A Dark Matter halo for every elementary particle in a Zwicky de Broglie synthesis. Abstract

Transcription

1 A Dak Matte halo fo evey elementay paticle in a Zwicky de Boglie synthesis E.P.J. de Haas (Paul) Nijmegen, The Nethelands (Dated: Septembe 20, 2015) Abstact In this pape I intoduce a new Dak matte hypothesis. I assume that evey elementay paticle has a Dak Matte halo. Given a est mass m 0 at = 0, it will have an additional spheical Dak Matte halo containing an exta mass in the sphee with adius as m DM = the constant Dak Matte adius m 0 with having a measued value somewhee in between 10 kpc and 20 kpc, so appoximately once o twice the adius of an aveage luminous galaxy. The total est mass of an elementay paticle contained within a sphee with adius will then be given by m = m 0 + m 0. Inside a sphee of adius 15kpc, the total mass of an elementay paticle at est wil be twice the oiginal est mass at = 0. ρ DM = The coelated mass density is m 0 4π 2. The new Newtonian gavitational enegy will be U g = GM 0m 0 GM 0m 0 esulting in an unchanged Newtonian foce of gavity but with a coect galaxy velocity otation cuve, due to the still applicable viial enegy theoem. The axiom is theoy of gavity neutal because it is a statement about mass and mass density distibution only. But it implies that WIMP s and the like aen t necessay to explain Dak Matte; my poposal isn t WIMP neutal. Beyond the scale of galaxy clustes the model becomes poblematic due to an exta halo halo inteaction tem becoming active at that scale. haas2u[at]gmail.com 1

2 I. PRINCIPLES OF THE DE BROGLIE MATTER WAVES OF REST MASSES Moden post-obital o post- Boh-Sommefeld wave quantum mechanics began with de Boglie s hypothesis of the existence of matte waves connected to paticles with inetial mass [1]. De Boglie stated with the assumption that evey quantum of enegy U should be connected to a fequency ν accoding to U = hν with h as Planck s constant [2],[3]. Because he assumed evey quantum of enegy to have an inetial mass m o and an inetial enegy U 0 = m 0 c 2 in its est-system, he postulated hν 0 = m 0 c 2. De Boglie didn t estict himself to one paticula paticle but consideed a mateial moving object in geneal [2]. This object could be a photon (an atom of light), an electon, an atom o any othe quantum of inetial enegy. If this paticle moved, the inetial enegy and the associated fequency inceased as ν i = γν 0. De Boglie assumed the inetial enegy of the moving paticle to behave as a wave-like phenomenon, so the inetial wave associated with a moving paticle not only had a fequency ν i but also a wave-length λ i, analogous to the fact that any inetial enegy U i of a moving paticle had a momentum p i associated to it accoding to the elation p = h/λ. De Boglie neve could specify what exactly oscillated in elation to m 0 with fequency ν and wavelength λ, but using these elations physicists could make majo pogess in theoy and expeiments. De Boglie s ideas egading matte waves suounding moving masses poved useful in the futhe development of quantum mechanics and the success of such chimeas is what finally counts in science, egadless of the ontological status of postulated ideas. Discussions egading expeimental eality of the fequency of est masses is ongoing [4], [5], [6]. II. THE PROBLEMS FOR WHICH THE DARK MATTER HALO OF GALAXIES WAS A SOLUTION In 1933 Dak Matte was mentioned as dunkle Mateie in a pape by Zwicky. Fitz Zwicky was studying the Coma Cluste of galaxies and found that his calculations fo obital acceleation and stella mass within it was off by a lage facto. He concluded that thee should be a much geate density of dak matte within the cluste than thee was luminous matte. Zwicky concluded that this constituted an unsolved poblem [7]. In 1937 Zwicky egaded his study on the Coma Cluste a test of Newton s law of gavity on the lagest 2

3 cosmological scale possible, by applying the viial theoem on a cluste of galaxies. He also mentioned in his 1937 pape the possibility to test the viial theoem by applying it to the otational velocities of the individual stas in the sepaate galaxies. But he concluded that this was technologically out of each [8]. The beakthough eseach of Rubin and Fod aound established beyond doubt the oute otational velocity cuves of individual galaxies, which tuned out to be flat [9]. This was in conflict with velocity cuves that esulted fom the application of the viial theoem to the luminous mass of these galaxies. Rubin and Fod cited colleagues who suggested the existence of a lage galactic halo of dak matte. In a 1980 pape pesenting futhe eseach they concluded that the fom of the otation cuves implied that significant non-luminous mass should be located at lage distances beyond the optical galaxy. The total mass of a galaxy should, fo lage distances, incease at least as fast as the distance fom the cente [9]. The thid majo evidence fo Dak Matte is the gavitational lensing effect of clustes of galaxies. The mass of stas and hot gas in clustes who collectively act as a gavitational lens is too small to bend the light fom the backgound galaxies so much. A lage density of dak matte in the cente of these cluste is needed to explain the stength of the obseved lensing effects. Fo ou pape, the galaxy otation cuve esults ae most elevant. All thee mentioned astophysical obsevations use the hypothesis of a galactic Dak Matte halo that extends fa beyond the luminous pat of the galaxies in ode to explain the esults of measuements. But whee the fist two effects elate to Newtonian gavity, gavitational lensing diectly involves Einstein s Geneal Relativity. Ou hypothesis will focus on the Newtonian viial theoem. III. THE DARK MATTER HALO AS A PROBLEM THAT HAS TO BE SOLVED Many solutions to the Dak Matte obsevations have been poposed. All these solutions ceate thei own poblems and scientific consensus is no whee in sight. The majo poblem with the Dak Matte elementay paticle hypothesis is that they cannot be found. The seach fo the elementay paticle constituting Dak Matte is in pocess to mimic the seach fo the ethe aound Some eseaches even mention 3

4 looking fo a Dak Matte wind, that should vay with the seasons. A smalle poblem is that the Dak Matte density aound the sun should be consideable and have possible tiny effects on the obits of ou planets. Howeve, no deviations fom Newtonian gavity have been found. Inteesting is the fact that a link has been established between the amount of luminous matte in a galaxy and the density of the Dak Matte in the halo. And the density of Dak Matte that could cause a flat otation velocity cuve has to fall of as the invese of the squae of the adius. The invese squae law indicates that the cental luminous galaxy might function as a souce of the Dak Matte. Some solutions have tied to explain the astonomical obsevations by coecting the elevant theoies of gavity instead of postulating Dak Matte. These solutions have poblems of thei own. MOND is a non-elativistic ad-hoc coection of Newtons foce of gavity, wheeas gavitational lensing is a GR phenomenon. GR in tun is specifically designed to coespond to Newtons theoy in the low field domain and that is exactly the galaxy and galaxy cluste viial theoem aea. The theoem that is in touble. Fom a ecent pape by Koopmans et.al. we quote: In both spial and elliptical galaxies with pominent bayonic components, thee appeas to be a conspiacy between dak-matte and bayons, leading to a nealy univesal total mass distibution out to the lagest measued adii that is vey close to isothemal (i.e. ρ 2 ), with only a small intinsic scatte between systems. [11] This is a key motivation fo ou poposed axiom as it is pesented in the next section. The obsevation in the quote indicates towads some kind of a souce like connection between bayons and thei Dak Matte halo. IV. THE DARK MATTER HALO AS AN AXIOM OF ELEMENTARY PARTICLE COSMOLOGY We will ty to tansfom the unsolved poblem egading the constitution of the Dak Matte halo into an axiom. So instead of us woking on the poblem we ty to let the poblem wok fo us. In the following we pesent ou poposed model. 4

5 Given a est mass m 0 at = 0, it will have an additional spheical Dak Matte halo containing an exta mass in the sphee with adius as m DM = m 0 (1) with the constant Dak Matte adius having a constant value somewhee in between 10 kpc and 20 kpc, so appoximately once o twice the adius of an aveage luminous galaxy. The total est mass of an elementay paticle contained within a sphee with adius will then be given by m = m 0 + m DM = m 0 + ( m 0 = m ). (2) The total mass of an elementay paticle at est inside a sphee of adius wil be twice the oiginal est mass at = 0. Distubances in the elementay Dak Matte halo due to changes in position and momentum at = 0 will tavel with matte wave velocity though the halo, with v wave = c2 v paticle (3) If the elementay paticle has velocity zeo, the distubances at the souce can tavel instantaneously though the entie halo. The DM halo functions as a medium fo the de Boglie matte waves, it is the halo that vibates when matte waves packages pass by. This diectly implies that the halo has local Bon-intepetation pobability-density popeties. Paticles that will be kicked out of an oiginal = 0, p = 0 position and momentum and acquie a velocity appoaching c will have a matte wave velocity appoaching c fom above and thee will be a consideable delay egading the Dak Matte halo adjustment to the new situation of the souce paticle. Consideable has to be intepeted as on galactic tavel and time scales with ly. In such cicumstances, lage etadation effects should be expected, diluting the conspiacy between dak-matte and bayons on cosmic scales. The identification of the Dak Matte halo with the de Boglie matte wave medium is necessay in ode to assues ou poposal to be in full accodance with Special Relativity and pe-spin Wave Quantum Mechanics. Ou theoy is a Relativistic Quantum Gavity without spin. The elementay paticle Dak Matte halo mass content has been deived fom a mass density that is invesely popotional to 4π 2. So the mass density of the halo dops of o dilutes at the suface of an eve lage sphee in the same way that all classical cental 5

6 souces do. We define a Dak Matte halo mass density as ρ DM = m 0 4π 2 (4) and then the spheically symmetic Dak Matte halo mass inside a sphee of adius is given by m = V ρ DM dv = ρ DM 4π 2 d = m 0 d = m 0 + m 0 (5) with the last facto as the obvious constant of integation, given the stating point of ou model that we have m = m 0 at = 0. The density and mass functions of the elementay paticle s halo ae chosen to match the values necessay to aive at the constant velocity otation cuve of galaxies. It is the astophysical expeimental input, especially the ρ 2 Dak Matte density distibution obsevation, see [11], that is tuned into axiomatic definitions egading the popeties of elementay est masses. V. THE CHECK WITH FACTS REGARDING THE VIRIAL THEOREM Given the definition of the gavitational potential as φ = GM (6) and the mass M as we get a gavitational potential at as M = M 0 + M 0 (7) φ = GM 0 GM 0 (8) Fo the esulting foce of gavity on a classical mass m we get the Newtonian esult F = m φ = GM 0m 2 ˆ. (9) This is due to the fact that the new facto vaies linea ove and thus esults in a additional potential tem that is constant. Ou Dak Matte halo acts as a gauge tem in the souce that poduces a constant tem in the potential and thus has zeo effect on the foce. 6

7 But the exta tem is effecting the gavitational enegy of a satellite mass m in the field of a souce mass M. This gavitational enegy is given by U g = m 0 φ = GM 0m 0 GM 0m 0. (10) Now we assume that the viial theoem is still valid, actually we assume that it is moe fundamental than the foce elation fom which it has been oiginally deived, giving 2U k = U g, so v 2 = φ fo obiting satellites and v 2 = φ = GM 0 + GM 0. (11) If we let then v 2 = GM 0, (12) which is a constant. This esult allows us to give an estimate of by applying this to an aveage galaxy. We get = GM 0 v 2 6, , (2, ) 2 = 3, m = 12kpc. (13) Actual galaxy velocity otation cuves vay consideably fom ou model with its point like mass distibution. Real galaxies have disk like o spheical like mass distibutions which cause deviations fom ou single paticle model. But given the luminous mass distibution, the associated galaxy Dak Matte halo as a summation ove all the elementay paticle Dak Matte halo s should be computable. Evey galaxy has it s own specific luminous mass distibution and then also a paticula Dak Matte halo. These calculations should eventually lead to a detemination of the value of the specific Dak Matte adius, the Dak Matte constant of my poposal. As fo galaxy clustes as fo example the Coma cluste studied by Zwicky, the same appoach should be used, assuming that all the luminous matte content of the cluste will fom one single halo. Given the fact that a typical cluste can have an extension adius of a 250 kpc and the galaxy otation cuve, and thus the galaxy Dak Matte halo, can stetch out as fa as 50 kpc, to teat the cluste in an identical way as a spheical galaxy should be justifiable. 7

8 VI. ABOUT QUANTUM MECHANICS AND GENERAL RELATIVITY The poblem with cold elementay Dak Matte paticles like WIMP s is that they cannot be found o detected. Ou poposal means that the Dak Matte halo isn t composed of sepaate elementay paticles, so to the extend that we ae coect, WIMP s and the like will not be found. In ou poposal, the Dak Matte halo is a stochastic quantum ethe, the de Boglie matte wave substatum with Bon-intepetation statistical popeties. The indiect quantum effects of this substatum have been amply detected, but the undelying substance itself is still eluding all detection. Ou poposition means that we assume that this local quantum stochastic substatum, accoding to de Boglie endowed with intinsic themodynamic popeties, is cosmologically showing gavitational mass popeties. Fom ou quantum mechanic pespective, tying to detect this halo locally is the same endeavou as tying to pove the de Boglie-Bohm pilot wave theoy coect by diectly measuing the piloting substance. The oiginal poblem with MOND o modified Newtonian gavity was that it wasn t elativistic. In the double sense of not in accodance with Special Relativity and neithe with Geneal Relativity. Ou poposal is embedded in SR fom the beginning because of the de Boglie connection. It is also easily incopoated in Geneal Relativity due to the fact that ou poposal is about the souce of GR effects, the density distibution of matte. Ou axiom only affects the souce of evey theoy of gavity, the amount and distibution of matte and matte density in the aea unde consideation. Ou axiom is theoy of gavity neutal because it is a statement about mass itself only. VII. RUNNING INTO TROUBLE WITH HALO-HALO INTERACTION AT THE SCALE OF THE UNIVERSE The gavitational enegy of a classical satellite in a Dak Matte halo was detemined using a simple m 0. But if we investigate the gavitational enegy of two elementay paticles with each a Dak Matte halo, at the distance between two sepaate galaxy cluste fom 8

9 each othe, say, we get U g = ( m 0 + m ) ( 0 GM 0 GM ) 0 = GM 0m 0 2GM 0m 0 (14) GM 0m 0. (15) RDM 2 This gives us a classical tem, a double constant tem and a new Dak Matte halo Dak Matte halo inteaction tem that inceases with. If we then detemine the foce using F = U g we get F = U g = GM 0m 0 ˆ + GM 0m 0 ˆ (16) 2 R 2 DM in which the fist tem is the Newtonian classical foce of gavity and the second tem is the halo-halo foce of gavity. That last tem has the invese sign and is a constant. Then in between galactic clustes with, the galaxy clustes will tend to epel each othe with a constant foce, supposing to be constant in time. Let s safely assume that we have eached the limit of applicability of ou poposal. At these distances, the expansion of the Univese intefees with gavity and Dak Enegy and Inflation come into play. Dak Matte effects will be oveuled by Dak Enegy, the dominant facto at that length-scale. [1] E. P. J. de Haas, The combination of de Boglies Hamony of the Phases and Mies theoy of gavity esults in a Pinciple of Equivalence fo Quantum Gavity, Les Annales de la Fondation Louis de Boglie, 29, , (2004) [2] L. de Boglie, Comptes Rendus 177, , (1923). [3] L. de Boglie, Recheches su la théoie des quanta, these, Pais, 25 novembe 1924; Annales de Physique 10, Tombe III, , (1925). [4] Mulle, H., Petes, A. and Chu, S. A pecision measuement of the gavitational edshift by the intefeence of matte waves. Natue 463, (2010). [5] Pete Wolf, Luc Blanchet, Chistian J. Bode, Sege Reynaud, Chistophe Salomon, and Claude Cohen-Tannoudji, Atom gavimetes and gavitational edshift, Comment to Natue, 466 doi : /natue09340 (2010) 9

10 [6] P. Wolf, L. Blanchet, C.J. Bode, S. Reynaud, C. Salomon, and C. Cohen-Tannoudji, Reply to comment on : Does an atom intefeomete test the gavitational edshift at the Compton fequency?, Class. Quantum Gav. 29, , ( 2012) [7] Zwicky, F., Die Rotveschiebung von extagalaktischen Nebeln, Helvetica Physica Acta 6: (1933). [8] Zwicky, F. On the Masses of Nebulae and of Clustes of Nebulae, Astophysical Jounal 86: , (1937) [9] Rubin, V. C., Thonnad, N., and Fod, W. K., J., ApJ 225 L107-L111 (1978) [10] V. Rubin, N. Thonnad, W. K. Fod, J, Rotational Popeties of 21 Sc Galaxies with a Lage Range of Luminosities and Radii fom NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc). Astophysical Jounal 238: 471 (1980). [11] Koopmans, L.V.E. et.al., Stong Gavitational Lensing as a Pobe of Gavity, Dak-Matte and Supe-Massive Black Holes, axiv: v2 [asto-ph.co] (2009) 10