Isomeric Dark Matter (IDM) and Cosmic Data That IDM Can Help Explain

1. Introduction

This unit (of this paper) places our work in a context of ninety years of cosmology and elementary-particle research.

Since the 1930s, people have discussed cosmological phenomena [1,2] that might not associate with visible matter. People use the two-word term ordinary matter to describe such visible matter. People use the term dark matter when discussing some possible explanations for the phenomena. People have yet to settle on a preferred description of dark matter [3,4]. Some proposed descriptions base dark matter on yet-to-be-found elementary particles. Some proposed descriptions feature copies or near-copies of standard model matter [5,6].

Our work posits that nature includes six so-called isomers, or near-copies, of a set of elementary particles. (An Appendix A, in this paper, regarding uses of the word isomer compares our use of the word isomer with other uses of the word isomer.) One isomer underlies ordinary matter. Five isomers underlie most or all dark matter. We embrace a notion of a set of IDM (as in isomeric dark matter) candidate specifications for dark matter. The notion that a set of IDM candidate specifications for dark matter elementary particles can have more than one member leaves room for helping to explain known data that we do not address or helping to explain future data. Explaining other known data or future data can lead to narrowing the set of IDM candidate specifications.

This paper motivates, discusses, and applies one IDM candidate specification for dark-matter elementary particles. The paper works, in steps, toward that specification. Our work posits that the six isomers, one ordinary-matter and five dark-matter, have similarities. For example, counterpart elementary particles have identical masses. Our work posits that the isomers have differences. For example, three isomers associate with left-handed elementary particles, and three isomers associate with right-handed elementary particles. Our work indicates differences between the stuff that the isomers underlie. For example, the stuff that associates with one dark-matter isomer includes atom-like objects and is more like ordinary-matter stuff than is the stuff that associates with the other four dark-matter isomers. The stuff that associates with those other four dark-matter isomers does not include significant amounts of atom-like stuff.

Our steps toward the IDM candidate specification point to needs to find explanations for some gravitational phenomena, including effects whereby, for each one of some pairs of objects, the two objects repel each other. Measurements that point to increases in the so-called rate of expansion of the universe point to the possibility of gravitational repulsion. Popular physics modeling includes theories of gravity that include gravitational repulsion [6,7]. Regarding explaining repulsion via so-called dark energy, popular modeling discusses reasons to find alternatives to invoking the cosmological constant [8]. This paper develops and uses multipole-expansion methods to develop notions of such two-object gravitational repulsion. These multipole-expansion methods have bases in Newtonian gravity, the motions of gravitationally interacting sub-objects within gravitationally interacting objects, and Lorentz invariance. Such two-object gravitational repulsion helps explain some dark-matter data as well as eras in the rate of expansion of the universe.

This paper indicates that our IDM candidate specification for dark matter can help explain data for which we are not aware of other explanations. Such data feature ratios of dark-matter presence to ordinary-matter presence and pertain to the observable universe, individual galaxy clusters, and individual galaxies. For such data about the observable-universe, the four-word term densities of the universe pertains [9].

2. Methods

2.1. Data for Which We Seek Underlying Explanations

This unit discusses data that motivate our work, that inform our work, and for which our work seeks to help provide underlying explanations.

The following items discuss observed ratios of not-ordinary-matter effects to ordinary-matter effects. We seek to help provide quantitative underlying explanations.

• 1:0+ – Amounts of stuff in some individual galaxies [10,11,12,13,14,15,16,17,18,19].
• 0+:1 – Amounts of stuff in some individual galaxies. (Popular modeling associates the symbol z with redshift. Popular modeling associates redshifts of zero with the present universe. Popular modeling associates larger redshifts with earlier times in the history of the universe.)

  −

  Redshifts of more than approximately seven [20,21].

  −

  Redshifts of approximately six [22].

  −

  Redshifts of less than six through redshifts of nearly zero [23,24,25,26,27,28,29,30].

• $\sim 4$:1 – Amounts of stuff in some individual galaxies [31,32].
• 5+:1 – Amounts of stuff in many individual galaxies [10,33].
• 5+:1 – Amounts of stuff in many individual galaxy clusters [33,34,35,36,37].
• 5+:1 – Densities of the universe [9].
• $\approx 0$:1 – Amounts of stuff in observed or optically observable solar systems.

  −

  Observations about our solar system [38].

  −

  Papers about other solar systems tend to feature possible methods for future observations and seem to imply that, presently, $\approx 0$:1 pertains regarding known solar systems other than our own solar system [39].

• 1:1 or 0:1 – Amounts of some depletion of cosmic microwave background radiation. (Ordinary-matter effects that associate with the depletion of cosmic microwave background radiation via hyperfine transitions in ordinary-matter hydrogen atoms might account for half of the observed depletion or for all the observed depletion. The case of half associates with 1:1. The case of all associates with 0:1.)

  −

  An observation [40,41,42] suggests 1:1.

  −

  Other pieces of research [43,44,45,46] suggest 0:1.

The following items summarize some observations about large-scale aspects of the universe. We seek to help provide qualitative insight that can underlie explanations for these data.

• Popular modeling suggests two observed multibillion-year eras regarding the so-called rate of expansion of the universe [47,48,49,50]. Chronologically, the first multibillion-year era associates with a positive rate of expansion that decreases as time increases. The second multibillion-year era associates with a positive rate of expansion that increases as time increases.
• Data and popular modeling might provide hints that the second multibillion-year era might be ending [51,52] and that a new era, which would associate with a positive rate of expansion that decreases as time increases, might be starting.

2.2. Six Isomers, of Which One Isomer Is a Set of Known Elementary Particles

This unit discusses aspects of our candidate description for dark matter.

We note that 5+:1 ratios associate with large-scale phenomena.

We posit that dark matter underlies the not-ordinary-matter effects.

We posit that nature includes six near-copies of a set of elementary particles and that one copy includes the set of all known fermion elementary particles. We posit that, across the six near-copies, counterpart particles have identical masses. These posits could underlie explanations for $\sim 5$:1 ratios that might associate with the known 5+:1 ratios.

We use the word isomer to associate with the notion of near-copy. Our use of the word isomer comports with typical use, in science, of the word isomer in situations in which science discusses the relevance of exact symmetries or approximate symmetries. We use the word isomer to denote each one of the six sets.

We use the word stuff to associate with occurrences of elementary particles, atomic nuclei, atoms, and so forth that exist in nature.

As an aside, we note that, because stuff that associates with one isomer interacts gravitationally with stuff that associates with other isomers, the possible graviton elementary particle would not be a member of the relevant set of elementary particles. At this point in our discussion, we leave open the question of the extent to which one might include elementary bosons in the relevant set of known elementary particles.

The number six, as in six sets, factors into a factor of two and a factor of three. We posit that each one of the two and the three associates with a symmetry or approximate symmetry.

We posit that the factor of two associates with at least one (and possibly all) of the following notions.

• Popular modeling notions that associate the one-element term left-handed with the set of known elementary particles [53,54]. (For this notion, at least one dark-matter isomer associates with right-handed counterpart-to-ordinary-matter elementary particles.)
• The popular modeling notion of matter-antimatter asymmetry (which is also known as baryon asymmetry) [55]. (For this notion, one dark-matter isomer underlies stuff that enables popular modeling to consider, in an adequately broad context, notions of matter-antimatter symmetry.)
• The 1:1 ratio that possibly pertains regarding some depletion of cosmic microwave background radiation. (For this notion, the stuff that associates with one dark-matter isomer includes hydrogen-like atoms that account for one-half of the relevant depletion of cosmic microwave background radiation.)

The appendix that discusses a possible symmetry regarding the factor of two discusses notions and details regarding this possible symmetry. We do not try to fully characterize the would-be symmetry or approximate symmetry. Regardless of the details, we use the term left-handed to describe the ordinary-matter isomer and we use the term right-handed to describe the dark-matter isomer that would be most like the ordinary-matter isomer.

We posit that the factor of three associates with at least one (and possibly all) of the following notions.

• Popular modeling notions of neutrino oscillations [56,57]. (Regarding this notion, popular modeling states that neutrino flavour-eigenstates to not fully align with neutrino mass-eigenstates.)
• Aspects of a formula that interrelates the masses of the three charged leptons (the electron, the muon, and the tau) and the geometric-mean masses for each of the three generations of quarks (Table 3.9.10 in [58] or Table 14 in [59]).
• A might-be approximate symmetry that would associate with matches between charged-lepton flavours and charged-lepton masses.

The appendix that discusses the possible approximate symmetry regarding the factor of three discusses notions and details regarding this possible approximate symmetry. We do not try to fully characterize the possible approximate symmetry.

We use the one-element term isomer-pair to describe each one of the three pairs of isomers for which the two isomers do not differ with respect to the might-be symmetry or approximate symmetry that associates with the factor of three.

We use the acronym SEA (as in significantly-electromagnetically-active) to describe aspects of the stuff that associates with the ordinary-matter isomer and aspects of the stuff that associates with the dark-matter isomer that associates with the isomer-pair that underlies ordinary-matter stuff.

We use the acronym MEA (as in marginally-electromagnetically-active) to describe aspects of the stuff that associates with the other four dark-matter isomers. (The appendix regarding the evolution of MEA dark-matter stuff proposes means by which the stuff that associates with each one of the four MEA isomers evolves to feature neutron-like analogs and proposes that the neutron-like analogs do not decay significantly into proton-analogs and electron-analogs.)

Table 1 discusses a numbering scheme for the posited six isomers of all known elementary fermions, specifications for the one ordinary-matter isomer and the five dark-matter isomers, and aspects of the stuff that associates with each isomer.

The stuff that associates with each dark matter isomer comports with the popular modeling acronym CDM (as in cold dark matter). The stuff that associates with the SEA dark-isomer comports with the popular modeling acronym SIDM (as in self-interacting dark matter).

We note, as an aside, the following. Some observational results [60,61,62] suggest that some dark matter might comport with popular modeling notions of self-interacting dark matter [3,63]. Some popular modeling results [64,65,66,67] point to possible benefits of considering that some dark matter is self-interacting dark matter.

Table 1 associates with a rather-well specified candidate description of dark-matter elementary fermions. Considering the notion of a multi-member set of candidate descriptions, we suggest that people might consider similar, but different, candidate notions regarding dark-matter elementary fermions. For example, except regarding the ordinary-matter isomer, the orderings of quark generations or the orderings of lepton flavors might vary from the respective orderings that Table 1 shows.

2.3. Objects, Interactions Between Objects, and Isomeric Reaches of Interactions

This unit posits values for integers that help describe the extent to which dark matter and ordinary matter interact with each other.

People and ordinary-matter equipment do not sense light that dark matter might emit. Table 1 indicates that stuff that associates with the DM (SEA) isomer is as electromagnetically interactive as is ordinary-matter stuff. (We note, as an aside, that DM (SEA) stuff might include atom-like objects, star-like objects, and solar-system like objects.) We posit that, at least to a first approximation, nature includes six instances of electromagnetism. We posit that each isomer associates with its own instance of electromagnetism. We say that, for each one of the six instances of electromagnetism, the interaction reach per instance is one isomer.

Popular modeling suggests that each nonzero-mass object can interact gravitationally with all other nonzero-mass objects. We posit that, at least to a first approximation, nature includes one instance of gravitation. We say that the reach per instance for some aspects of gravitation is six isomers.

We use the symbol $n_{in}$ to denote the number of instances of a type of interaction. (One type of interaction is so-called two-body gravitational monopole pull interactions.) We use the symbol $R_{/in}$ to denote the reach of an instance of the type of interaction. The reach is a number of isomers. Each one of $n_{in}$ and $R_{/in}$ is a positive integer.

We use the symbol $n_{in}$ to denote the number of instances of a property that objects exhibit. (Examples of properties include charge and blackbody temperature.) For example, blackbody temperature is a property that people and equipment observe regarding stars. We use the symbol $R_{/in}$ to denote the reach of an instance of the property. The reach is a number of isomers. Each one of $n_{in}$ and $R_{/in}$ is a positive integer.

We posit that, for each relevant aspect of electromagnetic interactions, for each relevant aspect of gravitational interactions, for each electromagnetic property, and for each gravitational property, Eq. (1) pertains.

$$n_{in}·R_{/in}=6$$

(1)

Eq. (1) associates with four potentially relevant solutions. Each solution associates with one of $R_{/in}=1$, $R_{/in}=2$, $R_{/in}=3$, or $R_{/in}=6$.

For a solution for which $n_{in}=3$ and $R_{/in}=2$, we posit that each one of the three instances associates with an isomer-pair and that, for each instance, the reach of two isomers associates with the two isomers that associate with the isomer-pair. For a solution for which $n_{in}=2$ and $R_{/in}=3$, we posit that each one of the two instances associates with a handedness and that, for each instance, the reach of three isomers associates with the three isomers that associate with one handedness.

We posit that listing an adequately robust set of interaction reaches per instance can associate with candidate explanations for some cosmic data.

The appendix regarding gravitational multipole expansions indicates that, for modeling that treats two objects as pointlike, the monopole component of the gravitational interaction associates with a force component that attracts (or pulls) the two objects toward each other; the dipole component of the gravitational interaction associates with force components that repel (or push) the two objects away from each other; and the quadrupole component of the gravitational interaction associates with force components that attract (or pull) the two objects toward each other.

Table 2 posits instances and reaches per instance for some components of some interactions.

The appendix regarding gravitational multipole expansions indicates the following notions regarding gravitational interactions. Non-monopole effects that associate with positions of sub-objects of objects associate gravitationally with one instance ($n_{in}=1$) and a reach per instance of six isomers ($R_{/in}=6$). For objects such as hadrons, modeling regarding gravitation can feature the masses and momenta of the objects and de-emphasize the masses and momenta of sub-objects such as quarks.

Possibly, physics will evolve Table 2 by changing terminology that the table uses, changing the number of rows in the table, changing the values of some instance-and-reach pairs, or determining values for which the table lists as to be determined.

Below, we indicate how the present Table 2 numeric values of instances and reaches per instance underlie steps forward regarding explaining observed ratios of presumed-dark-matter effects to ordinary-matter effects and regarding explaining eras in the rate of expansion of the universe.

3. Results

3.1. Galaxy Formation and Galaxy Evolution

This unit indicates that our work provides quantitative explanations for some observations regarding galaxy formation and evolution and that our work adds insight regarding galaxy formation and galaxy evolution.

Our work proposes that, early in the history of the universe, single-isomer objects clumped based on reach-1 effects of at least one of gravitational quadrupole pull and chromodynamics pull. Some smaller single-object clumps evolved into solar systems. Some larger clumps (that could have included multiple solar systems) evolved into early galaxies.

Our work proposes that the discussion above explains 1:0+ ratios that pertain to some early galaxies and 0+:1 ratios that pertain to some early galaxies.

Our work proposes that some later 1:0+ galaxies and some later 0+:1 galaxies retain their ratios from early in the evolution of the universe.

For each of some $\sim 4$:1 galaxies, our work proposes the following scenario. The galaxy started as a 0+:1 galaxy. Reach-2 dipole push contributions to gravity drove away some ordinary-matter stuff and the stuff that associated with one dark-matter isomer. Then, reach-6 monopole pull contributions to gravity attracted remaining nearby stuff. The galaxy evolved to a ratio of $\sim 4$:1.

For each of some $\sim 4$:1 galaxies, our work proposes the following scenario. The galaxy started as an MEA-isomer 1:0+ galaxy. Reach-2 dipole push contributions to gravity drove away some dark-matter stuff but essentially no ordinary-matter stuff. Then, reach-6 monopole pull contributions to gravity attracted remaining nearby stuff. The galaxy evolved to a ratio of $\sim 4$:1.

Many later galaxies are 5+:1 galaxies. Our work proposes that many 5+:1 galaxies resulted from mergers of smaller, previous galaxies. Our work proposes that such mergers associate with reach-6 monopole gravitational pull. Our work proposes that the earliest mergers that led to a 5+:1 galaxy could have been mergers that involved 1:0+ galaxies and 0+:1 galaxies.

3.2. The Fives in 5+:1 Ratios of Dark-Matter Effects to Ordinary-Matter Effects

This unit indicates that our work provides a quantitative explanation for the fives in some observed 5+:1 ratios of dark-matter effects to ordinary-matter effects.

Popular modeling proposes that 5+:1 ratios of dark-matter effects to ordinary-matter effects pertain for many galaxies, many galaxy clusters, and for densities of the universe.

Our work proposes that the notion of five dark-matter isomers explains the fives in such 5+:1 ratios of dark-matter effects to ordinary-matter effects.

3.3. The Pluses in 5+:1 Ratios of Dark-Matter Effects to Ordinary-Matter Effects

This unit indicates that our work provides a candidate qualitative explanation for at least some portions of the pluses in some observed 5+:1 ratios of dark-matter effects to ordinary-matter effects.

Popular modeling suggests that 5+:1 ratios of dark-matter effects to ordinary-matter effects pertain for many galaxies, many galaxy clusters, and for densities of the universe.

Our work proposes that the stuff that associates with either one of the two SEA isomers (one of which associates with ordinary matter and one of which associates with dark matter) associates with more electromagnetic energy than does the stuff that associates with any one of the four MEA isomers (each of which associates with dark matter). Table 2 provides an example of a reach-6 interaction component. Our work proposes that, at least early in the history of the universe, reach-6 or reach-3 interaction components might have enabled (via electromagnetic or other means) flows of electromagnetic energy between isomer-pairs. The net flows could have resulted in each MEA isomer having more stuff than each SEA isomer has. This notion of more stuff might explain all or some of the amounts that underlie the pluses in the 5+:1 ratios of dark-matter presence to ordinary-matter presence.

Our work does not necessarily rule out the possibility that some portions of (or the entireties of) the pluses in the 5+:1 ratios of dark-matter presence to ordinary-matter presence associate with axions, with other unfound elementary particles, or with other popular modeling suggestions regarding the nature of dark matter.

3.4. Our Solar System and Other Optically Observable Solar Systems

This unit indicates that our work provides quantitative explanations for some observations regarding solar systems.

Our discussion above regarding early galaxies proposes that components, including solar systems, of galaxies feature stuff that associates with just one isomer. Our work proposes that presently optically observable solar systems stem from early 0+:1 clumps and do or would usually measure as $\approx 0$:1.

3.5. Hyperfine Depletion of Cosmic Microwave Background Radiation

This unit indicates that our work provides a quantitative explanation for each of two possibilities regarding some depletion of cosmic microwave background radiation.

Regarding the depletion of cosmic microwave background radiation, popular modeling suggests that the second 1 in the possible 1:1 ratio or the only 1 in the possible 0:1 ratio associates with hyperfine effects of ordinary-matter hydrogen atoms.

Should popular modeling eventually settle on the 1:1 ratio, our work proposes that a reach per instance of at least two isomers pertains regarding hyperfine interactivity. Our work proposes that MEA-isomers do not underlie significant numbers of hydrogen-like atoms. Our work proposes that the first 1 in the 1:1 ratio associates with hyperfine effects of hydrogen-like atoms that associate with the SEA dark-matter isomer.

Should popular modeling eventually settle on the 0:1 ratio, our work proposes that a reach per instance of one isomer pertains regarding hyperfine interactivity.

3.6. Eras in the Rate of Expansion of the Universe

This unit indicates that our work provides qualitative insight for some observations regarding the rate of expansion of the universe.

Our work proposes that the relevant two or three eras in the rate of expansion of the universe associate with the moving apart from each other of neighboring, but not colliding, large objects.

Our work proposes that the rate of expansion of the universe associates with interactions between neighboring non-colliding large objects.

Our work proposes that the start of the first multibillion-year era associates with a transition to dominance, regarding interactions between many neighboring non-colliding large objects, including galaxy clusters, by gravitational quadrupole pull.

Our work proposes that the start of the second multibillion-year era associates with a transition to dominance, regarding interactions between many neighboring non-colliding large objects, including galaxy clusters, by gravitational dipole push.

Our work proposes that the possible start of a new era associates with a transition to dominance, regarding interactions between many neighboring non-colliding large objects, including galaxy clusters, by gravitational monopole pull.

4. Discussion

4.1. The Extent of the Assumptions That Our Work Makes and the Extent to Which Our Work Helps Explain Data

This unit compares the extent of the assumptions that underlie our work and the extent to which our work helps explain data.

Regarding large-scale presences (in many individual galaxy clusters and regarding densities of the universe) of dark matter and ordinary matter, our work helps explain ratios of about 5:1. The explanations associate with our assumptions regarding six isomers of a set of elementary particles. Our work possibly helps explain portions of the pluses in the ratios of 5+:1. The explanations associate with our assumptions regarding six isomers of a set of elementary particles.

Regarding galaxies, our work helps explain four ratios (1:0+, 0+:1, $\sim 4$:1, and 5+:1) of amounts of dark-matter stuff to ordinary-matter stuff. The explanations associate with the three rows in Table 2 that discuss two-body gravitational interactions.

Regarding known eras in the rate of expansion of the universe, our work helps explain the onset of two eras (the previous multibillion-year era of decreasing rate of expansion and the present multibillion-year era of increasing rate of expansion). The explanations associate with two of the three rows in Table 2 that discuss two-body gravitational interactions.

Regarding a possibly just-starting era in the rate of expansion of the universe, our work helps explain the onset of that might-be era (a possible multibillion-year era of decreasing rate of expansion). The explanation associates with one of the three rows in Table 2 that discuss two-body gravitational interactions.

Regarding optically observable solar systems, our work helps explain $\approx 0$:1 ratios of amounts of dark-matter stuff to ordinary-matter stuff. The explanations associate with rows in Table 2 that discuss two-body gravitational interactions.

Regarding possible dark-matter electromagnetism, our work helps explain the lack of observation, by people and ordinary-matter equipment, of light from dark matter. The explanation associates with two rows in Table 2. The rows associate with the sensing of one-body blackbody temperature and with two-body electromagnetic interactions.

One might say that comparing the extent of the data that our work helps explain with the extent of the assumptions that our work makes provides credibility for our work.

We note, as an aside, that our work might help close some possible gaps between data and popular modeling. To the extent that popular modeling and further data fail to resolve some so-called tensions between popular modeling and data regarding some large-scale phenomena, our work regarding instances and reaches per instance might help resolve remaining tensions. One such tension is the so-called Hubble tension [68,69,70]. Other such tensions associate with large-scale lumpiness [71,72,73,74,75,76,77,78,79] and include the so-called S8 tension. We propose that such tensions might associate with trying to extrapolate from popular modeling that works adequately well regarding phenomena that our work would associate with $R_{/in}=1$ gravitational quadrupole pull to estimate later phenomena that our work would associate with $R_{/in}=2$ gravitational dipole push. Such popular modeling extrapolations might, in effect, assume that gravitational dipole push associates with $R_{/in}=1$ and, thereby, underestimate gravitational push. The underestimates might associate with overestimating, compared to data, some clumping of stuff. These notions associate with two of the three rows in Table 2 that discuss two-body gravitational interactions.

4.2. Opportunities to Interrelate Physics Constants and to Reduce the Number of So-Called Fundamental Physics Constants

This unit discusses relationships, among data, that people might find useful for extending our work and possibly for reducing the number of physics constants that popular modeling assumes to be independent of each other.

Eq. (2) might associate with a relationship that links a strength of the electromagnetic interaction, a strength of the gravitational interaction, the masses of two elementary fermions, and the number of isomers [58,80]. $m_{\tau }$ denotes the mass of the tau. $m_e$ denotes the mass of the electron. The exponent 6 might associate with the number, six, of isomers. The right-hand side of the equation is the ratio of the electromagnetic repulsion between to electrons to the gravitational attraction between the same two electrons.

$$(4/3){(m_{\tau }^2/m_e^2)}^6=\left((1/\left(4\pi {ϵ}_0\right)){\left(q_e\right)}^2\right)/\left(G{\left(m_e\right)}^2\right)$$

(2)

An equation might interrelate the masses of all known non-neutrino fermion elementary particles. (Eqs. (A1), (A2), (A3), and (A4) in this paper associate with the equation. Table 3.9.10 in [58] and Table 14 in [59] discuss the equation.)

The following paragraphs discuss relationships regarding properties of boson elementary particles.

Regarding boson elementary particles, we define ${\left(N^{\prime }\right)}^2$ via Eqs. (3) and (4). $M^{\prime }$ denotes $m/(m_Z/3)$, in which m denotes the mass of an elementary boson and $m_Z$ denotes the mass of the Z boson. $S^{\prime }$ denotes S (as in the spin, in units of ℏ). $Q^{\prime }$ denotes the magnitude of the charge, in units of the magnitude of the charge of the W boson. (Popular modeling equates the magnitude of the charge of the W boson to the magnitude of the charge of the electron.) ${\mu }^{\prime }$ denotes the magnitude of the magnetic moment, in units of the magnitude of the magnetic moment of the W boson.

$${\left(N^{\prime }\right)}^2\equiv {\left(M^{\prime }\right)}^2+{\left(S^{\prime }\right)}^2+{\left(Q^{\prime }\right)}^2+{\left({\mu }^{\prime }\right)}^2-{\left(T^{\prime }\right)}^2$$

(3)

$${\left(T^{\prime }\right)}^2=1\iff M^{\prime }>0;{\left(T^{\prime }\right)}^2\iff M^{\prime }=0$$

(4)

Based on data [80], we propose that Eqs. (5) and (6) might pertain regarding all known boson elementary particles.

$$N^{\prime }\in \{0,1,2,3,4\}$$

(5)

$$N^{\prime }=4-S^{\prime }\ge 3\iff M^{\prime }>0;N^{\prime }=S^{\prime }\iff M^{\prime }=0$$

(6)

Eq. (7) comports with data and with Eq. (3).

$${\left(m_W\right)}^2:{\left(m_Z\right)}^2:{\left(m_{Higgs}\right)}^2::7:9:17$$

(7)

4.3. Potential Future Directions for Physics

This unit points to potential future directions for some aspects of cosmology, gravitation, and elementary-particle physics.

Future directions for physics might include the following. Determine the extent to which self-interacting-dark-matter stuff includes IGM (as in intergalactic medium), perhaps by analyzing data about collisions of galaxy clusters [81,82]. Determine the extent to which our candidate specification for dark matter is compatible with data that this paper does not discuss. Determine the extent to which adding, to simulations, our candidate specification for dark matter can be useful. Determine the extent to which gravitational wave signatures differ between collisions that involve same-isomer small-mass black holes and neutron (or dark-matter neutron-counterpart) stars and collisions that involve different-isomer small-mass black holes and neutron (or dark-matter neutron-counterpart) stars. Associating with perspective that general relativity has passed so-called precision tests [83] and that those tests seem not to involve dark matter, explore the extent to which general relativity might not be adequately accurate for circumstances in which the isomeric composition of stuff varies significantly between regions of the universe or for circumstances in which significant (or dominant) effective reaches per instance vary with time. (We note, as an aside, that popular modeling recognizes circumstances for which tests of general relativity have yet to be very precise and for which alternative theories of gravity might be appropriate [84].) Explore our notion of a possible symmetry (or approximate symmetry) regarding lepton masses and lepton flavours. Explore interrelationships among physics constants and possibly reduce the number of independent constants. Explore the extent to which our work can help focus and accelerate cosmology and elementary-particle research.

5. Conclusion

This unit summarizes results that our work achieves.

We propose elementary particles and dark-matter specifications that help explain otherwise seemingly unexplained data. We propose (via Eq. (1) and Table 2) a way to characterize components of interactions between objects. Regarding gravitational interactions, we propose and use a type of multipole expansion that considers the motions of sub-objects within gravitationally interacting objects. We interrelate some physics constants. We suggest potential future directions for some aspects of cosmology, gravitation, and elementary-particle physics.

Comparing the extent of the data that our work helps explain with the extent of the assumptions that our work makes seems to provide credibility for our work.

Appendix A.1. Uses of the Word Isomer

This unit compares our use of the word isomer with other uses of the word isomer.

Our use of the word isomer can associate with notions of symmetries, including chirality (or mirror image) symmetry, and with notions of approximate symmetries. Chemistry uses the word isomer regarding the notion that molecules can be mirror images of each other.

Our use of the word isomer does not directly associate with some other uses of the word isomer, for example regarding alternative geometric arrangements (other than those related to chiral symmetry) of atoms within molecules or regarding long-lived excited states of atomic nuclei.

We are not aware of attempts (to parallel, in elementary-particle physics, nuclear-physics notions of isomers and thereby ...) to use the word isomer in conjunction with the three flavour states of ordinary-matter charged leptons or the three generation-states of similarly-charge ordinary-matter quarks. Our work could embrace such uses of the word isomer; however, our work does not presently use the word isomer for such purposes.

Appendix A.2. Possible Symmetry or Approximate Symmetry Regarding the Factor of Two (That Associates with the Number, Six, of Isomers)

This unit discusses aspects related to the factor of two (that associates with the number, six, of isomers) and to right-handed counterpart-to-ordinary-matter dark-matter elementary particles, matter-antimatter asymmetry, and dark-matter hydrogen-like atoms.

The following notions might cover all three of right-handed counterpart-to-ordinary-matter dark-matter elementary particles, matter-antimatter asymmetry, and dark-matter hydrogen-like atoms.

• People, informally, use the one-element term left-handed to describe the ordinary matter isomer.
• In more technical terms, the following popular modeling notions pertain for the ordinary-matter isomer. Left-chiral components of matter elementary fermions and right-chiral components of antimatter elementary fermions associate, via the so-called the $SU{\left(2\right)}_L$ gauge group, with doublets and with interactivity via the weak interaction. Right-chiral components of matter elementary fermions and left-chiral components of antimatter elementary fermions associate, via the so-called the $SU{\left(2\right)}_L$ gauge group, with singlets and with no interactivity via the weak interaction.
• Popular modeling associates the technical aspects with the weak interaction and with a breaking of PC, as in parity and charge, symmetry by the weak interaction.
• We posit that popular modeling might consider that the following notions pertain for the counterpart-to-ordinary-matter dark-matter isomer. Right-chiral components of antimatter elementary fermions and left-chiral components of matter elementary fermions associate, via an $SU{\left(2\right)}_R$ gauge group, with doublets and with interactivity via the weak interaction. Left-chiral components of antimatter elementary fermions and right-chiral components of matter elementary fermions associate, via the $SU{\left(2\right)}_R$ gauge group, with singlets and with no interactivity via the weak interaction. Popular modeling would associate these technical aspects with the weak interaction and with a breaking of PC, as in parity and charge, symmetry.
• Possibly, popular modeling would associate the combination of the ordinary-matter-isomer aspects and the counterpart-to-the-ordinary-matter dark-matter aspects with a symmetry related to the weak interaction.
• Possibly, regarding the counterpart-to-the-ordinary-matter dark-matter stuff, nature would produce more antimatter stuff than matter stuff and popular modeling would associate the combination of the ordinary-matter-isomer aspects and the counterpart-to-the-ordinary-matter dark-matter isomer aspects with notions of matter-antimatter symmetry.
• The counterpart-to-the-ordinary-matter dark-matter stuff would include hydrogen-like atoms. While popular modeling regarding electromagnetism can feature two (as in left and right) orthogonal circular-polarization modes (or can feature two orthogonal linear-polarization modes), popular modeling notions of electromagnetic interactions do not necessarily depend on weak-interaction notions of left-chiral and right-chiral. One can leave to observational work the question as to whether counterpart-to-the-ordinary-matter dark-matter-stuff hydrogen-like atoms can absorb light that ordinary-matter stuff emitted.

We note, as an aside, that discussion above features specific (direct or implicit) reuses, from the ordinary-matter isomer to the counterpart-of-the-ordinary-matter dark-matter isomer, of dual-pairs such as matter and antimatter, more-prevalent and less-prevalent, $SU{\left(2\right)}_L$ and $SU{\left(2\right)}_R$, and positive charge and negative charge. Possibly, the number of dual-pairs is sufficiently large that alternative (to the choices we discuss above) choices (regarding some choices regarding some pairs) regarding the counterpart-to-the-ordinary-matter dark-matter isomer could be appropriate.

Appendix A.3. Possible Approximate Symmetry Regarding the Factor of Three (That Associates with the Number, Six, of Isomers)

This unit discusses aspects related to the factor of three (that associates with the number, six, of isomers) and to matches or mismatches, across isomers, between charged lepton flavours and quark generations.

We propose that each one of the following notions about ordinary matter is not incompatible with the notion of an approximate symmetry that would associate with and differentiate the three isomer-pairs.

• The three neutrino flavour eigenstates do not equal the three neutrino mass eigenstates. We propose that the mismatch associates with an approximate symmetry.
• Eqs. (A1), (A2), (A3), and (A4) pertain regarding the masses of the three charged leptons. Flavour-1 associates with the electron. Flavour-2 associates with the muon. Flavour-3 associates with the tau. Similar equations (with $k=0,+1, \mathrm{and} +2$) pertain regarding the geometric-mean masses for the three quark generations (Table 3.9.10 in [58] or Table 14 in [59]). We note, as an aside, that the notion that $\delta$ is not zero might associate with an approximate symmetry.

$$k=0, +2, \mathrm{and} +3, \mathrm{for} \mathrm{charged}\text{-}\mathrm{lepton} \mathrm{flavour}\text{-}1, \mathrm{flavour}\text{-}2, \mathrm{and} \mathrm{flavor}\text{-}3, \mathrm{respectively}$$

(A1)

$${\sigma }_k=0,+1,-1, \mathrm{and} 0, \mathrm{for} k=0,1,2, \mathrm{and} 3, \mathrm{respectively}$$

(A2)

$$\delta \approx 0.03668$$

(A3)

$$m_k/m_e\approx {(m_{\tau }/m_e)}^{(1/3)(k+{\sigma }_k\delta )}$$

(A4)

• The weak interaction associates with interactions in which charged leptons change flavour and with interactions in which quarks change generation. We propose that the mismatch associates with an approximate symmetry.

We posit that the approximate symmetry associates with, for some isomers, associations between lepton flavours and quark generations that differ from the associations between lepton flavours and quark generations that associate with the ordinary-matter isomer.

We do not try to fully characterize the possible approximate symmetry.

Appendix A.4. The Evolution of MEA (as in Marginally-Electromagnetically-Active) Dark-Matter Stuff

This unit discusses how the evolution of MEA (as in marginally-electromagnetically-active) stuff led to MEA stuff that features stable counterparts to ordinary-matter-stuff neutrons.

For the stuff that associates with each one of the six isomers, a ground-state singly-charged baryon that includes exactly three generation-3 quarks would be more massive than the counterpart, within the same-isomer stuff, ground-state zero-charge baryon that includes exactly three generation-3 quarks. For example, for ordinary-matter-isomer stuff, a ground-state nonzero-charge baryon that includes just two tops and one bottom would have a larger mass than would a ground-state zero-charge baryon that includes just one top and two bottoms. Popular modeling suggests that, for ordinary matter, W bosons play key roles regarding the decay of generation-3 baryons, such as possible generation-3 baryons to which the previous sentence alludes, into ground-state generation-1 baryons, namely the neutron and the proton [85]. Per Table 1, MEA-isomer flavour-3 charged leptons would be less massive than ordinary-matter flavour-3 charged leptons. When generation-3 quark states are much populated, the stuff that associates with an MEA-isomer would convert more charged baryons to zero-charge baryons than would the stuff that associates with the ordinary-matter isomer. Eventually, regarding the stuff that associates with the MEA-isomer, interactions that entangle multiple MEA-isomer W bosons would result in the stuff that associates with the MEA-isomer having more counterparts to ordinary-matter-stuff neutrons and fewer counterparts to ordinary-matter-stuff protons than does the stuff that associates with the ordinary-matter isomer. The sum of the mass of an MEA-isomer counterpart to the ordinary-matter proton and the mass of an MEA-isomer flavour-1 charged lepton would exceed the mass of an MEA-isomer counterpart to the ordinary-matter neutron. Compared to ordinary-matter neutrons, MEA-isomer neutrons would scarcely decay.

Appendix A.5. Gravitational Multipole Expansions

This unit develops the notions of gravitational monopole pull, gravitational dipole push, and gravitational quadrupole pull.

For each one of gravity and electromagnetism, popular modeling includes notions of multipole expansions that have bases in spatial distributions of a scalar property that can associate with sub-objects of an object. For gravity, the scalar property is mass. For electromagnetism, the scalar property is charge. For popular modeling multipole expansions regarding each one of charge and mass, the following notions pertain. The sum of the values of the scalar property for the sub-objects equals the value of the scalar property for the object. Popular modeling de-emphasizes the notion that the sub-objects might move within the object. Popular modeling de-emphasizes forces by which the sub-objects might attract or repel each other. Popular modeling de-emphasizes energies that might associate with keeping the sub-objects in their respective places.

Popular modeling regarding gravitation points to situations in which dipole contributions dilute monopole contributions and to situations in which dipole contributions augment monopole contributions. For example, consider an object that models as being two separated, equal-mass pointlike sub-objects. From the perspective of another body that lies along a line that runs through the center-of-mass of the object and is perpendicular to the line that runs through the two sub-objects, each sub-object is farther away than is the center-of-mass of the object; the body senses less gravitational pull than the body would sense if the two sub-objects existed at the center-of-mass point. However, if the other body lies along a line that runs through the two sub-objects and is farther away from the center-of-mass than is each sub-object, the body senses more gravitational pull than the body would sense if the two sub-objects existed at the center-of-mass point. From such notions, one might conclude that whether there is net dipole dilution of gravitational forces or net dipole augmentation of gravitational forces depends (for two objects) on details and (across several two-body interactions) on statistics.

For now (in this discussion), we de-emphasize the notion of spatial distributions of sub-objects. We focus instead on the motions of sub-objects and assume that modeling can treat the overall object as spatially pointlike.

Within cosmological (and other) objects, sub-objects can move. For example, solar systems move within galaxies. Popular modeling Lorentz invariance [86] suggests that a body that is remote from the object and that does not move relative to the object senses, for each sub-object that moves within the object and has a nonzero value of a scalar property, a larger (than if the sub-object did not move within the object) magnitude of the value of the scalar property.

We develop multipole expansion techniques that address the notion that, if sub-objects model as moving, a remote body likely (for electromagnetism) or always (for gravity) would sense a sum, across sub-objects, of the values of the scalar property (charge for electromagnetism or mass for gravity) that differs from the value of the scalar property that the body associates with the object.

Our development considers quantities that pertain in the rest frame of the body. Our development considers that the object models as approximately pointlike and that all sub-objects model as being (instantaneously) at the same point as the object. We de-emphasize notions that popular modeling associates with the two-word term retarded time.

We consider a case for which the object does not move relative to the body, the only significant interactions between the object and the body are electromagnetic, and the only significant interactions between the sub-objects and the body are electromagnetic. (Our discussions regarding electromagnetism echo popular modeling [87].)

We consider a sub-object that has a rest charge $q_0$ and that moves (relative to each of the object and the body) with a 3-vector velocity v. We define ${\gamma }^{*}$ by ${\gamma }^{*}\equiv 1+(\gamma -1)$, in which $\gamma ={(1-(\left|v\right|/c)}^2{)}^{-1/2}$ is the so-called Lorentz factor. The first term (as in the first 1) in ${\gamma }^{*}$ associates with $v=0$. The second term (as in $\gamma -1$) in ${\gamma }^{*}$ associates with $\left|v\right|>0$. Popular modeling associates the perceived scalar potential with a constant multiplied by $q_0{\gamma }^{*}/\left|r\right|$, in which r denotes a 3-vector distance from the sub-object. Popular modeling associates the perceived vector potential with a constant multiplied by $q_0{\gamma }^{*}(v/c)/\left|r\right|$. Based on aspects that are compatible with popular modeling, we replace the motion-related portion ($q_0(\gamma -1)/\left|r\right|$) of the scalar potential with a vector-potential term that is proportional to $q_0(\gamma -1)vt/\left|r^2\right|$, in which t denotes the temporal coordinate). (The relevant popular modeling equation is $E=-▽\varphi -\partial A/\partial t$, in which E is the 3-vector electric field, ▽ is the spatial-gradient operator, $\varphi$ is the scalar electromagnetic scalar potential, A is the 3-vector electromagnetic vector potential, and t is the temporal coordinate. Our work, in effect, replaces the motion-related, as perceived by the body, portion of $\varphi$ with a motion-related, as perceived by the body, component of A that does not contribute to the magnetic field perceived by the body.) Across all sub-objects, there are now the following three types of terms.

• Scalar potential terms that associate with rest charges and with monopole (as in $1/\left|r\right|$) spatial potentials.
• Vector potential terms that associate with $v\ne 0$ and that popular modeling can associate with dipole (as in $1/{\left|r\right|}^2$) spatial potentials.
• Vector potential terms that associate with $v\ne 0$ and with monopole (as in $1/\left|r\right|$) spatial potentials.

The type-1 terms comport with popular modeling notions of multipole expansions. The type-2 terms comport with popular modeling notions of zero contributions to magnetic fields and do not necessarily comport with popular modeling notions of multipole expansions that involve only scalar properties of objects. The type-3 terms comport with popular modeling notions of magnetic fields and are not necessarily relevant to our discussion (because our discussion pertains regarding the rest frame of the affected body).

From the standpoint of the affected body, type-2 terms and type-3 terms scale, for $\left|v\right|/c\ll 1$, proportionately to $\left|v\right|/c$. From the standpoint of the affected body, type-3 terms associate with notions of charge currents.

From the standpoint of the affected body, the type-2 (or $(\gamma -1)$) terms dilute the counterpart overall (or ${\gamma }^{*}$) scalar potential contributions.

We consider a case for which the object does not move relative to the body, the only significant interactions between the object and the body are gravitational, and the only significant interactions between the sub-objects and the body are gravitational. Compared to the above electromagnetic case, the following changes pertain. The word mass substitutes for the word charge. Rest masses are nonnegative. There are no gravitational analogs to electromagnetic cases in which interactions that associate only with scalar potentials associate with repulsion of the body away from the sub-objects or the object. From the standpoint of the affected body, type-3 terms associate with notions of momenta.

For gravitation, monopole components of interactions associate with pull (as in gravitational attraction of the body toward the object) and type-2 dipole components of interactions associate with push (as in gravitational repulsion of the body away from the object). Quadrupole components of interactions associate with pull (and can associate with considering that the body models as having sub-objects that move). Quadrupole effects can associate with considering that each one of the object and the body has moving sub-objects. Quadrupole effects correct for otherwise miscounting some dipole effects.

We note, as an aside, that our notions of pull and push do not necessarily encompass notions for which popular modeling use of the word torque would pertain.

We note, as an aside, the following possible associations between type-2 gravitational components and applications of general relativity in situations for which the energy flux is zero. Monopole might associate with energy density. Dipole might associate with pressure.

We note, as an aside, that we do not try to explore similarities and differences between gravitoelectromagnetism [88,89,90] and our work regarding gravitational multipole expansions.

We return (in this discussion) to the notion that gravitational modeling does not necessarily need to treat objects as pointlike.

One use above of pointlike associates with the notion that, for discussing aspects related to the motions of sub-objects, one can assume that, across the sub-objects, just one origin related to vectors r pertains. This notion is useful for simplifying our discussion. We posit that the one-origin simplification is not necessarily an oversimplification regarding aspects related to motions of sub-objects.

Regarding the instantaneous positions of sub-objects, the notion of just one origin related to vectors r (or the notion of pointlike modeling for the object) is, in general, not necessarily adequately accurate. Per discussion above, stationary multipole aspects can detract from or add to monopole aspects. For our work, we posit that, at least statistically across similar objects, one can de-emphasize net non-monopole effects that associate with positions of sub-objects compared to non-monopole effects that associate with velocities of sub-objects. We note, as an aside, that, for non-monopole effects that associate with velocities of sub-objects, net effects associate with gross effects.

Thus, we posit that the gravitational rows in Table 2 are appropriate for our work.

We note, as an aside, that net non-monopole effects that associate with positions of sub-objects or objects do not necessarily associate with needing modeling based on Lorentz invariance and would associate gravitationally with one instance ($n_{in}=1$) and a reach per instance of six isomers ($R_{/in}=6$).

We note, as an aside, that, for objects such as hadrons, modeling regarding gravitation can feature the masses and momenta of the objects and de-emphasize the masses and momenta of sub-objects such as quarks.