In a recent paper [Q. Wang, Z. Zhu, and W. G. Unruh, Phys. Rev. D 95, 103504 (2017)PRVDAQ2470-001010.1103/PhysRevD.95.103504] it was argued that, due to the fluctuations around its mean value, vacuum energy gravitates differently from what was previously assumed. As a consequence, the Universe would accelerate with a small Hubble expansion rate, solving the cosmological constant and dark energy problems. We point out here that the results depend on the type of cutoff used to evaluate the vacuum energy. In particular, they are not valid when one uses a covariant cutoff such that the zero-point energy density is positive definite.Fil: Mazzitelli, Francisco Diego. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; ArgentinaFil: Trombetta, Leonardo Giuliano. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentin

Mazzitelli, Francisco Diego

Trombetta, Leonardo Giuliano

English

arXiv

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arXiv:1801.00138v1  [gr-qc]  30 Dec 2017Comment on: “How the huge energy of quantum vacuumgravitates to drive the slow accelerating expansion of theUniverse”Francisco D. Mazzitelli and Leonardo G. TrombettaaCentro Atómico Bariloche and Instituto Balseiro,Comisión Nacional de Enerǵıa Atómica, 8400 Bariloche, Argentina.(Dated: January 3, 2018)AbstractIn a recent paper (Phys. Rev. D95, 103504 (2017)) it is argued that, due to the fluctuationsaround its mean value, vacuum energy gravitates differently from what previously assumed. Asa consequence, the universe would accelerate with a small Hubble expansion rate, solving thecosmological constant and dark energy problems. We point out here that the results depend onthe type of cutoff used to evaluate the vacuum energy. In particular, they are not valid when oneuses a covariant cutoff such that the zero point energy density is positive definite.a Present address: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, ItalyINFN - Sezione di Pisa, 56200, Pisa, Italy1In the traditional formulation of the cosmological constant problem, it is argued that thezero point energy density 〈ρ〉 associated to a quantum field is proportional to Λ4, whereΛ is an ultraviolet cutoff, of the order of the Planck energy EP lanck. Assuming that themean value of the stress tensor of the quantum field is covariantly regularized, one has〈Tµν〉 = −〈ρ〉 gµν , which corresponds to a cosmological constant of order E4P lanck, about 120orders of magnitude larger than the observed one.In Ref.[1] it was pointed out that the energy-momentum tensor associated to a quantummassless field φ has very large fluctuations around its mean value. Therefore, it is notcorrect to use 〈Tµν〉 as a source of the Einstein equations. When properly taken into account,these fluctuations lead to modified Einstein equations with a stochastic component. Moreconcretely, for a metric of the formds2 = −dt2 + a2(t,x)(dx2 + dy2 + dz2)(1)the evolution equation for the scale factor a(t, x) is that of a harmonic oscillatorä+ Ω2(t,x)a = 0 , Ω2(t,x) =4πG3(ρ+3∑i=1Pi)=8πG3φ̇2 , (2)where ρ = T00, Pi = Tii/a2. The quantity Ω2 is assumed to have a positive mean value〈Ω2〉, of order Λ4, and to have quasiperiodic stochastic fluctuations in a time scale of order1/Λ. Thus, due to parametric resonance, the scale factor has an exponential growth with aHubble rate H which is exponentially small in the limit Λ → ∞, solving the cosmologicalconstant problem.In this comment we would like to stress the following point: if the theory is regulatedby a Lorentz invariant cutoff in flat spacetime, then one has 〈p〉 = −〈ρ〉, and therefore〈Ω2〉 = −8πG〈ρ〉/3. Moreover, if the cutoff is such that 〈ρ〉 > 0, as usually assumed, then〈Ω2〉 < 0 and the whole picture of parametric resonance breaks down.Let us be more explicit. Wang et al first computed 〈Ω2〉 in Minkowski spacetime usinga non-invariant cutoff Λ such that |~p| < Λ, where ~p denotes the 3-momentum of the modesof the scalar field. In this case, both 〈ρ〉 and 〈p〉 are positive definite and proportional toΛ4. Note, however, that for this particular cutoff one has 〈p〉 = 〈ρ〉/3, breaking the Lorentzinvariance of 〈Tµν〉. This has been noticed long ago in Ref.[2]: a non-covariant cutoff cannotbe used to estimate the vacuum contribution to the cosmological constant (see also Refs.[3, 4]). If, in spite of this, one accepts the use of this cutoff, and assumes that the regularized2quantities have physical meaning, then the conclusions of Ref.[1] looks correct, although theinitial problem is different: 〈Tµν〉 does not describe a cosmological constant but a radiationfluid.Wang et al also computed 〈Ω2〉 using a Lorentz invariant procedure inspired in Pauli-Villars method. The particular implementation of this method used in Ref.[1] may give〈Ω2〉 > 0 (this is not completely clear from Eq.(195) in Ref.[1]). Once more, if this were thecase, the analysis of the dynamical equation for the scale factor in Ref.[1] would be correct,but at the price of regularizing the theory in such a way that 〈ρ〉 < 0. Clearly, the useof this particular Lorentz invariant cutoff would not be equivalent to the use of a cutoff in3−momentum space, since it produces a vacuum energy density with a different sign.But the situation is even worse: the Pauli-Villars method produces ambiguous resultsfor the polynomial divergences [5–7]. Only the logarithmic divergences are univocally de-termined by the method. We illustrate this fact with an example discussed in Ref.[5]. Theregularized energy momentum tensor in Minkowski spacetime, using Pauli-Villars method,is given by〈Tµν〉 = −gµν4N∑i=0CiM4i logM2iµ2, (3)where the masses Mi, i = 1, 2, ...N are the regulators, M0 = m is the mass of the field, µ isan arbitrary mass scale, C0 = 1, and the constants Ci, i = 1, 2, ...N satisfyN∑i=0Ci(M2i)p= 0 , (4)for p = 0, 1, 2. Due to these conditions, the result is independent of the scale µ. In principleone can add an arbitrary number N of regulator fields, the minimum being N = 3 to satisfythe above constraints. It has been shown that, for N = 3, the regularized version of thestress tensor produces a negative energy density. In the particular case Mi = Λ one has〈Tµν〉 = −gµν128π2[−Λ4 + 4m2Λ2 −m4(3 + 2 logΛ2m2)]. (5)This particular approach gives 〈ρ〉 < 0 and 〈Ω2〉 > 0. However, when including additionalregulator fields, there is a freedom in the choice of the constants Ci that can be used tofix the value of the quartic divergence at an arbitrary value, even zero. The introductionof additional regulator fields, needed in curved spacetimes, also give arbitrary values forthe polynomial divergences [7]. Other Lorentz invariant approaches, like inserting powers of3Λ2/(Λ2 − k2 − iǫ) in the divergent integrals, give only a quadratic divergence proportionalto m2Λ2, and no quartic divergence [5].In summary, if one regularizes the theory with the Pauli-Villars method, 〈Ω2〉 is notpositive definite, and becomes negative when one imposes the “physical” criterium that thevacuum energy density should be positive definite. In this case, it is not true that thefluctuations of the stress tensor around its mean value lead to a solution of the cosmologicalconstant problem, based on the parametric resonance mechanism proposed in Ref.[1]. Butmost importantly, in light of the fact that different cutoffs produce ambiguous results for thesign of 〈Ω2〉 and of the vacuum energy, the physical meaning of the regularized quantitiesin this context is doubtful.One could wonder whether the fluctuations around a negative 〈Ω2〉 could stabilize theupside-down harmonic oscillator in Eq. (2), through parametric stabilization [8], softeningthe effect of the cosmological constant. This seems difficult in the present model, giventhat the (quasi) frequency of the fluctuations is much smaller than (−〈Ω2〉)1/2. Moreover,this mechanism would suffer the same ambiguities pointed out in this comment, that is, itwould depend on the particular implementation of the regularization method. It would beinteresting to analyze the eventual suppression of the cosmological constant by parametricstabilization in the context of semiclassical stochastic gravity [9], by studying the effect ofnoise in the renormalized Einstein-Langevin equation.ACKNOWLEDGMENTSThis research was supported by ANPCyT, CONICET, and UNCuyo. We would like tothank D. López Nacir for discussions.[1] Q. Wang, Z. Zhu and W. G. Unruh, Phys. Rev. D 95, 103504 (2017).[2] E. K. Akhmedov, hep-th/0204048.[3] J. Martin, Comptes Rendus Physique 13, 566 (2012)[4] M. Eĺıas and F. D. Mazzitelli, Phys. Rev. D 91, 124051 (2015).[5] G. Ossola and A. Sirlin, Eur. Phys. J. C 31, 165 (2003).[6] R. H. P. Kleiss and T. W. Janssen, Eur. Phys. J. C 75, 46 (2015).4[7] M. Asorey, E. V. Gorbar and I. L. Shapiro, Class. Quant. Grav. 21, 163 (2003).[8] D. J. Acheson, Proc. R. Soc. Lond A 443, 239 (1993), and references therein.[9] B. L. Hu and E. Verdaguer, Living Rev. Rel. 11, 3 (2008).5

Comment on "how the huge energy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe"

CONICET Digital

arXiv:1801.00138v1  [gr-qc]  30 Dec 2017Comment on: “How the huge energy of quantum vacuumgravitates to drive the slow accelerating expansion of theUniverse”Francisco D. Mazzitelli and Leonardo G. TrombettaaCentro Ato´mico Bariloche and Instituto Balseiro,Comisio´n Nacional de Energ´ıa Ato´mica, 8400 Bariloche, Argentina.(Dated: January 3, 2018)AbstractIn a recent paper (Phys. Rev. D95, 103504 (2017)) it is argued that, due to the fluctuationsaround its mean value, vacuum energy gravitates differently from what previously assumed. Asa consequence, the universe would accelerate with a small Hubble expansion rate, solving thecosmological constant and dark energy problems. We point out here that the results depend onthe type of cutoff used to evaluate the vacuum energy. In particular, they are not valid when oneuses a covariant cutoff such that the zero point energy density is positive definite.a Present address: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, ItalyINFN - Sezione di Pisa, 56200, Pisa, Italy1In the traditional formulation of the cosmological constant problem, it is argued that thezero point energy density 〈ρ〉 associated to a quantum field is proportional to Λ4, whereΛ is an ultraviolet cutoff, of the order of the Planck energy EP lanck. Assuming that themean value of the stress tensor of the quantum field is covariantly regularized, one has〈Tµν〉 = −〈ρ〉 gµν , which corresponds to a cosmological constant of order E4P lanck, about 120orders of magnitude larger than the observed one.In Ref.[1] it was pointed out that the energy-momentum tensor associated to a quantummassless field φ has very large fluctuations around its mean value. Therefore, it is notcorrect to use 〈Tµν〉 as a source of the Einstein equations. When properly taken into account,these fluctuations lead to modified Einstein equations with a stochastic component. Moreconcretely, for a metric of the formds2 = −dt2 + a2(t,x)(dx2 + dy2 + dz2)(1)the evolution equation for the scale factor a(t, x) is that of a harmonic oscillatora¨+ Ω2(t,x)a = 0 , Ω2(t,x) =4πG3(ρ+3∑i=1Pi)=8πG3φ˙2 , (2)where ρ = T00, Pi = Tii/a2. The quantity Ω2 is assumed to have a positive mean value〈Ω2〉, of order Λ4, and to have quasiperiodic stochastic fluctuations in a time scale of order1/Λ. Thus, due to parametric resonance, the scale factor has an exponential growth with aHubble rate H which is exponentially small in the limit Λ → ∞, solving the cosmologicalconstant problem.In this comment we would like to stress the following point: if the theory is regulatedby a Lorentz invariant cutoff in flat spacetime, then one has 〈p〉 = −〈ρ〉, and therefore〈Ω2〉 = −8πG〈ρ〉/3. Moreover, if the cutoff is such that 〈ρ〉 > 0, as usually assumed, then〈Ω2〉 < 0 and the whole picture of parametric resonance breaks down.Let us be more explicit. Wang et al first computed 〈Ω2〉 in Minkowski spacetime usinga non-invariant cutoff Λ such that |~p| < Λ, where ~p denotes the 3-momentum of the modesof the scalar field. In this case, both 〈ρ〉 and 〈p〉 are positive definite and proportional toΛ4. Note, however, that for this particular cutoff one has 〈p〉 = 〈ρ〉/3, breaking the Lorentzinvariance of 〈Tµν〉. This has been noticed long ago in Ref.[2]: a non-covariant cutoff cannotbe used to estimate the vacuum contribution to the cosmological constant (see also Refs.[3, 4]). If, in spite of this, one accepts the use of this cutoff, and assumes that the regularized2quantities have physical meaning, then the conclusions of Ref.[1] looks correct, although theinitial problem is different: 〈Tµν〉 does not describe a cosmological constant but a radiationfluid.Wang et al also computed 〈Ω2〉 using a Lorentz invariant procedure inspired in Pauli-Villars method. The particular implementation of this method used in Ref.[1] may give〈Ω2〉 > 0 (this is not completely clear from Eq.(195) in Ref.[1]). Once more, if this were thecase, the analysis of the dynamical equation for the scale factor in Ref.[1] would be correct,but at the price of regularizing the theory in such a way that 〈ρ〉 < 0. Clearly, the useof this particular Lorentz invariant cutoff would not be equivalent to the use of a cutoff in3−momentum space, since it produces a vacuum energy density with a different sign.But the situation is even worse: the Pauli-Villars method produces ambiguous resultsfor the polynomial divergences [5–7]. Only the logarithmic divergences are univocally de-termined by the method. We illustrate this fact with an example discussed in Ref.[5]. Theregularized energy momentum tensor in Minkowski spacetime, using Pauli-Villars method,is given by〈Tµν〉 = −gµν4N∑i=0CiM4i logM2iµ2, (3)where the masses Mi, i = 1, 2, ...N are the regulators, M0 = m is the mass of the field, µ isan arbitrary mass scale, C0 = 1, and the constants Ci, i = 1, 2, ...N satisfyN∑i=0Ci(M2i)p= 0 , (4)for p = 0, 1, 2. Due to these conditions, the result is independent of the scale µ. In principleone can add an arbitrary number N of regulator fields, the minimum being N = 3 to satisfythe above constraints. It has been shown that, for N = 3, the regularized version of thestress tensor produces a negative energy density. In the particular case Mi = Λ one has〈Tµν〉 = −gµν128π2[−Λ4 + 4m2Λ2 −m4(3 + 2 logΛ2m2)]. (5)This particular approach gives 〈ρ〉 < 0 and 〈Ω2〉 > 0. However, when including additionalregulator fields, there is a freedom in the choice of the constants Ci that can be used tofix the value of the quartic divergence at an arbitrary value, even zero. The introductionof additional regulator fields, needed in curved spacetimes, also give arbitrary values forthe polynomial divergences [7]. Other Lorentz invariant approaches, like inserting powers of3Λ2/(Λ2 − k2 − iǫ) in the divergent integrals, give only a quadratic divergence proportionalto m2Λ2, and no quartic divergence [5].In summary, if one regularizes the theory with the Pauli-Villars method, 〈Ω2〉 is notpositive definite, and becomes negative when one imposes the “physical” criterium that thevacuum energy density should be positive definite. In this case, it is not true that thefluctuations of the stress tensor around its mean value lead to a solution of the cosmologicalconstant problem, based on the parametric resonance mechanism proposed in Ref.[1]. Butmost importantly, in light of the fact that different cutoffs produce ambiguous results for thesign of 〈Ω2〉 and of the vacuum energy, the physical meaning of the regularized quantitiesin this context is doubtful.One could wonder whether the fluctuations around a negative 〈Ω2〉 could stabilize theupside-down harmonic oscillator in Eq. (2), through parametric stabilization [8], softeningthe effect of the cosmological constant. This seems difficult in the present model, giventhat the (quasi) frequency of the fluctuations is much smaller than (−〈Ω2〉)1/2. Moreover,this mechanism would suffer the same ambiguities pointed out in this comment, that is, itwould depend on the particular implementation of the regularization method. It would beinteresting to analyze the eventual suppression of the cosmological constant by parametricstabilization in the context of semiclassical stochastic gravity [9], by studying the effect ofnoise in the renormalized Einstein-Langevin equation.ACKNOWLEDGMENTSThis research was supported by ANPCyT, CONICET, and UNCuyo. We would like tothank D. Lo´pez Nacir for discussions.[1] Q. Wang, Z. Zhu and W. G. Unruh, Phys. Rev. D 95, 103504 (2017).[2] E. K. Akhmedov, hep-th/0204048.[3] J. Martin, Comptes Rendus Physique 13, 566 (2012)[4] M. El´ıas and F. D. Mazzitelli, Phys. Rev. D 91, 124051 (2015).[5] G. Ossola and A. Sirlin, Eur. Phys. J. C 31, 165 (2003).[6] R. H. P. Kleiss and T. W. Janssen, Eur. Phys. J. C 75, 46 (2015).4[7] M. Asorey, E. V. Gorbar and I. L. Shapiro, Class. Quant. Grav. 21, 163 (2003).[8] D. J. Acheson, Proc. R. Soc. Lond A 443, 239 (1993), and references therein.[9] B. L. Hu and E. Verdaguer, Living Rev. Rel. 11, 3 (2008).5

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