Dark matter, dark energy and gravitational proprieties of antimatter

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1 Dak matte, dak enegy and gavitational popieties of antimatte Dagan Slavkov Hajdukovi PH Division CEN CH-111 Geneva Abstat We suggest that the eventual gavitational epulsion between matte and antimatte may be a key fo undestanding of the natue of dak matte and dak enegy. If thee is gavitational epulsion, vitual patileantipatile pais in the vauum, may be onsideed as gavitational dipoles. We use a simple toy model to eveal a fist indiation that the gavitational polaization of suh a vauum, aused by visible (bayoni) matte in a Galaxy, may podue the same effet as supposed existene of dak matte. In addition, we ague that anelation of gavitational hages in vitual patile-antipatile pais, may be a basis fo a solution of the osmologial onstant poblem and identifiation of dak enegy with vauum enegy. Hene, it may be that dak matte and dak enegy ae not new, unknown foms of matte-enegy but an effet of omplex inteation between the quantum vauum and known bayoni matte. 1. Intodution Aoding to the uent onodane model of osmology, the Eath, the stas, the Galaxies and eveything we ae familia with (i.e. eveything made fom quaks and leptons as building bloks), make up less than 5% of the total matte and enegy in the Univese. The est of 95% is dak enegy and dak matte; oughly in popotion :1. Hene, the pesent day Univese is dominated by its dak side (it eminds me the human soiety). Howeve no one knows what dak enegy and dak matte ae. Thei natue is the geatest mystey of ontempoay osmology. We ae living in the age of peision osmology, i.e. in the time of the unpeedented peision of obsevations; we ae able to obseve what s happening but without a satisfatoy undestanding why it s happening. The simplest evidene fo unseen dak matte omes fom obsevation of spial galaxies. A spial galaxy is a disk of dust and stas (typially 1 11 stas) otating about a ental nuleus. The veloity v() of otation of stas in the galaxy an be appoximately detemined as a funtion of distane fom its ente. The supise oming fom obsevations is that, outside a adius that ontains the most of the visible mass of the galaxy, v() emains appoximately onstant as fa out as an be measued. But, if we have to tust the well established law of gavitation, one would expet the veloity v() at a adius to be elated to the mass M() inteio to that adius by a elation oughly like GM ( ) v ( ) GM ( ) i. e. v( ) (1) Thus, thee is a big onflit between expetations that v() fall off as --1/ and obsevation that v() is nealy onstant. One possible solution is ad ho assumption that visible matte of the Galaxy is suounded by a spheial halo of some yet unknown and invisible dak matte. If we want the seond of Equations (1) to agee with obsevations, the mass of dak matte M dm () inteio to a adius must be popotional to that adius. Hene, the adial mass distibution M dm (), and adial density pofile ρdm ( ) of dak matte ae haateized with popotionalities: M dm ( ) ; ρ dm ( ) () 1

2 Dak enegy was invoked afte the eent disovey [1], that the expansion of the univese is aeleating athe than slowing down. As well known [], the easiest way fo poduing theoetial models with aeleated expansion of the univese is to assume a positive osmologial onstant Λ in Einstein s field equations. In fat, in addition to the usual soue tem, popotional to the enegymomentum tenso, Einstein s equations fo the gavitational field an aommodate an additional T soue tem popotional to the meti field g 1 g itself: 8G T 4 + Λg whee and denote the ii and sala uvatues defined with the meti g. The key point is that suh an ad ho intodued osmologial onstant has the same physial effet as if an appopiate, onstant mass (enegy) density is attibuted to the vauum. Equivalently eveything that ontibutes to the mass (enegy) density of the vauum, ontibutes to the osmologial onstant though linea elations: 8G 8G Λ ρ E and Λ ρ 4 m (4) whee ρ E and ρ m, ae onstant enegy density and onstant mass density of the vauum. At fist sight, it looks like a welome meeting point between Geneal elativity and Quantum Field Theoies (QFT). In fat, fom the point of view of QFT, the vauum is not just an empty spae as in nonquantum theoies, but a still pooly undestood kingdom of vitual patile-antipatile pais and fields. Hene, in piniple, QFT an povide a mehanism poduing the vauum enegy density and estimate the numeial value of suh density. But what a atastophe! The onfontation of equation (1) with obsevations [], plaes an uppe bound on Λ, and onsequently on ρ E and ρ m : Λ < 1 m ; ρ m < 1 kg / m ; ρ E < 1 J / m (5) while QFT pedit values [] whih ae, minimum a few tens odes of magnitude lage than these obseved values. This damati disepany (alled osmologial onstant poblem) is the key obstale fo attative identifiation of dak enegy with the vauum enegy. In the pesent pape we pesent a adially new idea that dak matte and dak enegy ae not new, unknown foms of matte-enegy but an effet of omplex inteation between the quantum vauum and known bayoni matte. Ou speulations ae based on the onjetue, that thee is gavitational epulsion between patiles and antipatiles.. Conjetue of gavitational epulsion between patiles and antipatiles The gavitational popieties of antimatte ae still not known. The AEGIS expeiment [4], eently appoved at CEN would be pesumably the fist one in the human histoy to measue aeleation of antimatte in the gavitational field of the Eath. In fat it would measue the gavitational aeleation of atoms of the antihydogen, using intefeomety; a tehnique well established fo atoms of odinay matte. While a huge majoity of physiists believes that gavitational aeleation of patiles and antipatiles is the same, thee is oom fo supises. The biggest supise would be if gavitational aeleation of patiles and antipatiles just diffes in sign. Suh a possible supise (gavitational epulsion between matte and antimatte) is main assumption in ou pesent pape. In piniple (as it was aleady known to Newton) we must distinguish between inetial mass m i and two gavitational masses (it may be bette to say gavitational hages): the ative gavitational mass whih is the soue of the gavitational field, and the passive gavitational mass whih esponds to an aleady existing field. As Newton oetly undestood, beause of the univesality of the fee ()

3 fall (established by Galileo), inetial and passive gavitational mass an be onsideed as equal; while his thid law (equality of ation and eation) implies that ative and passive gavitational mass an be onsideed to be equal as well. So, the piniple of equivalene of the inetial and gavitational mass of a body (known as the Weak Equivalene Piniple) was bon. This impotant piniple has been used by Einstein as a onestone of geneal elativity and so of moden Cosmology. Today, the WEP is the oldest and the most tusted piniple of moden physis. Howeve the WEP is tested only fo odinay matte. No one knows is it valid fo antimatte, dak matte, dak enegy, the eventually existing supesymmeti patiles The existing expeimental evidene and ou assumption of gavitational epulsion between matte and antimatte may be summaized as: m m ; m m ; m + m (6) i g i i g g Hee, as usually, a symbol with a ba denotes antipatiles; while indies i and g efe to inetial and gavitational mass. The fist two elations in (1) ae expeimental evidene [5], while the thid one is ou assumption whih damatially diffes fom geneal onvition that m m. It would be pudent to onside ou assumption as valid fo quaks and leptons and thei antipatiles; and only on the basis of it to dedue gavitational mass (hage) of omposite patiles (fo instane positonium, quakonium and neutal pion must have zeo gavitational mass; while negative pi meson, with quak stutue d u, must have the gavitational mass signifiantly smalle than the inetial one ). Let s note that ou onjetue (6) says nothing about gavitational popieties of gauge and Higgs bosons. Fo instane, it is an open question if gavitational field an see diffeene between photons and antiphotons (uently onsideed as the same patile). Anothe unetainty is the elation between ative and passive gavitational mass, beause in Geneal elativity, it is not fixed by anything like Newton s thid law. Fo instane, we know that photons feel gavitational field, but even if we believe so, thee is no expeimental evidene that photons ae also soue of the gavitational field.. Majo onsequenes of the onjetue The usual statement that we live in the Univese ompletely dominated by matte is not tue in the ase of quantum vauum, whee (if the quantum field theoies ae ight) vitual matte and antimatte appea in equal quantities. Thus, ou hypothesis must have damati onsequenes fo the quantum vauum (a wold of vitual matte) and only indietly though it to ou wold of eal matte. We limit ouselves to point out thee majo onsequenes of the onjetue (6). Fist, it is immediately lea that a vitual patile-antipatile pai is a system with zeo gavitational mass and suh a anelation of gavitational masses might be impotant fo an eventual solution of the osmologial onstant poblem. By the way, a simila anellation of the opposite eleti hages of patile and antipatile in a vitual pai, leads to the zeo density of the eleti hage of the vauum. Seond, a vitual pai may be onsideed as gavitational dipole with the gavitational dipole moment h p md ; p md (7) Hee, by definition, the veto d is dieted fom the antipatile to the patile, and pesents the distane between them. As the distane between patile and antipatile is of the ode of the edued Compton wavelength D h m, we shall use the seond of Equations (7) attibuting to evey vitual pai a gavitational dipole moment independent of mass. g g

4 The oesponding enegy of the dipole in an extenal gavitational field haateized with aeleation g is ε p g, i.e. h ; h GM ε g ε (8) The seond of these equations onens enegy in the field of a spheial body with mass M. Hene, polaization of the vauum by a gavitational field might be possible. In ode to gasp the diffeene between the polaization by an eleti field and the eventual polaization by a gavitational field, let s emembe that, as a onsequene of polaization, the stength of an eleti field is edued in a dieleti. Fo instane, when a slab of dieleti is inseted into a paallel plate apaito, the eleti field between plates is edued. The edution is due to the fat that the eleti hages of opposite sign attat eah othe. Thee is no eason to think about it in eletostatis, but let s note that if, instead of attation, thee was epulsion between hages with opposite sign, the eleti field inside a dieleti would be augmented not edued. But, aoding to ou hypothesis, thee is suh epulsion between gavitational hages of diffeent sign. Consequently, outside of a egion in whih a etain bayoni mass M is onfined, the eventual effet of polaization should be a gavitational field stonge than pedited by the Newton s law. The most impotant question is if the gavitational polaization of the vauum an podue the same effet as pesumed existene of dak matte. We will tun bak to this question in setion 4. Thid, the vauum might pemanently adiate. In ode to undestand it let s emembe the illuminating example oming fom Quantum Eletodynamis: eation of eleton-positon pais fom the (Dia) vauum by an extenal (lassial i.e. unquantized), onstant and homogenous eletial field E. In this patiula ase of the unifom eleti field, the patile eation ate pe unite volume and time is known [6] exatly: dn e + e 1 eeh m m m exp (9) dtdv 4 m h h eeh It is evident that patile eation ate is signifiant only fo an eleti field geate than the itial value E m eh. The above phenomenon is due to both, the omplex stutue of the physial vauum in QED and the existene of an extenal field. In the (Dia) vauum of QED, shot-living vitual eletonpositon pais ae ontinuously eated and annihilated again by quantum flutuations. A vitual pai an be onveted into eal eleton-positon pai only in the pesene of a stong extenal field, whih an spatially sepaate eletons and positons, by pushing them in opposite dietions, as it does an eleti field E. Thus, vitual pais ae spatially sepaated and onveted into eal pais by the expenditue of the extenal field enegy. Fo this to beome possible, the potential enegy has to vay by an amount ee Δl > m in the ange of about one Compton wavelength Δ l h m, whih leads to the onlusion that the pai eation ous only in a vey stong extenal field E, geate than the itial value E. It is evident, that in the ase of gavitational epulsion between matte and antimatte, a unifom gavitational field, just as a unifom eleti field tends to sepaate vitual eletons and positons, pushing them in opposite dietions, whih is a neessay ondition fo pai eation by an extenal field. But while an eleti field an sepaate only haged patiles, gavitation as a univesal inteation may eate patile-antipatile pais of both haged and neutal patiles. 4

5 In the ase of a unifom gavitational field, haateized with aeleation g, Equations (9) tivially tansfoms to with D being the edued Compton wavelength. Distibution (11) has a maximum fo dn D 1 g 1 exp dtdv 4 D gd (1) D max (11) g Hene, in an extenal gavitational field g vauum should adiate and the spetum of adiation is dominated by D max. If so, the vauum, as evey adiating body, might be attibuted a tempeatue T h A h kt A g (1) D max with k being the Boltzmann onstant and A, a dimensionless onstant whih has to be detemined. The onstant A in equations (1) an be detemined using assumption that fo all adial distanes vauum adiates as a blak body. If so, it must be λ max T b, whee b is Wien displaement law onstant. Fom this ondition it follows: bk A (1) h If the soue of gavitation is a spheially symmeti body of mass M, and if the aeleation g is detemined by the Newton law ( g GM ), the Eq.(1) gives: kt h A S S kb ; S GM (14) Thus, we have attibuted a tempeatue T, depending on the distane, to the vauum aound a massive body. In the patiula ase S, we edisove the essential pat of the famous Hawking s tempeatue of a blak hole adiating as a blak body. But this time Hawking s tempeatue seems to be just a patiula ase of a moe geneal phenomenon. In fat Hawking s fomula may be onsideed as a fist hint, that in geneal, in an extenal gavitational field, vauum aquies a gavitational tempeatue and onsequently adiates. Namely, the mass of a blak hole is ollapsed, onfined deep inside the hoizon and suounded by vauum. Thee is nothing in the viinity of the Shwazshild adius, just vauum. Hene, Hawking fomula tells us that, unde influene of a gavitational field, quantum vauum in the viinity of the Shwazshild adius, adiates as a blak body of tempeatue T. But, thee is no speial eason to limit vauum adiation in the viinity of the Shwazshild adius; the phenomenon might exist fo geneal. By the way, fomula (14) suggests that, deep inside the hoizon of a blak hole, tempeatue (and adiation) must be highe. If the mass M is suounded by dak matte (o something poduing the same effet) it follows fom Eq.(), that at lage distanes, aeleation is dominated by a tem G dm dm g ρ, ρ (15) d 5

6 whee ρ is a onstant. The oesponding vauum tempeatue is A Gρ h kt (16) Let s note that attempts to intodue themodynamis in a Fiedmann-obetson-Walke osmologial model [7], suggest a onstant gavitational tempeatue of the vauum detemined by 1 h 1 h ktu Ω 1 ; H (17) H U whee /H is Hubble length. Tempeatue (16) beomes equal to the onstant tempeatue T u, fo some itial distane. Gρ H 4A (18) Ω 1 It povides a natual ut-off fo ; at distane lage than, the tempeatue is a onstant. 4. A Toy model fo vauum polaization As a toy model, let us onside quantum vauum as a gas of independent gavitational dipoles in an extenal gavitational field, podued by a spheial body with mass M. Mathematially it is equivalent to the poblem of independent eleti dipoles in an extenal eleti field E. The applied extenal field tends to oient dipoles in its dietion, while the themal agitation tends to andomize dipoles. Polaization (dipole moment pe unit volume) is given by Langevin equation: 1 P Np oth( x) x (19) whee N is the numbe of dipoles pe unite volume, p is the oesponding (gavitational o eleti) dipole moment. Fo a gas of eleti dipoles x pe kt. In the ase of a gavitational dipole, following Eq.(8), instead of pe, we put H pg hgm (g is the gavitational aeleation of the extenal gavitational field); and fo kt we use Eq.(16). The polaization is: h h M 4kb ρ P N tgh 4kb ρ h M whee is the oesponding unit veto. It stays to guess what may be the value of N. We know that in ou wold of eal matte dominant building bloks ae quaks and leptons fom the fist geneation. It seems easonable to suppose that the same is tue in the vauum. If so, the dominant gavitational dipoles ae vitual mesons. Thus N may be appoximated by 1 m + ± m m N ; m (1) λ h m Fo simpliity we have defined as the mean value of the masses of and. ± () 6

7 In full analogy with eletostatis, a gavitational mass density ρ P oesponds to the polaization (). In the ase of lage (what is the main ase of inteest) elementay alulations lead to 1 h h M ρ p ( ) P N () 1 kb ρ with oesponding mass inside sphee with adius equal to h h M M p ( ) N ( ) kb ρ whee is a adius at whih mass of dak matte an be onsideed to be zeo. A ompaison of Eq. () with M ) ρ ( ) immediately gives dm ( p () h m M ρ (4) 6kb λ and M dm h m M ( ) ( ) 6kb λ (5) The adial density pofile (4) and the adial mass distibution (5) have the same fom as given by Eq. () what ae the essential haateistis of dak matte. This is a fist indiation that what we all dak matte eventually may be esult of vauum polaization and fom now we make identifiation M dm () M p (). Aoding to Eq. (1) and (5), the iula veloity aound the ente of a galaxy is given by: V i G h m M ( ) M + ( ) 6kb λ 1 (6) Now, using the Eq. (4), the Eq. (18) an be tansfomed and intepeted as the adius of the spheial halo of dak matte. G kb 6h m M 8 λ H Ω 1 (7) Fom Eq. (5) and (7) follows that the total amount of dak matte in a galaxy is: M tot dm 4G H m M M dm ( ) (8) Ω 1 λ Let s note that in the above, vey simplified onsideations, we didn t inlude the fat that vitual pais of haged patiles have also an eleti dipole moment. 5. Disussion The nea futue will show if the pesent pape is useless o signifiant. But fo sue this pape may eventually be impotant just beause of the ideas pesented heein, while the extemely simplified onsideations must be pomptly eplaed by moe igoous alulations. Howeve, in spite of the fat 7

8 that ou equations annot be expeted to be auate, it is still impotant to ompae them with obsevation, with the hope that they an epodue the ight ode of obseved magnitudes. 41 Let s onside the ase of ou Galaxy taking M 1 kg. Fistly, ou fomula (6) may be ompaed with the vey eent estimates [8] of the Milky Way s iula veloity V i () up to distanes of 6 kp. These esults show that the iula veloity vaies slightly with adius, dopping fom ~ km/s at the sola adius to ~17km/s at 6 kp; the null hypothesis that V i () has a onstant value of km/s is ejeted with vey high statistial signifiane (at a level of.1). The numeial esultv i ( 6kp) 16km / s, oming fom ou fomula (6) is not only the ight ode of magnitude but in supisingly good ageement with ef. [8]. On the othe side, fom Eq. (7) and (8) we have: tot 4 5 kp; M dm.14 1 kg what is also lose to eality (See [8] and efeenes theein). In alulations, we have used the values fo Ω and H, fom the latest edition of the eview of Patiles Popieties [9]. A final emak onening dak enegy, whih has nothing to do with polaization of the vauum. The appopiate toy model might be to onside the vauum as an ideal gas of vitual mesons, at a onstant gavitational tempeatue detemined by Eq. (17). A pion may be onsideed as a system with n f 48 degees of feedom; 4 fo (with the stutue ( uu dd )/ ) and 1 degees fo eah of ± (with espetive stutues u d and u d ). If thee is gavitational epulsion between matte and antimatte, the gavitational mass of the vitual gas of pions is zeo; but it has tempeatue (17) and oesponding enegy pe pion ε n f ktu. Hene, the enegy density is n f h 1 9 ρ E Nn f ktu Ω J / m (9) lose to the ight value fo the osmologial onstant (See elation (5)). Dediation This pape is dediated to my fathe Slavko, and my hilden Ivan and Anja-Milia efeenes [1] A.G. eiss et al., The Astophysial Jounal 56, 49 (1) S. Pelmutte, Int. J. Moden Physis A 15S1, 715 () [] S. Weinbeg, eviews of Moden Physis 61, 1- (1989) P.J.E. Peebles, eviews of Moden Physis, 75, () [] M.Tegmak et al., Phys. ev. D 69, 151 (4) [4] A. Kellebaue et al. (AEGIS Poto-Collaboation) NIM B 66 (8) 51 [5] C.M Will, 199 Theoy and Expeiment in Gavitational Physis Cambidge Univesity Pess (199), Cambidge G. Gabielse et al. Phys. ev. Lett. 8, (1999) [6] J.S Shwinge, Phys. ev (1951) W. Geine, B. Mülle and J. afaelski, 1985, Quantum Eletodynamis of Stong Field Spinge-Velag, Belin V.P Folov and I.D Novikov, 1998 Blak Hole Physis Kluwe, Dodeht [7] S. Bayin, Gen. el. Gav., 179 (199) [8] X.X. Xue et al. The Astophysial Jounal 684, 114 (8) [9] Patile Data Goup, Physis Lettes B667 (8) H λ 8