Uni…ed Holeum Theory of Dark Matter and Dark Energy A. L. Chavdaa , L. K. Chavdab a
Bungalow No. 49, Mahatma Gandhi Society, City Light Road, Surat (India) - 395007.
b
Chavda Research Institute.
(Dated: November 18, 2016)
Abstract We prove the Holeum conjecture that stable gravitational bound states of Primordial Black Holes were created in the early universe. It is shown that the Holeum theory covers not only the Dark Matter but also the Dark Energy. A downward atomic transition of a Holeum results in the emission of a bundle of gravitational energy called an Einsteinon carrying entropy as an additional quantum number. Einsteinons obey an Entropic Exclusion Principle as well as the Holographic one. Dark Energy is a sea of mutually repelling Einsteinons. Emission of Einsteinons causes the acceleration of the expanding universe. This is a quantum e¤ect. PACS numbers: 04.70.Dy, 04.70.-m, 04.70.-s, 14.80.-j, 98.80.-k, 95.35.+d
Electronic address:
a
[email protected],
b
[email protected]
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I.
INTRODUCTION
Dark Matter (DM) and Dark Energy (DE) are the greatest enigmas of our times. Currently the most successful theory “The Standard Model”has no clue to them. Although its putative successor the Super Symmetry (SUSY) theory promised an ample supply of DM in the form of the Lightest SUSY Particle (LSP) it had nothing to o¤er for DE. Not only that the LHCb experiment has …nally ruled out an experimental indication of a bump supposed to represent the LSP at a mass 750 GeV [1]. And the Large Underground Xenon (LUX) experiment has dealt the decisive blow to the Weakly Interacting Massive Particle (WIMP) [2]. However our Holeum theory proposed in 2002 provides a new form of DM and a wellknown source for it [3]. In order to take cognizance of the large amount of Primordial Black Holes (PBHs) produced in the early universe we made the ansatz that they must have formed stable gravitational bound states called Holeums. This way they would play an important role in the constitution and evolution of the universe similar to that of the other primordial particles such as the quarks and the leptons. Under this ansatz we have studied the properties of the Holeums and have made predictions which have received experimental support from Cosmic Rays data, Gamma Ray Burst (GRB) data and the Ice Cube experiment [4]. The latter has ruled out the Fire Ball model but it supports the Holeum theory [4]. The main aim of this paper is to point out that when the full scope of the Holeum theory is unravelled it naturally includes the DE also. Thus we will show that the Holeum theory is a uni…ed theory of DM and DE. Another important aim is to prove the Holeum conjecture. In section 2 we present the proof of the Holeum conjecture. Summary of the predictions and the experimental support for the Holeum theory are presented here. In section 3 we discuss the emergence of the DE in the theory. We show that a new graviton-like particle called an Einsteinon carrying a quantum of entropy must mediate the atomic transitions of the Holeum atom. The relationship of the new particles with the DE is illuminated. The role of these particles in the accelerated expansion of the universe is high-lighted. Discussion and conclusions are presented in section 4.
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II.
PROOF OF THE HOLEUM CONJECTURE
This proof consists of four parts: (1) a condition for thermal equilibrium, (2) a condition for non-dissociation of the Holeums produced, (3) a condition for immunity to Hawking radiation, and (4) a condition for the formation of standing wave pattern of the matter waves.
A.
Kolb and Turner Lemma for Thermal Equilibrium [5]
If an interaction is mediated by a massless boson then it can thermalize the primordial brew at a temperature T
1. This reduces to eq. (1) which was to be proved.
1.
Thermal Equilibrium
Consider a Holeum of two constituent PBHs each of mass m = 1014 GeV. For this case one has
g
= 10
10
. The temperature for thermal equilibrium satis…es the condition T < 108 eV
(9)
During the thermal equilibrium the mass of a PBH remains constant. Therefore it can be treated as a particle of constant mass during the equilibrium. The Schrodinger equation for two body problem for the 1=r potential is exactly solvable. One can obtain the energy eigen values, the binding energy, the mass of the two-body bound state, the most probable radius of the bound state and the Schwarzschild radius of it. In particular for the case under consideration the binding energy of the Holeum is 112:44 eV.
2.
Non-dissociation
Unless the temperature of formation of a Holeum is at least two orders of magnitude less than its binding energy the Holeum cannot survive dissociation. This gives us the temperature of production and survival of these Holeums to be about 1 eV.
3.
Immunity from Hawking Radiation (HR)
First we present here a condition for the bound state to be a black hole. If the Schwarzschild radius of the bound state is greater than its most probable radius then the bound state is a black hole and it will be destroyed by the HR. Conversely if the most probable radius of the bound state is greater than its Schwarzschild radius then the bound state is NOT a black hole. And it will not emit the HR despite containing two PBHs. This reduces
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to m < mc
(10)
where mp = 0:8862mp (11) 2 If the mass of the constituent black holes of a Holeum satis…es this condition then that 1
mc = ( ) 2
Holeum will be immune to HR.
4.
Condition for the formation of standing wave pattern of the matter waves
As shown in CQG [3] the condition for the formation of a standing pattern of the matter waves is also satis…ed for m < mc . Thus we have shown that all PBHs having their mass m < mc will form gravitational bound states that are not black holes. That is, they will be immune to HR. They will also satisfy the condition for standing waves. This proves the Holeum conjecture. A simple picture of a stable Holeum is as follows: If the HR emitted by the PBHs is strictly con…ned in a tube joining them and if each PBH emits as much HR as it absorbs then we have a stable bound state. The tube rotates about its midpoint. A tri-Holeum is made of three tubes of equal length and equal cross section. Curiously, a similar picture obtains in quarkonia also. But in the latter case the tube contains the lines of force. Let us summarize the properties and the predictions of the Holeum theory [4]. (1) A Holeum is a stable atom of two or more micro-black holes. It does not emit the HR despite containing two or more black holes in it. (2) Holeum occupies space just like particles of ordinary matter. (3) A holeum possesses a segregation property vis-à-vis the particles of ordinary matter. (4) Because of the segregation property and the rotation of galaxies Holeums accumulate as Cold Dark Matter (CDM) in the halos of galaxies. (5) Cosmic Rays arise from occasional break up of Holeums due to collisions among themselves. The Ultra High Energy Cosmic Rays, too, arise in a similar manner except that these ultra high energy collisions are much rarer. (6) The presence of the Holeums in the halo of our galaxy partially …lls up the GZK cut-o¤ leaving only a small feature in it. Observations bear this out. (7) A Holeum emits quantized gravitational radiation due to its atomic transitions. (8) Al Dallal has explained the systematics of the Gamma Ray Bursts (GRBs) on the basis of the Holeum theory [6]. (9) The theory predicts that the Cosmic Rays will come to the earth 5
almost exclusively from the halo of our galaxy and practically nil from its disc. In 1983 Giler studied the arrival directions of the Cosmic Rays and arrived at exactly this conclusion [7]. (10) The has ruled out the Fire-Ball (FB) model because it predicts a simultaneous arrival of a gamma ray and a neutrino. But the experiment never detected such a coincidence. In the Holeum theory when a Holeum in the halo of our galaxy breaks up due to a collision two PBHs are liberated. They will emit two jets of HR which consist of high energy gamma rays, particles and antiparticles in unequal numbers because the HR does not obey the CP symmetry. It is a completely random process. The gamma rays emitted in HR are the primary ones. There are also the secondary gamma rays. The latter are emitted as follows. If a particle from one jet collides with its own antiparticle from the other jet they will annihilate each other and will produce a pair of gamma rays. These are the secondary gammas. They will not be accompanied by any other particles. In particular, they will not be accompanied by a neutrino. The secondary gamma rays will be much delayed and completely erratic because of the thermal nature of the HR. But each one of the primary gammas will be accompanied by a completely randomly emitted neutrino. Therefore the primary gammas will not arrive in co-incidence with a neutrino. This is in agreement with the …nding of the Ice Cube experiment [8].
III. A.
DARK ENERGY The Entropy of a Holeum
The entropy of a black hole is given by S = A=4. Here S is the entropy and A is the surface area of the black hole. We have seen above that if Rn > rn then the bound state is a black hole. Thus its entropy will be given by Sn = Rn2 . Here n is the principal quantum number of the Holeum. But when Rn < rn the bound state is not a black hole and yet we will take its entropy as Sn = Rn2
(12)
This is because entropy is the property of a black hole which is characterized by its Schwarzschild radius and not the most probable radius.
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B.
Graviton exchange provides the binding of a Holeum
Since we are considering neutral, spinless PBHs they will have only the gravitational interactions among themselves. This interaction arises from the exchange of a massless graviton having spin J = 2. This means that only those atomic transitions of Holeums are allowed whose principal quantum numbers di¤er by two . Therefore we have two in…nite towers of excited states: an Odd Tower with n = 1; 3; 5; 7; : : : and an Even Tower with n = 2; 4; 6; : : : There is no transition from the n = 2 state to n = 1 state. Thus, we have two ground states: n = 1 and n = 2. This is the …rst atom having two ground states.
C.
The boson responsible for the atomic transitions of a Holeum.
Since the Holeum is a gravitational bound state of two PBHs the exchanged particle, the graviton, must have J = 2, and mass m = 0. On the other hand, in an atomic transition from the (n + 2)
state to the n-state a particle of mass 0 and spin 2 is emitted. It carries
away two units of angular momentum, the energy di¤erence the entropy di¤erence
S = Sn+2
E = En+2
En as well as
Sn . The last fact makes the emitted particle di¤erent
from the graviton. We will call it the Einsteinon in honour of Einstein who has made the most profound contributions to gravity to-date. The Einsteinon is a massless particle and it moves at the speed of light. So how can it carry the entropy which requires an area to carry it? There is a simple answer to this question: The Einsteinon carries a zero-thickness disc of area 4 S held perpendicular to its direction of motion. It is instructive to recall that the photon, a bundle of electro-magnetic energy, has a zero mass and an angular momentum ~ and a magnetic vector B ~ both of which are mutually J = 1. It carries an electric vector E perpendicular and they are also perpendicular to the direction of motion of the photon. Both ~ and B ~ rotate in a thin disc of zero thickness and a …nite area that just contains both E ~ and E ~ Similarly an Einsteinon is a bundle of gravitational energy that carries an entropy B.
S
de…ned above on an area 4 S held perpendicular to its direction of motion. Because of the conservation of entropy it is clear that one cannot put two or more Einsteinons in the same place at the same time. We call this the Entropic Exclusion Principle (EEP). According to the Holographic Principle, entropy resides on a surface and not inside a volume. All the Einseinons emitted by the atomic transitions of Holeums move at the speed of light and end
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up on the inner surface of the observable universe. They continue to move on the surface at the same speed without over-lapping one another. Consider an epoch t = t when the inner surface of the observable universe is fully covered with Einsteinons. If a new Einsteinon is emitted by a downward transition of a Holeum then to comply with the Holographic Principle and to conserve entropy the surface area of the universe has to increase by the amount 4 S. The radius of the observable universe will have to follow suit. This very tiny expansion of the universe is over and above that mandated by General Relativity. Therefore it leads to the acceleration of the expansion. General Relativity causes the expansion of the universe. This is a classical e¤ect. But the acceleration of the expansion is caused by the downward transitions of Holeums. Hence the acceleration is a quantum e¤ect.
IV.
DISCUSSION AND CONCLUSIONS
We have presented here a uni…ed Holeum theory of Dark Matter and Dark Energy. Ours is a dynamical theory. It is based on the Principle of Primordial Democracy that gives meaning and substance to the existence of a vast quantity of the hither to neglected primordial black holes produced in the early universe. It predicts two new particles: Holeum and Einsteinon. The …rst is a stable gravitational bound state of two or more PBHs. Holeum is an atom of DM. It is the progenitor of DM, DE, Cosmic Rays of all energies, short duration gamma ray bursts and the Quantized Gravitational Radiation (QGR) among other phenomena. Of course, in this theory QGR is identical with the DE. The Einsteinon is a bundle of gravitational energy like the graviton but it carries a quantum of entropy. Consequently the Einsteinons obey an Entropic Exclusion Principle that gives them an antigravity property. Thus, DE is a sea of Einsteinons possessing antigravity. Every downward transition of a Holeum forces the universe to expand. This leads to an acceleration in the expansion of the universe. Quantization of entropy emerges naturally in this theory. Finally we would like to comment on another uni…ed theory of DM and DE by Ma and Wang [9]. (1) The most important di¤erence between their theory and ours is the ENTROPY which is conspicuous by its absence from their theory. It plays a stellar role in our theory. It endows the Einsteinons with an exclusion property. In our theory the origin of the mysterious antigravity possessed by the DE is laid at the door of the Einsteinons. Now we truly understand the dynamics of the enhanced rate of growth of the expansion: A 8
downward atomic transition of a microscopic Holeum drives the expansion of the macrocosm of the universe. This is, of course very tiny. But the number of Holeums in the universe is not quite small. (2) With the proof of the Holeum conjecture presented in this paper we have no assumptions except the Principle of the Primordial Democracy. (3) We have no free parameters. (4) As listed above the Holeum theory explains or predicts a vast array of cosmological phenomena such as the origin of the Cosmic Rays, Gamma Ray Bursts, etc. Ma and Wang’s theory has no such predictions. (5) Holeum theory predicts that the accelerated expansion began roughly soon after the formation of the Holeums. On the other hand, only those Holeums will survive in a galactic halo which have their ground state binding energy at least two orders of magnitude greater than the highest thermal energy available in the halo. Then there will hardly be any atomic transitions among the Holeums. Then the acceleration of the expansion will cease. Thus, the Holeum theory has much greater predictive power.
[1] Mehlhase, S. Search for Stable Massive Particles with the ATLAS detector in proton-proton p collisions at s = 13 TeV. ATL-PHYS-PROC-2016-185. [2] LUX collaboration. Phys. Rev. Letts. 112 (2014) 091303. [3] Chavda, L. K. and Chavda, A. L. Classical and Quantum Gravity 19(2002) 2927 [4] Chavda, A. L. and Chavda, L. K. Int. J. Sc. Eng.Res. 3(2012) issue 11. [5] Kolb, E. W. and Turner, M. S. Early Universe. Addison-Wesley Publishing Co., Menlo Park, Cal. 1989. [6] Al Dallal, S. Advances in Space Research 40, No.8 (2006) 1186- 1198. [7] Giler, M. J. Phys G, Nucl. Phys., 9(1983) 1139. [8] Ice Cube collaboration, Nature 484(2012) 351. [9] Ma Tian and Wang Shouhong. arXiv: 1206.5078v1 [gr-qc], 2012.
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