Dark-Matter and Gravitational-Force Details that Echo Familiar Physics and Explain Cosmic Data

1. Introduction

This unit provides context for our work, previews aspects of our work, and discusses data that our work seeks to explain.

1.1. Context

This unit discusses two 90-year-old challenges that provide context for our work.

Cosmology includes two 90-year-old challenges. Since the 1920s, people have discussed the extent to which large objects separate from one another [1,2]. Recently, people associate with the separation the term rate of expansion of the universe. People have not fully developed an explanation for the separation. For a mechanism that governs some aspects of the separation, people suggest the term dark energy. Since the 1930s, people have discussed phenomena that might not associate with the term ordinary-matter stuff [3,4]. 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 [5,6].

1.2. Preview of Our Work

This unit previews results and methods of our work.

Our work might offer insight and partial resolution regarding the two 90-year-old challenges. Our work has been ongoing for a decade [7,8]. Our work suggests the following. Dark matter exists. Dark matter elementary particles are like ordinary matter elementary particles. Some dark-matter stuff is like ordinary-matter stuff and comports with popular modeling notions of self-interacting dark matter. Most dark-matter stuff is unlike ordinary-matter stuff and comports with popular modeling notions of collisionless dark matter. A new use of multipole expansion mathematics helps describe two-body gravitation. Our techniques include means to catalog some interaction-properties of objects. Parallels exist between the combination of classical electrodynamics and Lorentz invariance and a combination of an expanded list of two-body gravitational properties of objects and gravitational multipole aspects. Uniting our dark-matter specification and our multipole gravitational expansions requires one new integer-based equation. Our approach offers candidate explanations for data. Our candidate explanations cover enough data for people to possibly find our explanations credible and compelling.

1.3. Seemingly Otherwise Unexplained Cosmic Data

This unit discusses seemingly otherwise unexplained cosmic data that our work seeks to explain.

A key goal for our work is to explain the following observed ratios of not-ordinary-matter effects to ordinary-matter effects.

• 1:1 – Amounts of some depletion of cosmic microwave background radiation [9,10,11].
• 1:0+ – Amounts of stuff in some individual galaxies [12,13,14,15,16,17,18,19,20].
• 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. Various online services associate popular modeling notions of time after a supposed so-called Big Bang and redshifts [21]. For example, $z=7$ associates with approximately 0.76 billion years after the supposed Big Bang and with approximately 13 billion years before now.)
  • Redshifts of more than approximately seven [22,23].
  • Redshifts of approximately six [24].
  • Redshifts of less than six through redshifts of nearly zero [25,26,27,28,29,30,31,32].

• $\sim 4$:1 – Amounts of stuff in some individual galaxies [33,34].
• 5+:1 – Amounts of stuff in many individual galaxies [12,35].
• 5+:1 – Amounts of stuff in many individual galaxy clusters [35,36,37,38,39].
• 5+:1 – Densities of the universe [40].

2. Methods

This unit discusses assumptions that our work makes and methods that our work develops and uses. The following statements summarize aspects of our methods.

(1)

Our methods include an assumption that most dark matter associates with new elementary particles and that the new elementary particles are like elementary particles that underlie ordinary matter. Most dark matter associates with popular modeling notions of collisionless dark matter. Some dark matter associates with popular modeling notions of self-interacting dark matter.

(2)

Our methods include applications of multipole-expansion mathematics that describe electromagnetic fields and gravitational fields that associate with multiple properties, such as electromagnetic charge and electromagnetic magnetic moment or such as gravitational mass and object-internal angular momentum, of single objects. This contrasts with popular modeling multipole expansions that associate with spatial distributions of single properties such as charge or mass. Our multipole expansions underlie characterizations of two-body electromagnetic interactions and characterizations of two-body gravitational interactions. Our methods suggest that attention to such two-body gravitational interactions can be key to explaining eras in the rate of expansion of the universe.

(3)

Our methods include one new, with respect to popular modeling, integer-based equation. The equation applies once for each electromagnetic property of objects and once for each gravitational property of objects. For each such application, one of four arithmetically possible solutions to the equation pertains. Our methods suggest a set of solutions that enables explaining cosmic data that pertain to galaxies, galaxy clusters, cosmic microwave background radiation, the expansion of the universe, and densities of the universe. The breadth of the data that we explain might associate with credibility for our methods and explanations.

2.1. Dark-Matter Specification

This unit discusses the candidate specification for dark matter that our work assumes and uses.

We assume that nature includes dark matter. We assume that not-ordinary-matter effects are effects of dark matter.

We consider the set of all known elementary particles other than the photon. Based on a ratio of five-plus to one for dark-matter density of the universe to ordinary-matter density of the universe, we suggest that nature includes five dark-matter isomers of the set of all known non-photon elementary particles. Based on a ratio of one to one for some depletion of cosmic microwave background radiation, we suggest that stuff that associates with one dark-matter isomer is similar to ordinary-matter stuff. Here, as in other areas of physics, the word isomer can associate with variations with respect to might-be symmetries. Here, possible symmetries or broken symmetries might associate with the handedness of elementary particles and with the non-alignment between neutrino flavour eigenstates and neutrino mass eigenstates [41]. We suggest that one symmetry associates with the handedness of elementary particles (such as fermion elementary particles) that exhibit handedness. We suggest that one might-be symmetry associates with matches between charged-lepton flavours and charged-lepton masses.

Table 1 suggests a numbering scheme for the six isomers, specifications for the one ordinary-matter isomer and the five dark-matter isomers, and aspects of the stuff that associates with each isomer.

We use the one-element term MEA-isomer to designate an isomer other than isomer-0 and isomer-3. We suggest that the fermion flavour-and-mass pairings for the MEA isomers (that is, isomer-1, isomer-2, isomer-4, and isomer-5) led to stuff that associates with the MEA isomers forming stable counterparts to isomer-0-stuff neutrons and to stuff that associates with the MEA isomers not forming significant numbers of counterparts to isomer-0-stuff atoms. For 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, ground-state zero-charge baryon that includes exactly three generation-3 quarks. For example, for isomer-0, a 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 isomer-0, 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 [42]. Per Table 1, MEA-isomer flavour-3 charged leptons would be less massive than isomer-0 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 isomer-0. 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 isomer-0 neutrons and fewer counterparts to isomer-0 protons than does the stuff that associates with isomer-0. The sum of the mass of an MEA-isomer-counterpart-to-isomer-0 proton and the mass of an MEA-isomer flavour-1 charged lepton would exceed the mass of a counterpart-to-isomer-0 neutron. Compared to isomer-0 neutrons, MEA-isomer neutrons would scarcely decay.

Regarding DM (SEA), we note that some observational results [43,44,45] suggest that some dark matter might comport with popular modeling notions of self-interacting dark matter [5,46]. Some popular modeling results [47,48,49,50] point to possible benefits of considering that some dark matter is self-interacting dark matter.

We note, as an aside, that similarities between isomer-0 and isomer-3 might provide a basis for popular modeling to better, than now, come to terms with the popular modeling notion of matter-antimatter asymmetry (which is also known as baryon asymmetry) [51].

2.2. Two-Body Gravitational Properties of Objects

This unit develops (by reusing popular modeling aspects regarding Newtonian gravity, electromagnetism, and Lorentz invariance) a list of suggested new two-body gravitational properties of objects.

We discuss interactions between an object-A and an object-P. The A in object-A associates with the two-word term active properties. Popular modeling associates active properties with the notion of properties about which fields, such as electromagnetic fields and gravitational fields, convey information. The P in object-P associates with the two-word term passive properties. Popular modeling associates passive properties with interactions, by object-P, with fields that associate with objects, such as object-A, other than object-P.

The following notions contrast aspects of Newtonian force equations and Lorentz-invariant interpretations of the values of properties of objects.

• In Newtonian force equations, the values of some properties, including mass [52] and charge [53], of an object-A are invariant to the motion of object-A with respect to an observer, which could be object-P, of object-A.
• Based on Lorentz invariance [54], the perceived values of some object-A properties, including mass and charge, can vary based on a choice of an observer, which could be object-P. The magnitudes of the variations, away from values that associate with a frame of reference in which object-A is at rest, can associate with a nonzero velocity and with nonzero values for the object-A property current that associates with a property of object-A. (For example, charge current associates with charge.)

The following themes associate with this discussion. Expand the list of two-body gravitational properties of objects to include properties other than mass. For each one of some two-body gravitational properties of object-A, discuss the extent to which two-body gravitational interactions between that property of object-A and the mass of object-P associate with attraction (or, pull) of object-P toward object-A or associate with repulsion (or, push) of object-P away from object-A. Point to situations in which the total (across two-body properties of object-A) gravitational push can exceed the total (across two-body properties of object-A) gravitational pull.

The following remarks characterize aspects of our approach.

We discuss some gravitational-force cases and electromagnetic-force cases that lead to so-called extended Newtonian gravity.

Throughout our discussion of gravity and electromagnetism, we de-emphasize the notion that object-P can experience a torque based on its interactions with object-A. We assume that the gravitational mass of object-P equals the inertial mass of object-P. Regarding gravity, we generally assume that the only adequately relevant property of object-P is mass. Regarding electromagnetism, we generally assume that the only adequately relevant properties of object-P are (the electromagnetic property of) charge and (the inertial property of) mass.

For much of our discussion of gravity and electromagnetism, we de-emphasize the notion that each one of object-A and object-P might change, for example via radiation, its internal state.

Throughout our discussion of gravity and electromagnetism, we de-emphasize the notion that object-A and object-P might collide with each other.

Throughout our discussion of gravity and electromagnetism, we exclude from the list of possible objects-A and from the list of possible objects-P popular modeling zero-mass objects (such as photons and gluons). Throughout our discussion of gravity and electromagnetism, we exclude from the list of possible objects-A and from the list of possible objects-P popular modeling objects (such as quarks and gluons) that popular modeling models as not existing individually.

Equation (1) provides notation for the 3-vector velocity of object-A relative to the position of object-P.

$$v_{oA,rel.to.oP}$$

(1)

For much of our discussion of gravity and electromagnetism, we consider aspects for which popular modeling could consider that object-P is at rest. In such contexts, $v_{oA,rel.to.oP}$ also associates with the 3-vector difference between the velocity of object-A and the velocity of object-P.

Our work might have some parallels to work that popular modeling associates with the word gravitoelectromagnetism [55,56,57,58].

2.2.1. Two-Body Gravity and Seventeenth Century Modeling

This unit discusses aspects of Newtonian gravity.

Equations (2), (3), and (4) describe aspects regarding the motion of object-P [52]. G is the gravitational constant. $m_{oA}$ is the mass of object-A. Mass is a scalar property. $m_{oP}$ is the mass of object-P. r is the 3-vector distance that object-P is away from object-A. ▽ is the gradient operator. ▽ produces a 3-vector from a scalar. $F_{oP}$ is the force that object-P feels. $F_{oP}$ is a 3-vector. Object-P might sense effects of that force via an accelerometer that associates with object-P. In equations such as equation (2), V is a scalar. Popular modeling associates with V the word potential. In equations such as equation (3), $r^{n_r}$ denotes the $n_r$-th power of the magnitude of the 3-vector r.

$$G{m}_{oA}{m}_{oP}(-\nabla V)=F_{oP}$$

(2)

$$V=-1/r^{n_r}$$

(3)

$$n_r=1$$

(4)

Popular modeling suggests that $F_{oP}$ associates with a rate of change, with respect to time, of the momentum of object-P. Relevant popular modeling suggests that the momentum of object-P equals the mass $m_{oP}$ of object-P times the velocity of object-P. Relevant aspects of our work assume that $m_{oP}$ does not change. For a constant mass $m_{oP}$, equation (5) pertains. $a_{oP}$ is the acceleration of object-P. $a_{oP}$ is a 3-vector.

$$F_{oP}=m_{oP}{a}_{oP}$$

(5)

Equations (2), (3), (4), and (5) describe, for example, motion in which one body orbits the other body. Newtonian physics includes the notion that each of the two objects can exhibit nonzero motion.

Equations (2), (3), (4), and (5) refer directly to some properties of objects (that is, the mass of object-A and the mass of object-P), a position of one object relative to and other object, and the notion of acceleration. Equations (2), (3), (4), and (5) do not refer directly to notions of velocity. Equations (2), (3), (4), and (5) do not refer directly to the notion of a velocity $v_{oA,rel.to.oP}$ of object-A with respect to the position of object-P. Equations (2), (3), (4), and (5) do not refer directly to the notion of a velocity of object-A with respect to a velocity of object-P.

2.2.2. Two-Body Electromagnetism and Eighteenth Century Modeling

This unit discusses aspects of eighteenth century two-body electromagnetism.

Equation (6) is an aspect of eighteenth century two-body electromagnetism [53,59]. ${\epsilon }_0$ denotes the vacuum electric permittivity. $q_{oA}$ is the charge of object-A. $q_{oP}$ is the charge of object-P. Equation (5) pertains.

$$-(1/\left(4\pi {\epsilon }_0\right))q_{oA}{q}_{oP}(-\nabla \left(V\right))=F_{oP}$$

(6)

The eighteenth century precedes the notion that object-P-perceived properties of object-A can vary based on a frame of reference that associates with object-P. Each one of $q_{oA}$ and $q_{oP}$ is invariant to a choice of a frame of reference.

We suggest that equation (6) provides an electromagnetic parallel to equation (2). $v_{oA,rel.to.oP}$ is not necessarily relevant regarding equations (3), (4), (5), and (6).

2.2.3. Two-Body Electromagnetism and Nineteenth Century Modeling

This unit discusses and extends aspects, including two-body electromagnetic properties of objects, of nineteenth century two-body electromagnetism.

Compared to equation (6), popular modeling added two two-body electromagnetic properties, magnetic moment and charge current. Popular modeling added Lorentz invariance and notions that perceived values of properties depend on frames of reference.

We propose a way to catalog two-body electromagnetic properties of objects. The way features two integers. One integer is $n_r$, as in equations (3) and (4). The other integer is $n_{3s}$. $n_{3s}$ denotes a number of so-called threesomes that appear directly or indirectly on the left-hand sides of equations such as equation (2) and equation (6). A threesome might be a 3-vector.

Table 2 lists some two-body electromagnetic properties that might associate with an object.

Popular modeling suggests that relevant quantities include $q_{oP,rest}$, $q_{oA,rest}$, $v_{oA,rel.to.oP}$, $q_{oA}$, ${\varphi }_{oA}$, and $A_{oA}$. q denotes charge. $q_{oP,rest}$ denotes the charge of object-P that associates with observations in a frame of reference in which object-P is not moving. $q_{oA,rest}$ denotes the charge of object-A that associates with observations in a frame of reference in which object-A is not moving. $q_{oA}$ denotes the charge that object-P associates with object-A. ${\varphi }_{oA}$ denotes the electromagnetic scalar potential that object-P associates with contributions to the electromagnetic field that associate with object-A. $A_{oA}$ denotes the electromagnetic 3-vector vector potential that object-P associates with contributions to the electromagnetic field that associates with object-A. In popular modeling, ${\varphi }_{oA}$ and $A_{oA}$ combine to form a Lorentz-invariant 4-vector.

Popular modeling provides the following equations. $E_{oA}$ denotes the electric field that object-P associates with contributions, to the electromagnetic field, that associate with object-A. $v_{oP}$ denotes the velocity of object-P. $B_{oA}$ denotes the magnetic field that object-P associates with contributions, to the electromagnetic field, that associate with object-A. Each one of $E_{oA}$, $v_{oP}$, and $B_{oA}$ is a 3-vector. $\partial .../\partial t$ denotes a partial derivative with respect to time. The equations are invariant with respect to a choice of a frame of reference. Values for a variable in an equation can vary, based on the choice of a frame of reference.

$$q_{oP}(E_{oA}+(v_{oP}\times B_{oA}))=F_{oP}$$

(7)

$$E_{oA}=-\nabla {\varphi }_{oA}-\partial A_{oA}/\partial t$$

(8)

$$B_{oA}=\nabla \times A_{oA}$$

(9)

Popular modeling suggests that the following equations pertain. $I_{oA}$ is the charge current that associates with object-A. $I_{oA}$ is a 3-vector. $I_{oA}$ associates with both the charge $q_{oA}$ and the velocity $v_{oA,rel.to.oP}$. $A_{oA,I}$ is a 3-vector. $A_{oA,I}$ contributes to $A_{oA}$.

$${\varphi }_{oA}\propto q_{oA}/r$$

(10)

$$A_{oA,I}\propto I_{oA}/r$$

(11)

In popular modeling, $q_{oA}$ and $I_{oA}$ combine to form a Lorentz-invariant 4-vector.

Equation (7) contrasts with equation (6).

Popular modeling associates equation (12) with Lorentz invariance. The symbol · associates with the notion of dot (or, scalar) product. The constant is invariant to changes in $v_{oA,rel.to.oP}$.

$$|E_{oA}{|}^2-c^2{\left|B_{oA}\right|}^2\equiv E_{oA}·E_{oA}-c^2{B}_{oA}·B_{oA}=constant$$

(12)

2.2.4. A Re-look at Two-Body Electromagnetism and Nineteenth Century Modeling

This unit explores relationships between aspects of eighteenth century two-body electromagnetism and aspects of nineteenth century two-body electromagnetism.

Regarding eighteenth century two-body electromagnetism, we assume that equation (13) adequately associates with equation (6).

$$-(1/\left(4\pi {\epsilon }_0\right))q_{oA,rest}{q}_{oP,rest}(-\nabla \left(V\right))=F_{oP}$$

(13)

We explore relationships between equation (6) and equation (7).

We discuss interpretations with respect to the rest frame that associates with object-P. Equation (14) pertains.

$$v_{oP}=0$$

(14)

We note, as an aside, that choosing this rest frame associates with standardizing some notions regarding the time t. This paper does not necessarily directly take advantage of such standardization. Choosing this rest frame associates with standardizing notions of lengths, such as distances r away from object-A.

We note that distances r may pertain to present times for object-P and earlier times for object-A. The popular modeling notion of retarded time would pertain. We suggest that, for the purposes of this paper, time delays regarding the propagation of information about object-A are not necessarily adequately important to warrant more careful attention to the delays.

We limit discussions to cases in which $q_{oP}$ is the only nonzero-value electromagnetic property that associates with object-P.

Based on equations (7) and (14), the value of $B_{oA}$ is not relevant regarding $F_{oP}$. However, equations (7) and (8) suggest that $A_{oA}$ can still have relevance regarding $F_{oP}$.

We use the symbol ${\mu }_{oA}$ to denote the magnetic moment that object-P associates with object-A. ${\mu }_{oA}$ is a 3-vector.

We discuss a so-called case-e1.

A theme for case-e1 is to try to recover equation (13) from equation (6). Recovery associates with notions that associate with replacing $q_{oA}$ with $q_{oA,rest}$. Recovery associates with notions of reverting to invariances that associate with equation (13). In particular, equation (13) associates with the notion that each one of $q_{oA,rest}$ and $q_{oP,rest}$ is invariant with respect to velocity.

For case-e1, we make the following assumptions. $q_{oA,rest}\ne 0$. ${\mu }_{oA,rest}=0$. $q_{oP,rest}\ne 0$. No two-body electromagnetic property, other than $q_{oP,rest}$, of object-P is nonzero.

Popular modeling that associates with equation (7) and with Lorentz invariance suggests that, if object-P would infer that the magnitude of $v_{oA,rel.to.oP}$ is nonzero, object-P would infer that $|\nabla {\varphi }_{oA}|/|\nabla {\varphi }_{oA,rest}|$ exceeds one, that $q_{oA}/q_{oA,rest}$ exceeds one, and that $|E_{oA}|/|E_{oA,rest}|$ exceeds one.

We use the definitions that equations (15) and (16) show. The subscript $oA,v$ associates with the notion that each one of $q_{oA,v}$ and $-\nabla {\varphi }_{oA,v}$ characterizes a contrast between popular modeling based on (essentially eighteenth century) equation (13) and popular modeling based on (nineteenth century) equation (7). In the subscript $oA,v$, the symbol v associates with the velocity $v_{oA,rel.to.oP}$.

$$q_{oA,v}\equiv q_{oA}-q_{oA,rest}$$

(15)

$$-\nabla {\varphi }_{oA,v}\equiv -\nabla {\varphi }_{oA}-(-\nabla {\varphi }_{oA,rest})$$

(16)

We suggest that equation (8) points to possibilities for popular modeling for which $q_{oA}=q_{oA,rest}$, $-\nabla {\varphi }_{oA}=-\nabla {\varphi }_{oA,rest}$, $E_{oA}=E_{oA,rest}$ and for which equation (17) pertains.

$$E_{oA}=-\nabla {\varphi }_{oA,rest}$$

(17)

We suggest adding a new component, $A_{oA,v}$, to the vector potential that associates with object-A. Equation (18) pertains.

$$A_{oA}=A_{oA,I}+A_{oA,v}$$

(18)

We require that $A_{oA,v}$ does not impact the magnetic field $B_{oA}$. That is, we require that $\nabla \times A_{oA,v}=0$.

The following paragraphs provide an example of such an $A_{oA,v}$ for which $\nabla \times A_{oA,v}=0$.

We assume that each one of $q_{oA}$, $v_{oA,rel.to.oP}$, and $I_{oA}$ is a constant with respect to the time t. Equation (19) defines a candidate $A_{oA,v}$.

$$A_{oA,v}=(-\nabla {\varphi }_{oA}-(-\nabla {\varphi }_{oA,rest}))t$$

(19)

Equation (20) restates equation (19).

$$A_{oA,v}=-\nabla \left(({\varphi }_{oA}-{\varphi }_{oA,rest})t\right)$$

(20)

Mathematics suggests that the curl of the gradient of a scalar field is zero. Except at $r=0$ (which is not necessarily physically relevant), the contributions to $B_{oA}$ are $\nabla \times A_{oA,v}$, which is zero. (We note, as an aside, that the $A_{oA,v}$ that equation (19) suggests associates with 3-vectors that exhibit radial spatial dependencies of $1/|r_{oA}|$, point along radii with respect to the position of object-A, and otherwise do not vary based on angular coordinates. $r_{oA}$ denotes the 3-vector distance from object-A.)

Within and beyond the above example an $A_{oA,v}$ for which $\nabla \times A_{oA,v}=0$, we suggest that, at least in the rest frame of object-P, equation (13) provides a useful basis for modeling for phenomena that associate with equation (7).

We also suggest that equation (12) associates with the notion that the reversion process does not associate with notions for which $|E_{oA}|/|E_{oA,rest}|<1$ or with notions that the push or pull sense of $a_{oP}$ changes (either from push to pull or from pull to push).

The following notions summarize discussion above. We suggest that, for the magnitude of $v_{oA,rel.to.oP}$ being nonzero, one can revert Lorentz-invariance-compliant modeling toward equation (13). The amount of reversion associates with $|v_{oA,rel.to.oP}|$ and does not depend on the direction that associates with $v_{oA,rel.to.oP}$. The reversion suggests that, from the perspective of object-P, for a specific value of $n_r$, effects that associate with $n_{3s}=n_r+1$ detract from effects that associate with $n_{3s}=n_r$.

We discuss a so-called case-e2.

A theme for case-e2 is to try to understand the extent to which object-P perceived values, that differ from a rest-frame value of object-A charge, of charges that associate with object-A might associate with object-P perceived nonzero values of object-A magnetic moment.

For case-e2, we make the following assumptions. $v_{oA,rel.to.oP}=0$. $q_{oA,rest}\ne 0$. ${\mu }_{oA,rest}\ne 0$. Object-A includes sub-objects that have charges that have the same sign as $q_{oA,rest}$. Object-A includes no sub-objects that have charges for which the signs are the opposite of the sign of $q_{oA,rest}$. Motions, within object-A, of charged sub-objects account for the entirety of ${\mu }_{oA,rest}$. No two-body $oA,rest$ electromagnetic property, other than $q_{oA,rest}$ and ${\mu }_{oA,rest}$, of object-A is nonzero. $q_{oP,rest}\ne 0$. No electromagnetic property, other than $q_{oP,rest}$, of object-P is nonzero.

Unlike for case-e1, for case-e2, the two-body relative velocity $v_{oA,rel.to.oP}$, which is zero, is not a variable 3-vector. For case-e2, the two-body 3-vector relative position r is relevant.

We use the symbol $v_{sub.object.of.oA,rel.to.oP}$ to denote the velocity, relative to object-P, of a charged sub-object of object-A.

While $v_{oA,rel.to.oP}=0$ pertains regarding object-A, $|v_{sub.object.of.oA,rel.to.oP}|>0$ pertains for each moving charged sub-object of object-A. Popular modeling suggests that charges add. Paralleling case-e1, we suggest that each moving charged sub-object of object-A contributes to the notion that object-P would infer that $|\nabla {\varphi }_{oA}|/|\nabla {\varphi }_{oA,rest}|$ exceeds one, that $q_{oA}/q_{oA,rest}$ exceeds one, and that $|E_{oA}|/|E_{oA,rest}|$ exceeds one. Popular modeling associates the factor $1/r^2$ with the ${\mu }_{oA}$-related contribution to the overall potential that associates with $F_{oP}$. We suggest that the amounts that the three ratios exceed one scale as $1/r^2$. We associate this scaling with the two-word phrase dipole effects.

We suggest that, from the perspective of object-P, effects that associate with nonzero magnetic moment ${\mu }_{oA}$ dilute effects that associate with nonzero charge $q_{oA}$. The magnitude of the dilution scales linearly with the magnitude of ${\mu }_{oA}$. The magnitude of the dilution does not depend on the direction of ${\mu }_{oA}$.

Regarding reversion toward equation (13), we suggest that, for case-e2, dipole effects dilute monopole effects.

We note, as an aside, that a case-e2 analog to the case-e1 $A_{oA,v}$ that equation (19) suggests associates with 3-vectors that exhibit radial spatial dependencies of $1/{\left|r\right|}_{oA}{|}^2$, point along radii with respect to the position of object-A, and otherwise do not vary based on angular coordinates.

Assuming that object-A and object-P are adequately small, there is a range of small r for which object-P accelerates in the direction opposite to the acceleration for case-e1 (for which there are no ${\mu }_{oA}$-related effects).

We note, as asides, the following notions. Popular modeling includes cases, such as for bar magnets, in which objects can model as having zero charge and nonzero magnetic moment. We do not explore such cases. We do not explore popular modeling notions that there might be more than one popular modeling definition of magnetic moment for such cases and that the definitions might not be equivalent regarding Lorentz-invariant transformations [60]. For an object-A that has more than one nonzero-charge sub-object, there can be many circumstances for which the $n_{3s}=n_r=2$ property associates with too much physical complexity to have associations just with, or perhaps even adequately meaningfully with, notions of a magnetic moment. For example, one can consider cases in which sub-objects of object-A have same-signed charges and the motions of the sub-objects are such that contributions toward a might-be object-A magnetic moment tend to cancel each other.

We discuss a so-called case-e3.

A theme for case-e3 is the notion that motions of charged sub-objects of object-A might be chaotic.

Compared to case-e2, case-e3 removes the following assumptions. ${\mu }_{oA,rest}\ne 0$. No two-body $oA,rest$ electromagnetic property, other than $q_{oA,rest}$ and ${\mu }_{oA,rest}$, of object-A is nonzero.

For case-e3, the following case-e2 assumptions remain. $v_{oA,rel.to.oP}=0$. $q_{oA,rest}\ne 0$. Object-A includes sub-objects that have charges that have the same sign as $q_{oA,rest}$. $q_{oP,rest}\ne 0$. Object-A includes no sub-objects that have charges for which the signs are the opposite of the sign of $q_{oA,rest}$. Motions, within object-A, of charged sub-objects account for the entirety of ${\mu }_{oA,rest}$. No electromagnetic property, other than $q_{oP,rest}$, of object-P is nonzero.

For case-e3, the motion of the charged sub-objects might be chaotic. For case-e2, the two-word term magnetic moment describes the $n_r=n_{3s}=2$ property. For case-e3, we suggest using the two-element term charge whirl-or-jitter to name the $n_r=n_{3s}=2$ property. For case-e3, we suggest using the two-element term charge-whirl-or-jitter current to name the property for which $n_r=2$ and $n_{3s}=3$.

We discuss a so-called case-e4.

Compared to case-e3, case-e4 removes the assumption that $v_{oA,rel.to.oP}=0$.

For case-e4, we suggest considering the notion that charge-whirl-or-jitter current corrects for otherwise possible double-counting regarding effects that would associate with the motions of object-A nonzero-charge sub-objects for which the velocities within object-A are not perpendicular to $v_{oA,rel.to.oP}$. Also, in the sense that object-P perceptions of object-A charge increase with object-P perceptions of increasing object-A charge current and with object-P perceptions of increasing object-A charge whirl-or-jitter, object-P perceptions of object-A charge decrease with object-P perceptions of increasing object-A charge-whirl-or-jitter current.

Within and beyond case-e1 through case-e4, we note the following statements. The notion of a binding energy that might be necessary to keep object-A intact is not necessarily relevant. Charges add across sub-objects. While notions that popular modeling might associate with potentials (or with potential energies) appear in discussions above, notions that popular modeling might associate with kinetic energy do not appear explicitly in discussions above.

2.2.5. Two-Body Gravity and Suggested Twenty-First Century Modeling

This unit suggests two-body gravitational properties, of objects, that might extend aspects of popular modeling.

We discuss and extend a so-called case-g1, which is a gravitational analog to case-e1, and a so-called case-g2, which is a gravitational analog to case-e2.

In popular modeling, mass is the gravitational analog to charge in electromagnetism. We note that, paralleling notions above regarding electromagnetism, considering object-A structurally internal energies might not be necessary. Also paralleling notions above regarding electromagnetism, considering object-A kinetic energies might not be necessary. (We note, as an aside, that notions that we are discussing here do not extend to some cases in which gravity does not provide an adequately dominant force. One such case involves quantum-chromodynamics interactions within hadrons. Also, in cases such as ones for which quantum chromodynamics pertain, the notion that mass might add across objects is not necessarily useful.)

Table 3 lists some two-body gravitational properties that might associate with an object.

We suggest that Table 3 associates with a new type of gravitational multipole expansion. In each of popular modeling multipole expansions and our multipole expansions, an expansion features a series of terms. Each term contributes via addition or subtraction to a notion of an overall spatial potential. In popular modeling, a multipole expansion tends to have a basis in a spatial distribution of one property such as charge or mass. Our multipole expansions feature one object that models spatially as somewhat pointlike or as small and somewhat spherically symmetric and physically as having at least one nonzero-valued property.

We note, as an aside, that an $n_{3s}$ that exceeds by two the counterpart $n_r$ might associate with a nonzero acceleration and with the notion that, from the perspective of object-P, object-A would model as part of a system, of objects, for which the system does not necessarily include object-P.

Popular modeling suggests that the object-property of mass is always nonnegative. In the context of two-body interactions, the contribution that associates with the mass of object-A interacting with the mass of object-P associates with a pull component of force. Pull associates with notions of attraction of object-P toward object-A. More generally, we suggest that equations (21) and (22) pertain for two-body gravitation. $n_{3s,oA}$ denotes an $n_{3s}$ for object-A. $n_{3s,oP}$ denotes an $n_{3s}$ for object-P. The symbol ↔ denotes the two-word phrase associates with. Push associates with notions of repulsion of object-P away from object-A.

$$(n_{3s,oA}+n_{3s,oP})iseven\leftrightarrow pull$$

(21)

$$(n_{3s,oA}+n_{3s,oP})isodd\leftrightarrow push$$

(22)

Table 4 lists some contributions, by an object-A, to gravitational forces, as perceived by an object-P. Table 4 extends Table 3.

For a pair of rows, in Table 4, that associate with the same RSD, we suggest that the pull (or push) that associates with an object-A $n_{3s}=n_r$ property dominates the push (or, respectively, pull) that associates with the counterpart object-A $n_{3s}=n_r$-property current (for which $n_{3s}=n_r+1$ pertains).

For a pair of rows, in Table 4, that associate with two different object-A non-current properties such that one row associates with pull and the other row associates with push, we suggest that dominance with respect to pull or push depends on r. For example, consider the object-A properties of mass and angular momentum. For adequately large values of r, pull dominates. For lesser values of r, push can dominate. (We note, as an aside, that for yet lesser values of r, the notion that the objects are not colliding might no longer pertain.)

We discuss a so-called case-g3, which is analogous to case-e3, and a so-called case-g4, which is analogous to case-e4.

For case-g3 and for case-g4, the motions of nonzero-mass sub-objects within object-A might be chaotic. For case-g2, the two-word term angular momentum describes the $n_r=n_{3s}=2$ property. For case-g3 and for case-g4, we suggest using the two-element term mass whirl-or-jitter to name the $n_r=n_{3s}=2$ property. For case-g3 and for case-g4, we suggest using the two-element term mass-whirl-or-jitter current to name the property for which $n_r=2$ and $n_{3s}=3$.

Within and beyond case-g1 through case-g4, we note the following statements. The notion of energies that might be necessary to keep object-A structurally intact is not necessarily relevant. Masses add across sub-objects. While notions that popular modeling might associate with potentials (or with potential energies) appear in discussions above, notions that popular modeling might associate with kinetic energy do not appear explicitly in discussions above. For one value of $n_r$, $n_{3s}=n_r+1$ effects cannot reverse the object-P push or pull sense that associates with $n_{3s}=n_r$ effects. For one value of $n_g$, $n_{3s}=n_r=n_g+1$ effects can reverse the object-P push or pull sense that associates with $n_{3s}=n_r=n_g$ effects if object-A and object-P are adequately close to each other and are not colliding with each other.

2.3. Instances of Properties of Objects, Plus Reaches per Instance of Contributions to Interactions Between Objects

This unit introduces a notion of instances of electromagnetic properties and of two-body gravitational properties and a notion of reach per instance. This unit suggests, for some key two-body gravitational properties of objects and some key electromagnetic properties of objects, instances and reaches per instance. This unit suggests that numeric values of instances and reaches per instance can be key to explaining some cosmic data.

Popular modeling suggests that ordinary-matter stuff scarcely, if at all, sees dark-matter stuff. We suggest that nature includes six instances of the property of charge. We suggest that each isomer associates with its own instance of charge. We say that, for each one of the six instances of charge, the reach per instance is one isomer. Similarly, we suggest (based on the notion that ordinary matter does not see dark matter stars) that each isomer associates with its own instance of the property of blackbody temperature and that the reach per instance for blackbody temperature is one isomer.

Popular modeling suggests that each nonzero-mass object can interact with all other nonzero-mass objects. We say that nature includes one instance of the property of mass. We say that the reach per instance for that one instance is six isomers.

We use the symbol $n_{in}$ to denote the number of instances of a property. 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 suggest that, for each electromagnetic property and for each gravitational property, equation (23) pertains.

$$n_{in}·R_{/in}=6$$

(23)

Equation (23) pertains regarding properties, such as properties that Table 3 lists, of individual objects.

For a solution for which $n_{in}=3$ and $R_{/in}=2$, we assume that each one of the three instances associates with an isomer-pair and that the reach of two isomers associates with the two isomers that associate with the isomer-pair. We assume that the $n_{in}=2$ and $R_{/in}=3$ solution is not relevant for our work.

Table 5 suggests instances and reaches per instance for some two-body gravitational properties of objects and for some electromagnetic properties of objects.

For a component, such as a component that Table 4 lists, of a force that associates with a two-body interaction, we suggest that one needs to calculate the $n_{in}$ and $R_{/in}$ based on the relevant $n_{in,property.of.oA}$, $R_{/in,property.of.oA}$, $n_{in,property.of.oP}$, and $R_{/in,property.of.oP}$. We suggest the following notions. If at least one of $R_{/in,property.of.oA}$ and $R_{/in,property.of.oP}$ is one, $n_{in}=6$ and $R_{/in}=1$. If one of $R_{/in,property.of.oA}$ and $R_{/in,property.of.oP}$ is two and the other one of $R_{/in,property.of.oA}$ and $R_{/in,property.of.oP}$ is at least two, $n_{in}=3$ and $R_{/in}=2$. For this case, each one of the three instances of $R_{/in}=2$ associates with an isomer-pair. If each one of $R_{/in,property.of.oA}$ and $R_{/in,property.of.oP}$ is six, $n_{in}=1$ and $R_{/in}=6$.

We suggest that the instances and reaches per instance that Table 5 features seem to be compatible with all the data that this study seeks to explain.

2.4. Cosmic Clumping of Stuff

This unit suggests that the formation and evolution of smaller cosmic objects occurred earlier and more quickly than did the formation and evolution of larger cosmic objects.

We assume, for discussion purposes, that, at some time in the evolution of the universe, stuff (ordinary matter and dark matter) had a uniform spatial distribution. We assume that, at that time, bunches of stuff were moving away from each other. We de-emphasize notions related to properties that associate with the word current.

We consider Newtonian notions. We consider two spatially non-overlapping, similarly-sized spherical regions of stuff. The amount of stuff in each region scales as the cube of the radius of the region. The monopole contribution to the gravitational force that one region exerts on the other region scales inversely as the square of the distance between the centers of the two regions. If one scales the two radii and the one distance similarly, the monopole aspect of gravitational force scales as $r^3{r}^3{r}^{-2}=r^4$. Dipole aspects scale as $r^3{r}^3{r}^{-3}=r^3$. Quadrupole aspects scale as $r^3{r}^3{r}^{-4}=r^2$. Octupole aspects scale as $r^3{r}^3{r}^{-5}=r^1$. Hexadecapole aspects scale as $r^3{r}^3{r}^{-6}=r^0$.

We suggest notions that might associate with observed and possible clumping, at various scale sizes, in the history of the universe. Hexadecapole attraction might be scale-invariant and might not necessarily be adequately relevant for this discussion of clumping. Across a range of sizes, octupole repulsion was dominant at some time. Smaller similar-sized clumps transited a sequence, of octupole repulsion to quadrupole attraction and onward, faster than did larger similar-sized clumps. Typically, pairs of neighboring solar-system-sized clumps transited to dominance by monopole attraction before pairs of neighboring galaxy-sized clumps transited to dominance by monopole attraction. Typically, pairs of neighboring galaxy-sized clumps transited to dominance by monopole attraction before pairs of neighboring galaxy-cluster-sized clumps might transit to dominance by monopole attraction.

3. Results

This unit suggests that Table 5 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.

3.1. Hyperfine Depletion of Cosmic Microwave Background Radiation

This unit suggests that our work provides a candidate quantitative explanation for an observation regarding some depletion of cosmic microwave background radiation.

Regarding the observed depletion of cosmic microwave background radiation, popular modeling suggests that the second 1 in the 1:1 ratio associates with hyperfine effects of ordinary-matter hydrogen atoms. We suggest, in Table 5, that a reach per instance of at least two isomers pertains regarding hyperfine interactivity. We suggest that MEA-isomers do not underlie significant numbers of hydrogen-like atoms. We suggest that the first 1 in the 1:1 ratio associates with hyperfine effects of hydrogen-like atoms that associate with isomer-3.

3.2. Galaxy Formation and Galaxy Evolution

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

The solar system that includes the Earth associates with the notion of a solar-system-size clump of stuff that associates essentially with just one isomer. Our solar system is not spatially spherical.

We suggest that many other such one-isomer, not-spatially-spherical, somewhat-solar-system-sized clumps exist and generally date to early in the history of the universe. We suggest that such clumps associate with nonzero moments of inertia. We suggest that, during some period early in the history of the universe, quadrupole gravitational attraction dominated regarding interactions between neighboring same-isomer, solar-system-sized clumps. Table 5 suggests that a reach per instance of one isomer pertains. We suggest that solar-system-sized clumps clumped to form the halos of galaxies.

We suggest that the discussion above explains 1:0+ ratios that pertain to some early galaxies and 0+:1 ratios that pertain to some early galaxies.

We suggest 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, we suggest the following scenario. The galaxy started as a 0+:1 galaxy. Table 5 suggests that a reach per instance of two isomers pertains for dipole interactions. Reach-2 dipole push contributions to gravity drove away some ordinary-matter stuff and the stuff that associated with one dark-matter isomer. That one dark-matter isomer is isomer-3. Table 5 suggests that a reach per instance of six isomers pertains for monopole interactions. 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, we suggest the following scenario. The galaxy started as a not-isomer-3 1:0+ galaxy. Table 5 suggests that a reach per instance of two isomers pertains for dipole interactions. Reach-2 dipole push contributions to gravity drove away some dark-matter stuff but essentially no ordinary-matter stuff. Table 5 suggests that a reach per instance of six isomers pertains for monopole interactions. 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. We suggest that many 5+:1 galaxies resulted from mergers of smaller, previous galaxies. We suggest that such mergers associate with monopole gravitational attraction. Table 5 suggests that a reach per instance of six isomers pertains. We suggest that the earliest mergers that led to a 5+:1 galaxy could have been mergers that involved 1:0+ galaxies and 0+:1 galaxies.

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

This unit suggests that our work provides a candidate quantitative explanation for the fives 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.

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

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

This unit suggests that our work provides a candidate qualitative explanation for 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.

We suggest that nature might include electromagnetic properties (of objects) that would associate with reach-6. Table 5 provides a possible example. Based on reach-6 properties, early in the evolution of the universe, more electromagnetic energy would flow from isomer-pair-0 stuff to isomer-pair-1 stuff than would flow from isomer-pair-1 stuff to isomer-pair-0 stuff. Similarly, based on reach-6 properties, early in the evolution of the universe, more electromagnetic energy would flow from isomer-pair-0 stuff to isomer-pair-2 stuff than would flow from isomer-pair-2 stuff to isomer-pair-0 stuff.

We suggest that the electromagnetic energy flow imbalances would result in the existence of more isomer-pair-1 stuff than isomer-pair-0 stuff and in the existence of more isomer-pair-2 stuff than isomer-pair-0 stuff. We suggest that the flows would not disturb a one-to-one ratio of presence of isomer-3 stuff to presence of isomer-0 stuff.

We suggest that these energy flow imbalances might have produced (or, at least, contributed to) 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 or with other popular modeling suggestions regarding the nature of dark matter.

3.5. Eras in the Rate of Expansion of the Universe

This unit suggests that our work provides candidate qualitative explanations for some observations regarding the rate of expansion of the universe and that our work might add insight regarding possible eras, in the evolution of the universe, for which there might not be much data.

Popular modeling suggests two observed multibillion-year eras regarding the so-called rate of expansion of the universe [61,62,63,64]. 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 [65,66] and that a new era, which would associate with a positive rate of expansion that decreases as time increases, might be starting.

We suggest that those two or three eras associate with the moving apart from each other of similarly sized neighboring, but not colliding, large objects. We suggest that above-discussed notions regarding cosmic clumping of stuff pertain.

We suggest that, generally, similarly sized neighboring, but not necessarily colliding, galaxies transited long ago from an era of gravitational dipole push to an era of gravitational monopole pull. We suggest that, generally, gravitational monopole pull dominates interactions between non-colliding galaxies within galaxy clusters.

We suggest that the rate of expansion of the universe associates with interactions between neighboring non-colliding galaxy clusters. We suggest that the start of the first multibillion-year era associates with a transition to dominance, regarding interactions between neighboring non-colliding galaxy clusters, by gravitational quadrupole attraction (or, pull). We suggest that the start of the second multibillion-year era associates with a transition to dominance, regarding interactions between neighboring non-colliding galaxy clusters, by gravitational dipole repulsion (or, push).

Table 6 suggests possible explanations for some phenomena regarding the evolution of the universe.

There may be no data that directly pertain to before or during the would-be inflationary epoch [67,68,69,70,71]. Our notions regarding before inflation, early in the inflationary epoch, and after the later multibillion-year era might comport with some popular modeling notions of bounce cosmology [72,73].

Regarding dominance by hexadecapole gravitational attraction, one might need to suggest a non-gravitational mechanism that associates with a transition from a Big Crunch [74] to a Big Bang. Without a notion such as the notion of a bounce based on fermion particles and the Pauli-exclusion principle, hexadecapole pull might lead toward effects that would associate with 32-pole aspects. The word dotriacontapole and the word triacontadipole associate with 32-pole. Assuming that hexadecapole gravitational attraction pertained, we suggest that the reach per instance for each one of hexadecapole gravitational attraction and octupole gravitational repulsion might be six isomers. Consequences of those reaches being six isomers might associate with notions that, throughout the known evolution of the universe, large-scale densities of stuff tend to be, to a first approximation, spatially uniform [75].

3.6. Recap of How Our Methods Seem to Explain Otherwise Unexplained Cosmic Data

This unit recaps phenomena that our work seeks to and seems to help explain. This unit points to some aspects of our work that contribute to the seeming explanations.

The following items recap phenomena that our work seeks to and seems to help explain. Each item names phenomena, notes an approximate characterization of the explanation that we suggest, and points to one or two relevant bases for the explanation. DM:OM denotes ratios of dark-matter effects to ordinary-matter effects. CMB denotes cosmic microwave background radiation. Isomers refers to the notion that nature includes six similar, but not necessarily identical, instances of each one of most elementary particles. Extended Newtonian gravity refers to notions that lead to and include Table 3 and Table 4.

• Dark-matter elementary particles: Quantitative (Isomers).
• Dark-matter stuff: Quantitative (Isomers).
• DM:OM regarding some depletion of CMB: Quantitative (Isomers).
• Galaxy evolution and DM:OM regarding some galaxies: Quantitative (Isomers, Extended Newtonian gravity).
• DM:OM regarding some galaxy clusters: Quantitative (Isomers).
• DM:OM densities of the universe: Quantitative (Isomers).
• Eras in the rate of expansion of the universe: Qualitative (Isomers, Extended Newtonian gravity).
• Gravitational phenomena, including so-called dark-energy: Qualitative (Isomers, Extended Newtonian gravity).

4. Discussion

This unit discusses relationships between our work and popular modeling. This unit suggests opportunities for observational work and for enhancing popular modeling.

4.1. Extended Newtonian Gravity, Isomers, and General Relativity

This unit suggests similarities and differences between aspects of our work and aspects of general relativity.

Our explanations for data do not use general relativity.

Our work above regarding extended Newtonian gravity has bases in properties of individual objects. Popular modeling includes two properties, energy and momentum, of individual objects that our work above somewhat scarcely mentions. General relativity has bases in a stress-energy tensor that has bases in properties (such as energy density and momentum density) that associate with regions.

We suggest that popular modeling might want to consider the extent to which each one of the following four statements pertains regarding gravitation and general relativity. Mass (as in gravitational $n_r=n_{3s}=1$) associates with general relativistic notions of energy density. Mass current (as in gravitational $n_r=1$ and $n_{3s}=2$) associates with general relativistic notions of energy flux and of momentum density. Mass whirl-or-jitter (as in gravitational $n_r=n_{3s}=2$) associates with general relativistic notions of pressure. Mass-whirl-or-jitter current (as in gravitational $n_r=2$ and $n_{3s}=3$) associates with general relativistic notions of shear stress and momentum flux.

More generally, popular modeling might want to consider the extent to which the following four statements pertain regarding gravitation and general relativity. Regarding two-body gravitational properties, the pair $n_{3s}\leftrightarrow odd$ and $n_{3s}=n_r$ associates with general relativistic notions of energy density. Regarding two-body gravitational properties, the pair $n_{3s}\leftrightarrow even$ and $n_{3s}=n_r+1$ associates with general relativistic notions of energy flux and of momentum density. Regarding two-body gravitational properties, the pair $n_{3s}\leftrightarrow even$ and $n_{3s}=n_r$ associates with general relativistic notions of pressure. Regarding two-body gravitational properties, the pair $n_{3s}\leftrightarrow odd$ and $n_{3s}=n_r+1$ associates with general relativistic notions of shear stress and momentum flux.

To the extent that the possible associations between extended Newtonian gravity and the general-relativity stress energy tensor pertain, the following statements might pertain. Our work suggests that general relativity might not be adequately accurate regarding circumstances for which at least one of the following sentences pertains. Non-gravitational effects, such as electromagnetic effects or chromodynamics effects, are significant. The isomeric composition of stuff varies significantly between regions. Significant (or dominant) effective reaches per instance vary with time.

Our suggested cautions about possible lacks of adequate accuracy regarding general relativity might not pertain significantly regarding circumstances for which general relativity has passed so-called precision tests [76]. For example, popular modeling regarding the perihelion precession of the orbit of the planet Mercury associates with just one isomer, namely isomer-0. Popular modeling of the deflection of light by the Sun also associates with the ordinary-matter isomer and with no other isomers.

4.2. Some Possible So-Called Cosmic Tensions

This unit discusses examples, such as the so-called Hubble tension, of possible mismatches between cosmic data and popular modeling. This unit discusses possibilities that, assuming further data does not resolve the tensions, our work points qualitatively to sources of the tensions.

Popular modeling uses the word tension to describe some possible mismatches between popular modeling and cosmic data. Some popular modeling suggests that early-multibillion-year-era values for the Hubble constant are not compatible with later-multibillion-year-era values for the Hubble constant [77]. Popular modeling associates the two-word term Hubble tension with this possible mismatch. Some popular modeling suggests that popular modeling overestimates large-scale lumpiness of stuff [78,79,80,81,82,83,84,85]. Popular modeling sometimes associates the two-element phrase S8 tension with some such possible overestimates.

Recent observations suggest that some such gaps between data and popular modeling might be smaller than previously thought or might be essentially nonexistent. Some recent observations push back against the notion of a Hubble tension [86]. Some recent observations push back against the notion of an S8 tension [87].

Our work suggests qualitative explanations for such possible mismatches. We suggest that the explanations might point toward total or partial closings, but not toward wider openings, of the possible gaps between modeling and data. Lacking means to quantify the would-be explanations, we do not try to suggest the usefulness of the explanations and we do not try to estimate the extents to which might-be-tension-related data might tend to confirm or refute our work.

4.2.1. The Hubble Tension

This unit suggests that our work might provide a candidate qualitative explanation for possible mismatches, known as the Hubble tension, between data and popular modeling regarding some large-scale phenomena.

Popular modeling estimates for a Hubble constant that would associate with the early universe might suggest a Hubble constant that is significantly less than estimates for a Hubble constant that would associate with the recent universe. Estimates tend to associate [88] with equation (24). c denotes the speed of light. $\rho$ denotes energy density and is nonnegative. P denotes pressure and is nonpositive.

$$-(c\rho +3P)$$

(24)

Popular modeling suggests that an increasing rate of expansion associates with equation (24) evaluating to a positive number. Popular modeling suggests that a decreasing rate of expansion associates with equation (24) evaluating to a negative number.

Table 6 suggests that interactions between galaxy clusters provide a basis for the rate of expansion. Regarding around the beginning of the multibillion-year era of decreasing rate, attraction (between neighboring non-colliding galaxy clusters) that associates with $n_r=n_{3s}=3$ two-body gravitational pull dominates regarding changes in the rate of expansion. Regarding around the beginning of the multibillion-year era of increasing rate, repulsion (between neighboring galaxy clusters) that associates with $n_r=n_{3s}=2$ two-body gravitational push dominates regarding changes in the rate of expansion.

Per discussion above, our notions of two-body-interaction reaches might apply to stress-energy-tensor components such as pressure. Here, our discussion assumes that notions of reaches pertain regarding pressure.

Table 7 suggests dominant contributions to pressure, relevant to popular modeling rate-of-expansion calculations, at selected times in the evolution of the universe. In Table 7, each one of the two numerically specified pairs of one $n_{in}$ and one $R_{/in}$ comports with Table 5 and with data about galaxy evolution.

We suggest that a reach per instance of one for pressure might associate with the start of the multibillion-year era of decreasing rate of expansion of the universe. We suggest that a reach per instance of two for pressure might associate with the start of the multibillion-year era of increasing rate of expansion of the universe. Based on such an increasing-with-time-from-one-to-two effective reach, we suggest that extrapolating from so-called early-universe pressures might lead to underestimations of so-called later-universe pressures. Underestimations of later-universe pressures might associate with possible underestimations of later-universe values of the Hubble constant.

4.2.2. Lumpiness Tensions

This unit suggests that our work might provide a candidate qualitative explanation for some possible mismatches, including a so-called S8 tension, between data and popular modeling regarding some large-scale phenomena.

We suggest that large-scale lumpiness associates, at least in part, with an $n_r=n_{3s}=4$-through-$n_r=n_{3s}=2$ sequence that we suggest above regarding galaxy evolution and regarding the rate of expansion of the universe. We suggest that popular modeling might underestimate repulsion that we associate with $n_r=n_{3s}=2$. We suggest that the underestimate might associate with possible overestimates, by popular modeling, of large-scale lumpiness.

4.3. Relationships Among Our Work, Data, and Popular Modeling

This unit summarizes, in a context of present data and popular modeling, aspects of our work.

Compared to state-of-the-art popular modeling, our work seems to offer the following strengths. Our work explains some otherwise unexplained data. Our work offers additional insight into galaxy formation. Our work offers a better-defined candidate specification for dark matter. Our work offers a perhaps more-promising basic description of two-body gravity. Our work might better identify possible bases for possible tensions between data and popular modeling.

Compared to state-of-the-art popular modeling, our work might seem to exhibit the following weaknesses. People have yet to develop means to perform calculations and simulations based on our work. Our work does not yet say much about contributions, to two-body gravitational interactions, for which neither property is mass. Our work does not yet include a many-body-physics analog to general relativity.

Our work seems to offer the following opportunities. Advance scientific understanding regarding dark matter, gravitation, and galaxy formation. Develop quantum gravitation notions that parallel quantum electrodynamics notions. Advance techniques regarding modeling and simulations.

4.4. Suggestions Regarding Cataloging Types of Cosmic Data That Physics Collects

This unit suggests means for characterizing and cataloging types of cosmic data that physics collects.

Table 2 and Table 3 use $n_r$ and $n_{3s}$ to catalog some two-body electromagnetic properties of objects and some two-body gravitational properties of objects.

We suggest that case-e3 and case-g3 point to possible advantages for using notions from characterization mathematics [89,90], along with or in place of using some popular modeling vocabulary, when discussing properties of objects. For example, the characterization gravitational property plus $n_r=2$ plus $n_{3s}=2$ might be more useful for some circumstances than the phrase object-internal angular momentum.

We suggest that popular modeling might want to consider using types of interactions, values of $n_r$, values of $n_{3s}$, and values of $R_{/in}$ to characterize observations.

Cosmological redshifts provide an example. Popular modeling suggests that the motion of a source of light relative to an observer of that light can pertain. One might characterize such so-called Doppler redshift effects as electromagnetic, with $n_{3s}=n_r+1$ (or, velocity-related), with $n_r$ not necessarily having relevance, and with $R_{/in}=1$. Popular modeling suggests that interactions between light and gravitational fields can pertain. One might characterize such so-called gravitational redshift effects as gravitational, with $n_{3s}=n_r$ (or, not-necessarily velocity-related), with $n_r$ having relevance, and with $R_{/in}$ possibly having relevance.

One might speculate regarding the possible relevance of object-properties for which $n_{3s}\ge n_r+2$. $n_{3s}=n_r$ associates with relative spatial position. $n_{3s}=n_r+1$ associates with relative spatial position and with relative velocity. By extrapolation, $n_{3s}=n_r+2$ might associate with relative spatial position, with relative velocity, and with relative acceleration. Based on (for example) time-lapse observations or multi-messenger observations or multi-object simulations, an object-P might be able to infer an object-A retarded-time charge acceleration $a_{oA.rel.to.oP}$.

4.5. Suggestions for Observational Work

This unit suggests some opportunities, to which our work points, for verifying or refuting aspects of our work and for pinpointing opportunities for observational or experimental work.

We suggest considering possible synergies between galaxy-evolution studies and our work. To what extent does our work comport with early-universe galaxy-formation and galaxy-evolution? Did nature form at least as many 1:0+ (or, dark-matter) galaxies as 0+:1 (or, ordinary-matter) galaxies? To what extent does it seem reasonable that some of today’s 1:0+ galaxies and 0+:1 galaxies maintained those ratios from early in the evolution of the universe? To what extent do our suggested scenarios for the formation of ∼4:1 galaxies comport with nature?

We suggest considering observational work, regarding the aftermath of galaxy-cluster collisions, that could help verify or refute our notions of significantly-electromagnetically-active dark matter. To what extent does isomer-3 intergalactic medium exist? To what extent does isomer-3 intergalactic medium lag other dark-matter stuff? More specifically, our specification for dark matter might not necessarily be incompatible with data about collisions, such as the Bullet Cluster collision, of two galaxy clusters [91,92]. Popular modeling suggests two types of trajectories for stuff. Most dark matter, from either one of the clusters, exits the collision with trajectories that are consistent with having interacted just gravitationally with the other cluster. Ordinary-matter stars, from either cluster, exit the collision with trajectories that are consistent with having interacted just gravitationally with the other cluster. However, ordinary-matter intergalactic medium, from either cluster, lags the cluster’s ordinary-matter stars and dark matter. Popular modeling suggests that the ordinary-matter intergalactic medium interacted electromagnetically with the other cluster’s ordinary-matter intergalactic medium, as well as gravitationally with the other cluster. We suggest that the stuff that associates with four isomers, namely isomer-1, isomer-2, isomer-4, and isomer-5, does not form much electromagnetically active intergalactic medium. We suggest that further analysis of data might help determine the validity of our notion that the stuff that associates with isomer-3 forms electromagnetically interactive intergalactic medium.

We suggest considering observational (and perhaps even experimental) work that could help verify, refute, or extend a table such as Table 5. Which reach pertains for hyperfine phenomena? Which reach or reaches pertain for atomic transitions that are not hyperfine transitions? To what extent might data suggest a reach for magnetic moment? To what extent do the reaches for $n_{3s}+1$ current properties comport with the reaches for the counterpart $n_{3s}$ properties?

We suggest considering experimental work that might detect dark matter of the types that we suggest.

We suggest considering work that would test general relativity for circumstances in which our work would suggest that dominant isomeric compositions change spatially. To what extent do gravitational-wave signatures from collisions differ between collisions of pairs of same-isomer-based neutron stars and collisions of pairs of different-isomer-based neutron stars?

We suggest considering work that would test general relativity for circumstances in which our work would suggest that dominant reaches change temporally or spatially. To what extent do gravitation-wave signatures from collisions differ among collisions of pairs of same-isomer-based neutron stars, collisions of pairs of different-isomer-based neutron stars, and collisions of pairs of supermassive black holes (for which each black hole might include roughly equal presences stuff that associates with each one of the six isomers)?

4.6. Suggestions for Enhancing Popular Modeling

This unit suggests some opportunities, to which our work points, for enhancing popular modeling.

We suggest considering how to evolve popular modeling to embrace notions that, while much dark matter might qualify as collisionless dark matter, significant amounts of dark matter might not qualify as collisionless dark matter.

We suggest considering how to incorporate notions regarding instances and reaches per instance into popular modeling and into numerical simulations.

We suggest considering the extent to which popular modeling might benefit by using the series gravitational scalar potential, gravitational vector potential, gravitational two-tensor potential, and so forth.

We suggest considering details regarding how to implement multipole expansions that feature potentials that associate with multiple properties, such as charge and magnetic moment or such as mass and object-internal angular momentum. We note, as an aside, that equation (25) might point [7,8] to some relationships between energies that associate with properties of the three known nonzero-mass boson elementary particles. S denotes the spin (or, object-internal angular momentum) of a particle divided by the spin of the Z boson or by the spin of the W boson. $M^{\prime }$ denotes the rest mass of a particle divided by one-third the rest mass of the Z boson. Q denotes the charge of a particle divided by the magnitude of the charge of a W boson. Data [93] do not rule out equation (25). We note, as a second-level aside, that, in some sense, the right-hand side of equation (25) might suggest that spin detracts from mass and that charge detracts from mass.

$${(4-S)}^2+1={\left(M^{\prime }\right)}^2+S^2+Q^2$$

(25)

We suggest considering how to integrate into popular modeling (for electromagnetism and for gravitation) methods that we use to develop and apply extended Newtonian gravity. We suggest considering how to extend such methods to apply to torques (as well as to forces).

We suggest considering the notion that transitions from gravitational-octupole push to gravitational-quadrupole pull to gravitational-dipole push to gravitational-monopole pull associate with or might extend popular modeling notions of phase transitions. We suggest considering the notion that such transitions seem to scale from interactions between neighboring non-colliding solar-system sized objects to interactions between neighboring non-colliding larger objects.

5. Conclusion

This unit summarizes key aspects of this study.

We suggest details about dark matter and gravity.

The details associate with suggested reuses of and suggested extensions to familiar popular modeling. For example, regarding dark matter, we reuse, with variations regarding details, a list of known elementary particles. Also, regarding gravity, we reuse, with variations, notions of multipole expansions.

Our work quantitatively explains dark-matter phenomena that seemingly no other work explains. Our work suggests insight about galaxy formation and about the large-scale evolution of the universe.

The data that we explain pertain to galaxies, galaxy clusters, cosmic microwave background radiation, the expansion of the universe, and densities of the universe. Based in part on the breadth of the data that we explain, people might find our explanations credible and compelling.

We suggest opportunities for new uses of extant data, for new observational work, and for upgrades to popular modeling.