It has been argued that the correct, i.e. positive, sign of quantum vacuum
energy density, or more properly, negative sign of quantum vacuum pressure,
requires not a very large number, e.g. ~100, of additional, undiscovered
fundamental bosonic particle species, absent in the standard model.
Interpretation of the new particle species in terms of dark matter ones permits
to qualitatively, and even quantitatively, connect all the three concepts given
in the title.Comment: 6 pages, 0 figure

Broda, Bogusław

Szanecki, Michał

ISRN Astronomy and Astrophysics

English

arXiv

It has been argued that the correct, that is, positive, sign of quantum vacuum energy density, or, more properly, negative sign of quantum vacuum pressure, requires not a very large, and to some extent model-independent, number,
for example, ∼100, of additional, undiscovered fundamental bosonic particle species, absent in the standard model. Interpretation of the new particle species in terms of dark matter ones permits to qualitatively, and even quantitatively, connect all the three concepts given in the title.</jats:p

Bogusław Broda

Michał Szanecki

Crossref

International Scholarly Research NetworkISRN Astronomy and AstrophysicsVolume 2011, Article ID 509836, 3 pagesdoi:10.5402/2011/509836Research ArticleVacuum Pressure, Dark Energy, and Dark MatterBogusław Broda andMichał SzaneckiDepartment of Theoretical Physics, University of Ło´dz´, Pomorska 149/153, 90-236 Ło´dz´, PolandCorrespondence should be addressed to Bogusław Broda, bobroda@uni.lodz.plReceived 3 November 2011; Accepted 27 December 2011Academic Editors: R. N. Henriksen and M. SazhinCopyright © 2011 B. Broda and M. Szanecki. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.It has been argued that the correct, that is, positive, sign of quantum vacuum energy density, or, more properly, negative signof quantum vacuum pressure, requires not a very large, and to some extent model-independent, number, for example, ∼100, ofadditional, undiscovered fundamental bosonic particle species, absent in the standard model. Interpretation of the new particlespecies in terms of dark matter ones permits to qualitatively, and even quantitatively, connect all the three concepts given in thetitle.Dark energy [1–3] and dark matter [4, 5] are two main con-stituents of our universe. Their contribution amounts toalmost 96% of the total energy-mass of the universe. Thenature of the both seems to be mysterious but completelydiﬀerent. Another mysterious and elusive constituent of ouruniverse, appearing in theoretical context rather than in acosmological one, is quantum vacuum [6]. But, recently,mainly due to advances in the Casimir eﬀect, the quantumvacuum is beginning to enter reality [7]. The old idea to iden-tify dark energy and quantum vacuum energy is theoreticallyvery attractive but the main diﬃculty is to reconcile thevalues of both energies [8–13]. That is a puzzle. But thereis another puzzle, usually not being mentioned. Since, as itis well known, the sign of quantum vacuum energy followsfrom the statistics of fluctuating fields, the right (positive)sign of vacuum energy corresponds to bosonic modes. Butthe number of diﬀerent fermionic particle species prevails inthe standard model, and that is the puzzle. Thus, actually, wehave the two independent puzzles related to the connectionbetween quantum vacuum energy and dark energy: the puz-zle of the huge (absolute) value of quantum vacuum energydensity and the puzzle of its sign.It appears, and this is the main subject of our letter, thatit is possible to solve the second puzzle, establishing a linkbetween the issue of dark energy and dark matter. Namely,not a very large, and to some extent model-independent,number of undiscovered bosonic fields should be includedin the fundamental set of particle species to obtain the rightsign of vacuum energy density. These new species are naturalcandidates for dark matter particle species.A typical approach to quantum vacuum energy yields thestandard formula [8]ρvac = 12∫ Λuv04π(2π)3c√(mc)2 + k2k2dk, (1)where m is the mass of a bosonic mode and k is its momen-tum. For a large ultraviolet (UV) cutoﬀ Λuv we have approx-imatelyρvac ≈ 1(4π)2Λuv42c. (2)Setting Λuv = ΛP, where the Planck momentumΛP =√c3G≈ 6.5 kgm/s (3)and G is the Newton gravitational constant, we obtain the“(in)famous” value (formula)ρvac ≈ c5(4π)2G2≈ 3.4× 1094 kg/m3 ≈ MPL3P, (4)whereMP and LP are the Planck mass and length, respective-ly. The fame of this formula rivals its absurdity. Not onlyis (4) some 10120 times greater than expected but evidently2 ISRN Astronomy and Astrophysicsthe sign is not as expected. For fermionic modes the sign of(4) will be reversed! Therefore, for presently known contentsof fundamental set of fields (with prevalent fermionicmodes) the sign will be wrong. Consequently, new bosonicspecies are urgently being looked for.One should emphasize that the diﬃculty with the signseems to be independent of the approach to the issue of thehuge value of quantum vacuum energy. In other words, addi-tional bosonic modes are presumably unavoidable providedquantum vacuum is supposed to have something to do withdark energy or, at least, with reality. Taking for granted thatnew bosonic particle species should enter the set of fun-damental fields any further estimate of their number couldalready depend on the assumed model of dark energy in theframework of the idea of quantum vacuum.In [14, 15], we have proposed a phenomenologicallypromising approach to solve the puzzle of the huge (abso-lute) value of “the quantum vacuum energy density.” In theframework of our approach the (absolute) value of the “vac-uum energy density” of a single mode is of the orderρ ∼ 0.01ρexp, (5)where ρexp is the experimental value of the energy density ofdark energy. Thus, (5) gives the result for a single mode. Ob-viously, the relation (5) is model dependent. In our approach[14, 15] it is just (5).Strictly speaking, the formula yielding the result (5), thatis, the lagrangian density [14, 15]∓141(4π)2G12(1− q)H2, (6)where the upper and lower signs correspond to a bosonic andfermionic mode, respectively, q is the present decelerationparameter, and H is the present Hubble expansion rate,corresponds to the pressure rather than to the energy density.In fact, the diagonal part of the energy-momentumtensor,Tμν = ∂L∂∂μφ∂νφ − gμνL, (7)reduces toT00 ≡ ρ = −g00L, for ∂0φ = 0, (8)orTii ≡ p = −giiL, for ∂iφ = 0, (9)where c = 1 and the signature of the metric is (+,−,−,−).Our case corresponds obviously to the second possibility,that is, (9), because our fields are homogeneous (∂iφ = 0)but time dependent (Friedmann-Lemaıˆtre-Robertson-Walk-er cosmological model).Coming back to our model∓141(4π)2G12(1− q)H2 ≈ L = − 1giip, (10)and thereforep ≈ ∓141(4π)2G12(1− q)H2, (11)which still conforms with our description of dark energy interms of quantum vacuum. For example, assuming tem-porarily an ad hoc barotropic relationp = wρ, w = −1, (12)we simply obtainρ ≈ ± 1128π2G(1− q)H2. (13)But (11) (for pressure) is more fundamental than (13) (forenergy density) because no equation of state needs to be pre-sumed.Consequently, as a next step, we should collect contrib-utions, of the type estimated by us, coming from all funda-mental physical modes. We can proceed in the spirit of thephilosophy of quantum- (vacuum-) induced interactions(see, e.g., [16–18]). Thus, the total pressure, coming from allfundamental modes, is according to (11) of the orderp ≈ − N0128π2G(1− q)H2, (14)and ρexp ≈ N0ρ, where N0 is the “alternated sum” of the fun-damental modes. Namely, we defineN0 ≡ nB − nF =∑k=0(−)knk/2, (15)where nB and nF are the number of fundamental bosonic andfermionic modes, respectively. By virtue of the spin-statisticstheorem we can rewrite N0 in terms of spin degrees offreedom, that is, n0, n1/2, n1, and so forth. One should notethat the particular case nB = nF corresponds to supersym-metry.Recapitulating, the puzzle related to quantum vacuumfluctuations consists in getting under control of the hugevalue of the vacuum energy density, strictly speaking, fromour point of view, the (absolute) value of the pressure. Oncethe value becomes reasonable in its size, the sign puzzleemerges. The sign assumes the expected value, for example,“minus” in (14), only provided the number of bosonic fun-damental modes is prevalent.We would like to stress, once more, that the conclusionconcerning bosonic species is not specific to our model ofdark energy, because only bosonic modes give contributionshuge or moderate but with right signs. Since the standardmodel contains greater number of fermionic particle species(leptons and quarks) than bosonic ones (mainly, gaugefields), we can conclude that there is a missing number ofinvisible bosonic modes. The bosonic modes do not enterthe standard model but they must interact gravitationally.Therefore, they are appropriate and natural candidates to therole of dark matter particles.ISRN Astronomy and Astrophysics 3More concretely, for the standard modelN0 = [4]− 2 · 3 · (3 + 3 · 4) + 2 · (1 + 3 + 8 + [1])= −(63± 3),(16)where in the first bracket we have included, as yet not dis-covered, Higgs modes, next there are leptons and quarks (2spins × 3 families × (3 leptons of the both helicities + 3colors × 4 quarks of the both helicities)), and, finally, gaugefields (2 spins × (photon + 3 weak bosons + 8 gluons)) withgraviton in the last bracket. From (14) and (16) (see, also (5)for a numeric value) it follows that the lacking number ofbosonic modes is of the order nB ∼ 100. This number ismodel dependent but the conclusion is not.Any possible, alternative, and independent solution ofthe puzzle of the huge (absolute) value of quantum vacuumenergy density also would require additional bosonic speciesbut their number could vary. For example, any potential so-lution directly yielding the right (absolute) value of the quan-tum vacuum energy density, meaning that the correct (abso-lute) value is recovered for a single mode, would require atuning between the number of bosonic and fermionic spe-cies, that is, nB − nF ∼ 1.Certainly, it would be very advantageous to present analternative and independent estimate of the number of thelacking fundamental bosonic modes nB. To this end we willmake use of the proposal given in [18], concerning quantumvacuum-induced gravitational action and quantum vacuum-induced gravitational (black-hole) thermodynamical entro-py. Assuming the number and kind of fundamental modesgiven in (16), we get nB ∼ 40 to properly induce the gravita-tional action (see equation (8) in [18]), and nB ∼ 200 toproperly induce the gravitational (black-hole) thermody-namical entropy (see equation (2) in [18]). Taking into ac-count an approximate character of our reasoning one shouldadmit that it qualitatively agrees with our present dark energyestimate, that is, nB ∼ 100.In this paper, we have proposed a consistent connectionbetween the following three concepts: quantum vacuumpressure, dark energy, and dark matter. First of all, we haveshown that, in the context of lagrangian approach, the quan-tum pressure naturally replaces the notion of quantum vac-uum energy density. Next, we have argued that, indepen-dently of an actual model of dark energy, additional, undis-covered fundamental bosonic particle species are necessary.Finally, applying our earlier model of dark energy [14, 15]we have estimated the number of the fundamental bosonicparticle species, namely, nB ∼ 100. Amazingly, it agrees withthe estimates being yielded by induced gravity [18].AcknowledgmentsThis work has been supported by the University of Ło´dz´grant and LFPPI network.References[1] A. G. Riess, A. V. Filippenko, P. Challis et al., “Observationalevidence from supernovae for an accelerating universe and acosmological constant,” The Astronomical Journal, vol. 116, no.3, pp. 1009–1038, 1998.[2] S. Perlmutter, G. Aldering, G. Goldhaber et al., “Measure-ments of Ω and Λ from 42 high-redshift supernovae,” TheAstrophysical Journal, vol. 517, pp. 565–586, 1999.[3] E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of darkenergy,” International Journal of Modern Physics D, vol. 15, no.11, pp. 1753–1936, 2006.[4] G. Bertone, D. Hooper, and J. Silk, “Particle dark matter: evi-dence, candidates and constraints,” Physics Reports, vol. 405,no. 5-6, pp. 279–390, 2005.[5] N. Fornengo, “Status and perspectives of indirect and directdark matter searches,” Advances in Space Research, vol. 41, pp.2010–2018, 2007.[6] G. E. Volovik, “Vacuum energy: myths and reality,” Interna-tional Journal of Modern Physics D, vol. 15, no. 12, pp. 1987–2010, 2006.[7] R. L. Jaﬀe, “Casimir eﬀect and the quantum vacuum,” PhysicalReview D, vol. 72, no. 2, Article ID 021301, 2005.[8] S. Weinberg, “The cosmological constant problem,” Reviews ofModern Physics, vol. 61, no. 1, pp. 1–23, 1989.[9] S. M. Carroll, “The cosmological constant,” Living Reviews inRelativity, vol. 4, pp. 1–78, 2001.[10] T. Padmanabhan, “Cosmological constant—the weight of thevacuum,” Physics Reports, vol. 380, no. 5-6, pp. 235–320, 2003.[11] T. Padmanabhan, “Dark energy: mystery of the millennium,”in Proceedings of the Albert Einstein Century InternationalConference, vol. 861 of AIP Conference Proceedings, pp. 179–196, Paris, France, 2005.[12] Y. B. Zel‘dovich, “Cosmological constant and elementary par-ticles,” Letters to Jounal of Experimental and Theoretical Physics,vol. 6, no. 9, pp. 316–317, 1967, Translated from ZhurnalEksperimentalnoi i Teoreticheskoi Fiziki, vol. 6, pp. 883-884,1967.[13] S. Nobbenhuis, “Categorizing diﬀerent approaches to thecosmological constant problem,” Foundations of Physics, vol.36, no. 5, pp. 613–680, 2006.[14] B. Broda, P. Bronowski, M. Ostrowski, and M. Szanecki, “Vac-uum driven accelerated expansion,” Annalen der Physik, vol.17, no. 11, pp. 855–863, 2008.[15] B. Broda andM. Szanecki, “Quantum vacuum and acceleratedexpansion,” in Dark Energy and Dark Matter: Observations,Experiments and Theories, E. Pe´contal, T. Buchert, Ph. diStefano, and Y. Copin, Eds., vol. 36 of EAS Publications Series,pp. 167–171, 2009.[16] A. D. Sakharov, “Vacuum quantum fluctuations in curvedspace and the theory of gravitation,” General Relativity andGravitation, vol. 32, pp. 365–367, 2000, Translated fromDoklady Akademii Nauk SSSR, vol. 170, pp. 70-71, 1967.[17] Y. B. Zeldovich, “Interpretation of electrodynamics as aconsequence of quantum theory,” ZhETF Pis ma Redaktsiiu,vol. 6, pp. 922–925, 1967, Letters to Jounal of Experimental andTheoretical Physics, vol. 6, pp. 345–347, 1967.[18] B. Broda and M. Szanecki, “Induced gravity and gauge inter-actions revisited,” Physics Letters B, vol. 674, no. 1, pp. 64–68,2009.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014High Energy PhysicsAdvances inThe Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014FluidsJournal of Atomic and  Molecular PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Advances in  Condensed Matter PhysicsOpticsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstronomyAdvances inInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014SuperconductivityHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Statistical MechanicsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014GravityJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstrophysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Physics Research InternationalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Solid State PhysicsJournal of Computational  Methods in PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Soft MatterJournal ofHindawi Publishing Corporationhttp://www.hindawi.comAerodynamicsJournal ofVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014PhotonicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofBiophysicsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014ThermodynamicsJournal of

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