Discovering Nature's Hidden Relationships, an Unattainable Goal?

E-mail: jrj2222@aol.com Abstract: Discovering hidden relationships, relationships that may reveal the secrets of nature, is a difficult if not impossible goal. This is primarily because the models of the two worlds, the Standard Model of Particle Physics (SMPP) and the Standard Model of Cosmology (SMC) have no elements in common, they are completely disjoint. Thus, relationships are hidden and finding them, may prove to be unattainable. However, attempts to identify "hidden relationships" must be based on dimensionless numbers and include constants and equations from both models. This article, by defining the dimensional and dimensionless constants within the SMPP and the SMC, illustrates the difficulty of finding relationships. Familiar ratios and a select group of ratios using Planck and Hubble constants are analyzed. The Schwarzschild black hole equation explains key relationships.


Introduction
No theory explains the secrets of nature-forces and constants are what they are. Why is it difficult to discover hidden relationships that might reveal these secrets? A complete picture of the universe must include both particle physics and cosmology. However, the Standard Model of Particle Physics (SMPP) and the Standard Model of Cosmology (SMC) are disjoint models. Most professional physicists and astronomers would say finding "hidden relationships" is a wild dream, a foolish, almost impossible goal. The noted string theorist, Ed Witten, once said that "he'd given up on ever predicting all constants of nature" (Tegmark, 2014). This article, by defining the dimensional and dimensionless constants within the SMPP and the SMC, illustrates the major difficulty. A quote from Paul Davies sets the stage and challenge:"Mathematics and beauty are the foundation stones of the universe. No one who has studied the forces of nature can doubt that the world about us is a manifestation of something very, very clever indeed" (Davies, 1986).
So why is nature disguised so well? Principally because the models of the two worlds, the Standard Model of Particle Physics (SMPP) and the Standard Model of Cosmology (SMC) (Scott, 2006) have no elements in common, they are completely disjoint. Thus, relationships are hidden and the task of finding them, may prove to be virtually unattainable. However, if secrets are to be found, dimensionless ratios using constants from both models is an appropriate approach.
After explaining the divergent characteristic of the SMPP and the SMC, the disjoint between the two models will be clear. Next, dimensionless numbers are explained, why they are required and how they are calculated. Then, constants and equations from both models are defined. And last, a select group of ratios using Planck and Hubble constants are presented. Is the large number coincidence documented (Valev, 2013) a hidden relationship or speculation? Disjoint Models: Standard Model of Particle Physics (SMPP), Standard Model of Cosmology (SMC) Overview Why are the SMPP and the SMC disjoint models? Table 1 provides an overview of the two models highlighting significant differences. The SMPP involves microscopic sizes (elementary particles, quarks and bosons); conversely, the SMC deals in the macroscopic world with planets, stars, galaxies and black holes. Differences in sizes are immense, for example, the radius of a hydrogen atom approximately 10 −15 meters is minuscule when compared with cosmological distances measured in light years, each light year about 10 16 meters. Both models contain successful theories: In SMPP, Quantum Electrodynamics (QED), Quantum Chromo Dynamics (QCD) and Special Relativity; and in the SMC, the Big Bang Theory, Inflation and General Relativity. Exact equations are the rule in SMPP, for example in QED, theoretical values match observation to many decimal places; but in the SMC, many calculations are approximate, for example, how exact is the number of baryons (protons and neutrons) in the observable universe which is estimated at 10 80 ?
Both models have numerous unique/mysterious features-the uncertainty principle and quantum weirdness for the SMPP and dark matter and dark energy for the SMC. Dimensional (Physical) constants applicable in each model have no relation to each other. Elementary particles with their associated mass and charge are quite dissimilar from the concepts of: Expanding space, CMB and critical density. With significant difference in characteristics, sizes, unique features and constants, the two models earn the right to be labeled disjoint.
The dimensionless constants play integral roles in their respective theories. In the SMPP, the fine structure constant (α) is essential to QED, with the strong (α S ) and weak (α W ) force strengths integral to Electroweak Theory and QCD respectively. In the SMC, the measure of homogeneity (Q) and gravitational force strength dictate the distribution of galaxies. However, no commonality is identified with dimensionless constants. One attempt to combine constants from both models, defined an equation (m pion = (ħ 2 H/(Gc)) 1/3 ) using the Planck constant (ħ) and the Hubble constant (H) (Weinberg, 1972). The resulting value is in units of mass, specifically a very small mass (1.06×10 −28 kg) -almost, but not exactly, equal to the Pion mass (2.50×10 −28 kg). Thus, the relationship is probably a coincidence. Another attempt used the same constants to calculate a "hypothetical" mass, actually the smallest possible mass (m H ), a minuscule value of 2.53×10 −69 kg (m H = ħH/c 2 ) (Valev, 2013). This hypothetical particle, possibly the energy of a graviton, can be used to create ratios with other masses as will be demonstrated later. These two equations may be the only good examples of equations that result in units of massby using both the Planck and Hubble constants. Next, why dimensionless numbers are required when searching for hidden relationships.

Dimensionless Numbers, Criteria
Since physical constants are defined by a system of units (for example, SI units-kilograms, meters and seconds), they do not represent inherent features of the universe-change the units and the comparison changes. Units are arbitrary standards. Thus, ratios where units cancel producing dimensionless numbers are necessary to discover symmetry or fundamental relationships. Using a dimensionless ratio like electron mass divided by proton mass, assures that in any system of units, the ratio would be the same, in this example, β equals 1/1836.12 or 5.45×10 −4 . Laws of physics are independent of arbitrary units and so are dimensionless ratios. As long as the system o units are the same for the numerator and denominator, the ratio of the two numbers represents an inherent feature of nature. The goal is to search for consistencies among many possible ratios.
But not all ratios reveal fundamental features, for example, one popular comparison is the mass of a typical star divided by the mass of an electron, a ratio of about 10 60 (Jordan,1947). This ratio does compare the micro and macro worlds, but since the mass of stars vary by a factor of one thousand, the ratio depends on the star chosen. Ideally, ratios should have some unique physical significance and identify a possible pattern with other ratios. More on this later, but first, a review of the familiar constants in each of the models.

Constants and Equations
Standard Model of Particle Physics

SMPP Constants
The SMPP physical constants include: speed of light, mass of electron and proton, electron charge, Planck constant and many others (reference Table 2).The dimensional constants are shown with SI units of measure. These physical constants occur in basic theories of physics (Quantum Mechanics, Newtonian Physics, Relativity and Electromagnetism) and have universally used symbols. They are a subset of approximately 200 defined fundamental physical constants which are virtually all dimensional.
The SMPP has few natural dimensionless constants, most are ratios like the fine structure constant. (On a more technical level, about twenty SMPP input parameters-coupling constants and matrix angles-exist, however when combined with the Higgs field the result is a unit of mass for particles) (Tegmark et al., 2006). Of the four SMPP dimensionless constants shown in Table 2, the two most useful and most mysterious are the fine structure constant (α) and the ratio of the electron to proton mass (β). The fine structure constant (which is a ratio itself) describes the strength of the electromagnetic interaction which determines the structure of atoms/molecules and the behavior of light. The constants α and β are the only two dimensionless constants required for the formulation of Quantum Electrodynamics (QED) and thus, reveal an underlying unity (Barrow and Tipler, 1986). The values of α and β appear to be unique, no one knows why are what they are -it is acomplete mystery. However, within the SMPP, their relationship is certainly not hidden.
In an attempt to predict fundamental particle masses, numerous mathematical analysis have been performed. One clever one, employs the electron mass and the fine structure constant-the muon, pion and kaon masses are "almost" exactly the electron mass times: 1.5/α, 2/α and 7/α respectively (Davies, 1986). Also, using coupling constant ratios and dimensional analysis, individual masses have been computed for selective particles (not all fundamental particles) (Valev, 2009;2010;Forsythe and Valev, 2014). Conversely, plotting the masses of nine fundamental particles (electron, muon, tauon and six quarks) produces a statistically random distribution rather than a logical pattern (Tegmark, 2014). Thus, physicist attempts to define a theory explaining fundamental particle (and subatomic) masses has been, to date, in vain.
The three SMPP forces-electromagnetic, weak and strong-are indeed strange and thus difficult to relate. They operate over distance in totally dissimilar ways, for example, the strong force increases with distance but only acts over a extremely short range, the weak force also acts over a short range but decreases with distance, the electromagnetic force decreases with distance squared but has unlimited range. How could they be more different?

SMPP Equations
The equations to calculate values for the electromagnetic force strength (the fine structure constant), the classical electron radius, the Bohr radius and the Planck entities are also shown in Table  2. These values can be used to formulate dimensionless ratios. The Planck mass is calculated from just three "universal" physical constants: speed of light, gravitation constant and Planck constant. By forming equations with these three constants, natural or Planck units are also created for length, time, temperature, energy and density. These values are primarily used in theoretical calculations. The Planck length is extremely small even when compared to nuclear sizes, but the Planck mass is relatively gigantic, entering the macro world with the mass of a grain of sand.

SMC Constants
The SMC constants address aspects of space (reference Table 3); they have virtually nothing in common and are not derived from the SMPP constants (Scott, 2006). The SMC dimensional constants shown are: Hubble constant (H), CMB temperature (T ɣ ), energy density (ρ C , ρ Λ , ρ B , ρ CDM , ρ ɣ ) and number density of photons and baryons (η γ , η B ) (Burles et al., 2001;Spergel et al., 2003). All except the dark energy density (ρ Λ ) actually vary over time but are considered constants because they change so slowly. The density of dark energy is predicted to remain constant per volume as the universe ages. Assuming the amount of baryon and cold dark matter is constant over time, the expanding universe dictates their decreasing density.  T PL = (m PL c 2 )/k B = 1.10×10 32 K Planck energy E PL = (ħc 5 /(2G)) 1/2 = 1.38×10 9 J Planck density ρ PL = 3c 5 /(16πħG 2 ) = 3.11×10 95 kg/m 3 Notes: 1. The strong force, α S , is sixty times α at a separation of 3×10 −17 m at low energy levels (SMPI, 2014) 2. The weak force, α W , is 10 −4 times α at a separation of 3×10 −17 m at low energy levels (SMPI, 2014) 3. The Electromagnetic force strengthis the ratio of electrostatic energy of repulsion between two elementary charges, e, separated by one Compton wavelength, to the rest energy of a single charge: k e e 2 /(ħ/(m e c))/(m e c 2 ) = k e e 2 /(ħc) 4. Planck length/mass is calculated by setting the reduced Compton wavelength, λ, equal to the gravitational radius (Schwarzschild): λ = ħ/(m PL c) = r s = 2Gm PL /c 2 , (Valev, 2013) The SMC also has few dimensionless constants. One well known is the measure of homogeneity in the universe, denoted by "Q" (Bennett et al., 1996;Smoot et al., 1992). It dictates how galaxies and clusters of galaxies form and is actually a ratio -the energy required to disperse cosmic structures (stars, galaxies and clusters of galaxies) divided by their rest-mass energy. Q is validated by the difference in the CMB radiation intensity, about two in 100,000 (Tegmark et al., 2006). Other familiar dimensionless constants are: Density ratios (matter, cold dark matter and dark energy), the number of baryons, the number of photons, the baryon-to-photon ratio and the gravitation force strength. The dimensionless density ratios (Ω Λ , Ω B , Ω CDM ) reflect current matter/energy density-as we know at very early times radiation dominated, then matter and now in our current epoch and in the future, dark energy dominates. The total density divided by the critical density equals one (within 1-2 percent), assuring a flat universe. The number of photons/baryons is calculated by multiplying their density times a volume based on the co-moving distance/radius, about 3.6×10 80 m 3 (Johnson, 2012). The gravitational force plays an essential role in cosmology but is not in the SMPP.

SMC Equations
The three equations for radius, mass/energy and critical density, are derived using three constants: speed of light (c), gravitational constant (G) and Hubble constant (H), reference Table 3. The mass and radius define a Hubble sphere, a finite space which encompasses the gravitationally connected universe. The radius of the Hubble sphere is 13.8 billion light years. A flat universe, requires the observed density to equal the critical density which amazingly it does. The gravitational force strength is calculated by the equation shown using the mass of a proton and electron.

Dimensionless Constants
In both the SMPP and SMC, there are few dimensionless constants to compare, reference Table 2 and Table 3. Three of the SMPP dimensionless constants reflect force strength ranging in value from about one to 7×10 −7 . These forces act in totally different ways as previously noted. The commonality with SMC, is that both the gravitational and electromagnetic force strengths decrease exactly the same, directly proportional to distance squared. This commonality is offset by the huge disparity of their magnitudes.
The SMC density ratios vary from 0.05 (Ω B -baryon density/critical density) to 0.68 (Ω Λ -dark energy density/critical density), a relatively small range which reflects the epoch of our time. However, there is no apparent relationship to SMPP constants. Two relatively close values are β (mass electron divided by mass proton) and Q (measure of homogeneity), 5×10 −4 and 2×10 −5 respectively. Is there a possible relationship between them? If so, it is not obvious.
The number of photons and baryons, which are large numbers, have no counterpart in SMPP. Thus, if there is any chance of finding hidden relationships, dimensionless ratios derived with dimensional values from both the SMPP and SMC must be employed.

Ratios with no Significance
Although not comprehensive, the data referenced in Table 2 and 3 provide an extensive source for creating ratios between the SCM and SMPP. Numerous ratios are possible but finding comparisons that result in more than a coincidence is challenging, for example, compare the ratio of the electromagnetic force strength divided by gravitational force strength (2.3×10 39 ) to the ratio of the universe radius (assuming a Hubble sphere) divided by classical electron radius (4.9×10 40 ). Although the values differ by more than ten times, the similar size of the exponents, 39 and 40, encourages an interpretation of a possible relationship-something more than a coincidence. However, this type of comparison and similar attempts have evolved into numerology, the unscientific manipulation of numbers to substantiate theories. Examples of these two and six other similar "large number" ratios are shown in Table 4. All have no apparent significance.

One Possibility with the Hubble and Planck Constants
One novel approach is based on only three fundamental constants (speed of light, gravitational constant, Planck constant) and the Hubble Constant. Six dimensionless ratios and their supporting equations (Forsythe and Valev, 2014) are listed in Table 5. The numerator and denominator, each from their respective models, have values with the same units (in each ratio, the numerator is the larger number eliminating negative exponents or small fractions).
The first four ratios based on the Hubble constant and the Planck constant produce an interesting result -an exact value of 6.05×10 60 -a number labeled "N". Also, critical density divided by Planck density (ρ C /ρ PL ) equals 36.6×10 120 or N 2 . Using the hypothetical mass with both the Planck and universe masses produce the same two values (N and N 2 ). Do the six ratios identify a hidden relationship or is there an explanation? How are the exact values of N and N 2 calculated? The next section addresses these questions.

Black Holes
How do black holes play a part in this coincidence game? The mass of a static non-rotating black hole is proportional to the Schwarzschild radius (m = r s c 2 /(2G)). The equations to compute both the universe mass/radius and the Planck mass/radius satisfy the Schwarzschild relationship (although expanding space is quite different than a black hole). Thus, as the chart in Fig. 1 shows, based on algebra, N equals: Universe mass divided by Planck mass (M U /m PL ), radius divided by length (R U /l PL ), age divided by time (T U / t PL ) and the square root of Planck Density divided by Critical Density (ρ pl /ρ C ). If the mass to radius proportionality is the same for any two masses, this relationship holds although the value of N is determined by the specific mass. Thus, since both mass/radius relationships are equal, algebraic correlation guarantees the other three relationships as shown in Fig. 1 -they are not unique coincidences, but rather a direct result of the Schwarzschild relationship.

Equations
Now for an explanation of why the ratios equal an exact value of N (6.05×10 60 ) or N 2 (36.6×10 120 ). It is because the ratios are derived from equations, reference Fig. 2 which defines relationships between three basic ratios: mass of the universe divided by Planck mass (M U /m PL ); Planck mass divided by the hypothetical mass (m PL/ m H ); and, mass of the universe divided by the hypothetical mass (M U /m H ). When we divide the numerator and denominator for each ratio, the resulting equation is: N = (c 5 /(2ħGH 2 )) 1/2 for the first two and N 2 = c 5 /(2ħGH 2 ) for the third ratio. The value of the first equation is a dimensionless number, 6.05×10 60 , the value of the second equation is 36.6×10 120 .
Other comparisons of large numbers, are not exact; but, these ratios, calculated from four constants of nature, are exact. Although we are now dealing with only three "basic" ratios, it is remarkable that they produce the same exact equation and equation squared (Valev, 2013) for a value of N and N 2 . Is this one unique calculation or does N have additional ramifications, does it reveal a hidden relationship in nature? The reader can decide.  Planck mass m PL = (ħc/(2G)) 1/2 Planck length l PL = (2ħG/c 3 ) 1/2 Planck time t PL = l PL /c = (ħG/c 5 ) 1/2 Planck density ρ PL = 3c 5 /(16πħG 2 ) Both Hypothetical mass m H = (ħH)/c 2

Conclusion
In our quest to discover hidden relationships, we have documented how the SMPP and SMC are disjoint models with significant differences. The SMPP has numerous exact dimensional constants. The SMC has only a few dimensional constants, most with approximate values and some varying with time like the Hubble constant. Both models contain only a few dimensionless constants. Thus, the search for hidden relationships is based on dimensionless ratios. Comparisons are difficult because of the diversity in the characteristics, objects, features and constants between the micro and macro environments. They are also suspect because the time dependence of key SMC constants. A number of familiar ratios, derived from micro and macro dimensionless constants have no significance. However, ratios created with four "constants" and supported by equations provide a possible inherent relationship based on a large number (N).
The framework defined provides perspective; but, have we identified how nature's secrets are so ingeniously disguised? Unfortunately, no. Why things "are the way they are" is still a mystery-discovery possibly unattainable via analysis of dimensionless numbers. However, from a mathematical point of view, if a Theory of Everything (TOE) is realized via String Theory or another theory, the answer may not be complex, quoting John Wheeler: "Behind it all is surely an idea so simple, so beautiful, that when we grasp it-in a decade, a century, or a millennium-we will all say to each other, how could it have been otherwise?"