According to general relativity, the present analysis shows on geometrical
grounds that the cosmological constant problem is an artifact due to the
unfounded link of this fundamental constant to vacuum energy density of quantum
fluctuations.Comment: 7 page

Triay, R.

English

arXiv

Talk given at : a) VII Asia-Pacific International Conference on Gravitation and Astrophysics (ICGA7) on the Occasion of the 90th Year of General Relativity November (2005 Na- tional Central University Chungli, Taiwan) b) Albert Einstein Century Conference, Paris July, 18-22, 2005According to general relativity, the present analysis shows on geometrical grounds that the cosmological constant problem is an artifact due to the unfounded link of this fundamental constant to vacuum energy density of quantum fluctuations

HAL AMU

An Issue to the Cosmological Constant ProblemR. TriayTo cite this version:R. Triay. An Issue to the Cosmological Constant Problem. 7th Asia-Pacific International Con-ference on Gravitation and Astrophysics, Nov 2005, Taipei, Taiwan. 2007. <hal-00005025v2>HAL Id: hal-00005025https://hal.archives-ouvertes.fr/hal-00005025v2Submitted on 10 Aug 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00005025, version 2 - 10 Aug 2006August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper135501AN ISSUE TO THE COSMOLOGICAL CONSTANTPROBLEMR TRIAY∗Centre de Physique The´orique ∗CNRS Luminy Case 907, 13288 Marseille Cedex 9, Francetriay@cpt.univ-mrs.frOn geometrical grounds, the cosmological constant problem turns out to bean artifact due to the unfounded link of this fundamental constant to vacuumenergy density of quantum fluctuations.Keywords: Cosmology; Vaccum Energy ; Cosmological Constant.1. IntroductionThe cosmological constant problem (CCP) is an enigma of modern cos-mology among the particles physics and cosmology communities. The mostcomprehensive contribution to this problem can be found in.29 It is a matterof fact that its origin is intimately related to the status of the cosmologicalconstant Λ. It was first assumed as a free parameter in the field equationswith the aim of accounting for a static Universe and then rejected becausea cosmological expansion was observed subsequently. Such an issue to thecosmological problem has provided us with (authority and/or simplicity)arguments in favor of Λ = 0 until acceleration of the cosmological expan-sion could not be avoided for the interpretation of recent cosmological data.However, the related estimate does not agree by hundred orders of magni-tude with its expected value as obtained from quantum field theories5,20,40by interpreting Λ/8πG as vacuum energy density of quantum fluctuations.The aim of the present contribution is to understand this problem on geo-metrical grounds.∗Unite´ Mixte de Recherche (UMR 6207) du CNRS, et des universite´s Aix-Marseille I,Aix-Marseille II et du Sud Toulon-Var. Laboratoire affilie´ a` la FRUMAM (FR 2291).August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper1355022. Status of the cosmological constantThe reason why the status of the cosmological constant Λ has long beendiscussed6,11,30,38 has an historical origin, which still contributes to a so-ciological debate at the present time. The key point is that either Λ is anuniversal constant ,35 as it is clearly established in General Relativity (GR),as similarly as Newton constant of gravitation G, or it is associated to aparticles/fields contribution to gravitation.2.1. Historical status of ΛFor solving the cosmological problem, Einstein’s goal was to obtain thegravitational field of a static universe, as it was supposed to be at thatepoch. Similarly to the necessary modification of Poisson’s equation fordescribing a uniform static distribution of dust in Newtonian gravitya, Λwas assumed in the gravitational field equations accordingly to GR. WithMach’s principle in mind (origin of inertia), a consistent cosmological solu-tion describing a spatially closed universe9 was derived, named as Einstein’smodel. A decade later, Friedmann’s model14 was used by G. Lemaˆıtre18 forpointing out the cosmological expansion from Hubble’s law,16 when Ein-stein’s model was shown to be unstable8b. This summarises very brieflythe state of the art as recorded in contemporary textbooks.43 Henceforth,Friedmann’s model with Λ = 0 was preferred because of Einstein’s def-inite renouncement from the point of view of “logical economy”,10 whatbecame the Standard world model .48 His confessionc to G. Gamov15 standsprobably for the historical reason why Λ was wrongly understood as a freeparameter in GR theory of gravitation by the majority of observational cos-mologists until recently. Although, Λ 6= 0 has been unsuccessfully envisagedin the 60’s for explaining observations.17,22,23,28,33 Nowadays, it is gener-ally believed that Λ ∼ 2h2 10−56 cm−2 (where h = H◦/100 kms−1Mpc−1)is required for interpreting the CMB temperature fluctuations1,19,32,34 andfor accounting of Hubble diagram of SN.24–26,31 Such a necessity was evi-dentd two decades earlier from statistical investigations based on quasarsaSuch an approach has been used by Neumann (1896) in Newtonian theory, see R.C.Tolman.43bi.e. in addition of suffering from a fine tuning problem on the values of Λ and thespecific density of energy of gravitational sources, any irregularity in their distributioncauses either a collapse or an expansion.cloc. cit. : “the biggest blunder of my life”.dIt was ignored because not representative of the general consensus at that epoch, whatis typical of the present days scholastic attitude.August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper135503and galaxiess.2,2,3,12,13,41,442.2. Geometrical status of ΛOn geometrical grounds, Principle of General Relativity applied to gravityprovides us with a unique interpretation of Λ. The gravitational field andits sources are characterized respectively by the metric tensor gµν on thespace-time manifold V4 and by a vanishing divergence stress-energy tensorTµν . The gravitational field equations must be invariant with respect to theaction of diffeomorphism group of V4,35,36 and therefore they readTµν = −A0F (0)µν +A1F (1)µν +A2F (2)µν + . . . (1)where F(n)µν stands for a covariant tensor of degree 2n, defined by means ofmetric tensor gµν and its derivatives, and An for a coupling constant , itsvalue is estimated from observations. The n = 0, 1 order terms are uniquelydefinedF (0)µν = gµν , F(1)µν = Sµν = Rµν −12Rgµν (2)where Rµν is the Ricci tensor and R the scalar curvature, whereas Fn≥2µνmust be derived from additional principles.Schwarzschild solution of Eq. (1) enables us to identify An=0,1 with New-ton approximation, what provides us with modified Poisson equation35divg˜ = −4πGρ+ Λ (3)where ~g stands for the gravitational acceleration field due to sources definedby a specific density ρ, and the following identification of constantsG =18πA1, Λ =A0A1(4)which shows their common status of universal constant .3. Modeling gravitational structuresThe space-time geometry is constrained by the presence of gravitationalsources by means of tensor Tµν in Eq. (1), each right hand terms contributesfor describing the geometry within its effective scale. A dimensional analysisof field equations provides us with an estimation of their correspondingmagnitudes, what is useful for modelling gravitational structures. Moreover,Newtonian approximation of gravitational field at large scale provides uswith a simple way for interpreting Λ effect.August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper1355043.1. Dimensional analysisAccording to GR, the speed of the light c = 1 (i.e. time can be mea-sured in unit of lengthe 1s = 2.999 792 458 1010 cm) and then G = 7.4243×10−29 cmg−1. Let us choose units of mass and of lengthf , herein denotedrespectively by M and L. The correct dimensional analysis of GR setsthe covariant metric tensor to have the dimension [gµν ] = L2, and thus[gµν ] = L−2, [Rµν ] = 1 and [R] = L−2. Since the specific mass density andthe pressure belong to T µν , one has [Tµν ] = ML−1. Hence, according toEq. (1), the dimensions of An are the following[A0] =ML−3, [A1] =ML−1, . . . [An] =ML2n−3 (5)which shows their relative contributions for describing the gravitational fieldwith respect to scale. Namely, the larger their degree n the smaller theireffective scaleg. Equivalently, the estimation of A0 demands observationaldata located at scale larger than the one for A1, etc. . . . This is the reasonwhy the Λ effect is not discernible at small scale but requires cosmologicaldistances.3.2. Newtonian gravity up to cosmological scalesThe observations show that gravitational structures within scales of orderof solar system can be described by limiting the expansion solely to Ein-stein tensor Sµν , when cosmology requires also the first term. The transitionscale between A0 and A1 is of order of 1/√Λ ∼ 7h−1 Gyr. Although GR ispreferred for investigating the dynamics of cosmic structures, Newton ap-proximation given in Eq. (3) provides us with an easier schema for realizingthe Λ effect. Hence, the acceleration field due to gravity around a pointmass m reads~g =(−Gmr3+Λ3)~r (6)Since Λ > 0, the gravity force is attractive at distance r < r◦ and repulsiveat r > r◦ with a critical distancer◦ =3√3mG/Λ (7)where the gravity vanishes.eThis is the reason why any statement on the variation of c is meaningless in GR.fOnly two fundamental units can be chosen, the third one is derived.gIn other words, the contribution of A0 dominates at scale larger than the one of A1,etc. . .August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper1355054. The cosmological constant problemLet us assume :(1) quantum fluctuations are sources of the gravitational field by means ofthe following stress-energy tensorhT vacµν = ρvac gµν , ρvac = ~kmax (8)in the field equationsEq.(1), where kmax stands for the ultraviolet mo-mentum cutoff up to which the quantum field theory is valid,5 one hasρEWvac ∼ 2 10−4, ρQCDvac ∼ 1.6 1015, ρPlvac ∼ 2 1089 (9)in units of g cm−3. The reason why such an estimation is not uniquecomes from the perturbative aspect of the theory for describing thequantum world, what can be understood as a weakness of this approach.(2) the cosmological term interprets as the contribution of quantum fluc-tuationsρΛ =Λ8πG(10)The difficulty of above hypotheses is that ρvac differs fromρΛ ∼ h2 10−29 g cm−3 (11)as obtained from astronomical observations, by 25–118 orders of magni-tude. Such an enigma is named the cosmological constant problem. Otherestimations of this quantum effect from the viewpoint of standard Casimirenergy calculation scheme49 provide us with discrepancies of ∼ 37 ordersof magnitude.7 A similar problem happens whenΛvac = 8πGρvac (12)is interpreted as a cosmological constant. Indeed, if the quantum field theorywhich provides us with an estimate of ρvac is correct then the distance fromwhich the gravity becomes repulsive in the sun neighborhood ranges fromrEW◦ ∼ 2 10−2h−2/3 a.u. down to rPl◦ ∼ 3 10−11h−2/3 A˚ depending on thequantum field theory, see Eq. (7). Obviously, such results are not consistentwith the observations.hThe usual picture which describes the vacuum as an isotropic and homogenous dis-tribution of gravitational sources with energy density ρvac and pressure pvac = −ρvac(although this is not an equation of state) is not clear and not necessary for the discus-sion.August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper135506Another version of the cosmological constant problem points out a finetuning problem. It consists on arguing on the smallness of Λ = Λvac + Λ◦,interpreted as an effective cosmological constant, where Λ◦ stands for abare cosmological constant.4.1. Understanding the recession of galaxiesThe observations show that the dynamics of the cosmological expansionafter decoupling era agrees with Friedmann-Lemaˆıtre-Gamov solution. Itdescribes an uniform distribution of pressureless matter and CMB radia-tion with a black-body spectra, the field equations are given by Eq. (1) withn ≤ 1. The present values of related densities are ρm = 3h2 10−30 g cm−3(dark matter included) and ρr ∼ 5h2 10−34 g cm−3. Their comparison to theexpected vacuum energy density ρvac shows that if quantum fluctuationsintervene in the dynamics of the cosmological expansion then their contribu-tion prevails over the other sources (by 26–119 orders of magnitude today).Such an hypothesis provides us with a vacuum dominate cosmological ex-pansion since primordial epochs. Therefore, one might ask whether suchdisagreements with observations can be removed by taking into accounthigher order n ≥ 2 terms in Eq. (1). With this in mind, let us describe thedynamics of structures at scales where gravitational repulsion (Λ > 0) isobserved. Since the values of universal constants G and Λ are provided byobservations, it is more convenient to use adapted units of time lg and ofmass mg defined as followslg = 1/√Λ ∼ h−1 1028 cm, mg = 1/(8πG√Λ) ∼ 4h−1 1054 g (13)herein called gravitational units . They are defined such that the field equa-tions read in a normalized formTµν = −gµν + Sµν +A2F (2)µν + . . . (14)i.e. A0 = A1 = 1, where the stress-energy tensor Tµν accounts for thedistribution of gravitational sources. It is important to note that, withgravitational units, Planck constant reads~ ∼ 10−120 (15)Indeed, such a tiny value as quantum action unit compared to ~ = 1 whenquantum units are used instead, shows clearly that Eq. (14) truncated atorder n ≤ 1 is not adapted for describing quantum physics.45,46 This isthe main reason why it is hopeless to give a quantum status to Λ.41 AsAugust 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper135507approximation, because of dimensional analysis described above, the con-tribution of higher order terms being the more significant as the densityis large, Eq. (14) can be splited up with respect to scale into two equa-tions systems. The first one corresponds to terms of order n < 2 (the usualEinstein equation with Λ) and the second oneT vacµν = A2F(2)µν + . . . (16)stands for the field equations describing the effect of quantum fluctuationson the gravitational field at an appropriated scale (quantum), interpretedas correction of the RW metric gµν . The identification of constants An (e.g.,A2 = ~) and the derivation of tensors F(n)µν with n ≥ 2 requires to modelgravitational phenomena at quantum scale, see e.g.39,42 Unfortunately, thestate of the art does not allow yet to provide us with a definite answer fordefining the right hand term of Eq. (16), see e.g.275. ConclusionSeeking for the contribution of quantum fluctuations into the cosmologicalconstant is a motivation that inherits from the previous attitude consist-ing on rejecting the cosmological term. On the other hand, to rescale thefield equations for describing the cosmological expansion prevents us to as-sume the vacuum acting as a cosmological constant. As a consequence, oneunderstands that such an interpretation turns to be the origin of the cos-mological constant problem. In other words, such a problem is the price topay for identifying Λvac to the cosmological constant. Because the under-standing of quantum gravity is still an ongoing challenge, the correct fieldequations describing the contribution to gravity of quantum fluctuationsare not yet established. However, the dimensional analysis shows that therelated gravitational effects are expected at small (quantum) scales anddo not participate to the general expansion of the universe according toobservations.References1. A. Benoit et al., Astron. Astrophys., 399, L25 (2003)2. G. Bigot, H.H.Fliche, R. Triay, Astron. Astrophys., 206, 1 (1988)3. G. Bigot, R. Triay, in The quest for the fundamental constants in Cosmology ,9th Moriond Astrophysics meeting, eds J. Audouze and J. Traˆn Thanh Vaˆn,Editions Frontie`res (1989)4. S.M. Carroll, W.H. Press, E.L. Turner, Annu. Rev. Astron. Astrophys. 30,499 (1992)August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper1355085. S.M. Carroll, Living Reviews in Relativity , 4,1 (2001)6. J.C. Charlton, M.S. Turner, Astrophys. J., 313,495 (1987)7. I.O. Cherednikov, Acta Phys. Pol. B , 33 (2002)8. A.S. Eddington, Month. Not. Roy. Astron. Soc., 90, 668 (1930)9. A. Einstein, Sitz. Preuss. Akad. wiss. Phys-Math, 142, 87 (1917)10. A. Einstein, Sitz. Preuss. Akad. wiss. Phys-Math, 235,37 (1931); Appendixto the 2nd edn. of The Meaning of Relativity ,(1945) on the “cosmologicalproblem”; reprinted in (all later editions) Princeton University Press (1955),p.10911. J.E. Felten, R. Isaacman Rev. Mod. Phys., 58,689 (1986)12. H.H. Fliche, J.M. Souriau, Astron. & Astrophys., 78, 87 (1979)13. H.H. Fliche, J.M. Souriau, R. Triay, Astron. & Astrophys., 108, 256 (1982)14. A. Friedmann, Z. Phys., 10,377 (1922); ibid. 21,326 (1924)15. G. Gamow, in My world line, Viking Press, New York (1970), p.4416. E.P. Hubble, Proc. Nat. Acad. Sci., 15, 168 (1927)17. N. Kardashev, Astrophys. J., 150, L135 (1967)18. G. Lemaˆıtre, Ann. Soc. Sci. Bruxelles., 47 A, 49 (1927); Month. Not. Roy.Astron. Soc., 91, 483 (1931)19. C.B. Netterfield et al., Astron. J., 571, 604 (2002)20. T. Padmanabhan, Phys. Rep., 380,235 (2003)21. J.V. Peach, Astrophys. J., 159, 753 (1970)22. V. Petrosian, E.E., Salpeter, P. Szekeres Astrophys. J., 147, 1222 (1967)23. V. Petrosian, E.E., Salpeter, Astrophys. J.., 151, 411 (1968)24. J.V. Perlmutter et al., Nature, 391, 51 (1998)25. J.V. Perlmutter et al., Astrophys. J., 483, 565 (1999)26. A.J. Riess et al., Astron. J., 116, 1009 (1998)27. C. Rovelli, Int. Jour. Mod. Phys. D , 12,1509 (2003)28. M. Rowan-Robinson, Month. Not. Roy. Astron. Soc., 141, 445 (1968)29. V. Sahni, A. Starobinsky, Int. Jour. Mod. Phys. D , 9, 373 (2000)30. A. Sandage, Ann. Rev. Astron. Astrophys., 26, 561 (1988)31. P.B. Schmidt et al., Astrophys. J., 507, 46 (1998)32. J.L. Sievers et al., Astrophys. J., 591, 599 (2003)33. I. Shklovsky, Astrophys. J., 150, L1 (1967)34. D.N. Spergel et al.; Astrophys. J. Suppl., 148, 175 (2003)35. J.M. Souriau, Ge´ome´trie et relativite´ (Hermann, Paris, 1964)36. J.M. Souriau, Ann. Inst. H. Poincare´, XX (4), 315 (1974)37. J.M. Souriau, in Ge´ome´trie symplectique et physique mathe´matique, Collo-ques Internationaux CNRS , 237, 59 (1974)38. J.M. Souriau, in Journe´es Relativistes – 1976 , ULB Bruxelles (1977)39. K.S. Stelle, Phys. Rev. D , 16,953 (1977)40. M. Straumann, in 1rst Sem. H. Poincare´. Paris (2002)41. J.M. Souriau, R. Triay, Grav. & Cosm., 3,(9),51 (1997)42. G.’t Hooft, M. Veltmann, Ann. Inst. H. Poincare´, (A) Phys. Th. XX (1), 69(1974)43. R.C. Tolman, in Relativity Thermodynamics and Cosmology , Oxford at theClarendon Press (1934), p.344August 10, 2006 17:32 WSPC - Proceedings Trim Size: 9in x 6in AICCP.hyper13550944. R. Triay, in Large Scale Structures and Peculiar Motions in the Universe,eds. D.W. Latham and L.N. da Costa, ASP conference series (1989)45. R. Triay, in The Cosmological Model , 37th Moriond Astrophysics meeting,Les Arcs, Editions Frontie`re (2002)46. R. Triay, in Proceedings of the Tenth Marcel Grossmann Meeting on GeneralRelativity , p. 2266 (Rio de Janeiro, Brazil 2003) . Edit. M. Novello & S.P.Bergliaffa, Ser. Edit. R. Ruffini. World Scientific (2006)47. R. Triay, Int. Journ. Mod. Phys. D , 1410,1667 (2005)48. S. Weinberg, Gravitation and Cosmology , Ed. John Wiley & Sons Inc.49. Ya. B. Zel’dovich, JETP Lett., 6, 316 (1967)50. Ya. B. Zel’dovich, I.D. Novikov, in The Structure and Evolution of the Uni-verse, Vol. 2, Ed. G. Steigman, The University of Chicago press (1983)

An Issue to the Cosmological Constant Problem

https://hal.archives-ouvertes.fr/hal-00005025

An Issue to the Cosmological Constant Problem

Abstract

Similar works

Full text

Available Versions

HAL AMU