Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture

Author(s)
Jody Geiger

ABSTRACT

A physical description of the orbital mechanics of stars around a galactic core has proved difficult. Notably, there is insufficient mass to account for observed star velocities. The mystery is one of few in modern science that defy the known laws of physics. It has been conjectured that there is a new form of matter that interacts gravitationally while otherwise remaining undetectable. In this paper we resolve the mystery. The expressions do not modify the known laws of physics, contain no free variables or fitting and are entirely classical in nature. Using the notion of counts of the fundamental measures—length, mass and time—it is shown that measure is bounded. Accounting for this bound and the expansion of space reveal that the conjecture is unnecessary thus resolving the dark matter mystery.

A physical description of the orbital mechanics of stars around a galactic core has proved difficult. Notably, there is insufficient mass to account for observed star velocities. The mystery is one of few in modern science that defy the known laws of physics. It has been conjectured that there is a new form of matter that interacts gravitationally while otherwise remaining undetectable. In this paper we resolve the mystery. The expressions do not modify the known laws of physics, contain no free variables or fitting and are entirely classical in nature. Using the notion of counts of the fundamental measures—length, mass and time—it is shown that measure is bounded. Accounting for this bound and the expansion of space reveal that the conjecture is unnecessary thus resolving the dark matter mystery.

KEYWORDS

Dark Energy, Dark Matter, Expanding Universe, Galaxy Dynamics, Galaxy Rotation, Gravitational Equilibrium

Dark Energy, Dark Matter, Expanding Universe, Galaxy Dynamics, Galaxy Rotation, Gravitational Equilibrium

1. Introduction

The focus of this paper will be a discussion of galactic rotation and the processes that affect and constrain gravity at galactic scales. The effect is an outcome of physically significant smallest units of measure, each of the three measures constrained by an upper and lower count bound with respect to the remaining two. A framework of countable units of measure―the fundamental measures―provides a mathematical foundation with which to describe phenomena with quantum precision. Most importantly, when a count bound is exceeded additional mass counts overlap; this is what constrains gravity at the galactic scale.

The idea of units of measure is most commonly known as Planck’s Units which we denote with a subscript p: length l_{p}, mass m_{p}, and time t_{p}. Planck’s Units differ slightly from that resolved with measurement quantization, the latter units referred to as fundamental units and distinguished with a subscript f.

By first describing gravity using the Pythagorean Theorem an approach to bounded measure may be applied. Resolving the upper bound to mass counts with respect to counts of the remaining two measures allows us to describe galactic orbital dynamics. Each relation is tightly constrained, a function of constants. When applied to the Milky Way, the minimum mass density, the crossover point between Newtonian and non-Newtonian behavior and the associated mass and velocity curves are resolved. The expansion of space is also integrated. Most importantly, a classical description is presented that does not require the presence of dark matter.

After applying the approach to existing Milky Way data, the expressions are then modeled with an even mass distribution to demonstrate what an average of thousands of galaxies would look like. As expected, orbital velocities flat-line. The magnitude of that velocity is correlated to the excess mass above the mass frequency bound (i.e. the upper count bound of mass with respect to time).

The presentation addresses the ΛCDM [1] dark matter distribution presently considered the leading candidate with respect to this phenomenon. Expressions for each distribution are presented, but ΛCDM is not used to resolve the distribution values. Instead measurement quantization [2] is used; an approach which differs from the Standard Model only in that it recognizes the physical significance of smallest units of measure. The advantage of this approach is that a base expression with no free variables may be resolved. The approach allows an inspection that resolves a concise understanding of distribution traits and differences.

Also addressed are existing proposals. For one, MOND models have provided a good correlation with observed star velocities. Alternatives such as that used by McCulloch modify inertial mass by assuming it is caused by Unruh radiation [3] . Each of these approaches incorporates some element of data dependence, but it is their dependence on less established mass distribution and expansion expressions (i.e. ΛCDM) that present conflicts. The expressions herein clarify the physical description of each mass distribution and why existing applications to galactic phenomena are in conflict.

Yet other approaches to describing dark matter may be demonstrated with extended theories of gravity, but importantly that landscape has increasingly been mitigated as a result of several runs at the LHC. In Corda’s paper, “Interferometric detection of gravitational waves: the definitive test for General Relativity” [4] , the field is further defined and reduced, specifically where Corda has presented constraints as a function of the interferometer response functions of a gravitational wave event. With respect to each of these observations we begin with a new approach to describing gravity that successfully avoids each of the concerns noted above.

2. Methods

2.1. Quantum Gravity

Quantum gravity is a consequence of measurement quantization. Informativity―a term that describes the application of measurement quantization to the description of phenomena―rests on evidentiary support for the physical significance of fundamental units of measure. This property of observation differs from what might be understood with respect to observations first proposed by Planck. Specifically, the fundamental measures do not imply that nature is discrete, only that measure is discrete. Thus, while nature is infinitely divisible in length, mass, and time, there are physically significant count bounds to what can be measured. Those bounds constrain the behavior of matter as much as they constrain the behavior of gravity.

We will discuss the evidence only briefly and refer the reader to the paper “Measurement Quantization Unites Classical and Quantum Physics” [2] for a more complete treatment of the subject. We also refer the reader to the paper “Measurement Quantization Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data” [5] for examples of the application of measurement quantization to the distortion of measure, quantum inflation, the transition event that ends quantum inflation, initiates expansion and marks the formation of a Cosmic Microwave Background (CMB). For those familiar with these papers you may skip directly to Section 3.

For those new to Informativity, we will review gravity as described in the first paper [2] . We begin with the premise, that there exists a physically significant smallest unit of measure (i.e. a reference), which will then be supported with observational data. A reference is the source thing used to ascertain and describe some other thing. By example, the fundamental measure of length l_{f} is the reference that may be used to describe any length. This is accomplished as a whole-unit count of the reference. A fractional count violates the definition of a reference indicating that the identified source is not the reference. In such a case, the newly identified target becomes the reference until no smaller candidates are found. We can describe this mathematically.

Consider that we wish to describe an unknown distance on side c of the triangle described in Figure 1 as a count of the reference. For long side c and short sides a = 1 (the reference) and b (a count of the reference) of any chosen integer count of a right-angle triangle, we may resolve a count representing the uncertain distance,

$c={\left(1+{n}_{Lb}^{2}\right)}^{1/2}{l}_{f}$ (1)

Any non-whole-unit count describes a change in distance and may be described by rounding up (repulsion) or down (attraction). The remainder lost to rounding will be denoted by Q_{L}. Notably, Q_{L} is less than half and thus attractive. The model describes a count of the reference that is closer by

${Q}_{L}={\left(1+{n}_{Lb}^{2}\right)}^{1/2}-{n}_{Lb}$ , (2)

Figure 1. Count of distance measures between an observer and target where n_{b} = 4.

at every instant in time. To demonstrate the math, if n_{Lb} = 4, then
${Q}_{L}/{n}_{Lb}=\left(\sqrt{17}-4\right)/4=0.1231/4$ . Because side c always rounds down, we find that n_{Lr} always equals n_{Lb}. As such, we will always refer to the “observed measure count” as n_{Lr}. Moreover, note that the reference measure against which all counts are measured is defined by n_{La} = 1. With this we conjecture that we have composed an expression for gravity such that the loss of the remainder relative to the whole-unit count is Q_{L}/n_{Lr}.

We proceed with that hypothesis by presenting the ratio in meters per second squared (m/s^{2}). We multiply by l_{f} for meters and divide by
${t}_{f}^{2}$ together describing the distance loss at the maximum sample rate of one sampling every t_{f} seconds per second,

$\frac{{Q}_{Lf}{l}_{f}}{{n}_{Lr}{t}_{f}^{2}}$ , (3)

Also note that this quantity is scaled and hence requires a scaling constant; we multiply by the speed of light c and divide by a scaling constant S. Setting r = n_{Lr}l_{f} and c = l_{f}/t_{f}, then

$\frac{{Q}_{L}{l}_{f}}{{n}_{Lr}{t}_{f}^{2}}\frac{c}{S}=\frac{{Q}_{L}{c}^{2}}{{n}_{Lr}{t}_{f}S}=\frac{{Q}_{L}{l}_{f}{c}^{2}}{{n}_{Lr}{l}_{f}{t}_{f}S}=\frac{{Q}_{L}{c}^{3}}{rS}$ . (4)

The ratio c/S may be understood as 1/kg or a maximum count of m_{f} per kilogram; it may also be thought of as the corresponding mass frequency associated with gravity. Where S = 3.26239, this expression is now equivalent to G/r^{2} to five significant digits for all distances greater than 10^{3}l_{f}. Where quantum differences are not a consideration, we may set the expression equal to G/r^{2} and thus

$\frac{{Q}_{L}{c}^{3}}{rS}=\frac{G}{{r}^{2}}$ , (5)

${Q}_{L}r{c}^{3}=GS$ . (6)

The expression may be reduced where the
${\mathrm{lim}}_{r}{}_{\to \infty}f\left({Q}_{L}{n}_{Lr}\right)=1/2$ as demonstrated in Appendix A.1. Such that r = n_{Lr}l_{f} then the expression becomes

$\frac{{c}^{3}}{G}=\frac{S}{{Q}_{L}r}=\frac{S}{{Q}_{L}{n}_{Lr}{l}_{f}}=\frac{2S}{{l}_{f}}$ . (7)

Our focus now turns to the scaling constant. What is it and how do we measure it? There are two physically significant phenomena where S may be measured. First, we may measure S as momentum; hence the units for these expressions will match accordingly. But, as described in Figure 2 S is also an angular measure and described by the expression

$S=\frac{{l}_{f}}{2}\left(\frac{{c}^{3}}{G}\right)=\frac{{l}_{f}}{2}\left(\frac{\hslash}{{l}_{f}^{2}}\right)=\frac{\hslash}{2{l}_{f}}$ (8)

It follows where S = ħ /2l_{f}, then the arc length of a circle of radius l_{f} and angle S is

$L=r\theta ={l}_{f}\left(\frac{\hslash}{2{l}_{f}}\right)=\frac{\hslash}{2}$ . (9)

with respect to this reference, the units for S are radians. Explicitly, the units for S depend on the frame of reference. For this reason we use θ_{si} throughout all Informativity expressions, not because the term always denotes a radian measure, but to emphasize that the value of θ_{si} is invariant for all frames of reference. The subscript s and the subscript i exist for historical purpose denoting the signal and the idler measures in the Shwartz and Harris quantum entanglement experiments [6] , both of which are precisely 3.26239 radians.

When θ_{si} is described with respect to other measures in the local inertial frame, either units of momentum or radians will apply depending on what is being measured. When θ_{si} is described with respect to a measurement bound (i.e. the age or diameter of the universe), the term is dimensionless. This is most evident in a unity expression for which an example will be presented later. In each case, the value of θ_{si} is the same. Most expressions are with respect to a bound, but where there is exception notes will be provided. A more complex example is examined in Appendix A.3.

Lastly, a notable example of cross-referenced expressions, combine both the momentum and angular expressions. Planck’s expression for Planck’s length is then

Figure 2. Arc length of a circle of radius l_{f} and subtending angle θ = S radians.

$\frac{{l}_{f}{c}^{3}}{2G}=\frac{\hslash}{2{l}_{f}}$ , (10)

${l}_{f}={\left(\frac{\hslash G}{{c}^{3}}\right)}^{1/2}$ . (11)

Planck’s mass and time expressions are also in the same class of cross-referenced expressions and as such all of his unit work is a derivative of two frames of reference. Mixing frames of reference may seem inappropriate. But, doing so also offers physically significant descriptions of nature. With that, care must be taken with each Informativity expression to track units and resolve them.

Evidence does not rest on one or even several experimental results. There are, at present, more than 20 verifiable predictions of the model [2] [5] in disciplines that include quantum physics (Table 1), quantum gravity Equation (6), classical physics, the distortion of measure (i.e. also described by relativity) ( [5] , Section 3.1), quantum inflation ( [5] , Section 3.14 - 3.15), expansion ( [2] , Section 3.12), and cosmology ( [2] , Section 3.10). One measure of θ_{si} is published in Shwartz and Harris’s 2011 paper, “Polarization Entangled Photons at X-Ray Energies” in Physical Review Letters [6] . Using Informativity, their measures can be described to the same precision as calculated in Table 1.

Most importantly, in recognition of physically significant units of measure, Informativity provides an approach that mathematically correlates measurement quantization to gravity. It follows, where bounds to measure are found, a corresponding bounding effect must also be found with respect to gravity. In the next section we will further explore the reference measures to build a toolset necessary for describing how gravity is bounded.

2.2. Fundamental Measures

The physical significance of fundamental units of measure is instrumental to describing galactic rotation. It is because the fundamental units are countable, having a smallest and greatest count with respect to the remaining two measures that gravity is constrained. A review of the fundamental units, their values and definitions provide the foundation for the expressions to follow. Thus, with Equation (7) and a measured value of θ_{si} equal to 3.26239 each of the fundamental measures can be resolved. When defined with respect to the fundamental measures, the units for θ_{si} are that of momentum kg×m/s. Thus,

Table 1. Angle setting in radians of the k vectors of the pump, signal, and idler for maximally entangled states at the degenerate frequency with corresponding Shwartz and Harris values (Ref. [6] ).

${l}_{f}=\frac{2G{\theta}_{si}}{{c}^{3}}=\frac{2\times 6.67408\times {10}^{-11}\times 3.26239}{{299792458}^{3}}=1.61620\times {10}^{-35}\text{\hspace{0.17em}}\text{m}$ , (12)

${t}_{f}=\frac{{l}_{f}}{c}=\frac{2G{\theta}_{si}}{{c}^{4}}=\frac{2\times 6.67408\times {10}^{-11}\times 3.26239}{{299792458}^{4}}=5.39106\times {10}^{-44}\text{\hspace{0.17em}}\text{s}$ , (13)

${m}_{f}={t}_{f}\frac{{c}^{3}}{G}=\frac{2{\theta}_{si}}{c}=\frac{2\times 3.26239}{299792458}=2.17643\times {10}^{-8}\text{\hspace{0.17em}}\text{kg}$ . (14)

To describe a count of l_{f}, m_{f} and t_{f} with respect to time divide the rate by the respective measure.

${n}_{L}=2.99792458\times {10}^{8}/{l}_{f}=1.85492\times {10}^{43}\text{units}/\text{s}$ , (15)

${n}_{M}=4.0371111\times {10}^{35}/{m}_{f}=1.85492\times {10}^{43}\text{units}/\text{s}$ , (16)

${n}_{T}=1/{t}_{f}=1.85492\times {10}^{43}\text{units}/\text{s}$ . (17)

The term mass frequency as used throughout describes a count of mass units relative to a count of time units. The upper count bound of mass units per second is 1.85492 × 10^{43}. The same count applies also to length frequency and frequency, the rate of time itself. Mass-to-length frequency is distinctly different and important to an understanding of galactic orbital dynamics.

Another often used expression in Informativity is the fundamental expression. This may be resolved from Equation (14) m_{f} = 2θ_{si}/c where c = l_{f}/t_{f},

${l}_{f}{m}_{f}=2{\theta}_{si}{t}_{f}$ . (18)

Lastly, while we have demonstrated the importance of θ_{si} in describing gravity, in resolving the fundamental units, in describing momentum, in defining Planck’s constant and in resolving Planck’s Unit expression for length, we haven’t specifically discussed the evidence for physical significant measure. To that end, consider Heisenberg’s Uncertainty Principle where applied to the position and momentum of a particle. Such that r = n_{Lr}l_{f} multiplied by Q_{L}n_{Lr} (i.e.
${\mathrm{lim}}_{r\to \infty}f\left({Q}_{L}{n}_{Lr}\right)=1/2$ ) to place distance measure in quantum form, m = n_{M}θ_{si}/Q_{L}n_{Lr}c generalized from the fundamental expression and v = n_{L}l_{f}/n_{T}t_{f}, then Heisenberg’s expression may be reduced to the counts n_{L}, n_{M}, n_{T}, and the length count between a target and a center of mass n_{Lr} such that

${\sigma}_{X}{\sigma}_{P}\ge \frac{\hslash}{2}$ , (19)

$f\left(r\right)f\left(mv\right)=\left({n}_{Lr}{l}_{f}2{Q}_{L}{n}_{Lr}\right)\left(\frac{{n}_{M}{\theta}_{si}}{{Q}_{L}{n}_{Lr}c}v\right)\ge {\theta}_{si}{l}_{f}$ , (20)

$\left(2{n}_{Lr}\right)\left(\frac{{n}_{M}v}{c}\right)\ge 1$ , (21)

$\left(2{n}_{Lr}\right)\left({n}_{M}\frac{{n}_{L}{l}_{f}}{{n}_{T}{t}_{f}}\frac{{t}_{f}}{{l}_{f}}\right)\ge 1$ , (22)

$2{n}_{M}{n}_{Lr}{n}_{L}\ge {n}_{T}$ . (23)

Before parsing, consider that gravity may be described as a loss of the fractional length count Q_{L} above and beyond the whole-unit count of the reference. The description of Q_{L} arises from the Pythagorean Theorem, a mathematical description of the measure of length. Thus, the interpretation implies two qualities. Firstly, nature is infinitely divisible or at least to the extent as described by all solutions to Q_{L}. Secondly, measure is discrete.

Consider now a description of light c = n_{L}l_{f}/n_{T}t_{f}, a whole-unit count of the reference such that
${n}_{L}={n}_{T}=1$ in Heisenberg’s reduced expression. It follows that the remaining counts of n_{M} = 1/2 and n_{Lr} = 1. The expression confirms the conjecture. Where we find support for the Heisenberg Uncertainty Principle, we also find the fundamental measures to be of physical significance, defining the threshold. The threshold between certainty and uncertainty is precisely at n_{M} = 1/2, n_{L} = 1 and n_{T} = 1 such that n_{Lr} = 1.

2.3. Nomenclature

Informativity uses a distinct nomenclature to describe length, mass, time, unit counts of those measures and the measure of several other quantities in the description of phenomena. Let us take this moment to discuss nomenclature.

The description of fundamental units with respect to the three measures are denoted as l_{f} for length, m_{f} for mass, and t_{f} for time. The description of counts of the fundamental measures is denoted with the symbol n, each measure recognized by a corresponding capitalized subscript, L for length, M for mass, and T for time. To avoid confusion between length descriptions of motion and those of gravitational fields, a subscript r (i.e. n_{Lr}) is used when describing a count of l_{f} between a static frame of reference and a center of gravity. Similarly, a subscript m (i.e. n_{Lm}) is used when describing a change in the count of l_{f} with respect to a target in motion to the observer.

With respect to mass distributions associated with the universe there are several categories. The total mass of the universe is distinguished with the term M_{tot}. The total may be divided into two parts, dark mass M_{dkm} and observable mass M_{obs}. The dark mass distribution is more commonly attributed to dark energy, but as presented in Section 3.1, is also the mass that can never be seen because it exists at such a distance that the expansion of the universe prevents light from ever reaching the observer.

Also important, if we now subtract the visible M_{vis} from the observable mass M_{obs}, we resolve that which will be observed, the unobserved mass M_{uobs}. The unobserved mass is that which will eventually be visible given sufficient elapsed time. The distribution is typically attributed to dark matter. There is also one more term, the fundamental mass M_{f}. This mass is associated with the mass frequency bound ( [2] , Equation (93))

${M}_{f}={A}_{U}{\theta}_{si}\frac{{m}_{f}}{{t}_{f}}$ , (24)

and is instrumental to the calculation of all the mass distributions. While the distribution values are the same as those resolved with ΛCDM, the two approaches differ significantly. The Informativity approach is an outcome, a prediction of Informativity implicit to physically significant quantized measure. There are no free variables and as such the precision is constrained only by the measure of θ_{si}, six significant digits. Each expression will be presented in Section 3.1 along with discussion as to their meaning, differences and why one may conclude that the only physical difference between the distributions is if and when mass is visible.

Lastly, the expansion of the universe can be described with respect to two measures. Stellar expansion, the measure of increasing distance between galaxies, follows the traditional understanding in modern theory. When discussing stellar expansion, we describe the effect using Hubble’s constant H_{o} which is quoted in kilometers per second per megaparsec. Conversely, universal expansion H_{U} describes the expansion of the universe when defined with respect to the universe. SI units are used, but the reference is fixed with respect to the age A_{U} and diameter D_{U} of the universe. Universal expansion describes an increasing space that is isotropic.

A listing of symbols used and there definitions may be found in Section 7.

2.4. Terminology

There are several terms often used when describing galaxies. As we have introduced the nomenclature for describing expansion, consider now the expression for universal expansion ( [2] , Equation (87)),

${D}_{U}=2{\theta}_{si}{A}_{U}=2\times 3.26239\times 13.799=90.035\text{\hspace{0.17em}}\text{bly}$ . (25)

The rate of expansion follows from definition 1/A_{U} and may be resolved in the customary units,

$H=\frac{\text{km}/\text{Mpc}}{{A}_{U}}=\frac{3.08567758\times {10}^{19}\text{\hspace{0.17em}}\text{km}/\text{Mpc}}{13.799\times {10}^{9}\text{\hspace{0.17em}}\text{y}\times \text{3}\text{.15576}\times {10}^{7}\text{\hspace{0.17em}}\text{s}/\text{y}}=70.860\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$ ,(26)

when defined with respect to the universe D_{U}/A_{U}, expansion is invariant ( [2] , Equation (81))

${H}_{U}=2{\theta}_{si}$ . (27)

Resolving a description of phenomena with respect to the universe can provide a perspective that is straight-forward with which to build a cohesive understanding of many presently unsolved physical phenomena.

We consider the universe, in this application, a frame of reference. As there is no outside reference to the universe, the universe is recognized as a self-defining frame. Terms that describe the universe are part of a class recognized as system bounds. For instance, the age and diameter of the universe describe the upper bound to elapsed time and length. Conversely, a thing defined relatively with respect to some other thing is called self-referencing. One’s choice of frame in no way identifies a physically significant difference. But, self-defining expressions are often invariant (i.e. H_{U} = 2θ_{si}). Self-referencing expressions often vary (i.e. H = ((km/Mpc)/A_{U}). And the units for θ_{si} depend on which frame is chosen.

While not as central to our discussion, it should be noted that the system constant 2θ_{si} is often present in physical description. The value is fundamental to the description of matter. For example, we may describe the expansion of the universe with respect to measure or as a function of 2θ_{si}. Use the fundamental expression to convert between them.

${\left({\left(\frac{{t}_{f}}{{l}_{f}{m}_{f}}\right)}^{1/3}\right)}^{2}+{\left(\frac{{n}_{Lm}}{{n}_{Lc}}\right)}^{2}=1$ , (28)

$\frac{1}{{\left(2{\theta}_{si}\right)}^{2/3}}+\frac{{n}_{Lm}^{2}}{{n}_{Lc}^{2}}=1$ . (29)

Many expressions are modifications of these unity expressions. There are two classes. Relations are expressions that may be reduced to the fundamental expression. Boundary expressions describe upper and lower count bounds relatively between measures.

It should not go unnoticed as to what anchors measure, the fundamental measures―(l_{f}m_{f}/t_{f}) = 2θ_{si}―or the corresponding rate of universal expansion n_{Lm}. This can be a difficult inquiry as measure is relatively defined. But their relation is fixed, distinguishing θ_{si} as perhaps the most fundamental constant. Many of the known constants may be reduced to include only θ_{si}, the fundamental measures or counts thereof. Several examples are ( [2] , Equation (36), Equation (49), Equation (81))

$\hslash =2{\theta}_{si}{l}_{f}$ , (30)

${E}_{f}=2{\theta}_{si}{l}_{f}/{t}_{f}$ , (31)

${H}_{U}=2{\theta}_{si}$ . (32)

As noted before, θ_{si} has units of kg×m×s^{−1} in the first two examples, but the later is a system bound and thus dimensionless. Conversely, the speed of light and the gravitational constant (see Appendix A.2) are examples of boundary expressions,

$c={l}_{f}/{t}_{f}$ , (33)

$G=\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{t}_{f}}{{m}_{f}}$ . (34)

Lastly, the terms, quantum, and, quantized, are often used. Neither should be understood as having a relation with respect to quantum mechanics. Rather, the term quantum is intended to mean small as in a few tens, hundreds or thousands of fundamental units of measure. The term quantized is intended to mean that expressions are composed of terms that are whole-unit counts of the fundamental measures.

A quantized expression possesses qualities that are immensely valuable in our effort to describe nature. For one, quantized expressions are defined for the entire measurement domain. Second, quantized expressions are nondimensionalized. Nondimensionalization is not in itself a valuable endeavor but demonstrating that all phenomena may be expressed entirely with nondimensionalized whole-unit counts of the fundamental measures contributes to a new understanding of measure that is finite and discrete.

A listing of terms used in Informativity may be found in Section 6.

3. Results

In the sections that follow we will use Informativity to present expressions describing the motion of stars in galaxies. As noted at the outset, when averaging hundreds or thousands of galactic rotational curves, the curve is nearly invariant at a given radius and outward. Star velocities are in conflict with Newton’s law of gravitation which describes a decreasing velocity with increasing distance.

A second anomaly concerns the magnitude of these velocities, a value that is significantly higher than expected. To describe these phenomena, incorporation of the effects of expansion and a new constraint to the behavior of matter will be entertained. While expansion is a seemingly straight-forward application, the constraint―mass frequency―is a new concept to modern theory. Like length frequency, c = l_{f}/t_{f}, mass frequency describes that bound where counts of m_{f} may no longer be distinguished, greater than 1.85492 × 10^{43} units per second, Equation (16).

The upper bound to mass frequency is physically significant and cannot be exceeded any more than a length frequency greater than 1 to 1 (i.e. n_{L}l_{f}/n_{T}t_{f} > c). As we work through an understanding of mass frequency we will demonstrate how counts above and beyond this bound correspond to measure smaller than the reference. Not only does a mass frequency above a frequency bound (i.e. a smaller value for m_{f} in the expression 1/m_{f}) describe a point in space-time subject to indistinguishable count of m_{f}, it also describes a faster-than-light relationship between length and time, identifiable using the fundamental expression, l_{f}m_{f} = 2θ_{si}t_{f} (i.e. a smaller value for m_{f} implies a larger value for l_{f} where c = l_{f}/t_{f} then a faster-than-light relation).

3.1. Mass Distribution

Galactic rotation follows classical theory with adjustments made for the effects described by relativity, the Informativity differential (Appendix A.1) and universal expansion. To simplify the expressions, the first two effects will not be integrated into the results. But, the third effect, expansion, is significant in magnitude. We begin with a review of expansion as described in the first paper followed by mass distribution.

Stellar expansion―the modern understanding of expansion―which is a function of universal expansion plus those forces of interaction since the earliest epoch will not be discussed. Universal expansion, conversely, describes the increasing space in the universe. The rate when defined with respect to the universe is Equation (27),

${H}_{U}=2{\theta}_{si}$ . (35)

The constant 2θ_{si} is referred to as the system constant. With it universal expansion may be described using familiar terms ( [2] , Equation (87)) such as the diameter D_{U} of the universe in billions of light-years and the age A_{U} of the universe in billions of years.

${D}_{U}=2{\theta}_{si}{A}_{U}=2\times 3.26239\times 13.799=90.035\text{\hspace{0.17em}}\text{bly}$ , (36)

with these parameters we may now summarize the mass distribution expressions starting with fundamental mass ( [2] , Equation (93)) which is then used to derive the distributions,

${M}_{f}={A}_{U}{\theta}_{si}\frac{{m}_{f}}{{t}_{f}}$ . (37)

Because our frame of reference is the universe, θ_{si} carries no units. A complete derivation is provided in the first paper ( [2] , Section 3.12). The advantage of this approach is that each distribution is clearly defined. The total is divided such that the dark mass M_{dkm} is that mass sufficiently distant that expansion prevents the light (i.e. information) from ever reaching the observer. The observable mass M_{obs} makes up the remainder. The observable may then be divided into two categories, that which is presently visible M_{vis} and the unobserved M_{uobs} which will be visible given sufficient elapsed time. Each distribution ( [2] , Equation (109), Equation (110), Equation (113), and Equation (115)) precisely matches the ∆CDM results. We learn here that each is invariant,

${M}_{dkm}=\frac{{\theta}_{si}^{2}-2}{{\theta}_{si}^{2}+2}=68.3624\%$ , (38)

${M}_{obs}=\frac{4}{{\theta}_{si}^{2}+2}=31.6376\%$ , (39)

${M}_{vis}=\frac{1}{2{\theta}_{si}}\frac{{M}_{obs}}{{M}_{tot}}=\frac{{M}_{obs}}{2{\theta}_{si}}=4.84884\%$ , (40)

${M}_{uobs}={M}_{obs}-{M}_{vis}=31.6376-4.84884=26.7888\%$ . (41)

In modern theory M_{dkm} is recognized as dark energy; M_{uobs} is recognized as dark matter. As neither reflects the calculations, the terms are accordingly replaced.

This brings us to an important observation as described in Equation (40),

${M}_{obs}=2{\theta}_{si}{M}_{vis}$ . (42)

If the visible is that which is presently visible and the unobserved is that which becomes visible with elapsed time, then with respect to the earliest epoch nearly all the visible we see today was previously unobserved (i.e. dark matter). The idea that dark matter is different than what we presently identify as visible is in conflict. Further technical details regarding the treatment of mass distributions are provided in Appendix A.6.

Consider now that the Informativity distributions precisely match the ∆CDM calculations. This is accomplished with only an understanding of fundamental mass M_{f}. Combining the distributions we find that ( [2] , Equation (118))

${M}_{f}=\frac{{M}_{tot}{M}_{obs}}{2{M}_{tot}-{M}_{obs}}$ , (43)

but this seemingly reveals a problem. If the expressions are invariant why are the distributions properly resolved while mass is moving from the unobserved to the visible? From yet another point of view, such that M_{f} is a function of time, must not M_{tot} increase? Yes, evidence that the total mass of the universe is increasing follows.

The CMB calculations are just one inevitable outcome of mass accretion M_{acr} ( [2] , Equation (135)). The age, quantity, density and temperature of the CMB may each be calculated such that

${M}_{acr}=\frac{{\theta}_{si}^{3}}{2}=17.3611\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}{m}_{f}/\text{unit}\text{\hspace{0.17em}}{t}_{f}$ . (44)

Most importantly, “there are no free variables”, in the calculation. Density and temperature, naturally, are a function of the elapsed time we identify as being the present,

${A}_{U}={\text{e}}^{\sqrt{3}{\theta}_{si}^{3}/2}=1.14652\times {10}^{13}\text{s}=363309\text{\hspace{0.17em}}\text{y}$ , (45)

${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}=1.50159\times {10}^{50}\text{\hspace{0.17em}}\text{kg}$ , (46)

$\rho =\frac{{M}_{tot}{c}^{2}}{{V}_{U}}=4.17041\times {10}^{-14}\text{J}/{\text{m}}^{\text{3}}$ , (47)

$T={\left(\frac{\rho}{a}\right)}^{1/4}=2.72468\text{\hspace{0.17em}}\text{K}$ . (48)

The calculations are a direct result of M_{acr}. The universe beings as a quantum bubble unable to expand at the speed of light because there exists no means to resolve a point outside of the bubble until the universe reaches a radius of
$\sqrt{3}{l}_{f}$ . This trigger ends quantum inflation precisely at 363,309 years, releases the accreted mass/energy (which occurs at the noted rate of
${\theta}_{si}^{3}/2$ units of m_{f} per unit of t_{f}) as CMB and initiates expansion as we see it today. There is no faster-than-light inflationary period and the results match our best observational data precisely. The calculations and details were published in the Journal of High Energy Physics, Gravitation and Cosmology ( [2] , Section 3.15) with additional explanation of the effects of measurement distortion following ( [5] , Section 3.6).

A graphical representation of the distributions is also presented in Figure 3. The mass values are constrained to the precision of the age of the universe, 13.799 billion years [7] , as our most accurate measure of the universe.

For a more complete list of mass distribution conversions refer to Appendix A.5.

Finally, to provide a reference for the expressions to follow we will use the Milky Way as our target. The calculations consider only the mass within the first 84,000 light-years. The corresponding value for observable mass is then

${M}_{obs}=8.56060\times {10}^{41}\text{\hspace{0.17em}}\text{kg}$ . (49)

Figure 3. Relative measure of mass.

All mass, density and velocity data for the Milky Way comes from Stacy McGaugh’s 2018 Milky Way mass models [8] .

3.2. Orbital Velocity Bound

Count bounds are an important and physically significant attribute in describing the behavior of matter. Length frequency is the most well-known count bound c = l_{f}/t_{f}; for each count of fundamental time there can be at most one count of fundamental length. Any count of l_{f} greater than t_{f} would correspond to a velocity greater than the speed of light. The physical significance of fundamental units of measure is what distinguishes measurement quantization from an unbounded description of nature.

There also exist upper and lower count bounds for m_{f}/t_{f} and m_{f}/l_{f}. We respectively call these bounds mass frequency and mass-to-length frequency. The orbital velocity of a star is subject to all three bounds in addition to the effects of expansion. A description may be resolved starting with the classical expression for orbital velocity,

$v=\sqrt{\frac{GM}{R}}=\sqrt{\frac{{l}_{f}^{3}}{{t}_{f}^{3}}\frac{{t}_{f}}{{m}_{f}}\frac{{n}_{M}{m}_{f}}{{n}_{Lr}{l}_{f}}}=\sqrt{\frac{{l}_{f}^{2}}{{t}_{f}^{2}}\frac{{n}_{M}}{{n}_{Lr}}}=c\sqrt{\frac{{n}_{M}}{{n}_{Lr}}}$ . (50)

As described in Appendix A.2 the upper mass-to-length count bound with respect to orbital velocity is 1 to 1, n_{M} < n_{Lr}. But, the relation we seek is the mass-to-length count bound with respect to the escape velocity

$2{n}_{M}<{n}_{Lr}$ . (51)

Consider now that the smallest count of m_{f} with respect to l_{f} may not be less than the precision offered by the reference
${m}_{f}=2.17647\times {10}^{-8}\text{\hspace{0.17em}}\text{kg}$ . To translate this to a whole-unit count of the reference scale the ratio,

$\frac{2.17643\times {10}^{-8}\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}{m}_{f}}{1\text{\hspace{0.17em}}\text{unit}\text{\hspace{0.17em}}{l}_{f}}=\frac{1\text{\hspace{0.17em}}\text{unit}\text{\hspace{0.17em}}{m}_{f}}{4.59468\times {10}^{7}\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}{l}_{f}}=\frac{1}{1/{m}_{f}}={n}_{Mb}$ . (52)

Combining both bounds the ratio is then 2 units of m_{f} per unit of l_{f} where 1/(1/m_{f}). Thus, 2(1/(1/m_{f})) = 2m_{f}. Where n_{Mb} and m_{f} are equal in value and have no units, then the classical velocity bound is

${v}_{bc}=c\sqrt{\frac{{n}_{M}}{{n}_{Lr}}}=c\sqrt{2{m}_{f}}$ . (53)

The expression does not account for the expansion of space H_{U} = 2θ_{si}, Equation (27). Such that H_{U} is relative to the diameter of the universe, divide by 2. The radial expansion respective of orbital and escape velocity may be written in two ways using the fundamental expression to convert between them:

${v}_{b}={\theta}_{si}c\sqrt{2{m}_{f}}=204.054\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ , (54)

${v}_{b}=c{m}_{f}\sqrt{{\theta}_{si}c}=204.054\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ . (55)

As a reminder, both θ_{si} and our substitution of m_{f} for n_{Mb} carry no units. This is the velocity bound corresponding to the upper count bound of m_{f} that may be discerned at a point in space. To resolve a corresponding mass bound set v_{b} equal to the same as expressed with Newton’s expression and reduce with the fundamental expression. The derivation may be found in Appendix A.3 along with an explanation of units,

${M}_{b\text{-}f\left(R\right)}=R{\theta}_{si}\frac{{m}_{f}^{3}}{{t}_{f}}$ . (56)

The mass bound M_{b-f}_{(R)} is a function of the mass within the target orbital radius f(R). By example, a galaxy with a radius of 84,000 light-years
$R=7.94157\times {10}^{20}\text{\hspace{0.17em}}\text{m}$ would need more than

${M}_{b\text{-}f\left(R\right)}=4.95454\times {10}^{41}\text{\hspace{0.17em}}\text{kg}$ (57)

of mass, 2.49 × 10^{11} solar masses to display behavior associated with a measurement quantization bound. Such a mass reflects
$2.49\times {10}^{11}/4.30\times {10}^{11}=57.9\%$ of the estimated mass of the Milky Way. Equation (54) describes the upper bound to measurable mass unadjusted for total mass and a mass density profile. If mass density exceeds this bound, the upper mass count bound will exceed the mass frequency bound causing additional mass count to be indistinguishable.

Lastly, consider what a higher or lesser velocity bound implies. We may demonstrate by reorganizing the fundamental expression m_{f}l_{f} = 2θ_{si}t_{f} into a form that resolves the length count presented in the denominator of Equation (52), 1/m_{f} = 4.59468 × 10^{7}. Thus, a count 100 units greater implies a corresponding speed of

$\frac{{l}_{f}}{{t}_{f}}=c=2{\theta}_{si}\frac{1}{{m}_{f}}\approx 2{\theta}_{si}\left(4.59468\times {10}^{7}+100\right)=299793110\text{\hspace{0.17em}}\text{m}\cdot {\text{s}}^{-1}$ . (58)

a 652 m/s increase above the speed of light. The increase also corresponds to a velocity bound of

${v}_{b}={\theta}_{si}c\sqrt{2{n}_{Mb}}\approx {\theta}_{si}c\sqrt{\frac{2}{\left(1/{m}_{f}\right)+100}}=204.053\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}$ , (59)

a decrease of 1 m×s^{−1}. This does not mean that the speed of a star may not fall below 204.054 km×s^{−1}. The expression describes an upper bound with which to discern mass counts and as such an upper bound to the gravitational pull on a star. When the mass count exceeds the mass count bound, the target is unable to distinguish additional mass and as such the gravitational effect of mass on a star reaches a maximum.

This investigation also does not imply that stars cannot have velocities greater than 204.054 km×s^{−1}. While these expressions are invariant, we have not integrated the effects of an uneven mass distribution typical of a galaxy. This will be the subject of the next section.

3.3. Galactic Rotation and the Milky Way

Using the velocity bound, an expression may now be developed as a function of mass distribution in a galaxy. The relation follows the same form as that which describes visible M_{vis} and unobserved M_{uobs} (aka dark matter) mass, Equation (A.5.6) from Appendix A.5,

${M}_{uobs}={M}_{vis}\left(2{\theta}_{si}-1\right)$ . (60)

Replacing the dimensionless speed parameter θ_{si} with the ratio of observed over bound v_{o}/v_{b} (i.e. in relativity then v/c) will provide the corresponding relation between the effective and mass bound. But, an understanding of the geometry of the substitution is difficult. For that reason, we will follow an algebraic approach that resolves the speed parameter β as a relative percent difference ∆%_{o-b} of the bound.

$\Delta {\%}_{o\text{-}b}=\frac{\left({v}_{o}-{v}_{b}\right)}{{v}_{b}}=\frac{{v}_{o}}{{v}_{b}}-1$ . (61)

To reduce, also needed is the velocity bound v_{b} from Equation (54), v_{b} = θ_{si}c(2m_{f})^{1/2}. The expression for mass is then the product of the mass bound M_{b-f}_{(R)} and 2∆%_{o-b}, hereafter referred to as the effective mass M_{e-f}_{(R)}. The symbol f(R) in subscript indicates that the mass considered is the mass within the orbital radius R from a galactic center. Like the relation presented in Equation (54), the speed parameter ∆%_{o-b} is doubled to describe mass in terms of the bound for escape velocity.

${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\Delta {\%}_{o\text{-}b}+1\right)={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-2+1\right)$ , (62)

${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ . (63)

when incorporating expansion we realize that the observer’s view of the universe is skewed; the effect suggests the presence of more mass than is actually present. In Figure 4 the observable M_{o-f}_{(R)}, bound M_{b-f}_{(R)}, and effective M_{e-f}_{(R)} mass are displayed.

Where the effective mass is less than the bound, the orbital velocity of stars follow a classical behavior. Conversely, an effective mass greater than the bound presents a mass count greater than the mass frequency bound. Some count of m_{f} will be indistinguishable leading to a constraining effect on gravity and corresponding star velocities. The crossover from classical to non-classical behavior occurs at 9.32848 × 10^{3} light-years.

Notice that the observable and effective mass differ by a factor of 3.9 at

Figure 4. Galactic mass corresponding to actual (green), mass frequency bound (red) and relative mass frequency bound (purple).

$R=84\times {10}^{3}$ light-years; 74% of the mass is missing. The magnitude of this effect depends on the total mass of the galaxy or galaxies considered. A second notable factor regards mass distribution. As discussed, excess mass count is indistinguishable creating a mitigating gravitational effect. Which mass counts are lost? This is presently unknown, but also less significant in a well-organized system such as a galaxy. Conversely, consideration of an uneven distribution (i.e. several galaxies) will present a center-of-mass offset respective of the indistinguishable mass count.

Both effects are notably evident in the Bullet Cluster. For one, the cluster exhibits a missing mass of approximately 90%. The cluster also exhibits a center-of-mass offset as would be expected with a lost mass count, the latter being of considerable interest for future research.

Using Newton’s expression for velocity, ${v}_{b}={\left(G{M}_{b\text{-}f\left(R\right)}/R\right)}^{1/2}$ and the expression for the mass bound ${M}_{b\text{-}f\left(R\right)}={\theta}_{si}{m}_{f}^{3}R/{t}_{f}$ from Equation (56), we may now resolve the effective star velocity

${v}_{e}={\left(\frac{G{M}_{e\text{-}f\left(R\right)}}{R}\right)}^{1/2}={\left(\frac{G{M}_{b\text{-}f\left(R\right)}}{R}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)\right)}^{1/2}$ , (64)

${v}_{e}={\left(\frac{G}{R}{\theta}_{si}\frac{{m}_{f}^{3}}{{t}_{f}}R\left(2\frac{{v}_{o}}{{\theta}_{si}c\sqrt{2{m}_{f}}}-1\right)\right)}^{1/2}$ , (65)

${v}_{e}=2{\theta}_{si}{\left(2\frac{{v}_{o}}{\sqrt{2{m}_{f}}}-c{\theta}_{si}\right)}^{1/2}$ . (66)

while it may seem more appropriate to use a mass or mass density dataset the choice is irrelevant. One may modify the expression to enter velocity, mass or mass density and still arrive at the same expression. For example, as resolved in Appendix A.4, we may substitute the observable velocity v_{o} in Equation (66) for this equivalent function written in terms of the effective mass,

${v}_{o}=\sqrt{\frac{{m}_{f}}{2}}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{R}\frac{{l}_{f}}{{m}_{f}^{3}}+{\theta}_{si}c\right)$ . (67)

More importantly, Newton’s expression for velocity does not produce the observable velocity curve. Informativity succeeds because the expression for effective velocity is a function of the mass count bound, Equation (54), an invariant expression with no free variables. To highlight that fact, we retain the corresponding velocity bound v_{b} in Figure 5 to demonstrate the natural tendency for stars to approach the bound when the mass count reaching a star exceeds the effective bound. The remaining curves are as follows. The effective velocity v_{e} is plotted in red. The observable velocity v_{o} is plotted in green. And the classical velocity v_{c} is plotted in blue; that’s Newton’s expression.

There are two points of view in conflict. That is, the classical velocity implies that what we observe is moving too fast. The curve also suggests that there is a missing dark matter holding the stars in orbit. At the same time, the observable velocity suggests correspondence to variations in mass density.

The Informativity approach resolves the discrepancy describing an effective velocity that follows the bound when the effective mass exceeds the mass bound. When effective mass does not exceed the bound, orbital velocities follow a classical behavior.

Although the bound is invariant―204.054 km×s^{−1}―variations in galactic mass density do affect the gravitational pull on a star. These effects may be evened out when taking an average of thousands of galaxies. Except near the galactic core where the crossover between classical and Informativistic behavior varies from one galaxy to the next, the velocity curve levels out reflecting an averaging of mass profiles.

An unexpected effect of mass count bounds is apparent between 4 and 8 thousand light-years where star velocities level out until otherwise affected by increasing mass density. The cause of this effect is a subject of interest. Perhaps physically insignificant, but star velocity may favor classical behavior at the crossover between the effective and mass bound.

The mass bound delineates two behaviors. Recall from Equation (56),
${M}_{b\text{-}f\left(R\right)}=R{\theta}_{si}{m}_{f}^{3}/{t}_{f}$ that the mass bound is a function of how much mass is within a given radius. Variations in mass density imply increases or decreases in the spherical space described by R for a fixed amount of mass. If we fix R in consideration of a region of greater mass density, then the effective velocity will be higher, describing measured velocities that rise above the bound (i.e. 204.054 km×s^{−1}). The opposite effect applies for less dense regions such that velocities lesson.

To further demonstrate this effect, consider Figure 6 where a model galaxy with the same mass as the Milky Way is presented, but mass distribution has been evened as though we were averaging the mass profile of thousands of galaxies. To be clear, a mass equal to that for R < 1000 light-years of the Milky Way center is taken. Then the remaining mass (where the total considered is only the

Figure 5. Stellar velocities corresponding to observed (green), relative mass frequency bound (red), mass frequency bound (purple) and Newton’s expression (blue).

Figure 6. Stellar velocities corresponding to actual (green), an even mass distribution (orange) and the mass frequency bound (red).

mass in the first 84,000 light-years) is evenly divided across the remaining 83 thousand light-years. The corresponding effective velocity (orange) is drawn. As expected, the curve levels out just above the bound velocity (purple) with a magnitude that is in proportion to the excess mass above the bound. An average of thousands of galaxies will demonstrate a level velocity curve with a magnitude that corresponds to the mass in excess of the mass bound.

As a final note, separation of the velocity term in Equation (63) from the data can be challenging. It is the mass density data that characterizes the galaxy under consideration. The argument may be extended to demonstrate that it is also irrelevant what dataset is chosen: mass, mass density or velocity. As each measure is mathematically related, an argument for data independence by favoring any dataset over another cannot be made.

But, there are two remaining considerations that are data independent. Notably, an expression must describe a phenomenon with the correct magnitude. The Informativity expression properly accommodates the effects of a mass count bound in an expanding universe. Where Newton’s expression does not provide the observed magnitude in describing orbital velocity, the Informativity expression does.

Also providing support is the bound itself, the purple line denoting an invariant velocity of 204.054 km×s^{−1}. The bound expression contains no measurement data, no free variables and as such no “fitting”,
${v}_{b}={\theta}_{si}c{\left(2{m}_{f}\right)}^{1/2}$ . Referring to Figure 5, star velocities favor the bound. But, that will not always be clearly evident. What is clear is that the bound is the baseline measure from which the magnitude of the Informativity expression is calculated. If the bound were not physically significant, the magnitude would be incorrect and the resulting curve would not match the observational data.

Returning to our initial discussion our goal was to develop a mass expression defined with respect to a bound. To this we can compare the effective and unobserved mass expressions, each taking the form M_{1} = M_{2}(2β − 1).

${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (68)

${M}_{uobs}={M}_{vis}\left(2{\theta}_{si}-1\right)$ . (69)

The details of the speed parameter depend on the masses being compared. In the case of orbital velocity, the parameter is found on the right-side of this expression ( [5] , Equation (68))

${\beta}^{2}=\frac{{v}^{2}}{{c}^{2}}={\left(\frac{{n}_{Lm}{l}_{f}}{{n}_{T}{t}_{f}}\frac{{n}_{T}{t}_{f}}{{n}_{Lc}{l}_{f}}\right)}^{2}=\frac{{n}_{Lm}^{2}}{{n}_{Lc}^{2}}=2\frac{{n}_{M}}{{n}_{Lr}}$ (70)

which predicts and demonstrates equivalence between the phenomena of motion and gravitation. Respecting the difference between orbital motion v = (GM/R)^{1/2} and escape velocity v = (2GM/R)^{1/2}, we remove the factor of 2. Then Equation (50) may be completed

$v=\sqrt{\frac{GM}{R}}=c\sqrt{\frac{{n}_{M}}{{n}_{Lr}}}=c\sqrt{\frac{{v}^{2}}{{c}^{2}}}=v$ (71)

and recognized as the same speed parameter β commonly found in relativistic expressions. And finally, comparison of Equation (50) and Equation (68) successfully confirm the correlation between motion and gravitation (i.e. mass) as expected.

3.4. What Does the 26.7888% Distribution Describe

In this section, we will discuss why the dark matter phenomenon has been so closely associated with the ΛCDM distribution also distinguished by the same name. We will not be using the ΛCDM approach to discuss mass distribution but instead use the Informativity expressions, such that each distribution is a function of one physical constant θ_{si}. Theta has been accurately measured to 6 significant digits [6] .

We begin with the unobserved mass

${M}_{uobs}={M}_{obs}-{M}_{vis}=31.6376-4.84884=26.7888\%$ , (72)

which describes the mass that will be observable M_{obs} Equation (39), but is not presently visible M_{vis} Equation (41). This is one interpretation. Using Equation (40) M_{obs} = 2θ_{si}M_{vis}, we can also resolve this distribution as

${M}_{uobs}=2{\theta}_{si}{M}_{vis}-{M}_{vis}$ . (73)

Such that H_{U} = 2θ_{si} Equation (27), then
${M}_{uobs}={M}_{vis}\left({H}_{U}-1\right)$ . The dark matter distribution M_{uobs} is then the energy of expansion as a function of the visible mass H_{U}M_{vis} minus the energy associated with the visible mass M_{vis}.

The two interpretations―mass and energy―while mathematically equivalent have led to significant confusion. Additionally, mass distributions are defined with respect to the universe. But, the rate of expansion is much less than 2θ_{si} in a region the size of a galaxy. As such, application of a distribution such as dark matter to the description of a galaxy is questionable.

With respect to existing observational support, Informativity does not imply that the mass we measure in a galaxy is all the mass present. There are studies that suggest there is additional non- or low-light-absorbing fine dust [9] . While there is a great deal to learn about galactic mass composition, Informativity constrains the magnitude of this mass to the observable distribution M_{obs}. It should be added that gravitational lensing studies are not an indicator of missing mass. Rather, these studies will need to incorporate effective mass which accounts for expansion and the mass frequency bound. With this approach the bending of light conforms to the effective mass as is demonstrated by the effective velocity curve.

3.5. Interpretation of Mass

At this point we have a better understanding of the unobserved distribution and its relation to expansion, but have not resolved a clear understanding of the bound.

We present three expressions each describing the mass bound against which effective mass is measured. Equation (54) set equal to Newton’s velocity expression describes the mass bound in terms of the fundamental measures (Appendix A.3).

${v}_{b}={\theta}_{si}c\sqrt{2{m}_{f}}$ , (74)

${v}_{b}={\left(\frac{G{M}_{b\text{-}f\left(R\right)}}{R}\right)}^{1/2}$ , (75)

${M}_{b\text{-}f\left(R\right)}={v}_{b}^{2}\frac{R}{G}=2{\theta}_{si}^{2}{c}^{2}{m}_{f}\frac{R}{G}=\left(\frac{{l}_{f}{m}_{f}}{{t}_{f}}\right){\theta}_{si}{c}^{2}{m}_{f}\frac{R{m}_{f}}{{c}^{3}{t}_{f}}$ , (76)

${M}_{b\text{-}f\left(R\right)}={\theta}_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (77)

For the next two expressions, consider one point on the mass bound curve such that the radius is that of the universe (Appendix A.5, Equation (A.5.20)),

${M}_{Ub}={M}_{f}{m}_{f}^{2}{\theta}_{si}$ . (78)

In a similar fashion, consider the mass bound in terms of mass distributions (Appendix A.5, Equation (A.5.10)),

$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (79)

${M}_{Ub}={\theta}_{si}{m}_{f}^{2}\frac{{M}_{tot}{M}_{obs}}{2{M}_{tot}-{M}_{obs}}={\theta}_{si}{m}_{f}^{2}\frac{{M}_{tot}{M}_{obs}}{{M}_{tot}+{M}_{dkm}}$ . (80)

while each approach offers opportunity to present the mass bound as a function of mass, energy or physical constants, Equation (77) presents the clearest description, a line. The relation demonstrates that geometry is at work.

With the bound more clearly understood, return to Equation (63) and resolve the effective mass,

${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (81)

${M}_{e\text{-}f\left(R\right)}={\theta}_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}\left(2\frac{{v}_{o}}{{\theta}_{si}c\sqrt{2{m}_{f}}}-1\right)$ , (82)

${M}_{e\text{-}f\left(R\right)}={v}_{o}2R\frac{{m}_{f}^{3}}{{l}_{f}\sqrt{2{m}_{f}}}-{\theta}_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (83)

Built on Equation (77), velocity v_{o} is the only new variable, a data dependent value that characterizes the target. The result is quantum in detail and valid for the entire measurement domain. When effective mass rises above or falls below the mass bound so does the velocity. When the effective and bound masses are equal, then the velocities are as well.

We may summarize effective mass as having one of two states. The first serves as the reference, defined where the effective and bound mass are equal, a purely geometric description ${M}_{b\text{-}f\left(R\right)}={\theta}_{si}R{m}_{f}^{3}/{t}_{f}$ . The second state we call the offset Equation (83). Collectively the two states describe observed velocity as a function of the effective mass that characterizes the target.

Several studies of galaxies and galaxy clusters have suggested the presence of a gravitational force that does not coincide with the visible matter. Notably, the effective mass of a given matter field describes force that is unexpected from our point of view. While we will not review the specific calculations of existing investigations, it is expected with respect to this model that the effects of a mass frequency bound when integrated with that of expansion must produce an offset and that offset will be even more pronounced when describing disorganized targets.

Importantly, expansion does not explain the dark matter mass discrepancy because mass alone does not determine orbital velocity as Newton had surmised. Several effects are at work. Thus, Newton’s expression is correct so long as expansion, mass frequency, measurement distortion (i.e. also described by relativity) and the Informativity differential are not significant factors.

Modern physical descriptions use mass to describe gravity, but the effective mass is significantly greater in magnitude than the observed mass described by Newton. The bound describes a geometric reference with an offset swinging from one side to the other like a weight on a rubber band. While the Informativity and Newton expressions coincide for systems having less mass density, velocity is not solely a function of mass. Thus, the question, where is the missing mass, is not valid.

Bounded gravity may also be applied to the early universe when mass density was significant. When expressions that incorporate bounded gravity are used some doors may open with early universe modeling. While not the focus of this paper, a detailed account of quantum inflation, the trigger event that ends this epoch and the ensuing expansion are described in the first paper ( [1] , Section 3.15) with additional explanation of the effects of measurement distortion described in the second paper ( [5] , Section 3.6). Notably, the solution as presented in Equations (45)-(48) is a function of one physical constant.

A final question is why should the measured mass of a galaxy be attributed to the observable and not the visible distribution? The answer is primarily subjective as mass distributions may not be used to describe a galaxy. That said, one may note with elapsed time that a specific amount of observable mass becomes visible. Because the mass of a galaxy does not increase, the label “observable” is more appropriate.

In practice, the issue with scaling distributions is that the scaling process changes the properties that the distributions are defined against. To succeed any application must retain each property of the initial definition. For instance, the scaling would require that the outer edge of the galaxy expand at the speed of light. Loss of this property is immediately obvious. For one, dark mass cannot even exist. As well, the visible and observable distributions are always the same.

3.6. Kinetic Energy

As a follow up to mass frequency, we may provide one final confirmation of our understanding of n_{M}/n_{Lr} = 2m_{f} by reducing the Informativity interpretation to demonstrate the equation for kinetic energy. Notably, the classical expression does not include the radial expansion parameter θ_{si} which is defined with respect to a bound and thus carries no units. So, we start with the static radial form. Such that m_{f} = 2θ_{si}/c from the fundamental expression and the expression for half a fundamental unit of mass E_{f} = 2θ_{si}c ( [2] , Equation (49)), then the static velocity bound is

$v=c\sqrt{2{m}_{f}}=c\sqrt{\frac{4{\theta}_{si}}{c}}=\sqrt{4{\theta}_{si}c}=\sqrt{2{E}_{f}}$ . (84)

$v=c\sqrt{2{m}_{f}}=c\sqrt{\frac{2}{1/{m}_{f}}}=c\sqrt{\frac{2}{{n}_{M}}}=\sqrt{\frac{2{c}^{2}}{{n}_{M}}}$ , (85)

$v=\sqrt{\left(\frac{2{\theta}_{si}}{c}\frac{1}{{m}_{f}}\right)\frac{2{c}^{2}}{{n}_{M}}}=\sqrt{\frac{4{\theta}_{si}c}{{n}_{M}{m}_{f}}}=\sqrt{\frac{2E}{m}}$ , (86)

and may then be reduced to resolve the kinetic energy associated with any mass,

$E=\frac{m{v}^{2}}{2}$ . (87)

One may compare the first and last velocity expressions and wonder why the latter has a mass value in the denominator. The mass value is what generalizes the expression for any mass, velocity and energy. The initial expression is an invariant description of the smallest unit of energy E_{f} corresponding to a mass count bound of n_{M} = 1/m_{f}. That ratio is precisely 1 leaving us with 2E_{f} under the square root operator.

4. Discussion

Perhaps the most significant outcome of this research is not a model of galactic orbital dynamics, but the inclusion of expansion, the mass frequency bound and mass density into a single description of orbital velocity, since the time of Newton mass has been considered the primary factor describing the effects of gravitation. There have been modifications to that understanding (i.e. relativity), but such modifications have been a fine-tuning of the broader expressions set forth by Newton, mass and radial distance being the variables that determine orbital motion. But, with the expressions set forth here, mass is one of several factors. The mass frequency bound now designates the demarcation point; it is quantum and valid for the entire measurement domain.

Finally, where there have been several proposals describing solutions to galactic orbital dynamics [10] , the traditional approach is one of resolving data dependent expressions. Informativity takes a uniquely different view of the universe, that physical expression is an outcome of bounds to measure. The mass frequency bound is an outcome of this axiom, a geometric expression that identifies a reference and an offset against which the effective mass is resolved. Mathematics of counts of the fundamental measures is all that is needed to unravel the motions of the stars.

Acknowledgements

We thank Edanz Group (https://www.edanzediting.com/?utm_source=ack&utm_medium=journal) for editing a draft of this manuscript.

Appendix

A.1. Numerical Limits to Q_{L}n_{Lr}

The term Q_{L}n_{Lr} is referred to as the Informativity differential in recognition of the central role it plays in describing how fractional values less than the reference measure reflect a distorting effect in distance measurement. Knowing the limits to Q_{L}n_{Lr} is essential in resolving the fundamental measures.

Q_{L}n_{Lr} is Equation (2) multiplied by n_{Lb}.

${Q}_{L}{n}_{Lr}=\left(\sqrt{1+{n}_{Lb}^{2}}-{n}_{Lb}\right){n}_{Lb}$ . (A.1.1)

Note, what is measured always equals a whole-unit count of a fundamental measure, and with a = 1 we find that n_{Lb} = n_{Lr} for all values. This is easily verified in that the highest value for Q_{L} is obtained for n_{Lb} = 1 where
${\left(1+{1}^{2}\right)}^{0.5}-1=0.414$ and the “observed” distance of c presented as a count n_{Lr} is always rounded down to the highest integer value equal to the count n_{Lb} with Q_{L} = 0.414 at its highest and quickly approaching 0 with increasing n_{Lb}. Therefore,

${Q}_{L}{n}_{Lr}=\left(\sqrt{1+{n}_{Lr}^{2}}-{n}_{Lr}\right){n}_{Lr}$ . (A.1.2)

The lower limit where n_{Lr} = 1 is easily produced,
${\mathrm{lim}}_{r=1}f\left({Q}_{L}{n}_{Lr}\right)=\sqrt{2}-1$ . Conversely, if we divide by n_{Lr}, then add n_{Lr}, square, subtract
${n}_{Lr}^{2}$ , and divide by 2, we find that

$\frac{{Q}_{L}^{2}}{2}+{Q}_{L}{n}_{Lr}=\frac{1}{2}$ . (A.1.3)

Q_{L} decreases with increasing n_{Lr} until the left term drops out. Distance does not need to be significant to reduce the Informativity differential. At just 10^{4}l_{f}, Q_{L}n_{Lr} rounds to 0.5 to nine significant digits.

A.2. Upper Bound Relationship between Length and Mass

To resolve the upper bound relation between length and mass, we begin with the expression for escape velocity, set velocity equal to the speed of light denoting the upper bound and then substitute fundamental units for each of the terms. Notably, the expression for G follows from Equation (6) as

$G=\frac{{Q}_{Lf}r{c}^{3}}{{\theta}_{si}}=\frac{{Q}_{Lf}{r}_{Lf}{l}_{f}{c}^{3}}{{\theta}_{si}}=\frac{{c}^{3}{l}_{f}}{2{\theta}_{si}}=\frac{{c}^{3}{t}_{f}}{{m}_{f}}=\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{l}_{f}}{{t}_{f}}\frac{{t}_{f}}{{m}_{f}}$ . (A.2.1)

Likewise, a generalized mass count n_{M} of m_{f} follows from the fundamental expression l_{f}m_{f} = 2θ_{si}t_{f}. Where the
${\mathrm{lim}}_{r\to \infty}f\left({Q}_{L}{n}_{Lr}\right)=1/2$ as resolved in Appendix A.1, then

${m}_{f}=\frac{2{\theta}_{si}}{c}=\frac{{\theta}_{si}}{{Q}_{Lf}{r}_{Lf}c}$ . (A.2.2)

and where c = l_{f}/t_{f}, the expression for escape velocity may be reduced to show that

$v={\left(\frac{2GM}{r}\right)}^{1/2}$ , (A.2.3)

$c>{\left(\frac{2}{r}\frac{{Q}_{L}r{c}^{3}}{{\theta}_{si}}\frac{{n}_{M}{\theta}_{si}}{{Q}_{L}{n}_{Lr}c}\right)}^{1/2}>{\left(\frac{2{n}_{M}{c}^{2}}{{n}_{Lr}}\right)}^{1/2}$ , (A.2.4)

${n}_{Lr}>2{n}_{M}$ . (A.2.5)

Using escape velocity the upper bound of a count of n_{M} with respect to n_{Lr} is resolved. Conversely, for orbital velocity, the expression is v = (GM/r)^{1/2}. The relation differs by a factor of two,

${n}_{Lr}>{n}_{M}$ . (A.2.6)

A.3. Observable Mass Bound

The observable mass may be resolved by setting the bound velocity equal to the classical velocity and reducing. Where G = c^{3}t_{f}/m_{f}, then

${\theta}_{si}c\sqrt{2{m}_{f}}=\sqrt{\frac{G{M}_{b\text{-}f\left(R\right)}}{R}}$ , (A.3.1)

${M}_{b\text{-}f\left(R\right)}=2{\theta}_{si}^{2}R{c}^{2}{m}_{f}\frac{1}{G}=2{\theta}_{si}^{2}R{c}^{2}{m}_{f}\frac{{m}_{f}}{{c}^{3}{t}_{f}}$ , (A.3.2)

${M}_{b\text{-}f\left(R\right)}=2{\theta}_{si}^{2}R{c}^{2}{m}_{f}\frac{{m}_{f}}{{c}^{3}{t}_{f}}=2{\theta}_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}$ , (A.3.3)

${M}_{b\text{-}f\left(R\right)}=2{\theta}_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}=2{\theta}_{si}R\frac{{m}_{f}{l}_{f}}{2{t}_{f}}\frac{{m}_{f}^{2}}{{l}_{f}}$ , (A.3.4)

${M}_{b\text{-}f\left(R\right)}={\theta}_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}$ . (A.3.5)

Recall, the left portion of the v_{b} expression in Equation (A.3.1) has a value of m_{f} which is a dimensionless substitute for n_{Mb}. There are no units. This is fine until Equation (A.3.4) where R in meters cancels with l_{f} in meters leaving one of the two
${m}_{f}^{2}$ with a single kilograms describing M_{b-f}_{(R)}. But in Equation (A.3.1) we introduce the dimensionless expression θ_{si} = m_{f}l_{f}/2t_{f}. Several cancellations leave both R, t_{f} and an additional m_{f} each dimensionless. The result is kilograms,

${M}_{b\text{-}f\left(R\right)}=2{\theta}_{si}^{2}R\frac{{m}_{f}^{2}}{{l}_{f}}m\frac{kg}{m}={\theta}_{si}R\frac{{m}_{f}^{3}}{{t}_{f}}kg$ . (A.3.6)

A.4. Resolving Effective Velocity as a Function of Mass

The effective and observed velocities can be the same in value, yet each term is identified separately. This calls into question the use of one to identify the other if they are not physically different.

The two terms operate as a limit where the estimated value for v_{o} constrains the resulting value for v_{e}. The correct physical description is where the modeled value for v_{o} produces a value for v_{e} that is closer than any other combination.

${v}_{e}=2{\theta}_{si}{\left(2\frac{{v}_{o}}{\sqrt{2{m}_{f}}}-c{\theta}_{si}\right)}^{1/2}$ . (A.4.1)

There may exist theoretical argument that use of an observed velocity to resolve the effective velocity is still in principle problematic. For that reason, an alternative is offered whereby the observable velocity is replaced by a function with effective mass as the only free variable,

${M}_{e\text{-}f\left(R\right)}={M}_{b\text{-}f\left(R\right)}\left(2\frac{{v}_{o}}{{v}_{b}}-1\right)$ , (A.4.2)

$\frac{{v}_{o}}{{v}_{b}}=\frac{1}{2}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{{M}_{b\text{-}f\left(R\right)}}+1\right)$ , (A.4.3)

$2{v}_{o}={v}_{b}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{{M}_{b\text{-}f\left(R\right)}}+1\right)$ , (A.4.4)

$2{v}_{o}={\theta}_{si}c\sqrt{2{m}_{f}}\left(\left({M}_{e\text{-}f\left(R\right)}/{\theta}_{si}\frac{{m}_{f}^{3}}{{t}_{f}}R\right)+1\right)$ , (A.4.5)

${v}_{o}=\sqrt{\frac{{m}_{f}}{2}}\left(\frac{{M}_{e\text{-}f\left(R\right)}}{R}\frac{{l}_{f}}{{m}_{f}^{3}}+{\theta}_{si}c\right)$ . (A.4.6)

while it is not possible to produce a data independent expression that characterizes any target galaxy, the initial expression can now be resolved as a function of the effective mass and that can be resolved as a function of observed mass. In short, the effective velocity may be resolved as a function of the observed mass.

A.5. Mass Distribution Conversions

Following are a list of commonly used mass distribution conversion expressions. Several are resolved from the first paper ( [2] , Equation (113), Equation (110), Equation (109), and Equation (108)). Notably, many of the expressions in the first paper are percentage expressions of a total mass. To resolve distribution values in kilograms, multiply the distribution percentage by Mtot in kilograms.

${M}_{obs}=2{\theta}_{si}{M}_{vis}$ , (A.5.1)

${M}_{obs}={M}_{tot}\frac{4}{{\theta}_{si}^{2}+2}$ , (A.5.2)

${M}_{dkm}={M}_{tot}\frac{{\theta}_{si}^{2}-2}{{\theta}_{si}^{2}+2}$ , (A.5.3)

${M}_{tot}={M}_{obs}+{M}_{dkm}$ , (A.5.4)

${M}_{uobs}={M}_{obs}-{M}_{vis}$ . (A.5.5)

These are resolved from the prior,

${M}_{uobs}={M}_{vis}\left(2{\theta}_{si}-1\right)$ , (A.5.6)

$2{M}_{tot}={M}_{vis}{\theta}_{si}\left({\theta}_{si}^{2}+2\right)$ , (A.5.7)

${M}_{dkm}={M}_{obs}\frac{\left({\theta}_{si}^{2}-2\right)}{4}$ , (A.5.8)

${M}_{dkm}={M}_{vis}\frac{{\theta}_{si}\left({\theta}_{si}^{2}-2\right)}{2}$ . (A.5.9)

And from the first paper ( [2] , Equation (118)) we may also resolve

$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (A.5.10)

${M}_{tot}{M}_{f}={\theta}_{si}{M}_{vis}\left({M}_{tot}+{M}_{f}\right)$ , (A.5.11)

${M}_{f}=\frac{{M}_{tot}{\theta}_{si}{M}_{vis}}{{M}_{tot}-{\theta}_{si}{M}_{vis}}$ . (A.5.12)

We may also derive the relationship between the total and fundamental mass using the expression for total mass ( [2] , Equation (134)) and the expression for fundamental mass ( [2] , Equation (128)),

${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}$ , (A.5.13)

${M}_{f}={n}_{Tu}{m}_{f}{\theta}_{si}$ , (A.5.14)

${M}_{tot}={M}_{f}\frac{{\theta}_{si}^{2}}{2}$ , (A.5.15)

$\frac{{M}_{f}}{{M}_{tot}}=\frac{2}{{\theta}_{si}^{2}}$ . (A.5.16)

Such that the fundamental mass from Equation (24) is reduced with
${D}_{U}=2{R}_{U}=2{\theta}_{si}{A}_{U}$ from Equation (25) and set equal to the bound mass in Equation (77), then the mass bound for the universe M_{Ub} is

${M}_{f}={A}_{U}{\theta}_{si}\frac{{m}_{f}}{{t}_{f}}={R}_{U}\frac{{m}_{f}}{{t}_{f}}$ , (A.5.17)

${M}_{b\text{-}f\left(R\right)}=R{\theta}_{si}\frac{{m}_{f}^{3}}{{t}_{f}}$ , (A.5.18)

$\frac{{M}_{Ub}}{{m}_{f}^{2}{\theta}_{si}}=R\frac{{m}_{f}}{{t}_{f}}={M}_{f}$ , (A.5.19)

${M}_{Ub}={M}_{f}{m}_{f}^{2}{\theta}_{si}$ . (A.5.20)

Lastly, given the observable v_{obs} and visible v_{vis} velocity and Equation (A.5.1), then

$\frac{{v}_{obs}}{{v}_{vis}}=\frac{\sqrt{G{M}_{obs\text{-}f\left(R\right)}/R}}{\sqrt{G{M}_{vis\text{-}f\left(R\right)}/R}}$ , (A.5.21)

$\frac{{v}_{obs}}{{v}_{vis}}=\sqrt{\frac{{M}_{obs\text{-}f\left(R\right)}}{{M}_{vis\text{-}f\left(R\right)}}}=\sqrt{\frac{2{\theta}_{si}{M}_{vis\text{-}f\left(R\right)}}{{M}_{vis\text{-}f\left(R\right)}}}=\sqrt{2{\theta}_{si}}$ . (A.5.22)

In terms of mass visible corresponds to the 4.84884% distribution as described in Equation (40). The observable corresponds to the 31.6376% distribution as described in Equation (39) and incorporates universal expansion, M_{obs} = H_{U}M_{vis}.

A.6. Clarifying Interpretation of Mass Distributions

Distribution expressions may take a percentage or mass value. An expression demonstrating percentages may be converted to kilograms by multiplying the result by M_{tot} in kilograms. Depending on the substitutions elected, a resulting expression can lead to an incorrect interpretation. To demonstrate the issue, consider Equation (42),

$2{\theta}_{si}=\frac{{M}_{obs}}{{M}_{vis}}$ , (A.6.1)

$\frac{2}{{\theta}_{si}^{2}}=\frac{1}{{\theta}_{si}^{3}}\frac{{M}_{obs}}{{M}_{vis}}$ . (A.6.2)

Then set the two expressions equal to one another,

$\frac{{M}_{f}}{{M}_{tot}}=\frac{1}{{\theta}_{si}^{3}}\frac{{M}_{obs}}{{M}_{vis}}=\frac{1}{{\theta}_{si}^{3}}\frac{2{\theta}_{si}{M}_{vis}}{{M}_{obs}/2{\theta}_{si}}=\frac{4}{{\theta}_{si}}\frac{{M}_{vis}}{{M}_{obs}}$ , (A.6.3)

${\theta}_{si}{M}_{obs}{M}_{f}=4{M}_{vis}{M}_{Tot}$ . (A.6.4)

And finally where

${M}_{tot}={n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}$ . (A.6.5)

is a known function of time ( [2] , Equation (134)), we may reduce Equation (A.6.4) such that time is the only free variable.

$2{M}_{tot}{M}_{f}={M}_{obs}\left({M}_{tot}+{M}_{f}\right)$ , (A.6.6)

$2{M}_{tot}2\frac{{M}_{Tot}}{{\theta}_{si}^{2}}={M}_{obs}\left({M}_{tot}+2\frac{{M}_{Tot}}{{\theta}_{si}^{2}}\right)$ , (A.6.7)

${\theta}_{si}^{2}{M}_{tot}{M}_{obs}+2{M}_{Tot}{M}_{obs}-4{M}_{Tot}^{2}=0$ , (A.6.8)

${\theta}_{si}^{2}{n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}{M}_{obs}+2{n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}{M}_{obs}-4{n}_{Tu}^{2}{m}_{f}^{2}\frac{{\theta}_{si}^{6}}{4}=0$ , (A.6.9)

${n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{5}}{2}{M}_{obs}+{n}_{Tu}{m}_{f}{\theta}_{si}^{3}{M}_{obs}-{n}_{Tu}^{2}{m}_{f}^{2}{\theta}_{si}^{6}=0$ , (A.6.10)

$\frac{{\theta}_{si}^{2}}{2}{M}_{obs}+{M}_{obs}-{n}_{Tu}{m}_{f}{\theta}_{si}^{3}=0$ , (A.6.11)

${M}_{obs}\left(\frac{{\theta}_{si}^{2}}{2}+1\right)={n}_{Tu}{m}_{f}{\theta}_{si}^{3}$ , (A.6.12)

${M}_{obs}=2{n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{{\theta}_{si}^{2}+2}$ . (A.6.13)

with elapsed time n_{Tu} one might assume that the observable mass distribution M_{obs} is increasing. This is not a complete picture. The observable and total mass (A.6.5) are both increasing while the distributions remain invariant,

${M}_{obs}=\left({n}_{Tu}{m}_{f}\frac{{\theta}_{si}^{3}}{2}\right)\frac{4}{{\theta}_{si}^{2}+2}$ , (A.6.14)

${M}_{obs}={M}_{tot}\frac{4}{{\theta}_{si}^{2}+2}$ . (A.6.15)

The result was demonstrated in the first paper ( [2] , Equation (110)).

Glossary of Terms

Framework

A frame of reference against a system of measure is applied. Frameworks are commonly discussed in Informativity and are typically either that of the observer’s inertial frame, the observed target or that of the universe.

Fundamental Expression

The simplest expression correlating the three fundamental measures, l_{f}m_{f} = 2θ_{si}t_{f}.

Fundamental Mass

The fundamental mass of the universe distinguishes a specific amount of mass whereby from a point in space-time additional mass would cause overlapping mass events that could not be distinguished due to physically significant bounds to the measure of fundamental units of mass. Understanding and resolving fundamental mass in turn allows one to solve for all the mass distributions presently understood only with ΛCDM.

Fundamental Measure

One of the measures length l_{f}, mass m_{f}, and time t_{f} along with their correlation called the fundamental expression. Using measurement data from the Shwartz and Harris experiments in combination with Heisenberg’s Uncertainty Principle, each are macroscopically defined and physically significant.

Informativity Differential

The Informativity differential Q_{L}n_{Lr} describes a new form of length contraction associated with the lower bound to measure. The loss of immeasurable space at each increment of t_{f} describes gravity.

Observable Mass

The observable mass includes the mass which is visible in the present and the mass which will be visible at some point in the future. The observable mass represents all the mass that can be known in the universe. This is as opposed to mass that exists sufficiently distant that it is beyond the horizon and as such, due to the expansion of the universe, the light from that mass will never reach the observer.

Quantum

The term quantum is intended to mean a small measure such as a few tens, hundreds or thousands of fundamental units of measure.

Quantized

The term quantized is intended to mean that expressions are composed of terms that are whole-unit counts of the fundamental units and that those units are physically significant.

Visible Mass

The visible mass is that mass which is presently visible. In relation to the universe this would be the mass of those stars, dust or other forms of mass that are visible in the present as opposed to the mass corresponding to light that will be visible in the future.

Symbol Definitions

H_{U} is the expansion of the universe defined with respect to the universe (diameter). This differs slightly from stellar expansion (i.e. Hubble’s description).

l_{f}, m_{f} and t_{f} are effectively Planck’s Units for length, mass, and time, but not precisely the same.

θ_{si}, is 3.26239 radians or kg×m×s^{−1} (momentum) or no units at all a function of the chosen frame of reference. This is a new constant to modern theory and exists in nearly every equation of the model. It may be measured macroscopically given specific Bell states necessary for quantum entanglement of X-rays such as those carried out by Shwartz and Harris.

β is the speed parameter typically found in relativistic expressions. The parameter varies depending on the measures being compared.

A_{s-ref} is the dilated age of the universe as measured from our point of view inside an expanding universe.

A_{s-def} is the non-dilated age of the universe as would be measured if the universe were not expanding.

M_{vis} is the mass that is presently seen from a point in space.

M_{obs} is the mass that is presently or will eventually be seen from a point in space.

M_{dkm} is the mass that is beyond the observable mass, mass which will never be seen from a given point in space.

M_{uobs} is the mass that will eventually be seen from a point in space, but has not presently in view.

M_{tot} is all the mass in the universe.

M_{f} is the fundamental mass. Mass in excess of the fundamental mass exceeds the number of mass events per unit of time that can be distinguished at a point in space.

M_{acr} is the rate of mass accretion with respect to the universe.

M_{o-f}_{(R)} is the observable mass within a given radial orbit of a target galaxy

M_{e-f}_{(R)} is the Informativity effective radial mass within of a target galaxy. The value incorporates Newton’s expression and the effects of universal expansion.

M_{b-f}_{(R)} is the Informativity mass frequency bound radial mass which corresponds to upper mass bound of mass events that equals but does not exceed the upper mass-to-length frequency bound.

M_{e} is one solar mass.

A_{U} is the age of the universe.

R_{U} is the radius of the universe.

D_{U} is the diameter of the universe.

H_{U} is the rate of universal expansion with units light-years per year.

n_{Mu} is a count of m_{f} equal to the total of mass/energy in the universe.

n_{Tu} is a count of t_{f} equal to the age of the universe.

n_{Lu} is a count of l_{f} equal to the diameter of the universe.

n_{Lo} is a count of l_{f} that is being observed.

n_{Lr} is a count of l_{f} from the observer to a center of gravity.

n_{Ll} is a count of l_{f} as measured in the local frame of reference.

n_{Tl} is the count of t_{f} as measured in the local frame of reference.

n_{To} is the count of t_{f} that is being observed.

n_{Lm} is the change in position of the target as a count of l_{f} as measured in the local frame of reference.

n_{Lc} is the change in position of light as a count of l_{f} as measured in the local frame of reference.

n_{M} is a count of m_{f} representing the mass corresponding to a gravitational field.

n_{L} is a count of l_{f} representing the length between an observer and the target.

n_{T} is a count of t_{f} representing the time elapsed between two events.

n_{Lf} is a known count of l_{f} typically used when describing distance with respect to an observer.

Q_{L} is the fractional portion of a count of l_{f} when engaging in a more precise calculation.

n_{Lb} is a known distance, a count of the reference l_{f}.

v_{n} is the radial velocity of a star plotted with respect to Newton’s expression for gravity.

v_{o} is the observed radial velocity of a star when accounting for all well-established effects.

v_{e} is the Informativity effective velocity of a star in orbit around a galactic core. The expression may resolve using Newton’s expression and the effective radial mass for a given radius.

v_{b} is the Informativity mass frequency bound velocity which corresponds to upper mass bound of mass events that equals but does not exceed the upper mass-to-length frequency bound.

G is Newton’s gravitational constant.

S is the symbol assigned to the unknown constant when resolving a description of gravity. The symbol is replaced with θ_{si}.

c is the speed of light which may also be written as c = l_{f}/t_{f}.

v is velocity measured between an observer and a target.

r is some unknown distance between an observer and a target.

h is Planck’s constant adjusted to reflect the quantum effects of the Informativity differential.

ħ is Planck’s reduced constant adjusted for the Informativity differential as a function of distance to target.

σ_{x} is a description of the uncertainty in the position of a particle.

σ_{P} is a description of the uncertainty in the momentum of a particle.

k is the Boltzmann constant.

ρ is the energy density of mass/energy accumulated at a given age of the universe.

a is the total energy radiated as described with respect to blackbody radiation (i.e. the Stefan-Boltzmann law).

T is the temperature of the Cosmic Microwave Background.

Cite this paper

Geiger, J. (2019) Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture.*Journal of High Energy Physics, Gravitation and Cosmology*, **5**, 473-506. doi: 10.4236/jhepgc.2019.52028.

Geiger, J. (2019) Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture.

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[1] Lonappan, A.I., Ruchika and Sen, A.A. (2017) Is It Time to Go beyond CDM Universe?

https://arxiv.org/abs/1705.07336

[2] Geiger, J.A. (2018) Measurement Quantization Unites Classical and Quantum Physics. Journal of High Energy Physics Gravitation and Cosmology, 4, 262-311.

https://doi.org/10.4236/jhepgc.2018.42019

[3] McCulloch, M.E. (2012) Testing Quantised Inertia on Galactic Scales. Astrophysics and Space Science, 342, 575-578.

https://doi.org/10.1007/s10509-012-1197-0

[4] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://doi.org/10.1142/S0218271809015904

[5] Geiger, J.A. (2018) Measurement Quantization Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data. Journal of High Energy Physics Gravitation and Cosmology, 4, 655-694.

[6] Shwartz, S. and Harris, S.E. (2011) Polarization Entangled Photons at X-Ray Energies. Physical Review Letters, 106, Article ID: 080501.

https://doi.org/10.1103/PhysRevLett.106.080501

[7] Allen, C.W. and Cox, A.N. (2002) Allen’s Astrophysical Quantities. Springer, Berlin, 294.

[8] McGaugh, S.S. (2008) Milky Way Mass Models and MOND. The Astrophysical Journal, 683, 137-148.

https://doi.org/10.1086/589148

[9] Madhavacheril, M., et al. (2015) Evidence of Lensing of the Cosmic Microwave Background by Dark Matter Halos. Physical Review Letters, 114, Article ID: 151302.

https://doi.org/10.1103/PhysRevLett.114.151302

[10] Hossenfelder, S. and McGaugh, S.S. (2018) Is Dark Matter Real? Scientific American.

https://www.scientificamerican.com/article/is-dark-matter-real/