion close to linear.Although the horizon problem is absent for such a model, the Hubble con-stant H plays a role of a typical scale for the evolution of perturbations.In particular, the perturbations with a wave number k < H decrease dur-ing the Universe evolution, while the perturbations with k > H increase.According to numerical estimations, there is no need in the dark matterfor perturbations growth, because of the perturbations increase intensivelyat the small scales and enter into the nonlinear regime. It seems that theMilne’s viewpoint on the necessity to proceed from a “cosmological picture”and “descent” to a local theory of gravitation still could be more relevantthan it usually considered. Namely one should describe the physical andcosmological properties of vacuum fluctuations first, and only then introducelacking pieces like dark matter and energy.Despite the active latest debates on the Milne’s-like cosmologies (“freelycoasting universe”, “ R h = ct -universe,” etc.), the discourse is staying onthe natural philosophy level until now. This paper aims to divert this dis-cussion into physical context. Namely, the vacuum ES unifying the darkenergy/matter and the system of equations for the perturbations evolution Equation˙of˙a˙vacuum˙state printed on June 23, 2020 provides the necessary calculational paradigm for the quantitative compar-ison with the standard model. One has to note that the nonlinear evolutionof perturbations is much more tricky for analysis because it could requirethe consideration of nonlinear operators evolution for the energy density ofquantized fields. REFERENCES [1] Y. B. Zeldovich, Sov. Phys. Usp. , 216 (1981).[2] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[3] V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D , 373 (2000).[4] S. M. Carroll, Living Rev. Rel. , 1 (2001), arXiv:astro-ph/0004075.[5] T. Padmanabhan, Phys. Rep. , 235 (2003).[6] A. Chernin, Phys. Usp. , 253 (2008).[7] M. Li, X.-D. Li, S. Wang, and Y. Wang, Comm. Theor. Phys. , 525 (2011).[8] N. D. Birrell and P. C. W. Davis, Quantum Fields in Curved Space (CambridgeUniversity Press, Cambridge, 1982).[9] S. V. Anischenko, S. L. Cherkas, and V. L. Kalashnikov, Nonlin. Phenom.Compl. Syst. , 16 (2009), arXiv:0806.1593.[10] S. Cherkas and V. Kalashnikov, Theor. Phys. , 124 (2017).[11] S. L. Cherkas and V. L. Kalashnikov, Proc. Natl. Acad. Sci. Belarus, Ser.Phys.-Math. , 83 (2019), arXiv:1609.00811.[12] P. A. M. Dirac, Nature , 906 (2051).[13] V. F. Zolotarev, Soviet Physics Journal , 51 (1985).[14] M. C. Bento, O. Bertolami, and A. A. Sen, Phys. Rev. D , 043507 (2002).[15] P. T. Silva and O. Bertolami, Astr. J , 829 (2003).[16] L. Amendola and S. Tsujikawa, Dark energy: Theory and Observations (Cam-bridge University Press, Cambridge, 2010).[17] S. I. Blinnikov and A. D. Dolgov, Phys. Usp. , 529 (2019).[18] B. S. Haridasu, S. L. Cherkas, and V. L. Kalashnikov, A reference levelof the universe vacuum energy density and the astrophysical data, 2019,arXiv:1912.09224.[19] J. J. van der Bij, H. van Dam, and Y. J. Ng, Physica A , 307 (1982).[20] F. Wilczek, Phys. Rep. , 143 (1984).[21] W. G. Unruh, Phys. Rev. D , 1048 (1989).[22] E. Alvarez, Journal of High Energy Physics , 002 (2005).[23] M. Henneaux and C. Teitelboim, Physics Letters B , 195 (1989).[24] L. Smolin, Phys. Rev. D , 084003 (2009).[25] M. Visser, Particles , 138 (2018), arXiv:1610.07264. quation˙of˙a˙vacuum˙state printed on June 23, 2020 , 1997 (2008).[27] S. L. Cherkas and V. L. Kalashnikov, Eicheons instead of black holes, 2020,arXiv:2004.03947.[28] E. J. Copeland, Annalen. der Phys. , 62 (2015).[29] S. L. Cherkas and V. L. Kalashnikov, JCAP , 028 (2007), arXiv:gr-qc/0610148.[30] S. L. Cherkas and V. L. Kalashnikov, Plasma perturbations and cosmic mi-crowave background anisotropy in the linearly expanding Milne-like universe,in Fractional Dynamics, Anomalous Transport and Plasma Science , edited byC. H. Skiadas, chap. 9, Springer, Cham, 2018.[31] S. L. Cherkas and V. L. Kalashnikov, Universe driven by the vacuum ofscalar field: VFD model, in
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