On a physical description and origin of the cosmological constant
aa r X i v : . [ g r- q c ] O c t On a physical description and origin ofthe cosmological constant
S. Viaggiu ∗ Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni,Universit´a degli Studi Guglielmo Marconi, Via Plinio 44, I-00193Rome, Italy. Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”,Via della Ricerca Scientifica, 1, I-00133 Roma, Italy. INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, Via Cintia Edificio 6, 80126 Napoli, Italy.October 18, 2018
Abstract
In this paper we use and extend the results present in [1, 2, 3, 4] and inparticular in [4] to obtain a statistical description of the cosmologicalconstant in a cosmological de Sitter universe in terms of massless exci-tations with Planckian effects. First of all, we show that at a classicallevel, the cosmological constant Λ > T → PACS Number(s): 95.36.+x, 98.80.-k, 04.60.Bc, 05.20.-y, 04.60.-m ∗ [email protected] and [email protected] Introduction
The nature of the dark energy, representing about the 68% of the actualenergy content of the universe, is perhaps one of the biggest issue in modernphysics. In general relativity, dark energy is depicted in terms of the cosmo-logical constant Λ, with energy density ρ Λ = Λ c πG and constant equation ofstate p Λ = − c ρ Λ . The physical origin of Λ is still obscure. Since the cosmo-logical constant enters in Einstein’s equations as T µν = − Λ g µν , its naturalexplanation is in term of a vacuum energy. Nevertheless, naive quantumfield theory computations suggest for Λ a value looking like Λ ∼ L P , with L P the Planck length, a value about 10 orders greater than the one reallyobserved [5, 6, 7, 8, 9, 10]: this is named ’vacuum catastrophe’. In fact, afterintroducing a cutoff energy scale E c , we have ρ vac ∼ E c with the effectivemeasured cosmological constant Λ and the bare one Λ related byΛ = Λ + Λ vac , (1)where Λ vac = 8 πGρ vac . In order to have the observed value Λ = 10 − /m ,a magic cancellation of about 10 orders is required, but no physicallyrealistic realizations have been yet obtained and serious doubts exist onthe possibility that such a cancellation is physically possible according togeneral relativity. Moreover, supersymmetry is required, but this beautifullmechanism has not been at present day observed at LHC.In the literature, to overcome these issues, a lot of dark energy models witha time-dependent equation of state p ( t ) = − γ ( t ) ρ ( t ) c with γ > / f ( R ) models [11, 12, 13]) or adding ad hoc potentials tothe energy-momentum tensor. However, at present day, the description ofthe dark energy in terms of the cosmological constant (the simplest fieldsatisfying the Klein-Gordon equation) within the general relativity is stilllacking. In any case, the physical origin of Λ, its smallness and the roleof the vacuum energy in the dynamic of the universe are yet fundamentalunsolved problems.Recentely, I have proposed [1, 2, 3, 4] a way to depict the black holeentropy in terms of trapped gravitons [1, 2] together with the logarithmicentropy corrections [3]. The treatment has been extended in [4] to any mass-less excitation. In particular, in [3, 4] a possible mechanism transforminga radiation field into a one with a linear equation of state is presented asa theorem. In [3] it has been shown that this mechanism naturally gives,after considering quantum gravity corrections, the well known logarithmic2orrections to the black hole entropy. In [4], it has been shown that, con-trary to the black hole case, macroscopic configurations with an equationof state with γ = − As a first step, we must have at our disposal a suitable expression for the(quasi-local) energy in Friedmann spacetimes in comoving coordinates: ds = − c dt + a ( t ) dr − kr + a ( t ) r (cid:0) dθ + sin θ dφ (cid:1) . (2)In the background (2), as well known, this quasi-local energy, reducing inthe Newtonian limit to the matter-energy content of the spacetime togetherwith its gravitational energy, is provided by the Misner-Sharp mass M ms [14] with associated energy E ms = M ms c . After defining L = a ( t ) r wehave: E ms ( L ) = c L G (cid:16) − h ab ∂ a L ∂ b L (cid:17) , (3)where h ab is the two metric spanned by the coordinates ( t, r ). In terms ofthe apparent horizon L A with L A = c q H + ka t ) , where H denotes the Hubblerate, we have: E ms = c G L L A . (4)Expression (4) can be seen as the quai-local energy within a proper vol-ume V ( L ) = 4 πL /
3. We are interested in a de Sitter flat expandinguniverse obtained from (2) with k = 0 and a ( t ) = e ct q Λ3 together with3 ( t ) = H Λ = c q Λ3 .After specifying the expression for the quasi-local energy within a volumeof proper areal radius L , we must specify the macroscopic model. To thispurpose, consider a spherical region of proper volume V ( L ): the contribu-tions to E ms in (4) are supposed to be provided by massless modes (perhapsgravitons or also photons) with proper wavelengths λ such that λ ≤ L . Asa consequence, concerning the allowed discrete spectrum, for the angularfrequencies ω (0) we can use the following expression, in line with [4]: ω (0) n = a cnL , a ∈ ℜ + , n ∈ N . (5)Note that, when quantum fluctuations are taken into account in section 4,we have a natural ultraviolet cutoff given by the Planck length L P , with L ≥ L P .We can identify E ms as the internal energy U (0) of massless excitations inside . With the (5), we can calculate the partition function Z (0) T = Z (0) N for N excitations: Z (0) = + ∞ X n =0 e − β ~ ω (0) n = 11 − e − ac ~ βL , β = 1 K B T , (6)where T is the temperature and K B the Boltzmann constant. As usual wehave U (0) = − (ln Z T ) ,β (comma denote partial derivative): U (0) = ca ~ NL (cid:16) e ac ~ βL − (cid:17) , (7)with the constraint U (0) = c G L L A = ca ~ NL (cid:16) e ac ~ βL − (cid:17) . (8)Note that in the classical limit L >> ac ~ β , we obtain the classical result U (0) = N K B T . For the free energy we have F T = − N K B T ln( Z (0) ) withthe pressure p (0) given by F T (0) ,V = − p (0) . From (6) and (7) we obtain In [15] to a Friedmann flat spacetime at the apparent horizon we associated a zerointernal energy. This result is not in disagreement with the asumption of this paper. Infact, in [15], the internal energy appearing in the first law at the apparent horizon is asummation bewteen the Misner-Sharp mass and the negative contribution due to the nonstatic dinamic of the universe and these contributions cancel out. (0) V = U (0) , i.e. as expected for (5) a radiation field [3, 4]. In order todepict the cosmological constant equation of state p Λ = − c ρ Λ , we mustobtain the suitable frequency spectrum. As in [3, 4], we can write, withoutloss of generality, the frequency ω in the form ω ( n, L ) = ω (0) n + Φ( L ) N , (9)where φ ( L ) is a function to be determinated. Moreover, we have F T = − N K B T ln( Z T ) = F (0) T + ~ Φ( L ). It is worth to be noticed that, in practice,the added term due to Φ it gives a contribution to the energy that is inde-pendent on the temperature T . This is reminescent of solid state physics,where for the free energy, to the summation over the oscillation frequen-cies of the atoms, a further term is added, namely ǫ ( N/V ) independenton the temperature and depending only on the density of the solid: thisadded term depicts the energy of the atoms of the solid in the equilibriumconfiguration. In order to describe the cosmological constant in a physicallysound way within general relativity, the energy density must be constant intime and space and given by ρ Λ = Λ8 πG . In fact, this is what we obtain bytaking ρ Λ = E ms /V , with E ms given by (4). The next step is to adopt andmodify the theorems quoted in [3, 4] in a form suitable for a cosmologicalcontext. To this purpose, since the energy is fixed by the general relativisticexpression (4) and as a result we have at our disposal the expression for theinternal quasi-local energy U ( L ), the point is to obtain the exact expres-sion for the spectrum of the excitations given by (9) leading to U ( L ) viathe partition function Z T = (cid:0)P ∞ n =0 e − β ~ ω ( n,L ) (cid:1) N . The following result stillholds: Proposition:
Let ω , given by the (5), denote the angular frequency of N masless excitations within a volume of proper areal radius L . The excita-tions with energy ~ ω = ~ ω + ~ Φ( L ) N have a linear equation of state P V = γU provided that the differentiable function Φ( L ) satisfies the following equation ~ [ L Φ ,L ( L ) + Φ( L )] = U ( L )(1 − γ ) , (10) together with the condition U ( L ) − ~ Φ( L ) > . (11) Proof.
With the usual relation U ( L ) = − ln( Z T ) ,β , we obtain U = U (0) + ~ Φ( L ) . (12)5ince from the (7) we have U (0) >
0, condition (11) follows. For the freeenergy we have F T = − N K B T ln( Z T ) = F (0) T + ~ Φ( L ). Moreover F T ,V = ~ Φ ,L L ,V + L ,V F (0) T ,L = − P, (13)with L ,V F (0) T ,L = − P (0) and P (0) V = U (0) . Hence, from (13) we get ~ L V Φ ,L − U (0) V = − P. (14)After using the (12) with P V = γU ( L ), from (14) we obtain the equation(10).Note that for a radiation fiels, i.e. γ = 1 /
3, the right hand side of (10)becomes zero. The solution Φ( L ) ∼ /L is solution of the homogeneousequation (10) giving a radiation field equation of state. By requiring thatfor γ = 1 / , by the Misner-Sharp energy (3); this fact is oftenmissing in the literature.It is interesting that the so obtained proposition is in fact independent onthe explicit form of the potential U (0) and also on the explicit (discrete) ex-pression for ω (0) n in (5). The fundamental assumption is that U (0) representsa radiation field. Thus, in principle, also massless fermionic excitations rep-resenting a radiation field could contribute to U (0) , with U (0) representingthe whole contribution due to fermions and bosons and with obviously adifferent expression for (6) but with the solution for Φ( L ) left unchanged,provided that the expression for U ( L ) is fixed by Misner-Sharp expression. .However, we stress that our approach for the cosmological constant is dif-ferent from usual ones present in the literature and based principally onsupersymmetry. As will be shown in section 4, the equation of state for thecosmological constant only emerges when Planckian fluctuations come intoaction, without introducing the usual vacuum made of all kind of particles.Hence, we refer to ”vacuum” as Planckian fluctuations, and massless exci-tations seem to be more appropriate at Planckian scales. Without quantum gravity motivated corrections considered in section 4 To this purpose, in principle also massive fermions can be considered.
To start with, we must solve the equation (10) with U ( L ) = E ms ( L ) = c G L L A . We obtain: ~ Φ( L ) = (1 − γ ) c G L L A . (15)With the solution (15), the existence condition (11) it gives γ > − γ = − T = 0 for our system of massless excitations. Hence, the cosmo-logical de Sitter spacetime with a positive cosmological constant Λ can bedepicted in terms of massless excitations at the absolute zero temperature.This fact is in agreement with the idea that the cosmological constant is amanifestation of vacuum energy. In fact, for T = 0 we have U (0) ( L ) = 0and consequently U ( L ) = E ms ( L ) = ~ Φ( L ), with Φ( L ) given by (15) with γ = − T . For T = 0, according withthe third law of thermodynamics, we have zero entropy S ( T = 0) = 0. It isworth to be noticed that T → V is theMisner-Sharp one E ms representing a non vanishing cosmological constant,exactly what physically we expect. For the reasonings above, in the follow-ing we denote with Λ = Λ( T = 0) the ’bare’ cosmological constant. Thecalculations of this section clearly show that, in order to understand the truenature of the cosmological constant, quantum Planckian corrections must beconsidered. In the next section we will ’dress’ Λ by introducing quantumgravity motivated modifications to the expression (8) for U ( L ).7 Cosmological constant with quantum gravity mo-tivated corrections
In this section we depict the cosmological constant by considering correctionsdue to quantum fluctuations that are expected in a quantum gravity regime.Our approach is in some sense ’phenomenological’. This means that, in orderto apply the (10), we must obtain an expression for U ( L ) with quantum-gravity motivated corrections. We may suppose that at a quantum gravitylevel the spacetime cannot longer be depicted with a classical metric as the(2), but the spacetime granularity comes into action. There, a quantumspacetime [16] motivated by the non-commutativity of the spacetime coor-dinates can be quoted. Physically motivated spacetime uncertainty relations(STUR) can be found in [15] in a Newtonian approximation and further gen-eralized in [17] and in [18] in a Friedmann flat background. In particular,in [18] physically motivated STUR have been obtained in spatial Cartesiancoordinates { x i } by introducing proper coordinates given by η i = a ( t ) x i . Inthe coordinates η i , the Heisenberg uncertainty relations can be written, inany allowed state s , as∆ s E ∆ s t ≥ ~ , ∆ s p η i ∆ s η i ≥ ~ , ∀ i = 1 , , . (16)STUR in a Friedmann flat quantum spacetime [18] can be obtained interms of the ratio ∆ s A/ ∆ s V , where, ∆ s A = P i,j,i ≤ j ∆ s η i ∆ s η j and ∆ s V = Q i ∆ s η i : √ ∆ s A √ s ( H )∆ s A c ≥ L P s A ∆ s V , (17) c ∆ s t (cid:18) √ ∆ s A √ s ( H )∆ s A c (cid:19) ≥ L P . (18)For the spacetime (2) with k = 0 representing a de Sitter expanding universe,the Hubble rate H is constant, and as a consequence the mean value s ( H )is a constant in a de Sitter spacetime. Moreover, thanks to quantum effects,in (17) and (18) we can substitute the bare cosmological constant Λ withthe one ’dressed’ by quantum interactions Λ, i.e. s ( H Λ ) = c q Λ3 . Note that,since 1 / √ Λ is of the order of the dimension of the apparent horizon (Hubbleradius of the universe) the following condition is expected to hold: s ( H Λ ) p ∆ s A ≤ c. (19)8oreover, the state such that the STUR (17)-(18) are satured are calledmaximal localizing states [16]: they are state (spherical) with all uncertain-ties of the same magnitudo, i.e. c ∆ s t ∼ ∆ s η i ∼ ∆ s η . Hence, the STUR(17)-(18) do imply that ∆ s η i ≥ χL P , with χ of the order of unity. As aconsequence, the (16) for ∆ s E becomes ∆ s E ≥ χ c ~ s η . In terms of ourproper variables L = a ( t ) r we have ∆ s L ∼ ∆ s η i and thus ∆ s E ≥ χ c ~ s L . (20)In this section we treat the parameter χ as a constant that it is expectedof the order of unity. Expression (20) physically motivates the followingexpression for U ( L ) in a proper volume V = 4 πL / U ( L ) = c G L L A + χ c G L P L . (21)The next step is to use expression (21) for U ( L ) in (10) with the solution ~ Φ( L ) = (1 − γ ) c G L L A + χ (1 − γ ) c G L P L ln (cid:18) LL (cid:19) , (22)with L a positive constant. For χ >
0, the condition (11) becomes:1 − (1 − γ ) ln (cid:18) LL (cid:19) > . (23)We are interested in the cosmological constant case γ = − L < L e . Since in our model we have a natural constant Λ, we maytake L ∼ / √ Λ ∼ L A and as a consequence L can be taken also of theorder of Hubble radius. As we will show in the next section, in order tobe in agreement with general relativity without introducing quintessenceexotic dark energies, the proper radius L must be interpreted as an effectivephysical length-scale.Another possibility, explored in [4], is to take χ <
0. In this case a macro-scopic configuration is possible with a minimum radius L with L >
0, butwith L = sL P and s of the order of unity or greater. However, as we willshow in the next section, for our interpretation of the dependence on L of Note that in the spherical case the STUR implies that ∆ s L ≥ L P , thus representingan ultraviolet cutoff. In general relativity, in a Friedmann flat cosmology (2), a fluid with an equation ofstate with γ = − ρ Λ .
9, where the range of L is requested at the Planck length up to macroscopicscales where decoherence and classicality are expected to hold, the proposal χ > T = 0. This result is in agreementwith physical reasonability for a bare cosmological constant. However, it iscustomary to associate to the apparent horizon of a given classical Fried-mann universe a temperature T h , namely T h = c ~ πK B L A . This temperaturelooks like an Unruh temperature for an ingoing radiation from the apparenthorizon. In [15] it has been shown that, in a de Sitter universe, since theapparent horizon is static, one can associate to the dark energy a temper-ature given by T h . The point is: it is this temperature the one effectivelymeasured by a thermometer or is a temperature arising to satisfy the firstlaw of thermodynamics ? It is not easy to ask to this question in a rigorousway. For example, in [19] it has been advanced the possibility that the Un-ruh temperature is not the one measured by a thermometer, thus it does notrepresent an exchange of heat with a surrounding gas, but rather it is causedby quantum effects generated by a local coupling between the thermometerand the vacuum state. When cosmological constant is dressed with quantumcorrections, one expect that a non-zero temperature T Λ can arise. For thereasonings above and present in [19], this temperature T Λ could be differentfrom T h . If we are wilings to accept, thanks to the holographic principle,that to a dark energy is associated a non-vanishing entropy proportional tothe area of the apparent horizon , then we can set T Λ = bT h , with b ∈ ℜ + of the order of unity or less. These reasonings are in agreement, on generalgrounds, with the first law of thermodynamic in the usual form T Λ dS Λ = dU + P dV. (24)In fact, the cosmological constant equation of state requires that, by inspec-tion of (24), T Λ dS Λ = 0: for the bare case with T Λ = S Λ = 0 this equationis trivially satisfied. For T Λ >
0, we must have S Λ = k , with k ∈ ℜ + . Sinceapparent horizon of a de Sitter cosmological universe is constant in time,the entropy of a de Sitter spacetime is expected to be constant in time atthe apparent horizon. With and only with the choice T Λ = bT h we havean entropy proportional to the area of the apparent horizon together withlogarithmic corrections, thanks to the logarithmic term in (20). In this way,the entropy of a de Sitter spacetime at the apparent horizon becomes the See for example [1, 2, 15, 20] and references therein T Λ with the related expression for the entropyplay a marginal role in the paper. In this section we study the formula (21) for the quasi-local energy insidethe region of proper areal radius L . Formula (21) implies that ρ Λ = c Λ8 πG + 3 χc πG L P L , (25)that in turn implies Λ = Λ + 3 χL P L . (26)It should be stressed that the meaning of (26) is that the observable cosmo-logical constant Λ is composed of two contributions: the former due to Λthat is the contribution of the radiation massless field , while the latter isthe one dressed by Planckian effects. As stated by formula (10), this termis fundamental in order to have the equation of state suitable for a cosmo-logical constant.A naive interpretation of the formula (26), since of the dependence on L ,is in terms of quintessence dark energy. A quintessence model requires avariating equation of state, while in our setups the equation of state is con-stant in time and space with γ = −
1. To obtain a physical understandingof the (18), it should be noted that, thanks to the results of sections 3 and4, the cosmological constant case with γ = − ∼ /L . Hence, the proper-length parameter L can be cor-rectly interpreted as a parameter depicting the scale at which the physics isconsidered (averaged). In fact, we may consider the spacetime metric h ab asa fluctuating quantity depending on general spatial coordinates { x a } and atime coordinate t , with line element ds = − c dt + h ab ( t, x a ) dx a dx b . (27) In sections above we have named this contribution bare cosmological constant sinceit can be also seen as the cosmological constant at T = 0 . t could be considered as an average time of fluctuatingmetrics on a given spherical box of proper areal radius L . We can assumethat when classicality is recovered, the time t is nothing else but the cosmicone t present in the classical metric (2): in a quantum spacetime t becomes[18] an essentially self-adjoint operator satisfying the STUR (17) and (18).Concerning the spatial metric h ab , we can adopt, after some suitable adjust-ment, the well known Buchert formalism [21]. There, we can consider a sliceof constant t with t = s ( t ) = k ∈ ℜ , with s a quantum allowed state withrespect to the (17)-(18).To start with, consider a proper volume V ( L ). For any scalar quantity ψ ( s ( t ) , x i ), the average with respect to the volume V ( L ) is: < ψ ( s ( t ) , x i ) > V ( L ) = 1 V L Z V ( L ) ψ ( s ( t ) , x i ) q g (3) d x, (28)where g (3) denotes the determinant of the three metric h ab on the slice a s ( t ) = const. and V L = Z V ( L ) q g (3) d x. (29)Moreover, for the averaged expansion rate < θ > V ( L ) we have < θ > V ( L ) = ˙ V L V L = 3 ˙ a V ( L ) a V ( L ) , (30)where dot denotes time derivative with respect to t and the dimensionlesseffective scale factor a V ( L ) ( s ( t )) is given by a V ( L ) ( s ( t )) = (cid:18) V L ( s ( t )) V L ( s ( t )) (cid:19) , (31)with s ( t ) an initial time. For the averaged Hubble flow we have H = <θ> V ( L ) . Since on average, in order to obtain the de Sitter spacetime, weexpect a spatiallty flat metric, we can set to zero the averaged curvature R .As a consequence, we have an effective averaged metric given by: ds = − c dt + a L ( s ( t )) (cid:2) dr + r (cid:0) dθ + sin θdφ (cid:1)(cid:3) , (32)where we used the more short notation a V ( L ) ( s ( t )) = a L ( s ( t )). With theabove definition, the relevant equations for our purposes for an irrotational As shown in [18], in a spacetime with big bang, t can be symmetric with a uniqueself-adjoint extension, while in a de Sitter spacetime with no big bang t can be self-adjoint. L for the above mod-ified Buchert equations are:3 ˙ a L a L = c Λ + 3 c χL P L − Q L , (33) Q L = 23 (cid:2) < θ > − < θ > (cid:3) − < σ > . (34)In (34) Q L is the kinematical backreaction, while σ represents the shear.In a cosmological context [22] the shear is negligible on scales also smallerthan the scale of homogeneity. In a similar manner, in our context wherethe scales are Planckian or microscopic, it is reasonable that on scales soonafter the Planck one this term is also negligible. Moreover, the kinemati-cal backreaction term Q L is also expected to give small [22] values and asa consequence it gives a small contribution for the cosmological constant.Under these assumptions, formal integration of (33) at the scale L it gives a L ( s ( t ) = a L ( s ( t )) e (cid:18) c R s ( t ) s ( t q Λ eff dt (cid:19) , (35)with Λ eff = Λ + 3 χL P L − Q L . (36)The term Q L in (36) as stated above, is expected to be negligible on lengthscales above the Planck one, and also is expected to rapidely decreases fortime-scales t > t p , with t p the Planck time. In fact, the integrability condi-tion for the system (33)-(34) is6 Q L ˙ a L + a L ˙ Q L = 0 , (37)showing that Q L scales as Q L ∼ a L ( s ( t )) /a L ( s ( t )) . The (35) depicts anew view to look to the cosmological constant problem: instead of a naivesummation over the different vacuum energy contributions (Planck scale,QCD scale...), the cosmological constant is provided from an average over agiven physical length scale L . For a bigger and bigger L , the vacuum contri-bution proportional to 1 /L becomes smaller and smaller and for L >> L P it becomes practically negligible thus representing a solution, at least, to the’old’ cosmological constant problem.It should be noticed that, in the usual cosmological background, there existsa scale L o , named scale of homogeneity, such that an average on scales L greater than L > L o is obviously trivial since the metric factor a L is in factno longer dependent on L . What means this reasonings translated in the13anguage of an effective quantum gravity theory ? On general grounds,for a quantum system, it is expected that a decoherence scale emerges (seefor example [23, 24] in a quantum gravity context) and so also this scale isexpected to arise for the quantum gravity regime. This is obviously a com-plicated matter and to treat the problem in a suitable manner we need thequantum gravity theory that it is not actually at our disposal. In particularwe may think to a decoherence length L D such that for L > L D quantumPlanckian corrections become irrelevant and the transition to classicalitycomes into action. At such a scale, the term Q L D in (35) can be neglectedan as a result a de Sitter phase emerges with a L D ( s ( t ) ∼ a L D ( s ( t )) e c ( s ( t ) − s ( t )) q Λ3 , (38)where at the transition at the classicality we have s ( t ) = t = t and Λ givenby Λ = Λ + 3 χL P L D . (39)After performing an average on scales L > L D , since the classicality isreached, the average acts trivially as obviously acts on the classical metric(2).In the new approach presented in this paper to the cosmological constantproblem, the smallness of Λ is due to the existence of a decoherence scale L D . In our phenomenological approach, some numerical examples can dedone.To start with, it is interesting to calculate the scale L D in such a waythat the term χL P L D is of the same magnitudo expected for the cosmologicalconstant, i.e. ∼ − /m : we found, after taking χ of the order of unity, L D ∼ − m . This could look as a rather big value, but it should be notedthat decoherence in a gravitational field can be rather huge [23]. In any case,in our calculations we have supposed that the constant χ present in (21) isthe same as the one present in the STUR (20). Hence, we can alleviate thisassumption and suppose that the effective energy U is provided by U ( L ) = c G L L A + ξ c G L P L , (40)where ξ ∈ (0 , L A in the semiclassical term of U in (40) can be confusing. It must be stressed again that our approach is phenomenological in the sense that nounderlying quantum gravity theories are advanced and our procedure is not a proposal toquantize the gravity.
14o avoid any possible confusion for the reader, the following changement canbe consistently given. First of all, we introduce a new parameter, namely Γand we substitute L A in (40) with L A = : in practice we have made thesubstitution Λ → Γ in order to stress that L A refers to a bare cosmologicalconstant Γ. The first term in (40) is the one due to a massless radiationfield with constant density ρ r = c Γ8 πG and the second one is due to Planckianfluctuations. The energy expression (40) depicts the physics of our systemat and above Planck scale with the cutoff L inf ∼ L P dictated by the STUR(17)-(18). As far as the classicality is reached, the averaged metric evolvesaccording equation (38). For Λ from (40) we obtainΛ = Γ + 3 ξL P L . (41)How can we physically mark the crossover to the classicality ? Quite remark-ably, our phenomenological model can provide a physically sound answer.To this purpose, we expect that at the critical decoherence scale-length L D the quantum system undergoes a transition phase and the value of the mea-sured dressed cosmological constant Λ remains, for length-scales L > L D frozen to the value at L D , i.e. for L > L D → Λ( L ) = Λ( L = L D ). The point L = L D must thus be an absolute minimum for U ( L ), i.e. the configurationat the length-scale L = L D is at the lowest state enegy configuration: L D = (cid:18) ξL P Γ (cid:19) . (42)Thanks to (42), for Λ( L = L D ) we get:Λ( L = L D ) = Γ + 3 ξL P L D = 4Γ . (43)Equation (43) relates the phenomenological parameter Γ to the measuredcosmological constant. As a consistence check, we expect that for L ≥ L D the system evolves with the cosmological constant Λ and with the classicalexpression of the Misner-Sharp energy. In fact, thanks to (42) and (43) wehave: c L D GL = c G L D L A + ξ c G L P L D , (44)where L = 3Λ = 34Γ . (45)15s a cosequence of the (44), for L > L D classicality is reached and a pure deSitter spacetime with dressed cosmological constant (the one we measure)Λ and with the usual Misner-Sharp energy E ms ( L ) = c L GL arises.The crossover to classicality can also be understood in a thermodynamicallanguage. In fact, suppose that the temperature at the length-scale L isgiven by T = T ( L ). For the specific heat C = dUdL from (40) we obtain: C = c G (cid:18) L Γ − ξL P L (cid:19) dLdT ( L ) . (46)It is interesting to note that exactly at L = L D , the critical length-scale, wehave C = 0. Hence, at L = L D , i.e. at the minimum of U ( L ), the systemis thermodynamically dead and as a result in this vacuum state a pure deSitter spacetime emerges. Moreover, by supposing T ∼ /L , as happens fora black hole or for the Unruh temperature at the apparent horizon, we have dT /dL < L < L D we have C >
0, while for
L > L D we obtain C <
0, according to the expectation that in a classicalself gravitating system the specific heat is negative. This could represent anew way to look to the decoherence scale in a cosmological context.As a further consideration, note that expression (43) can be written as:Λ = 4 ξ L P L D . (47)The formula (47) is interesting because it gives the measured cosmologicalconstant ( ∼ − /m ) in terms of our parameters L D and ξ . The deco-herence length-scale L D from quantum gravity regime to classicality couldbe, at least in principle, measured in future experiments, while ξ could becalculated from models on quantum Planckian fluctuations of the energyin a non-commutative spacetime. In any case our phenomenological modelpredicts an expression for Λ in terms of quantum gravity motivated quanti-ties.Also note that the technology presented above can be easily extended to thecase where other kind of matter-energies are present. As an exapmle, if dustis present as in the concordance model, this contribution must be includedin the modified Buchert equations (33)-(34): as a result a decoherence scale L D emerges representing the crossover to classicality and the metric of theconcordance ΛCDM model is obtained.We expect that if classicality emerges near the Planck scale, the dimensione-less parameter ξ must be ξ << E ql within a spherical box V ( L ) looking like E ql ∼ L . In order to depictthis negative vacuum energy, we must have an energy U ( L ) with a sta-tionary point at the decoherence length-scale L D . This can be done with U ( − Λ) = − U ( L ), with U ( L ) given by (40) together with Λ = − U ( − Λ) has a local maximum at L = L D given by(42), and thus L = L D does not represent a true vacuum state with thelowest energy and the system is thus instable under quantum fluctuationsand as a consequence the transition to classicality cannot happens in themanner depicted above.We have presented a physically sound phenomenological model that is ca-pable to depict the cosmological constant from a statistical point of view, togive a solution to the old cosmological constant problem and also to give aphysical mechanism explaining the origin of our small cosmological constantas an averaged spacetime at the length-scale L D representing the decoher-ence from a quantum gravity regime to classicality. The critical length-scale L D is defined as the one giving an absolute minimum for the energy U ( L )thus representing the lowest-state energy, i.e. a true vacuum state that isfrozen for scales L > L D .In this regard, our approach is completely new since it indicates a morephysical and serious way to treat the vacuum energy, by means of a length-scale dependent cosmological constant in complete agreement with generalrelativity and quantum mechanics.An alternative interesting approach to the vacuum energy problem can befound in [25]. There, the vacuum energy is considered fluctuating and ratherinhomogeneous over the whole spacetime, whit the fluctuating spacetime ob-tained as a stochastic field of inhomogeneous metrics. Parametric resonanceis invoked to obtain an emerging spacetime with a small cosmological con-stant. Although the physical mechanism to explain the origin of a classicalde Sitter universe presented in [25] is different from the one presented in thisarticle, the paper [25] has the merit to present an approach to the cosmo-logical constant problem very different from the current ones present in theliterature based on supersymmetry, where unfortunately no supersymmetryhas been at present day detected at LHC and thus different scenarios mustbe explored. In this line, it should be also noticed the recent paper [26],where a dynamical cosmological constant is introduced, in a backgroundobtained by means of an extension of the general relativity in terms of theAshtekar variables, where a new uncertainty relation between the dynamical17 and the Chern-Simons time emerges. In the usual approach to the cosmological constant problem, the vacuumenergy is depicted as a summation over all possible vacuum contributionsfrom different energy scales present in physics (Planckian fluctuations, QCDcontributions and so on.), since thanks to the equivalence principle, all formsof energies do gravitate. Severe cancellations and the presence of supersym-metry is invoked in order to obtain the fine tuning necessaryy to solve the socalled ’vacuum catastrophe’. However, it should be noticed that when theequivalence principle is invoked, vacuum energies cannot merely summedas happens in a flat spacetime and the explicit expression for the vacuumenergy leading to the cosmological constant equation of state is constrainedby general relativity. In fact, the natural arena for the equivalence principleis provided by general relativity, where the equations are not linear and thevacuum energies modify the spacetime metric in a non-trivial way. A secondimportant question is that in the usual treatment of the cosmological con-stant as vacuum energy it is not often clear if the vacuum contributions con-sidered effectively satisfy the suitable equation of state p Λ = − c ρ Λ and notfor example the radiation one, while this fact should be carefully checked. Asan example, for an harmonic oscillator, the vacuum energy looks like ~ ω/ L wehave ω ∼ c/L leading, thanks to (10) to a radiation field equation of staterather than to the one of the cosmological constant. In our approach theequation of state for Λ is fixed from the onset by formula (10). According tothe line present in [25, 26], a new way to treat the vacuum energy is urgent.In this paper, we propose a new approach, based on the results present in[3, 4] in particular. The basic idea is that the cosmological constant has aquantum origin and can be depicted only when Planckian fluctuations arecorrectly taken into account. In this regard, a statistical description of thecosmological constant Λ emerges with a dependence on the scale at whichthe spacetime is averaged. To this purpose, the spacetime at Planckianscale is fluctuating and we depict these fluctuations in terms of an averageover a proper volume V ( L ) composed of inhomogeneous metrics, by usinga suitable modification of the Buchert formalism and without introducinga quintessence field. Another important ingredient in our modeling, oftenmissing in the literature, is that general relativity provides an expression forthe quasi-local energy within a sphere L given by the Misner-Sharp mass18 ms . In particular, in usual thermodynamical systems within a volume V ,there is not a priori relation between the energy E and the volume V . Con-versely, general relativity it gives a further geometric information providing,by means of E ms , a relation between E ms in a spherical box and the area ofthe box. As an example, the ADM black hole mass M is related to the arealradius R by the famous relation R = 2 GM/c . These constraints must betaken into account for a sound description of the classical de Sitter universe:this is what we have done for example in equation (8). As a consequenceof our setups, the classical de Sitter spacetime arises at the decoherencescale L = L D , representing an absolute minimum for our ’phenomenologi-cal’ expression for U ( L ). This absolute minimum represents the crossover toclassicality. Moreover, since L = L D is an absolute minimum for U ( L ), thevalue for Λ we observe is fixed exactly at this decoherence length-scale, ex-pressed in terms of the two phenomenological parameters, namely { L D , ξ } ,by equation (47) an can be thus an explanation to the small value for Λ weobserve. As a final remark, it should be noted that the mechanism depictedin the paper for the origin of the cosmologcial constant can in principle bealso applied at the primordial inflation. There, the proper dimension of theuniverse before the inflation has been very small, of the order of the Planckscale and thus the potential (40) should have gained an absolute minimumabove the Planck scale, thus representing the begin of the primordial infla-tion in presence of other kind of matter-radiation. This can be certainlymatter for next works. References [1] Viaggiu S 2017
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