Preface

Dark matter is a subject confined by the observational context. It is currently only indirectly observed as the puzzling motions of distant galaxies and the unpredictable rotational velocities of stars in neighboring galaxies. Attempts to observe dark matter as traditional sub-atomic particles always fail, leaving this worldly topic a major observational problem. This is very different from ordinary matter, which can be mutated, split into components, destroyed, and resurrected at will by physicists, given enough energy, time, and money.

Dark matter was first postulated by astronomers in the mid-twentieth century to explain the seemingly non-Newtonian character of galactic dynamics. A discrepancy was observed by Fritz Zwicky for the relative velocities of three spiral galaxies circulating within a distant, small galactic group. To calculate the relative velocities of these galaxies, Zwicky made the reasonable assumption that the galactic masses are proportional to their luminosities. In that case, the galaxies behaved as if they were much more densely packed with matter than one could explain in simple terms. Since that report, it has been found that dark matter of some form is often required to properly explain the paths and velocities of large galaxies within galactic groups. Though an intriguing observation, I don't think Zwicky necessarily meant that a new type of matter was warranted but rather that the observed rotational velocities are much faster than expected and much of the matter in these galaxies is not luminous.

Dark matter remained just a thought, not seriously considered a problem for many years, until the publication of the relative velocities of some bright stars of the Andromeda galaxy. The velocities of very luminous stars, which are outside the dense interior so could be observed without interfering with light, do not follow Newton's gravitational law. According to Kepler and Newton, as supported by observations of our planetary neighbors, objects more distant from a large central object should travel more slowly than those closer to the galactic center. This correlation does not hold for Andromeda though; astronomers found that past a critical distance from the center, all stars circulate at about the same angular velocity – meaning distant stars exhibit greater instantaneous velocities than those closer to the galaxy center. This is a real surprise. Similar observations of star motions in other spiral galaxies confirming this have been interpreted as necessitating the presence of something like dark matter. To bring this mystery closer to home, the Dutch astronomer Oort observed that nearby stars in our Milky Way seem to be traveling too fast when one considers only the masses calculated for our luminous neighbors.

The outstanding problem with dark matter is how to relate these interstellar and intergalactic observations to something on Earth. Answering this question has been the goal of many young particle physicists with dozens of aspiring theoreticians proposing a bevy of suggestions often in the form of new microscopic particle types. Up to now, observations of any new, special particle that only interacts via gravity but not electromagnetically, as required to be dark matter, have not been confirmed.

The only observations supporting a new microscopic particle were made by the DAMA/LIBRA experiment. This sensitive method uses underground detectors of thallium doped-sodium iodide crystals housed in a radiation-dampening enclosure to detect novel events over many months. The investigators observe a seasonal signal variation that can be simply explained by the annual difference in the earth's velocity as we circle the sun while traveling through our galaxy. It was reported since that even these results cannot be duplicated. If these counterclaims are true this means that all concrete, worldly evidence attributing dark matter to microscopic particles has evaporated. There are several chapters in this book that go beyond the now-discredited claims for a particle type to explain dark matter observations.

The chapter by E. T. Tatum proposes a direct explanation for dark matter, which should be given consideration. He thinks it is likely that dark matter is the abundant cold dark hydrogen that remains undetected, therefore unaccounted, because we lack the technical ability for detection. He makes the reasonable claim that the total mass of hydrogen, being the most abundant species in our universe, has been and is still being seriously underestimated. He provides several references to support his proposition.

In our chapter about dark matter in spiral galaxies, we hypothesize that the effects attributed to dark matter can be modelled by assuming the translational decay of omnipresent axions. The gravitational redshift due to the orbital decay of innumerable axions in a galaxy is translated into a gravity-like effect. The hypothesis is checked against the behaviours of highly luminous stars in two galaxies with well-mapped translations and a good fit is reported between the model and data.

A. N. James reminds us that the Friedmann approximation of the Einstein Equation, as the FLRW interpretation, should perhaps not be taken too seriously. The level where one considers the universe as isotropic and homogeneous should not include the distances from here to "closer" supernovae emissions. He presents a new model that mitigates some of the problems with the FLRW assumptions and offers a counter explanation for the supernovae emission data; not requiring dark matter nor dark energy. He also correctly points out that the Ωk term of this FLRW model is overinterpreted as representing spacetime curvature rather than the more conservative, mathematical view that it only presents the remainder of the Ωm term after normalization, a view shared by others.

In his chapter, A. K. Sinha deals with the likely possibility that dark matter really is very dark; much is located in black holes. This suggestion seems reasonable since black holes are a strong "magnet" for matter and LIGO/Virgo observations suggest black holes are more common than previously thought. He asks the question of how black holes containing dark matter will evolve with time and presents a considered hypothesis for the eventual fate of some dark matter.

P. D. Morley presents a mathematical description of non-Keplerian orbits that should be observed when objects are circulating about non-spherical or multiple centers. He posits that the most abundant particles in the universe, neutrinos, are unlikely to be produced isotropically in galaxies. Such situations could be observed as stars (and planets) following non-Keplerian orbits due to the unequal gravitational pull by these abundant particles. He then presents his mathematical solution to this special solution.

**V**

G. I. Burde considers the processes accompanying cosmic ray propagation on cosmological scales. He develops the basic concepts of "relativity with a preferred frame" from the basis of particle dynamics, electrodynamics, and general relativity.

universe shall we search?"

A. Bogomolov presents his argument that dark matter and dark energy are as much a problem of information as astronomy and physics. He considers the philosophies underlying these two concepts and concludes by merging philosophy with cosmology. This editor tends to agree with the ideas of Bogomolov since now that microscopic particles are off the menu, our question is, "What shall we do next, where in our

**Michael L. Smith, Ph.D., M.D.Sc.**

Umeå University, Umeå, Sweden G. I. Burde considers the processes accompanying cosmic ray propagation on cosmological scales. He develops the basic concepts of "relativity with a preferred frame" from the basis of particle dynamics, electrodynamics, and general relativity.

A. Bogomolov presents his argument that dark matter and dark energy are as much a problem of information as astronomy and physics. He considers the philosophies underlying these two concepts and concludes by merging philosophy with cosmology. This editor tends to agree with the ideas of Bogomolov since now that microscopic particles are off the menu, our question is, "What shall we do next, where in our universe shall we search?"

> **Michael L. Smith, Ph.D., M.D.Sc.** Umeå University, Umeå, Sweden

#### **Chapter 1**

## Introductory Chapter: Introduction to Dark Matter

*Michael L. Smith*

#### **1. Introduction**

Dark matter is a subject defined by the observational context. It, whatever it may be, is currently something only indirectly observed as the puzzling motions of distant galaxies and the unpredictable rotational velocities of stars in neighboring galaxies. Attempts to observe dark matter as something like traditional particles almost always fail leaving this earthly topic as a major observational problem. This is a very different situation from that of ordinary matter that can be mutated, split into components, destroyed and resurrected at will by physicists, given enough energy, enough time and plenty of money.

Dark matter was first postulated by astronomers in the mid-twentieth century to explain the seemingly non-Newtonian character of intergalactic dynamics. The first, widely broadcast, discrepancy was observed by Fritz Zwicky for the relative velocities of three spiral galaxies circulating within a distant, smallish galactic group. To calculate the relative velocities of these galaxies, Zwicky made the assumption that the galactic masses are proportional to the luminosities. In this case, the galaxies behave as if they are much more densely packed with matter than one can explain in simple terms. Since that report, it seems that dark matter of some form is often required to properly explain the paths and velocities of large galaxies within galactic groups [1]. Though intriguing, I do not think Zwicky necessarily meant that a new type of matter was warranted but only that the observed rotational velocities are much faster than expectation and that much of the matter in these galaxies is not luminous.

Dark matter remained just a thought, not even considered a serious question for many years, until publication of the relative velocities of some bright stars of the Andromeda galaxy [2]. The velocities of very luminous stars, which are outside the dense interior so could be observed without interfering light, do not follow the Newton-Kepler laws as these rotate around the galactic center. According to classical gravity, as supported by observations of our planetary neighbors, objects more distant from a large central object should travel more slowly than those closer to the galactic center. This correlation does not hold for stars of the Andromeda galaxy though; astronomers found that past a critical distance from the center, all stars circulate at the similar angular velocities—meaning distant stars travel faster, in the relative sense, than those closer to the galaxy center. This is a real surprise (and seemingly an insurmountable challenge). This and more recent observations of star motions in other spiral galaxies have been interpreted as necessitating the presence of something like dark matter [3]. To bring this mystery closer to home, the Dutch astronomer Oort observed that nearby stars in our Milky Way seem to be traveling too fast when one considers only the mass calculated for our luminous neighbors.

It was when these latter observations were published that the problem of dark matter woke up some astronomers and physicists. The outstanding, current problem with dark matter is how to relate these interstellar and intergalactic observations to something on earth. Answering this question has been the goal of many young particle physicists with dozens of aspiring theoreticians proposing a bevy of suggestions often in the form of new microscopic particle types. These ideas include many brands of super-symmetry, where the borders of the standard model of particle physics are enlarged to encompass the new particle types. The most important type might (or may not) be WIMPS (Weakly Interacting Massive ParticleS). But other than exhibiting gravitational attraction, predictions of WIMPS properties are nearly impossible to constrain.

Unfortunately, observations of any new, special particle that only interacts *via* gravity but not electromagnetically, as required to be WIMPS, are very rare and have not been confirmed. A few observations supporting a new microscopic particle type have been claimed by the DAMA/NaI experiment of Italy, which have been searching for new microscopic particles for the past two decades [4]. This sensitive method uses underground detectors of Th-doped-NaI crystals housed in a radiation dampening enclosure to detect novel events over many months. In this respect, the detector(s) is designed in a manner similar to neutrino detectors, expecting very rare, low-energy interactions with normal matter. DAMA has spawned related experiments, ANAIS-112 in Spain and COSINE-100 in Korea, in related attempts to observe WIMPS. Every once in a while, DAMA reports successful observations of signals that indicate the presence of some type of WIMPS [5].

It has been pointed out by others that DAMA investigators observe a seasonal signal variation that can be simply explained by the annual difference in the earth's velocity as we circle the sun—loosely analogous to the Michelson-Morley experiment. It has also been reported that these claims of particle collisions with the NaI detector cannot be duplicated. If these serious counterclaims are true, this means that concrete, worldly evidence attributing dark matter to microscopic particles has evaporated [6]. Good-bye WIMPS.

There are other mechanisms whereby the motions of stars and galaxies attributed to dark matter can be explained. The exaggerated gravitational redshifts might be the results of orbital decay of innumerable axions in a galaxy, which is translated into a gravity-like effect. One may check the validity of such models, presuming the presence of axions, against the behaviors of highly luminous stars in galaxies with well-mapped translations and observe if the model(s) properly describe the data.

Another very likely explanation for dark matter, which seems rather obvious, is abundant dark H and He, which remains undetected, therefore unaccounted, residing in intra- and intergalactic space. Because most of this H and He reside in the ground state, both are dark and remain undetected, except by gravity, because we lack the technical ability for detection. In line with this thought is the likely possibility that dark matter really is very dark; much is located in black holes. This suggestion seems very reasonable since black holes are a strong "magnet" for matter, perturbing the spacetime within and between galaxies rather than indicating the presence of dark matter. Perhaps both mechanisms are in play?

Perhaps, we take the Friedmann approximation of the Einstein Equation, as the FLRW interpretation, too seriously and too direct, and this is part of the problem. The level of accuracy with which one considers the distances from here to many supernovae emissions is exaggerated invalidating the FLRW model. A universe that is not isotropic and homogeneous should not be expected to be predictable. Also, the Ωk term of the FLRW model is often overinterpreted as representing spacetime curvature. The Ωk term rather only presents the remainder of universe after the Ωm term is calculated and not really spacetime curvature [5, 7]. That is, we cannot trust our distance estimates or trust the claim of flat spacetime and violation of the Newton-Kepler gravitational laws should rather be attributed to poor distance estimates.

#### *Introductory Chapter: Introduction to Dark Matter DOI: http://dx.doi.org/10.5772/intechopen.101679*

Finally, recent observations of many black holes of much smaller mass than well-known supermassive black holes, by LIGO/VIRGO, may require more explicit math than the Kepler approximation to explain the stellar orbital motions. These observations also indicate that "smaller" black holes are much more abundant than previously thought. Such a situation should be observed as stars (and planets) follow non-Keplerian orbits due to the unequal gravitational pull within galaxies. This should also be observed as more rapidly rotating galaxies within galactic groups than predicted from luminosity measurements.

### **Author details**

Michael L. Smith Umeå University, Umeå, Sweden

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Zwicky F. Republication of: The redshift of extragalactic nebulae. General Relativity and Gravitation. 2009;**41**:207. DOI: 10.1007/ s10714-008-0707-4

[2] Rubin VC, Ford WK. Rotation of the Andromeda Nebula from a spectroscopic survey of emission regions. The Astrophysical Journal. 1970;**159**:379. DOI: 10.1086/150317

[3] Persic M, Salucci P, Stel F. The universal rotation curve of spiral galaxies I. The dark matter connection. Monthly Notices of the Royal Astronomical Society. 1996;**281**:27. DOI: 10.1093/mnras/278.1.27

[4] Daley J. Testing DAMA. 2019. Available from: https://www. symmetrymagazine.org/article/ testing-dama

[5] Baum S, Freese K, Kelso C. Dark matter implications of DAMA/LIBRAphase2 results. Physics Letters B;**789**:262. DOI: 10.1016/j. physletb.2018.12.036

[6] Cho A. Is the end in sight for famous dark matter claim? Science. 2021;**374**:805. DOI: 10.1126/ science.acx9611

[7] Oztas AM, Smith ML. Re-evaluation of Ωk of the normalised Friedmann-Lemaître-Robertson-Walker model: Implications for Hubble constant determinations. New Astronomy. 2021;**88**:101609. DOI: 10.1016/j. newast.2021.101609

#### **Chapter 2**

## The Case for Cold Hydrogen Dark Matter

*Eugene Terry Tatum*

#### **Abstract**

The novel 'Cold Hydrogen Dark Matter' (CHDM) theory is summarized in this chapter. Special attention is paid to the fact that current technology prevents us from directly observing extremely cold ground state atomic hydrogen when it is of sufficiently low density in deep space locations. A number of very recent observations in support of this theory are summarized, including cosmic dawn constraints on dark matter. The importance of the Wouthuysen-Field effect as a probable mechanism for CMB decoupling of hydrogen at cosmic dawn is also stressed. This mechanism does not require a non-baryonic dark matter intermediary. Several predictions for this theory are made for the coming decade of observations and simulations.

**Keywords:** dark matter, CHDM, primordial hydrogen, cosmic dawn, Wouthuysen-Field effect, cosmology theory, Milky Way, galactic evolution, interstellar medium, hydrogen snow clouds

#### **1. Introduction and background**

This chapter will familiarize the reader with the author's 'Cold Hydrogen Dark Matter' (CHDM) theory and summarize a wealth of recent observational data in support of this theory. A discussion of a likely cosmic and galactic evolution scenario for primordial hydrogen will follow, and the chapter will end with several predictions for the coming decade of observations and simulations.

#### **1.1 The theory**

In May of 2019, the author was invited to attend a dark matter workshop as part of the World Science Festival. There he had the opportunity to present his CHDM theory to colleagues, after which the theory was published [1], and then first summarized in a chapter of IntechOpen's book entitled *Cosmology 2020 - The Current State* [2].

The theory holds that a certain deep space interstellar, circumgalactic, and intergalactic species of cold hydrogen can be particularly difficult to detect, much less accurately quantify. Specifically, extremely low density cold atomic hydrogen in its lowest ground state (i.e., ground state H I with anti-parallel electron spin) *cannot* emit light and does not collide sufficiently often with its neighboring atoms to create a luminous cloud readily visible from Earth or telescopic satellites. Furthermore, its deep space location vastly reduces its frequency of photon absorption that could be visible from Earth or satellites.

In light of this theory, it may seem paradoxical that a soft glow of H I line (21-cm) emissions can be readily detected throughout our Milky Way galaxy (MW) by radio astronomy [3, 4]. The ultrafine 21-cm line indicates a spin flip transition of the hydrogen electron in its ground state. The lower energy spin state is one in which the electron spin orientation is anti-parallel to the hydrogen nuclear spin orientation. Not surprisingly, 21-cm H I emissions are readily detectable in relatively dense and dynamic (cold, warm or hot) hydrogen clouds, which are particularly concentrated within the galactic bulge and the spiral arms of the disk. These 21-cm radio waves readily pass through dense galactic clouds which block most visible light. In fact, our knowledge of the number and distribution of the spiral arms of the MW and other galaxies is largely predicated on radio astronomy H I line mapping. However, while some of these H I emissions are undoubtedly coming from cold (i.e., slow-moving) hydrogen, that does not fully meet the author's definition of CHDM, precisely because it is relatively *concentrated* atomic hydrogen and, therefore, observable. To put it simply, not all cold hydrogen is dark. Only the very low density, lowest ground state, H I species near the cosmic microwave background (CMB) radiation temperature is dark.

When we make observations well away from the spiral arms, and along lines roughly perpendicular to the MW disc, it is estimated from spectral analysis that the vacuum density of deep interstellar space averages roughly one atom per cubic centimeter [5–7]. Similar observations in lines-of-sight to distant light sources have resulted in an average intergalactic vacuum density estimate of roughly one atom per cubic meter. According to the theory presented here, it is in these interstellar, circumgalactic and intergalactic deep space regions where the majority of the difficult-to-detect, extremely cold, and lowest density hydrogen (i.e., CHDM) must reside.

#### **2. Observational support for the CHDM theory**

#### **2.1 MW observations and halo mass calculations**

Posti & Helmi reported using *Gaia* data on globular cluster proper motions to calculate relative masses of dark matter and visible matter within a 20 kpc (65,200 light-years) radius halo sphere centered on the MW [8]. **Figure 1** shows this author's schematic representation of the Posti & Helmi sphere, according to their description.

This diagram is roughly to scale, since the MW disk has a radius of approximately 50,000 light-years and an average thickness of approximately 1,000 lightyears. The halo is denoted in black, and the rare stars in the halo outside the disk and bulge are also schematically represented.

Posti & Helmi provided data which allow one to calculate a dark matter-to-visible matter ratio of 2.54 within their 20 kpc radius sphere. Normalizing the visible mass (in the form of stars, gas clouds and dust) of the MW disc and bulge to a current best estimate [9] of 250 billion ʘ (solar masses), allows one to calculate a dark matter mass within the Posti & Helmi sphere of 635 billion ʘ. This number is obtained by multiplying 250 billion ʘ by 2.54.

To compare the CHDM theory with these observations, one can assume that a great majority of the estimated single atoms per cubic centimeter of the 20 kpc halo sphere deep space are cold hydrogen atoms. This conservatively implies an average vacuum matter density of 1.67 x 10−21 kg.m−3 within a spherical volume of 9.85 x 1062 m3 . Multiplying these two numbers together gives a total halo vacuum mass of 1.645 x 1042 kg. This amounts to roughly 827 billion ʘ within the halo

**Figure 1.** *Posti & Helmi 20 kpc halo sphere and MW galaxy.*

vacuum *outside* the thin slice representing the visible galactic disk and bulge. This estimate is obtained by dividing 1.645 x 1042 kg by the solar mass of 1.989 x 1030 kg. Even allowing for only 0.77 such atoms per cubic centimeter of the halo outside the visible galaxy, the Posti & Helmi ratio of 2.54 can be met. This is because 827 billion ʘ multiplied by 0.77 is 636.8 billion ʘ. So, an exceedingly low average halo vacuum density of approximately 0.77–1.0 hydrogen atom per cubic centimeter can *dwarf* the 250 billion ʘ of the visible MW stars, gas clouds, and dust! This makes the CHDM-defined species of atomic hydrogen a serious candidate for the 'missing matter' we are currently referring to as dark matter.

#### **2.2 Cosmic Dawn observations of the redshifted H I line**

The early cosmic dawn (reionization epoch) spin temperature of primordial neutral atomic hydrogen can be studied by measuring the redshifted 21-cm H I line at frequencies at or near 78 MHz. These line measurements (specifically '21-cm brightness temperatures') correspond to redshifts in the range of 15 < *z* < 20. By performing this analysis in the EDGES study, Bowman, et al. [10] have accumulated a wealth of data on the temperature of cosmic dawn primordial hydrogen. They have determined that cosmic dawn hydrogen was temporarily chilled to the low single digits of the Kelvin scale, reaching a nadir at around *z* = 17 (about 180 million years after the big bang). Thus, primordial hydrogen during early cosmic dawn, from about 100–250 million years after the big bang, was temporarily *decoupled* from its usual equilibration with the CMB radiation. This is what made it *visible*.

Barkana [11] has used the Bowman data to *significantly* constrain the mass *m*x of dark matter particles according to a baryon-dark matter (b-DM) particle interaction theory. This theory holds that, for *non-baryonic* dark matter to have interacted with baryonic matter in the early universe, it needed to *first* decouple from the CMB temperature, chilling to the point where it could *then* decouple primordial hydrogen from the CMB temperature.

Barkana's dark matter constraints are nicely graphed in Figure 3, page 9 of his review. This finding was a great disappointment to WIMP theorists, since the dark matter constraints effectively eliminate most, if not all, weakly-interactive massive particles. It *also* rules out baryons, except for atomic hydrogen, molecular hydrogen and perhaps He-3. This is because Barkana's dark matter constraints effectively rule out a dark matter particle much greater than 2–3 GeV, which strongly supports the case for atomic hydrogen!

To comprehend the significance of these dark matter constraints, the reader should scrutinize Figure 3 of [11]. They should pay particular attention to the dark matter particle mass *m*x corresponding to a cross-section σ1 value of 10−20 cm2 and a 21-cm brightness temperature log10 value (in mK) of 2.32. Note that these values correspond to a cold dark matter particle matching neutral atomic hydrogen, with a similar low velocity scattering cross-section and a mass-energy of 0.938 GeV. Furthermore, it should be remembered that the redshifted cosmic dawn 21-cm H I line is the *signature* of atomic hydrogen in its ground state.

#### **2.3 McGaugh's argument for a 'purely baryonic universe'**

In March of 2018, around the time of the Bowman, et al. EDGES publication, physicist Stacy McGaugh published a brief note [12] which made a cogent and compelling argument for baryonic dark matter. He noted that the high intensity of Bowman's redshifted H I line was a problem for the standard model of cosmology assumption of a non-baryonic b-DM interaction at cosmic dawn. McGaugh correctly pointed out that current atomic theory would indicate such a signal has the maximum *T*21 intensity when the neutral hydrogen fraction *X*HI is 1 and the spin temperature *T*S is equal to the Kelvin temperature. Although McGaugh did not explicitly state that a primordial atomic hydrogen self-interaction (i.e., decoupled colder atomic hydrogen interacting with CMB-equilibrated atomic hydrogen) was a more reasonable explanation for the EDGES study results, his statement that these observations would be 'expected for a purely baryonic universe' clearly carried this implication. He made the point that a dark matter particle outside of the standard particle model (i.e., a non-baryonic particle) was unnecessary to explain the EDGES study findings.

#### **2.4 A cosmic Dawn H I mechanism (the Wouthuysen-Field effect)**

The Bowman, Barkana and McGaugh publications, while opening up the possibility for a light baryon to fit within the current dark matter constraints, did not clearly specify a mechanism by which a baryon could have decoupled primordial hydrogen from the cosmic dawn CMB temperature. However, as detailed in the author's original CHDM publications, those mysterious cold baryons could well have been the first of the primordial hydrogen atoms chilled by the first stars of cosmic dawn.

The first stars of cosmic dawn are believed to have been massive blue stars emitting a great deal of ultraviolet (UV) radiation, including Lyman-alpha (Lyα) waves. Anyone with a knowledge of UV radiation effects on atomic hydrogen would have learned of the Wouthuysen-Field effect (WFE) discovered in the 1950s [13].

#### *The Case for Cold Hydrogen Dark Matter DOI: http://dx.doi.org/10.5772/intechopen.97557*

In short, the author proposes that the WFE is the likely mechanism by which the first primordial hydrogen atoms were decoupled from the CMB during cosmic dawn. A non-baryonic intermediary would not have been required.

In the laboratory, Lyα radiation has the correct energy to trigger, by a multistep process, the otherwise 'forbidden' transition (i.e., parallel to anti-parallel electron spin orientation) in ground state atomic hydrogen. As mentioned in the Introduction, the anti-parallel electron spin state has the lower energy level. This effect would have decoupled H I from the CMB temperature. Absorption and re-emission of Lyα photons effectively causes a redistribution of the balance between the 21-cm hydrogen ground states, such that a higher percentage of H I has its electron in the anti-parallel electron spin orientation with respect to the proton spin orientation. The net effect would have been to chill the Lyα-absorbing H I well below the CMB temperature and to make its redshifted ultrafine 21-cm line highly visible (whereas, CMB-equilibrated H I is *invisible* [14]).

Thus, the author's CHDM theory incorporates the WFE mechanism as a very reasonable alternative to the standard ΛCDM cosmology theory that *non-baryonic* dark matter chilled first during the dark cosmic epoch (which immediately preceded cosmic dawn) and then interacted with (i.e., chilled) primordial H I during cosmic dawn. The author's proposed WFE mechanism is a more simple and direct explanation. More importantly, it better explains the measured intensity and timing of the redshifted 21-cm signal at the beginning of cosmic dawn. The author proposes that it was the first *stars* which triggered the chilling of CMB-equilibrated H I, rather than exotic non-baryonic dark matter particles. To put it simply, it appears likely that *the 'cold dark matter' interacting with cosmic dawn CMB-equilibrated H I was, in fact, the first of the atomic hydrogen to be chilled by the Lyα radiation of the first stars.*

#### **2.5 The hydrogen snow cloud model**

For readers to become comfortable with the idea that cold hydrogen (molecular as well as atomic) could still be invisible and, therefore, missing from the baryonic budget, a brief description of Walker and Wardle's hydrogen snow cloud model [15] is in order. In their model, molecular hydrogen intermixed with small amounts of atomic hydrogen and helium can theoretically condense to a cold, high-density regime where solid or liquid hydrogen can form. By a complex mechanism, an inverted entropy gradient forms, allowing for the cloud outer layers to maintain an equilibrium state such that they are *below* the CMB temperature. Crucially, Walker and Wardle calculate specific model luminosities to be so low that such hydrogen snow clouds are, effectively, 'a type of baryonic dark matter' (their words). While the supercooled outer envelope of these snow clouds is mostly in the form of molecular hydrogen, Walker and Wardle presume there to be less cold H I in the snow cloud interiors. Therefore, effectively, the atomic and molecular hydrogen in these snow clouds is invisible to direct observation (i.e., dark). Moreover, there is, as of yet, no good reason why such hydrogen snow clouds could not be in large quantity within and around galaxies. This is CHDM in a somewhat different configuration, but dark hydrogen nonetheless.

#### **2.6 The new galactic pin scintillation method for observing otherwise dark baryonic matter**

In January of 2021, Wang et al. [16] reported first results of an ingenious method to observe otherwise invisible baryonic matter in the MW. By using the Australian Square Kilometer Array Pathfinder (ASKAP), they were able to track radio scintillations of such matter by using distant galaxies as 'locator pins' to map their extent. A degrees-long filamentous cloud was observed. As described by Wang, 'This gas is undetectable using conventional methods, as it *emits no visible light of its own and is just too cold for detection via (usual) radio astronomy.'* Artem Tuntsov, another of the authors, suggested the possibility that their cloud could be a hydrogen snow cloud having undergone tidal disruption by a nearby star [17]. This is a new method which will undoubtedly be used to identify a great deal more MW hydrogen dark matter than has been possible to measure otherwise.

#### **3. Discussion**

Given the CHDM theory presented here, it is useful to consider a reasonable cosmic evolution scenario following the CMB emission epoch. We begin with the CMB anisotropy pattern created when the universe was transitioning from a 3,000 K plasma ball to a hot neutral gas of mostly atomic hydrogen.

The denser regions within the CMB anisotropy pattern presumably followed the positive feedback of gravity to become the galaxies, galaxy clusters, and filaments we observe. Meanwhile, the fractal distribution of less dense regions of the CMB anisotropy pattern intricately interlaced with the denser regions presumably became much less dense, owing to ongoing universal expansion and cooling. These cooler primordial hydrogen regions ultimately became the interstellar, circumgalactic, and intergalactic deep space we see today. To summarize, the denser regions of the CMB anisotropy pattern became warm to hot gas clouds and hot stars, while primordial hydrogen atoms of deep space became increasingly cold, following the descending CMB temperature.

During the early cosmic dawn this otherwise *invisible* atomic hydrogen became highly visible while it was decoupled from the CMB temperature by the Lyα UV radiation of the first stars. However, continuing cosmic expansion and cooling eventually brought the increasingly-separated primordial hydrogen atoms back into CMB equilibrium. Therefore, it appears likely that this mostly invisible primordial hydrogen is still present in deep space in great abundance. Its low kinetic energy, low average vacuum density, tiny scattering cross-section, and remoteness from light sources makes it the most difficult chemical species in the universe for us to locate and measure. One can certainly understand why, to this point, *indirect* observations, such as galactic rotation curves, have been necessary to provide support for the existence of dark matter.

A discussion of the cosmic evolution of primordial hydrogen would be incomplete without a specific focus on its role in galactic evolution and dynamics. By the end of the cosmic dark age, the universe, according to the widely-accepted big bang theory, was at a CMB radiation temperature of a little less than 100 K. The first stars of cosmic dawn had not yet ignited. In their place were rapidly-accumulating concentrations of warming primordial hydrogen interspersed with lesser amounts of primordial helium and lithium. Intimately interlaced with these gravitating and warming hydrogen clouds and 'pre-stars' were zones of hydrogen still equilibrated with the CMB temperature and destined to be, with further cosmic expansion, the cold atomic hydrogen of interstellar, circumgalactic, and intergalactic deep space. According to the CHDM theory, this was the slowly-cooling primordial hydrogen destined to be what we are currently calling 'cold dark matter.'

Not surprisingly, because deep space is interlaced between today's stars, galaxies, galaxy clusters and filaments, deep space primordial hydrogen undoubtedly has a gradient of concentrations and temperatures (i.e., velocities). This depends

#### *The Case for Cold Hydrogen Dark Matter DOI: http://dx.doi.org/10.5772/intechopen.97557*

upon its precise interstellar, circumgalactic or intergalactic location. Thus, the deep space interstellar and circumgalactic vacuum density averages approximately one atom per cubic centimeter, and the deep space intergalactic vacuum density averages approximately one atom per cubic meter. Presumably, in the vicinities of particularly dynamic and concentrated star formation, the interstellar primordial hydrogen is warmed up by energetic photons and concentrated stellar winds, to the point where it can become either visible or highly depleted within these zones. This mechanism may account for observations of a 'cored' (i.e., relatively depleted) dark matter, as hydrogen, distribution near the centers of active starforming galaxies.

The study by Read et al. [18] of nearby dwarf galaxies showed that those which stopped forming stars more than 6 billion years ago tended to be cuspier than those with more recent 'bursty' star activity. The more recently-active galaxy centers tended to show more pronounced dark matter coring. The Read findings agree well with models where dark matter (baryonic or non-baryonic) is heated up and/or redistributed by concentrated active star formation. This interpretation fits nicely with the CHDM theory concerning the galactic evolution of primordial hydrogen. If this is the correct interpretation, then many bizarre properties of non-baryonic dark matter become unnecessary.

With the cosmic and galactic evolution scenario presented above, it should be apparent that the theorized non-baryonic dark matter is not required in order to understand the structural distribution of visible matter we see today. The opposing processes of gravitational clustering and cosmic expansion provided for a divergence of outcomes within the primordial hydrogen. Presumably, approximately one-sixth of this primordial matter gravitationally collapsed sufficiently to become the visible matter portion of our universe, and the remainder of this primordial matter became the CHDM of deep space.

As for predictions having to do with CHDM, the following appear to be likely:

#### **3.1 Improved methodologies for detecting baryonic dark matter**

There will be many more creative methodologies invented over the next decade which will indirectly locate and quantify otherwise invisible collections of cold hydrogen in the MW and its circumgalactic medium. Some of these methodologies may even be applicable to nearby galaxies.

#### **3.2 Tightening constraints on dark matter**

There will be tightening dark matter constraints around a particle *m*x value of 0.938 GeV (i.e., the mass-energy of neutral atomic hydrogen).

#### **3.3 Computer simulations of CHDM**

Computer simulations of galaxy formation and evolution which incorporate the CHDM theory presented herein will show excellent correlations with observations, including the coring effect of heating and/or ejecting cold interstellar hydrogen from active galactic centers with bursty star formation.

#### **3.4 No exotic non-baryonic dark matter**

No exotic non-baryonic dark matter fitting the observed qualitative and quantitative dark matter constraints will ever be discovered.

#### **4. Summary and conclusions**

An explanation has been given as to why very low density, deep space, primordial atomic hydrogen in its lowest energy electron spin ground state should be largely invisible, especially when close to the CMB temperature. Calculations made on the 20 kpc radius Posti & Helmi halo sphere support this theory, especially when one realizes that an average halo vacuum density of only 0.77–1.0 hydrogen atom per cubic centimeter dwarfs the visible matter of the MW galactic disk and bulge.

Further support for the CHDM theory is provided by the EDGES study of the redshifted 21-cm H I line corresponding to cosmic dawn. The cosmic dawn dark matter constraints given in Barkana's review, and McGaugh's arguments for a 'purely baryonic universe,' fit nicely with the theory presented herein. The most reasonable mechanistic explanation for the intensity and timing of the cosmic dawn signal appears to be the Wouthuysen-Field effect. There appears to be no necessity for an exotic non-baryonic intermediary in the creation of this signal.

Exciting recent theoretical work on the hydrogen snow cloud model also provides some rationale for an abundance of extremely cold hydrogen invisible to direct observation. The new galactic pin scintillation method for indirect observation of 'baryonic dark matter' not reliant upon gravitational phenomena is also exciting. These new ideas and methods bode well for additional inventive studies over the ensuing decade.

The discussion section has focused on providing reasonable cosmic and galactic evolution scenarios for primordial atomic hydrogen which clearly do not require a *pre-existing* structural scaffold of non-baryonic dark matter. Primordial hydrogen created the structure we see. A reasonable CHDM explanation for a correlation between 'cored' galactic dark matter distribution and active galactic centers has also been given. This is consistent with observations reported by Read, et al.

Finally, several predictions having to do with CHDM have been made for the coming decade of observations and simulations.

In conclusion, it is worth asking the following question:

'If interstitial cold atomic hydrogen in its lower ground state is qualitatively and quantitatively sufficient to explain dark matter observations to date, do we really need to spend more of our time and money continuing to look for anything else?'

#### **Author details**

Eugene Terry Tatum Independent Researcher, Bowling Green, KY, USA

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Tatum, E.T. (2019). My C.G.S.I.S.A.H. Theory of Dark Matter. J. Mod. Phys. 10: 881-887. https://doi.org/10.4236/jmp.2019.108058

[2] Tatum, E.T. (2020). Dark Matter as Cold Atomic Hydrogen in its Lower Ground State. In Michael L. Smith (Ed.). *Cosmology 2020: The Current State.* (pp. 93-102). London: IntechOpen. https://dx.doi.org/10.5772/ intechopen.91690

[3] Kalberla, P.M.W. and Kerp, J. (2009). The H I Distribution of the Milky Way. Annu. Rev. Astron. Astrophys. 47:27-61. doi:10.1146/annurev-astro-082708- 101823

[4] Wikipedia contributors. "Hydrogen line." *Wikipedia, The Free Encyclopedia*. Wikipedia, The Free Encyclopedia, 26 Mar. 2021. Web. 26 Mar. 2021.

[5] Mammana DL. (2000). *Interstellar Space*. Popular Science, New York, p. 220.

[6] Chaisson E and McMillan S. (1993). *Astronomy Today*. New York. Prentice Hall, p. 418.

[7] Pananides NA and Arny T. (1979). *Introduction to Astronomy.* Second Edition. Wiley & Sons, New York, p. 293.

[8] Posti L and Helmi A. (2019). Mass and Shape of the Milky Way's Dark Matter Halo with Globular Clusters from *Gaia* and Hubble. A&A. 621: A56. DOI:10.1051/0004-6361/201833355

[9] Watkins LL, et al. (2019). Evidence for an intermediate-mass Milky Way from *Gaia* DR2 Halo Globular Cluster Motions. The Astrophysical Journal. 873:118-130. DOI:10.3847/1538-4357/ab089f

[10] Bowman JD, et al. (2018). An Absorption Profile Centered at 78 Megahertz in the Sky-Averaged

Spectrum. Nature. 555, 67 DOI:10.1038/ nature25792

[11] Barkana R. (2018). Possible Interactions Between Baryons and Dark Matter. arXiv:1803.06698v1 [astro-ph.CO]

[12] McGaugh S.S. (2018). Strong Hydrogen Absorption at Cosmic Dawn: The Signature of a Baryonic Universe. arXiv:1803.02365v1 [astro-ph.CO] DOI:10.3847/2515-5172/aab497

[13] Wikipedia contributors. "Wouthuysen–Field coupling." *Wikipedia, The Free Encyclopedia*. Wikipedia, The Free Encyclopedia, 14 Dec. 2020. Web. 26 Mar. 2021.

[14] Baek, S., et al. (2010). Reionization by UV or X-ray sources. A&A. 523: A4. arXiv:1003.0834. DOI:10.1051/0004- 6361/201014347.

[15] Walker, M.A. and Wardle, M.J. (2019). Cosmic Snow Clouds: Selfgravitating Gas Spheres Manifesting Hydrogen Condensation. ApJ. 881:69. https://doi.org/10.3847/1538- 4357/ab2987

[16] Wang, Y., et al. (2021). ASKAP Observations of Multiple Rapid Scintillators Reveal a Degrees-long Plasma Filament. MNRAS. 502: 3294- 3311. https://doi.org/10.1093/ mnras/stab139

[17] University of Sidney Press Release. (Feb 04, 2021). Student Astronomer Finds Galactic Missing Matter. https://sciencecodex.com/studentastronomer-finds-galactic-missingmatter-666476

[18] Read, J.I., et al. (2019). Dark matter heats up in dwarf galaxies. Mon. Not. Roy. Astro. Soc. 484: 1401-1420. arXiv:1808.06634v2 [astro-ph.CO] https://doi.org/10.1093/mnras/sty3404

#### **Chapter 3**

## Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons

*Firmin Oliveira and Michael L. Smith*

#### **Abstract**

Several recent attempts to observe dark matter with characteristics similar to atomic or subatomic particles as Weakly Interacting Massive Particles (WIMPs) have failed to detect anything real over a wide energy range. Likewise, considerations of large, non-emitting objects as the source of most dark matter fall short of expectations. Here we consider the possibility that massless gravitons suffering slow redshift may be responsible for the properties of spiral galaxies attributed to dark matter. Particles such as gravitons will be extremely difficult to directly detect; the best we can envision is measuring this influence on stellar and galactic motions. Since the motions of stars and galaxies are non-relativistic, we can apply our idea to describe the expected large-scale motions using only Newtonian mechanics. Using our assumption about the importance of the graviton, we here describe the well-known Tully-Fisher relationship of spiral galaxies without resorting to hypothesizing exotic WIMPs or invoking modifications of Newtonian dynamics (MoND).

**Keywords:** dark matter, spiral galaxies, galaxy dynamics, baryonic Tully-fisher relation, Newtonian mechanics, gravitons, BTFR

#### **1. Introduction**

The first hint for the existence of dark matter (DM) between galaxies was the observation over eight decades ago from a study of the luminosities and centrifugal velocities of three galaxies, a galactic "triplet" of the Coma Cluster [1]. Fritz Zwicky noted that the motions of these galaxies about a central point could not be properly described using his presumed values for the galaxy matter densities. He, like all astronomers, estimated matter density assuming that the average density is proportional to the observed luminosity. These three galaxies were behaving as if each contained much more matter than calculated from the luminosities. Zwicky proposed that these three galaxies behaved as if under the influence of a type of plentiful, non-radiating matter, hence DM.

*Stellar motions in spiral galaxies* The existence of significant DM was later required to explain another interesting problem of astronomy—the intragalactic motions of stars within the Andromeda galaxy [2]. For what seems a general rule, the rotational motions of stars about the center of the galactic plane, of spiral-type galaxies, do not follow Kepler's laws of planetary motion. Instead, the more distant stars travel at much greater velocities relative to those of the interior [3]. Distant stars circulate at a more frantic pace than those closer to the center contradicting the planetary laws of Kepler; objects more distant from the center should be traveling slower than those closer. Follow-up observations by Rubin and coworkers of the stellar velocities within many other spiral galaxies confirm this observation as a generality [4]. This is strongly supported by another group in a study of over 1000 galaxies [5]. It still seems uncertain if this grand observation by Rubin also holds for globular and other types of galaxies because observation of individual star motions is more difficult for these galaxy types. Investigators also pointed out that presumed DM explains the rotational velocities of stars in low luminosity galaxies better than for luminous galaxies [5]. The reader should understand that the quandaries of both Zwicky and Rubin depend on the belief that the quantity and distribution of "normal" matter can be correctly estimated from the galactic luminosity.

Proper explanation of these anomalies remains without a satisfactory answer after more than five decades. An answer to this conundrum is considered as utmost importance to both physics and astronomy. Many investigators now call for either new physics or for the presence of considerable DM, of some type, to explain the stellar motions in these galaxies [6]. A solution of Rubin's observations was proposed years ago by utilizing Einstein's General Relativity under the condition of cylindrical geometry [7] but was quickly shown deficient for describing other important properties of spiral galaxies [8]. Another and a more credible explanation of the baryonic-Tully-Fisher relation (BTFR) has been made using Carmeli General Relativity (CGR), which does not require new particle physics nor DM [9].

*Dark matter identity* If DM is the explanation, the exact nature still defies description after all these years. We divide the proposed solutions into three suspects. The first type would be DM as small atomic or subatomic particles, which are often termed **W**eakly **I**nteracting **M**assive **P**article**s** or (WIMPS) [10, 11]. We include exotic particles, as those predicted by weak-scale supersymmetry, in this group [12]. The terms massive, WIMPS refer to particles with energies significantly above the common atomic and subatomic energies. To be effective DM such particles would obviously have to exhibit measurable mass, be non-relativistic and hence *cold* in nature. They would also have to be dark, that is, not interacting with photons. WIMPS must also exhibit elastic collisions with themselves and with baryons (no clumping allowed), and there would have to be a lot of them—about 4–6 times the total mass of ordinary matter. In addition, because WIMPS are noninteractive, cold and nearly frictionless, a novel explanation is required to explain how the velocities were significantly reduced after the Big Bang. This is a long list of special conditions and novel physics.

Many attempts have been made to detect DM, as WIMPS, in terms of particles at near atomic or subatomic energies. We list some of the latest attempts to detect such a particle here. Years of attempting underground detection of entities from intergalactic origin with sensitive probes have been without any success [13]. Attempted detection of particles resulting from WIMP self-annihilation in our sun also ended without positive results [14, 15]. Detection of signals in large Xe-filled detectors over long periods (time) has also not reported positive results [16, 17]. A recent summary of several programs that have failed to detect WIMPS is available [18]. There is one recent event recorded by the Particle and Astrophysical Xenon TPC (PandaX) collaboration, which might be evidence for WIMPS, which obviously needs to be reproduced [19]. In spite of these several failures, there are proposed programs hopeful to deliver positive results. It is questionable if these searches will be funded.

#### *Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons DOI: http://dx.doi.org/10.5772/intechopen.101130*

One proposal will use the planned International Linear Collider, which might be built in Japan, to detect heavy WIMPS up to 3 Tev in size [20]. Another suggestion is to rather concentrate on detecting the lightest neutralino of supersymmetry. According to some people, this hypothesized particle remains the best candidate for dark matter with decent detection prospects [21]. On the other hand, faith in small mass WIMPS around the mass of neutrinos as DM candidates has been ruled out from examination of supernovae observations showing the total neutrino and neutralino mass of the Universe is simply too small to be DM [22]. The best chance for detecting massive WIMPS may come from detecting products from cosmic ray collisions, which are high enough energetically to produce particles with energies from 150 and 500 GeV [23]. But the final argument against WIMPS is the need for special pleading around the existence of WIMPS, which supposedly make up about 80% of matter; about 20% of our Universe. It is *special* that WIMPS are everywhere but not in our solar system, an argument contrary to the Cosmological Principle of Einstein [24]. So we can believe in WIMPS and discard the Cosmological Principle or hold this principle true and discard the notion of WIMPS, but not both.

The second explanation is that matter of the common variety is responsible for the effects of DM, and there is much more common matter than we can detect in spiral galaxies. These dark objects are often referred to as **M**assive **A**strophysical **C**ompact **H**alo **O**bject**s** (MACHOS). This group includes massive objects such as brown dwarfs, Jupiter-sized planets, neutron stars, and "small" black holes [25]. Some investigators also include white and red dwarfs in this group since those inhabiting other galaxies are invisible to our telescopes but would exert gravitational attraction nonetheless [26]. One should also include large clouds of known atomic particles, such as neutral hydrogen and helium. Large clouds of atoms and small molecules may well be undetected in unlit regions of intra- and intergalactic space. Investigations into the MACHO hypothesis have waned the past few years but might revive if data from two planned satellites support the MACHO scenario. There are two programs may uncover data critical to the MACHO hypothesis. The first is the James Webb satellite, rescheduled (again) for launch later this year. This satellite will be able to detect objects emitting light in the near and mid-infrared range, thus allowing a better assay of the population of dark stars and Jupiter-sized planets [27]. Another satellite with a detection range in the visible and near-infrared regions is the Nancy Grace Roman space telescope. This platform will be able to collect data over a much greater volume than any current instrument, be able to detect new Jupiter sized planets and so should also add to our knowledge of MACHOS and hence DM [28]. The planned launch date is sometime in 2025, so it may be collecting data before the James Webb telescope. Finally, it has been pointed out that the methods previously used to calculate the MACHO density have not been optimal, with better observational methods and more satellite data, this explanation of DM may rebound in popularity [29].

One key relationship associated with the presence of luminous matter in galaxies is the Tully-Fisher relation (TFR), which is something like

$$L \propto V\_{\theta}^{5.5-4} \tag{1}$$

where *L* is the apparent luminosity of the stars in the galactic plane, and *V<sup>θ</sup>* is the average rotational velocity of these stars. This holds true even when midinfrared and microwave radiations are assayed [30]. This relationship is sometimes written as

$$M \propto V\_{\theta}^{3.5-4} \tag{2}$$

where *M* is the estimated mass of luminous stars and modeled gas in that galaxy. This empirical finding is thus a broad correlation between the rotational velocities as determined by the Doppler shift and the baryonic matter content of many spiral galaxies [31]. This empirical correlation is sometimes termed the baryonic-Tully-Fisher relation (BTFR).

The third explanation is gravitons. This particle was first proposed by Feynman to modulate the interactions of gravity in analogy to the photon, which modulates interactions between ordinary matter [32]. A graviton is the agent of the gravitational field having zero rest mass traveling at the speed of light in a vacuum. Since a graviton has never been observed, the bulk properties may be like those of axions or even neutralinos [33]. These are all hypothesized "particles" not exhibiting properties of ordinary or DM other than gravitational attraction [34]. The graviton has also been hypothesized as a type of heavy particle outside the standard model (SM) of physics [35]. In general, not much is known about these particle types placing our culture in a situation similar to our understanding of WIMPS. Unlike the frustrating studies of WIMPS, the study of gravitons has been generally overlooked. Here we suggest a simple description of gravitons in action to better explain the BTFR and hopefully stir up more interest in this line of inquiry.

Finally, there are several people who have expressed their concern that the searches for new particles, WIMPS, and other gravity sources have failed and that future investigations are not likely to do any better [36]. In the meantime, WIMPS and MACHOS are suggested solutions over which many investigators lavish time and imagination, some can even directly profit from broadcasting their ideas [37]. Dark matter and dark energy have made their way into YouTube pop-science culture, which helps support some younger investigators [38]. The topic of dark matter has even been unofficially designated as the calendar day: October 31.

To properly model the Zwicky and Rubin observations, people have proposed theories that do not invoke any particle type but rather modify the fundamental equation of gravity. Some look to modified gravity (MoND) as a revolutionary answer to explain problems on scales larger than our solar system—the galactic scale [39]. Some of the problems with MoND have been addressed for public consumption [40]. This topic, however, is outside the interest of this chapter.

#### **2. Theory**

We suggest that gravitons, as agents of the gravitational field, may be modeled as a dense soup of massless particles that nonetheless interact with massive particles. When gravitons travel in a gravitational field, modeled by the equivalence principle as an accelerating system [41], over a short time period *δt* ¼ *δr=c*, where the acceleration *<sup>a</sup>* at a point *<sup>r</sup>* in the field is given by *<sup>a</sup>* ¼ �*GM=r*2, they must experience an energy loss of *δ*Ξ due to motion in that field. We express this differentially as

$$
\delta\Xi = -\left(mc^2\right)\frac{\delta v}{c} = \left(mc\right)a\,\delta t = -\frac{GMm}{r^2}\,\delta r,\tag{3}
$$

where *m* is the total relativistic graviton mass, *δv* is the change in velocity of the system observed from an inertial reference frame, *G* is Newton's gravitational constant, *M* is the baryonic mass of the field source, *r* is the distance between the center of the source and the location of the moving gravitons, *δt* is the short travel time of the gravitons at speed *c* over distance *δr*. The energy change is a loss (negative) because *δv* is in the same direction as the motion of the gravitons, so that for an inertial observer moving with velocity *δv*, the energy of the graviton is redshifted.

*Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons DOI: http://dx.doi.org/10.5772/intechopen.101130*

Integrating Eq. (3) between radial positions *r*<sup>0</sup> and *r*, where *r*<sup>0</sup> ≤*r*, we obtain the total graviton energy change ΔΞ, given by

$$
\Delta\Xi = \int\_{v\_1}^{v\_2} -\left(mc^2\right)\frac{\delta v}{c} = \int\_{r\_0}^r -mc\left(\frac{GM}{r^2}\right)\frac{\delta r}{c} = -GMm\left(\frac{1}{r\_0} - \frac{1}{r}\right).\tag{4}
$$

Eq. (4) describes the gravitational redshift of the graviton energies as these travel from a position *r*<sup>0</sup> of lower (more negative) to a position *r* of higher (less negative) potential energy consistent with energy conservation. For simplicity, we assume here that the mass *M* is a point mass.

Consider the energy equation for a galaxy of baryonic mass *M* interior to position *r*<sup>0</sup> with a small baryonic mass *m* in a circular orbit of radius *r*. The gravitons traversing the distance from the interior mass to the orbiting mass at speed *c* have a decrease in energy ΔΞ given by Eq. (4). The total orbital energy is given by

$$\frac{1}{2}mv^2 - \frac{GMm}{r} + K\_\lg\Delta\Xi = \frac{1}{2}mv^2 - \frac{GMm}{r} - K\_\lg GMm\left(\frac{1}{r\_0} - \frac{1}{r}\right) = E,\tag{5}$$

where *Kg* >0 for *r*≥*r*<sup>0</sup> and *Kg* ¼ 0 for *r*< *r*0. *Kg* is a galaxy dependent dimensionless coefficient, which is necessary to fit the theory with observation, but the actual physical reason for it is not known at this time.

Multiplying Eq. (5) by 2*=m* and moving all terms except *v*<sup>2</sup> to the right-hand side with the total orbital energy *E* ¼ �*GMm=*2*r*, we obtain

$$v^2 = \frac{GM}{r} + 2K\_\xi GM \left(\frac{1}{r\_0} - \frac{1}{r}\right). \tag{6}$$

The properties of a typical spiral galaxy are more easily observed than other types and can be described by the final radial distance *rf* , at which the final rotational velocity is *v <sup>f</sup>* and the orbiting mass is *m*. Substituting *v <sup>f</sup>* for *v* and *rf* for *r* into Eq. (6) and solving for the viral mass *Mvir*, we obtain

$$\mathbf{M}\_{vir} = \frac{v\_f^2 r\_f}{G} = \mathbf{M} \left( 2K\_\mathbf{g} \left( \frac{r\_f}{r\_0} - \mathbf{1} \right) + \mathbf{1} \right),\tag{7}$$

from which we get the apparent dark matter *Md* in the galaxy,

$$\mathbf{M}\_d = \mathbf{M}\_{vir} - \mathbf{M} = 2MK\_{\mathbf{g}} \left( \frac{r\_f}{r\_0} - \mathbf{1} \right). \tag{8}$$

Solving Eq. (7) for *r*0, and simplifying yields,

$$r\_0 = \frac{2\mathcal{K}\_\mathcal{g} r\_f}{\left(\left(v\_f^2 r\_f\right)/\mathcal{G}\mathcal{M}\right) + 2\mathcal{K}\_\mathcal{g} - 1}.\tag{9}$$

These results ideally require virtually all the galaxy baryonic mass *M* to be interior of the radial distance *r*0, with only small test masses beyond *r*0. Additionally, determining the actual mass interior to *r*<sup>0</sup> and the velocity *v <sup>f</sup>* is not an exact science. Thus, proving the validity of Eq. (9) for determining *r*<sup>0</sup> can be difficult.

Here we relate *r*<sup>0</sup> to the baryonic Tully-Fisher relation (BTFR) [10]. At *r* ¼ *r*0, by squaring Eq. (6) and putting it into an inverted form of the BTFR, we get

$$v^4 = MA^{-1} = GM \left(\frac{GM}{r\_0^2}\right) = GMa\_0,\tag{10}$$

where *A*≈50*M*<sup>⊙</sup> *km*�<sup>4</sup> *s* <sup>4</sup> and *M* in solar mass units [42, 43]. For *v* in *m s*�<sup>1</sup> , *M* in *kg* and *<sup>r</sup>*<sup>0</sup> in *<sup>m</sup>*, *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> *GM=r*<sup>2</sup> <sup>0</sup> <sup>¼</sup> ð Þ *GA* �<sup>1</sup> <sup>≈</sup>1*:*<sup>507</sup> � <sup>10</sup>�10*m s*�2, where we again emphasize that we are assuming that *M* is the mass interior to *r*0, which is an approximation. However, the BTFR is also an approximation, albeit a very good approximation from a large amount of galaxy rotation data.

For galaxies in general, we model the mass density function *ρ*ð Þ*r* as spherically symmetric. The mass *M r*ð Þ of the galaxy within the radial distance *r* is given by

$$M(r) = \int\_0^r 4\pi\rho(u)u^2 du. \tag{11}$$

With mass density *ρ*ð Þ*r* , the graviton energy loss Eq. (4) between *r*<sup>0</sup> and *r* is expressed by

$$\Delta\Xi(r) = -\int\_{r\_0}^{r} m \, GM(u) \frac{du}{u^2} = -\int\_{r\_0}^{r} m \, G\left(\int\_0^u 4\pi\rho(s)s^2 ds\right) \frac{du}{u^2}.\tag{12}$$

Using Eqs. (11) and (12), Eq. (6) becomes

$$v^2 = \frac{GM(r)}{r} - \frac{2K\_\mathcal{g}\Delta\Xi(r)}{m} = \frac{G}{r}\int\_0^r 4\pi\rho(s)s^2ds + \int\_{r\_0}^r \frac{2K\_\mathcal{g}G}{u^2} \left(\int\_0^u 4\pi\rho(s)s^2ds\right)du. \tag{13}$$

To obtain *r*0, solve the following equation iteratively,

$$r\_0^2 = \frac{G}{a\_0} \left( \int\_0^{r\_0} 4\pi\rho(s)s^2 ds \right). \tag{14}$$

Solving Eq. (13) for *Kg* yields the equation,

$$K\_{\mathcal{g}} = \frac{v\_{\text{flat}}^2 - \text{GM}/r\_f}{\int\_{r\_0}^{r\_f} \left(\frac{2G}{u^2}\right) \left(\int\_0^u 4\pi \rho(s)s^2 ds\right) du},\tag{15}$$

where *vflat* is the velocity of the flat part of the rotation curve.

#### **3. Results**

We performed nonlinear modeling of the graviton model, Eqs. (13)–(15), with the velocity rotation curve data for spiral galaxies NGC 3198 and the Milky Way. The mass density function *ρ*ð Þ*r* that we use is defined by,

$$\rho(r) = \left(\frac{M}{(4/3)\pi}\right) \sum\_{j=1}^{n} \left( \left(\frac{b\_j}{a\_j^3}\right) \exp\left(-\frac{r}{a\_j}\right) \right),\tag{16}$$

where the total baryonic mass equals *M* over the total radial range of the observational data, *a <sup>j</sup>* are scale lengths, and *b <sup>j</sup>* are weights, where the weights sum to unity. Determinations for the mass *M r*ð Þ ,*r*<sup>1</sup> within the positions *r*<sup>1</sup> and *r*, where 0≤*r*<sup>1</sup> ≤*r*, must be normalized, so the mass is computed from Eq. (16) in the form

$$M(r, r\_1) = \left(\frac{M}{\int\_0^{r\_f} 4\pi\rho(u)u^2 du}\right)\Big|\_{r\_1}^r 4\pi\rho(s)s^2 ds,\tag{17}$$

where *rf* is the largest radial distance of the data: the galaxy *edge*.

*Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons DOI: http://dx.doi.org/10.5772/intechopen.101130*

*NGC 3198* For the analysis of NGC 3198 rotation data, with stellar mass <sup>2</sup>*:*<sup>3</sup> � <sup>10</sup><sup>10</sup> <sup>M</sup><sup>⊙</sup> and mass of gas 0*:*<sup>63</sup> � 1010 <sup>M</sup>⊙, we have a total baryonic mass of *MN*<sup>3198</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>93</sup> � <sup>10</sup><sup>10</sup> <sup>M</sup><sup>⊙</sup> [44]. For the rotation curve data of NGC 3198, we use data from the thesis of Begeman [45]. We set *n* ¼ 4, with the scale lengths *a*<sup>1</sup> ¼ 1*:*1 kpc, *a*<sup>2</sup> ¼ 1*:*27 kpc, *a*<sup>3</sup> ¼ 1*:*35 kpc, *a*<sup>4</sup> ¼ 1*:*5 kpc, and the weights are *b*<sup>1</sup> ¼ 0*:*4, *b*<sup>2</sup> ¼ 0*:*3, *b*<sup>3</sup> ¼ 0*:*2 and *b*<sup>4</sup> ¼ 0*:*1. By solving Eq. (14) we obtain *r*<sup>0</sup> ¼ 4*:*254 kpc, the radial distance where the acceleration equals *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>507</sup> � <sup>10</sup>�10*m s*�<sup>2</sup> . Solving Eq. (15), we obtain a value of *Kg* ¼ 0*:*421. The result using the NGC 3198 data is presented in **Figure 1**. The fit, not necessarily optimal, has a Mean Absolute Error of *MAE* <sup>¼</sup> <sup>4</sup>*:*322*km s*�1, expressed by *MAE* <sup>¼</sup> ð Þ <sup>1</sup>*=<sup>n</sup>* <sup>Σ</sup>*<sup>n</sup> <sup>j</sup>*¼<sup>1</sup>∥*O <sup>j</sup>* � *P <sup>j</sup>*∥, where *O <sup>j</sup>* are the observation values and *P <sup>j</sup>* are the model predictor values. The square of the correlation coefficient between the observed *vobs* and modeled *vmod* rotational velocities is *<sup>r</sup>*<sup>2</sup> <sup>¼</sup> *corr vobs* ð Þ ð Þ , *vmod* <sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*986.

*Milky Way Galaxy* A fit of the graviton model to the velocity rotation curve data for the Milky Way is shown in **Figure 2**. The mass for our galaxy is approximated by the BTFR given by Eq. (10) with a flat rotational velocity of *<sup>v</sup>* <sup>¼</sup> <sup>202</sup>*km s*�1, which yields a mass of *MGalaxy* <sup>¼</sup> <sup>8</sup>*:*<sup>328</sup> � <sup>10</sup>10M⊙. The nonlinear curve fit to the rotation curve data [46] is made with *n* ¼ 6, with the scale lengths given by *a*<sup>1</sup> ¼ 0*:*11 kpc, *a*<sup>2</sup> ¼ 0*:*7 kpc, *a*<sup>3</sup> ¼ 1*:*71 kpc, *a*<sup>4</sup> ¼ 1*:*82 kpc, *a*<sup>5</sup> ¼ 1*:*83 kpc, and *a*<sup>6</sup> ¼ 1*:*84 kpc. The weights used are *b*<sup>1</sup> ¼ 0*:*095, *b*<sup>2</sup> ¼ 0*:*1, *b*<sup>3</sup> ¼ 0*:*2, *b*<sup>4</sup> ¼ 0*:*3, *b*<sup>5</sup> ¼ 0*:*3, and *b*<sup>6</sup> ¼ 0*:*005, where the weights sum to unity. Solving Eq. (14), we find the radial distance where the acceleration to be *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>507</sup> � <sup>10</sup>�<sup>10</sup>*m s*�<sup>2</sup> is *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*:*157 kpc. Then, using this value for *r*0, we solve Eq. (15) to obtain a value of *Kg* ¼ 0*:*418. The fit of the graviton model, not necessarily optimal, has an error of *MAE* <sup>¼</sup> <sup>6</sup>*:*33*km s*�<sup>1</sup> for the fit up to *<sup>r</sup>*<sup>0</sup> and for the entire range of the data up to 1*:*<sup>28</sup> � <sup>10</sup><sup>3</sup> kpc, the error is

**Figure 1.**

*The velocity vs. radial distance from the galactic center of NGC 3198. The data points and errors are from [45]. The solid curve is a good fit to the data with the graviton model, Eqs. (13–15).*

#### **Figure 2.**

*The velocity vs. radial distance from the galactic center of the milky way galaxy. The data points and errors are from [46]. The solid curve is the fit to the galaxy rotation curve with the graviton model, Eqs. (13–15).*

*MAE* <sup>¼</sup> <sup>13</sup>*:*25*km s*�1. For the radial distance up to *<sup>r</sup>*0, the square of the correlation coefficient between the observed and modeled velocities is *<sup>r</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*136. For the full radial distance up to 1*:*28 Mpc, the square of the correlation coefficient between observed and modeled velocities is *<sup>r</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*171.

The results for NGC 3198 and the Milky Way galaxies are summarized in **Table 1**. Our graviton model of the stellar velocity data of these two galaxies exhibits a decreasing rotational velocity for increasing distance from the galactic center, which is a result of the lingering of the Newtonian velocity component in the model, using Eq. (6), expressed proportionally by

$$w(r) \propto \sqrt{\frac{1 - 2K\_{\rm g}}{r} + \frac{2K\_{\rm g}}{r\_0}},\tag{18}$$

for *r*<sup>0</sup> ≤*r*. This effect may mimic the external field effect observed in spiral galaxies as reported by [47]. Use of the graviton model to analyze more galaxies is necessary to quantify this effect.

For comparison, we also have modeled the stellar rotational velocity *vs* distance from the galactic centers as a version of this empirical relationship,

$$V\_{\theta} = \frac{V\_{m} D\_{L}}{K\_{\theta} + D\_{L}} \tag{19}$$

where *V<sup>θ</sup>* is the rotational velocity in the galactic plane, *Vm* is the average maximum velocity, *DL* the observed binned distance from the galactic center, *K<sup>θ</sup>* is some constant, which should not be confused with *Kg* . The general form of this relationship is well known in physical chemistry as the Langmuir equation

**Table 1.**

*Graviton model results from fits of Eq. (13) to the NCG 3198 and milky way full range binned data. Note that r*<sup>2</sup> *is the correlation coefficient corrected for parameter number and not the value for any radius.*

*Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons DOI: http://dx.doi.org/10.5772/intechopen.101130*

**Table 2.**

*Results from nonlinear regression of galaxy binned data with the Langmuir equation. Note that r*<sup>2</sup> *is the correlation coefficient corrected for parameter number and not the value for any radius.*

#### **Figure 3.**

*The rotational velocity (km s*<sup>1</sup>*) vs. radial distance (kpc) from the galactic center of NGC 3198. The solid curve is a good fit to the data with the Langmuir model, Eq. (32). The errors shown here and used for modeling are the standard deviations from the observations [45].*

describing a substrate-saturation curve of a binding site-limited system. Note that we allowed both *Vm* and *K<sup>θ</sup>* to be the two free parameters. We select this model because the data exhibit an initial quick increase in *V<sup>θ</sup>* followed by a long progression of stars with fairly similar *V<sup>θ</sup>* as evidenced by many barred galaxies [4]. We present the results from nonlinear regression modeling Eq. (19) with the NGC 3198 and the Milky Way data in **Table 2** with all three analyses including the position of the galactic center, velocity, 0,0 without error. Note the results of Milky Way (113) including all data and the results of Milky Way (99) where the 14 data pairs closest to the galactic center are not used because the data are very noisy. This model fairly describes the Milky Way data as the values for *r*<sup>2</sup> are nearly 1.

**Figure 3** presents the best fit to the binned NGC 3198 data using the Langmuir model. It appears that the Langmuir equation is a fair model to the NGC 3198 data for stars at and beyond 4 kpc from the galactic center and might be useful to estimate the average rotational velocity of stars in spiral galaxies, which are not too close to the center. Another thing to note is that the rotational velocity of the Milky Way estimated by both the graviton and Langmuir models is similar at about 200 km s1.

#### **4. Conclusions**

We suggest an alternate explanation to the usual for the Zwicky and Rubin observations without resorting to the special pleading required by the WIMP and MACHO scenarios. Though there have been several well-intentioned attempts to observe WIMPS through many different energy ranges, we have not read of any experiment or observation that one might call success. The search continues, however, with a portion of the high energy time of the CERN facility dedicated to such experiments [48]. More astronomical observations confirming or denying, indirectly, DM are on the horizon, involving hundreds of people [49]. We wonder if the results of these will make any difference to the opinions of astronomers since observations to data of distant galaxies yielding results that both confirm and deny DM are continually reported to this day; the opinions of astronomers seem to be data-independent [50, 51].

Our results from modeling the rotational data of spiral galaxy NGC 3198 resulting from graviton physics are in close agreement to observations. Examination of **Figure 1** and the results in **Table 1** reveals our graviton model to be an excellent description of this galaxy. The results of using this model with the Milky Way data, **Figure 2**, are not nearly as good. This is probably due to two different sources. First, the Milky Way data is quite noisy, which is to be expected since astronomers are collecting information that necessarily travels through densely packed stars and gas clouds. Second, the large number of parameters needed in the mass density model to properly use our graviton model makes it very difficult to properly select the best values for modeling. A more robust mass density model is needed, rather than the rudimentary density of Eq. (16), to better match the density in the Milky Way and other complex spiral galaxies.

Our results from modeling the data from these two galaxies using the Langmuir equation are good, as seen in **Figure 3** and **Table 2**. We do not claim this model properly describes the rotational velocities of either galaxy. It may be of use to astronomers for selecting the best average velocity of stars a fair distance from the galactic center.

Although the fits to the Milky Way rotation data were weak for the graviton model, due to the poor density model, this does not diminish its potential. The graviton model is based on the physical principle that gravitons are bosons and thus should behave like photons in a gravitational field. Gravitons traveling away from a massive source should lose energy due to gravitational redshift just as observed for photons. This process takes place for gravitons within galaxies, between galaxies, and within galactic clusters and throughout the Universe. Taking proper account for gravitons can go a long way to explain dark matter and even dark energy and orbital decay of binary stars [41]. This effect use of the graviton model has been overlooked for nearly a century.

*Cosmological Principle* Finally, we draw the readers' attention to the commonly ignored fact that there is no evidence for WIMPS on earth (or in our solar system). The bald fact is that the properties required by DM demand WIMPS should be concentrated at the center of the earth. Few have bothered to investigate this obvious requirement; those who have made some observations have not uncovered evidence for this [52]. Dark matter proponents of WIMPS must resort to special pleading that WIMPS are "out there" but not here. This requirement is simply ignoring the cosmological principle. That is, if WIMPS are common in other galaxies including Andromeda, but not in the Milky Way, then the cosmological principle is not true. If so, astronomy is left without any principles at all.

#### **Conflict of interest**

The authors declare no conflict of interest.

*Dark Matter in Spiral Galaxies as the Gravitational Redshift of Gravitons DOI: http://dx.doi.org/10.5772/intechopen.101130*

#### **Thanks**

We thank K.G. Begeman for kindly allowing us use of his NGC 3198 data.

### **Abbreviations**

#### **Author details**

Firmin Oliveira1† and Michael L. Smith<sup>2</sup> \*†

1 Asian Observatory/James Clerk Maxwell Submillimetre Telescope, Hilo, Hawaii, United States

2 Umeå, Sweden

\*Address all correspondence to<EMAIL_ADDRESS>
†These idea, derivation and modeling were conceived and written by Oliveira, Smith wrote the introduction and conclusions.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 4**

## The Most Probable Cosmic Scale Factor Consistent with the Cosmological Principle, General Relativity and the SMPP

*Arthur N. James*

### **Abstract**

Current literature on the evolution of the cosmic scale factor is dominated by models using a dark sector, these all involve making many conjectures beyond the basic assumption that the Cosmological Principle selects a space–time metric of the Friedmann–Lemaître–Robertson–Walker type through which ordinary Standard Model of Particle Physics matter moves according to General Relativity. In this chapter a different model is made using the same basic assumptions but without making extra conjectures, it depends on following the idea introduced by Boltzmann that when physically meaningful concepts fluctuate the value which will be observed is the one which has the highest probability. This change removes the mathematically incorrect procedure of averaging the matter density before solving Einstein's Equation, the procedure which causes the introduction of many of the conjectures. In the non-uniform era the changes are that the evolution of the scale factor is influenced by the formation of structure and removes the conjecture of having to use two inconsistent probability distributions for matter through space, one to calculate the scale factor and one to represent structure. The new model is consistent from the earliest times through to the present epoch. This new model is open and matches SNe 1a redshift data, an observation which makes it a viable candidate and implies that it should be fully investigated.

**Keywords:** cosmology, gravitation, dark matter, dark energy

#### **1. Introduction**

This first section introduces the motivational background to the study described in this chapter. The study is a response to the difficulties found by an academic physicist trying to upgrade from an amateur cosmologist, who just followed conclusions published in the scientific literature, to a more professional stance by finally studying General Relativity late in life. In 1956 before starting nuclear physics research for a doctoral degree a lifelong interest in the cosmos was triggered by Martin Ryle's course in Radio Astronomy where he described how using Malmquist bias on the C2 Catalogue source counts he could demonstrate that the cosmos was evolving [1].

The teaching of physics in the early 1950s was largely in the style of natural philosophy which means that nature was observed and then modelling of the phenomena was made by searching for some appropriate mathematics. The current position in cosmology is different. For many years through the middle of the twentieth century the simple assumption of a flat space Friedmann–Lemaître–Robertson–Walker (FLRW) metric was used despite the absence of any direct observational evidence supporting this choice. Observations of individual objects in the sky are satisfactorily described using such flat space but there are unobservable consequences such as horizons present in the associated cosmology. Later the mathematical invention of Cosmic Inflation to overcome the horizon problem associated with the flat cosmologies appears to have converted the flatness assumption into an approved folk lore not to be questioned.

Since that time cosmology has included many conjectures required to match the real observed Universe, each of these should carry with it an unknown improbability weighting. Because of the accumulation of such weightings the old fashioned way of choosing between different models describing the same observations would have been to quote Occam's Razor and select the model with the least conjectures so as to improve the odds of being correct.

The study described here is an attempt to use only well authenticated physics and observations in a return to basics and a natural philosophers method for constructing a model for the evolution of the cosmos. This leads to a new model for the cosmological scale factor which is essentially free from additional conjectures.

#### **2. Background**

A hundred years ago Friedmann combined the cosmological principle with Einstein's Equation to predict ways in which a cosmic metric could change with time, his initial model filled the cosmos with a uniform non-relativistic distribution of matter which slowed down any existing expansion of the cosmos. The symmetry described by the cosmological principle enables the modelling of an expanding evolving cosmos with curved space–time because these conditions imply that the background space–time metric must be an FLRW metric. At low matter densities Friedmann's solution has open space sections, as the density increases the solution appears to change smoothly through one special solution with a flat space section into the high density region where the space sections are closed and the Universe collapses back to a point. That description is misleading, in more general situations when other fluids are also present in the cosmos there are three disconnected families of solutions, open, flat and closed. Within each family there are many variations in the way that the cosmic scale factor can change with time depending on the particular mixture of substances filling the cosmos. Against any assumed background metric for the Universe the Universe's content of ordinary matter can be modelled through the formation of structures using conventional physics, see e.g. Peebles' textbook [2].

The present epoch of the Universe is characterised by a non-uniform distribution of matter and a simple Friedmann solution is not possible because Einstein's Equation is non-linear and the procedures of solving and smoothing must be carried out in that order, they do not commute (see e.g. page 452 in Padmanabhan's book [3]). When the wrong order of procedures is used the source distribution does not properly represent the natural distribution so that the solution to Einstein's Equation is that for a non-existing situation, it must be nonsense in the context of representing nature. The Dark Sector cosmologies which are currently widely used incorporate this mathematical misdemeanour, the cosmology becomes the

*The Most Probable Cosmic Scale Factor Consistent with the Cosmological Principle, General… DOI: http://dx.doi.org/10.5772/intechopen.99325*

solution to a problem where its dust component has to be in two places at once, both in galaxy clusters and simultaneously everywhere else, this property of the dust material defies relativity. Being aware of these difficulties suggests questioning all the conjectures which form the essential starting point for Dark Sector models.

The study described here has two themes, one is the blunt rejection of the Dark Energy models, the second theme is a proposal using only well established physical concepts that an open cosmology is highly probable and should be examined further by groups with the appropriate knowledge, skills and computing resources.

In Section 3 the essential physical knowledge and observations which provide a common basis of knowledge for both the new and the Dark Sector models is described, this knowledge is used to make clear those situations where an additional conjecture is necessary to make further progress. All of the science used is well described in many textbooks, the ones quoted here are Padmanahaban's [3] and Peebles' [2]. The notation commonly used in applying such basic knowledge to the formation of cosmological models is introduced by describing the uniform radiation era. Section 4 will introduce the proposed new method using probability density distributions using a simple description of the present epoch, the resulting model is an open cosmology. Section 5 describes how the new method may also be used to describe the whole of the evolution of the cosmic background metric from the radiation era to the present day. Section 6 indicates many of the conjectures which have to be made to construct the Dark Sector models. Section 7 concludes by advocating development of the open model and lists some questions which have to be addressed and answered before continuing to use the Dark Sector models.

#### **3. Basic knowledge common to all models**

The symmetry described by the cosmological principle implies that the cosmos has a background space–time metric of the FLRW type. The three families of these metrics are distinguished by their space sections being either open, closed or flat, the cosmic scale factor of these metrics will be increasing with time to describe the expansion. The 3D curvature of the space of each time slice is determined by the geometry of the FLRW metric, the flat space section family have infinite radius of curvature whereas the open and closed families have a radius of curvature equal to the scale factor, see e.g. Padmanabhan [3]. This mathematical fact is currently being widely ignored in fitting procedures such as that of the Planck collaboration [4] where small deviations from "flatness" are being interpreted as indicating non flat space sections. Such small curvatures would indicate departures from the cosmological principle which is the most essential assumption of the entire conception of modelling background space-times.

Observations and laboratory experiments confirm that the only directly observable substances in the Universe are made from components of the Standard Model of Particle Physics (SMPP). If in addition to the cosmological principle the contents of the universe are also uniform across each time slice then Einstein's Equation of General Relativity can be used to determine the way in which the cosmic scale factor changes with time. There are two types of fluid which can be made from the particles of the SMPP, particles which are moving with relativistic speed make a radiation fluid with significant pressure whereas particles moving slowly make a dust fluid with zero pressure. The equation of state relation between the energy density and pressure influences the rate of change of the cosmic scale factor through Friedmann's Equations which combine General Relativity and FLRW metrics.

The energy density and pressure of particulate fluids are thermodynamic intensive quantities and will be fluctuating, they should correctly be described using

the statistical methods of physics and probability distributions. The convention for pressure and temperature introduced by Boltzmann but now used throughout physics for fluctuating quantities is to use the most probable value as the value to represent overall behaviour with the fluctuations being assumed relative to that value. If electromagnetic signalling is used as an example then two situations must be considered, when many particles are involved the fluctuations are thermal noise, symmetrical around the mode of the distribution which equals the average, but when only very small numbers are involved the fluctuations are shot noise with the mode being zero and with fluctuations in just one direction. Probability distributions where the most probable value is zero will be used in this study when considering non-uniformity such as the matter density in the Universe in the present epoch.

Through the second half of the twentieth century it became apparent that the early universe could be modelled using a radiation fluid uniformly filling an expanding cosmos, the radiation era of the Big-Bang. The cosmic density of light elements predicted by Big-Bang nucleosynthesis verifies both the expansion rate as being that of a radiation era and that the density of SMPP matter is close to the value estimated from observations of astronomical objects. The importance of the Cosmic Microwave Background Radiation (CMBR) to cosmology cannot be exaggerated, it is the most reliable demonstration of the cosmological principle whilst simultaneously justifying the assumption that before its emission at the recombination temperature for atomic Hydrogen the density distribution was uniform. The fluctuations in the CMBR imply that the cosmological principle must be used in a form where the density probability distribution is the same at every spacial point in a particular time slice.

The CMBR is an important boundary in the development of cosmic expansion, before that time uniformity was ensured by radiation mixing throughout the ionised plasma. The emission of the CMBR signals the end of ionisation and the cessation of the mixing, the universe becomes non-uniform. That the universe is non-uniform is obvious in the present epoch, the cosmos is populated by many galaxies in complex structures, the way in which these can form from the minute fluctuations in the CMBR can be explained using ordinary physics [2].

Towards the end of the uniform era the dominating fluid of SMPP particles filling the universe changes in character from the earliest relativistic radiation fluid to a non-relativistic dust fluid with zero pressure. Using the normal Friedmann description the equation showing the relationship between Hubble's constant *H*(*t*) and the cosmic scale factor *a*(*t*) adjusted so that these take the values *H*o and *a*o(*t*) = 1 at the present time is

$$H^2 = H\_\text{o}^{-2} \left[ \Omega\_\text{R} a^{-4} + \Omega\_\text{gM} a^{-3} - kc^2 \left( H\_\text{o} a \right)^{-2} \right] \tag{1}$$

The ΩR and ΩBM are cosmic densities of radiation and baryonic matter normalised in the usual way to that of the flat Friedmann matter only cosmological model. The curvature term where *k* equals −1, 0 or + 1 has a different character, it is part of the FLRW geometry and represents the relation between the absolute value of the cosmic scale factor and the spacial curvature of the particular chosen geometry, it has nothing to do with Einstein's Equation but stems only from the symmetry of the Cosmological Principle [3].

At early times in the expanding universe when the cosmic scale factor is smallest the radiation density ΩR term dominates but when the baryonic matter density ΩBM deduced from the Big-Bang nucleosynthesis is used the non-relativistic matter term will have just become relevant at the emission of the CMBR and a simple Friedmann matter only model suggests an open cosmos. The curvature term will always be insignificant during the uniform era which is therefore insensitive to its value and whether *k* is −1, 0 or + 1 can not be deduced from observations of the uniform era.

#### **4. The present epoch and the open model**

Observations of the local environment have shown that it is characterised by an array of galaxies arranged in structures which are separating according to the Hubble flow. At each point in space Einstein's Equation ensures that it is only the local density of matter which will control the rate of change of volume of a small element of space around that point, the local matter density changes the local scale factor. Remote massive objects will distort the space element as they approach but will return it to its original state after they have passed, not affecting the local scale factor. The matter density probability distribution at each point can be estimated by considering the density distribution over space, it is peaked strongly at zero. The vast majority of points have zero density which means that the local scale factor will be expanding progressively faster than anywhere where the matter density is positive. It is this mechanism where dense regions expand more slowly than empty regions which is essential to trigger the formation of structure. However the main consequence in the context of the local cosmology is that emptiness is becoming ever more frequent. Using a maximum probability algorithm that the best estimate of the cosmic scale factor must be the most frequent local scale factor sets the cosmic scale factor as that for emptiness.

The FLRW metric for empty space is Friedmann's well known empty universe solution, a useful demonstration of this is given by Vishwakarma [5], he also shows that this metric and a Dark Sector model metric have equally good fits to SNe 1a redshift data. The empty universe solution has a metric like that of the cosmology proposed by Milne [6] in 1935, open and expanding with its cosmic scale factor being the product of the velocity of light and the age of the universe. The galaxies are in free fall and simply drift apart making the Hubble flow. At any time Hubble's constant is the inverse of the age of the universe and its present value is the only parameter of this cosmology. The obvious conclusion must be that Milne's metric, a good solution of Einstein's Equation of General Relativity and the Cosmological Principle, is the best metric with which to approximately model the present day epoch.

In this argument for the present epoch it is assumed that Einstein's Equations have been exactly solved by Nature before smoothing the solution is attempted, it is then completely free from the commutation misdemeanour inherent in every dark energy model. The resulting model, Milne's metric, is just a normal FLRW model of a smooth universe neglecting blemishes such as the contamination by many massive galaxies, a situation which also occurs for all other models. Einstein's Equation is fully respected in this open model, both in the scale factor solution and in the next weak curvature approximation which has to be made to describe the interrelated motions of all the massive objects through the use of Newton's Laws of Motion.

#### **5. The open model from the radiation era to the present day**

If the mode of the density probability distribution rather than the average is used for the uniform era then modelling the scale factor using Eq. 1 is unaffected because the uniformity ensures only a narrow distribution of densities and the

mode of the density distribution will be almost identical to the mean in this case. After the emission of the CMBR the density distribution becomes non-uniform and changing with time. Using the mode of the density distribution for the non-uniform era produces a model where the cosmic scale factor responds to the formation of structure by modifying the Friedmann equation as time evolves.

The model is generated by imagining a stepwise numerical time integration technique where the cosmic scale factor change through a step will be calculated using Friedmann's Equation with the density mode from the start of the step while the changes to the density distribution function during the step are calculated non relativistically in the usual way. At the end of the step the density mode will be different so the Friedmann's Equation which should be used in the next step to compute the scale factor will be different. In this way the initial conditions for each step match the solution and the nearness of the approximation to a continuous integration with smoothing following solution will increase as the step size decreases. An idiotically simple example of using the wrong order of procedures in a non-linear problem and then applying such an integration procedure to evade its effects is given as an appendix.

In this stepwise process the Friedmann equation to be used is shown below including explicitly the curvature term for an open cosmology (where *Ha = c*)

$$H^2 = H\_o^{\
u} \left[ \Omega\_{\mathbb{R}} a^{-4} + \Omega\_{\mathbb{BMC}} \left( t \right) a^{-3} + c^2 \left( H\_o a \right)^{-2} \right] \tag{2}$$

The term ΩBMC(*t*) represents that Baryonic Matter Component which affects the cosmic scale factor at each time, this term changes from its full value ΩBM at the CMBR time to zero for the present day the details depending on how the mode of the matter distribution function evolves with time. Because ΩR and ΩBMC(*t*) are small for the present epoch the curvature term *c* 2 (*H*o*a*)−2 for the Milne metric dominates. In any fitting procedure for the CMBR and structure formation the only parameters will be the properties of the CMBR fluctuation distribution, the Hubble constant *H*o and the baryonic matter content ΩBM, all parameters having clear physical meaning in relation to the SMPP and General Relativity and related to observations.

If the early value for Hubble's constant predicted from the CMBR does not match the late value determined by the SNe 1a data then the value for the matter term required to fit the data may be larger than ΩBM, perhaps dark matter such as that suspected from studies of the Bullet cluster [7] will have to be included. This dark matter must behave in the same way as SMPP matter according to the rules of General Relativity and should not be considered in any way as being similar to the cosmic dust component of Dark Energy models.

A qualitative description of what happens as the Cosmic Baryonic Matter term reduces to zero is simply that isolated discrete lumps of matter are unable to influence the scale factor of the whole of a time slice at once, it should be obvious that such an influence violates relativity principles. As stars and galaxies form they remove the matter from its cosmic role leaving the smaller curvature term to finally take over control of the expansion.

#### **6. The problems of the concordance model**

The simplest Dark Energy models introduce four conjectures into their model for the cosmic scale factor, flat space sections, cosmological Dark Matter (DM),

*The Most Probable Cosmic Scale Factor Consistent with the Cosmological Principle, General… DOI: http://dx.doi.org/10.5772/intechopen.99325*

Dark Energy (Λ) and weird properties for the dust component. The Friedmann equation on which all these models are based is shown here with a zero *a*−2 curvature term because of the flatness assumption

$$H^{2} = H\_{\text{o}}^{\;\;\!\;\!} \left[ \Omega\_{\text{R}} a^{-4} + \left( \Omega\_{\text{BM}} + \Omega\_{\text{DM}} \right)\_{\text{WD}} a^{-3} + \Omega\_{\text{A}} \right] \tag{3}$$

In this equation both matter terms, baryonic ΩBM and dark ΩDM, have been bracketed together because the physical properties for this Weird Dust (WD) are very strange. The Weird Dust appears to have two different density distributions across a time slice, the modeller chooses to use one or the other depending on context, a smooth distribution to compute the cosmic scale factor evolution but then a non-uniform one to compute structure formation. This inconsistency within the model makes it bad science due to the illogicality where the premise of uniformity for the matter content does not match the outcome of the model calculations which predict the destruction of uniformity.

Another way of describing the situation is that in addition to the normal physical properties of an isolated concentrated lump of dust these models conjecture an extra weird property for that isolated concentrated lump of dust, that of being able to instantly affect the universe's cosmic scale factor uniformly throughout the universe. Such behaviour is not allowed by General Relativity which respects the restrictions of a finite velocity of light.

Each of the conjectured Dark components comes with a quantity of substance and an equation of state all of which are artefacts of the model. These two quantities and two functions provide enough flexibility in the fitting procedures for the solution to respond to the attraction towards flatness imposed on the models by using a zero cosmic curvature term in the Friedmann equation. The quantity 'Ωk' = (1 - ΩM - ΩΛ) which is seen for example in the fitting procedures for Planck data [4] cannot represent curvature in a FLRW metric, its value is generally found to be near zero which must simply be a measure of the accuracy within which the fit has approached flatness.

One more conjecture is made in Dark Energy models because the horizon introduced by the flatness means that Inflation must also be conjectured to ensure the cosmological principle is present through both the uniform and non-uniform eras.

#### **7. Summary and conclusion**

A new technique to find an approximate solution to Einstein's Equation for the cosmic scale factor in a non uniform universe has been found by going back to basic physics, SMPP plus General Relativity, and following Boltzmann's use of maximum probability concepts in physics. This technique causes the cosmic scale factor to be affected by the formation of structures in the universe. The resulting open cosmological model is very different from conventional cosmologies, its simplest predictions are that evolution from the early radiation era to the present epoch produces an empty cosmos with a small contamination of massive galaxies drifting apart in accordance with the observed SN1a redshift data. There appears to be no obvious contradictory observational data to this new cosmological model. This open cosmology has a lower density than flat models, the expansion through the CMBR epoch and the structure forming era will be slower giving extra time for the formation of early astrophysical objects. Detailed structure calculations to see how this open model fits the CMBR and the Baryonic Acoustic Oscillation data are required

to validate it further, computations which can be justified by the simplicity of all the physical concepts required to establish it.

The empty universe's expanding cosmic scale factor can only be modelled by an open cosmology implying that the cosmos has always been open and causally connected. A causally connected model of the universe does not require inflation to establish the uniformity of the cosmological principle. Such a model implies little about the most primordial universe, it must be open, contain the SMPP and a radiation spectrum to match the details of the CMBR.

Examination of Dark Energy models of the cosmos suggests several questions about the conjectures used in the models which should be answered successfully before proceeding to further investigations. The main objection to the Dark Energy models must be the mathematical commutation of procedures misdemeanour, that this is a problem has not gone unacknowledged but the problem has not been confronted directly, it has only been circumvented by additional conjectures which do not eliminate the misdemeanour (see [8] and references therein). Apart from this misdemeanour but perhaps in consequence of the misdemeanour a list of questions requiring answers are:-.

#### **Appendix on solving and smoothing in non-linear systems**

The Dark Energy models for the cosmic scale factor average the source term in Einstein's Field Equations before solving, but that problem is non-linear meaning that a mathematical misdemeanour has been included right at the beginning of the modelling, the wrong initial conditions have been used to solve the problem and the answer must be wrong. A peculiar feature of this problem is that the equations are correct and nature provides the correct solution, a consequence is that the Dark Energy models introduce artefacts to correct the wrong solution towards the correct solution, these are Dark Matter and Dark Energy. The following idiotically simple problem is given here to illustrate how this has been done.

Consider a problem where the answer is known from other considerations to be 1/3, the question is:-.

"What is the average value of *y* where *y* = *x*<sup>2</sup> over the range -1 < *x* < +1?"

Averaging then solving gives the answer as zero, wrong because solving and averaging are performed in the wrong order. The initial conditions have been altered from the correct situation to an incorrect one. It is possible to adjust this answer by conjecturing an arbitrary parameter which can be adjusted to match the known answer. This arbitrary parameter will be adjusted to the value 1/3, this parameter is an artefact of the method used to solve the problem but has no real meaning.

#### *The Most Probable Cosmic Scale Factor Consistent with the Cosmological Principle, General… DOI: http://dx.doi.org/10.5772/intechopen.99325*

Now split the range up into small segments, averaging first for each segment then solving, a final averaging of all the intermediate steps gives an answer quite close to the previously known correct value. By responding to the changing situation as the value of *x* increases the error has been vastly reduced and no arbitrary constant is required. The error will depend linearly on the step size enabling its size and effect to be detected.

In modelling the development of structure in cosmology the solution is too complicated for normal integration but essentially Dark Energy models do the whole problem without acknowledging the changing situation. The models have to introduce artefacts such as extra dust and Dark Energy, with arbitrary constants and functions, in order to fit observations. The new method using the mode of the matter probability distribution leading to the open cosmology will have reduced the effect of the mathematical misdemeanour by responding to the changing situation, the smaller the step size the smaller the error. No extra artefacts will have to be introduced.

#### **Author details**

Arthur N. James Department of Physics, University of Liverpool, UK

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Ryle, M. The Observatory. 1955; **75**: 137

[2] Peebles P. J. E., Principles of Physical Cosmology. Chichester: Princeton University Press; 1993.

[3] Padmanabhan T., Gravitation, Foundations and Frontiers. Cambridge: Cambridge University Press; 2010.

[4] Ade P. A. R. et al, Planck 2015 results, XIII. Cosmological parameters. Astron. Astrophys, 2016; **594 A13**

[5] Vishwakarma R. G., Mysteries of the Geometrization of Gravitation. Res. Astron. Astrophys. 2013; **13:** 1409-1422

[6] Milne E. A., Relativity, gravitation and world-structure. Oxford: Clarendon Press; 1935

[7] *Clowe D., Gonzalez A. and Markevich M., Weak-Lensing Mass Reconstruction of the Interacting Cluster 1E 0657-558: Direct Evidence for the Existence of Dark Matter. Astrophys. J. 2004;* **604** *(2): 596-603*

[8] Buchert T., Mourier P. and Roy X., On average properties of inhomogeneous fluids in general relativity III: general fluid cosmologies. Gen. Rel. Grav; 2020: **52** 27

#### **Chapter 5**

## Black Holes as Possible Dark Matter

*Aloke Kumar Sinha*

#### **Abstract**

Black holes and Dark matter are two fascinating things that are known very little. They may have non gravitational interactions, but those are definitely extremely feeble in comparison to their gravitational interactions. Nowadays some people think that one may contain the other. In this chapter we will see that some black holes may contain the dark matter. These black holes decay under Hawking radiation, but do not vanish completely. They produce stable end states due to both quantum gravitational effects and thermodynamic reasons. These end states are the replicas of what we call dark matter. We will develop the complete theory for decay of such black holes, starting from some scheme independent assumptions for the quantum mechanical nature of the black holes. We will then consider explicit examples of some black holes to show that they indeed produce replicas of dark matter at their end states. Thus this chapter is going to be a manuscript for theoretical development of black hole decay from a quantum mechanical perspective and its consequences for producing replicas of dark matter.

 Introduction**

Einstein had first shown, with the help of his classical field equations of general theory of relativity, that black holes accreted everything surrounding them [1, 2]. Hence they are expected to grow in size in an unbounded manner. His theory was entirely classical. But Hawking later invoked quantum mechanics in the context of black hole [3], to study its interaction with matters surrounding it. He proved explicitly that black holes could radiate and as a consequence they decayed away.

Hawking considered only matters as quantum entities, but spacetime was still classical in his theory. Hence in his theory, black holes were still classical. Thus this theory was semi classical as matters were treated differently in comparison to black holes. We had resolved this issue in our earlier works [4, 5]. Semiclassical analysis claimed the thermal instability of asymptotically flat, non extremal black holes under Hawking radiation. They are unstable as their specific heat is negative [6, 7] and have been deduced from semiclassical facts based on their classical metric. Their temperature increases as they lose mass, indicating a complete thermal run away process. It is to be noted that semiclassical analysis explicitly depends on the classical metric of a black hole. Hence it is inherently a 'case-by-case' analysis. This shortcoming implies

that such semiclassical analysis cannot give general results about the thermal stability of generic black holes under Hawking radiation. Semiclassical analysis predicted the thermal instability of asymptotically flat black holes from the negativity of their specific heat, defined semi classically from their metric. But this result does not say anything in general about an arbitrary black hole. It is of course true that gravity is yet to be quantized fully. But we realistically expect certain symmetries for that theory [4]. These symmetries are sufficient for us to construct the grand canonical partition function of a generic black hole, if we assume the black hole to be in contact with the rest of the universe, that acts as a heat bath. We derived the criteria for thermal stability of a generic black hole with arbitrary number of parameters in any dimensional spacetime, based on the convergence of the grand canonical partition function [5]. These criteria appeared as a series of inequalities, connecting second order derivatives of black hole mass with respect to its parameters.

These criteria imply that AdS black holes with fixed cosmological constant are stable under Hawking radiation for a certain range of their parameters [4]. We have also noticed that asymptotically flat rotating charged black holes satisfy some of the stability criteria, but not all together, in certain regimes of spacetime [4, 5]. Thus although these black holes decay, they are different from unstable black holes, like asymptotically flat Schwarzschild black holes. These black holes are named as "Quasi Stable" black holes. We will later see that AdS black holes with varying cosmological constant are also quasi stable under Hawking radiation.

We had calculated the fluctuations for the parameters of a stable black hole and they were expectedly turned out to be very small [8]. These tiny fluctuations are actually the indications of the stability for a black hole. We did the same for quasi stable black holes and it resulted in tiny fluctuations for some parameters [9], like stable black holes, in a certain regime of parameter space. This is as quasi stable black holes satisfy some of the stability criteria. This makes them slow down their decay rate in certain regimes of their parameter space [9].

Black holes, like ordinary thermodynamic systems, also have different phases. Stable and unstable black holes respectively possess stable and unstable phases in possible allowed regimes of their parameter spaces. The respective examples are AdS black holes with fixed cosmological constant and asymptotically flat Schwarzschild black holes. Unstable black holes remain in the same phase during their decay. Stable black holes likewise stay in a stable phase, maintaining equilibrium with their surroundings and hence they do not decay under Hawking radiation. But things are changed entirely for quasi stable black holes. We had already shown that quasi stable black holes also have various different phases. The quasi stable black holes undergo phase transitions among these phases during their decay process. The nature of fluctuations change from one phase to another phase. In this way quasi stable black holes decay under Hawking radiation. But at the end states, most of these black holes become tiny balls of the order of Planck size.  On the other hand, some other parameters of certain quasi stable black holes settle down to their macroscopic values at the end states. Thus we see that these black holes become thermodynamically stable, preventing further decay under Hawking radiation. Hence they stop interacting with the rest of the universe, except gravitational interaction. Thus these black holes seem to behave like dark matter (the way we call it). In fact some of these black holes may have electric charge as well. Hence it may correspond to charged dark matter. But some unknown mechanism must be there to prevent it from interacting with the universe through known electrical interaction.

This chapter is organized as follows: A detailed discussion on thermal stability of black holes is done in Section 2. In the next section, we have discussed quasi stability and phase transitions of quasi stable black holes. In the following section, we have

considered some examples of quasi stable black holes and have discussed their quasi stability and hence the possible connection with dark matter. We finished in the next chapter with a special note.

#### **2.  These four quantities are related by a relation on the horizon. Thus these parameters are expected to be promoted as operators if black hole can be treated as a quantum system. Three out of these four parameters are independent and the remaining one depends on the other three. It is certainly not possible to have charged rotating black hole without any mass and horizon area. Thus *Q*^ and ^*J* have to play the status of primary operators i.e. role of fundamental observables. We choose *A*^, *Q*^ ,^*J* � � to be the primary operators and *M*b to be the secondary operator. Hence *M*b as an operator becomes *<sup>M</sup>*<sup>b</sup> <sup>¼</sup> *<sup>M</sup>*<sup>b</sup> *<sup>A</sup>*^, *<sup>Q</sup>*^ ,^*<sup>J</sup>* � �. Now horizon area, like electric charge, is invariant under SO(3) rotations beside its invariance under U(1) gauge transformation. *SO*ð Þ3

generates angular momentum while global gauge group *U*ð Þ1 generates electric charge. These give the following commutation relations,