The cosmic microwave background radiation, CMB, is fundamental to observational cosmology, and is believed to be a remnant of the thermal radiation from the Big Bang [1] - [13] . It is considered a landmark test of the Big Bang model of the Universe. There has recently developed questioning if this is the only possible interpretation [14] . Utilizing quantum methods perhaps a Big Bang singularity can be avoided. This perspective is not well established, and not the standard approach.

CMB measurements have been made with a series of probes including: NASA’s Cosmic Background Explorer (COBE) satellite that orbited in 1989-1996, the ESA (European Space Agency) Planck Surveyor in 2009, and Planck cosmology probe all-sky map data from 2013. The experimental value of the CMB peak spectral radiance ν_max is approximately 160 GHz. The most recent reported associated temperature is 2.72548 ± 0.00057 K. The CMB cannot be derived by the Standard Model(SM) or String Theory (ST) [15] .

Planck time, t_P, is an important unique unifying composite cosmology constant that interlinks the speed of light c, Planck’s constant h, and the Newtonian gravitational constant G that are classical, quantum, and cosmologic constants. The units of t_P are seconds, inverse Hz [16] The Hubble constant, H₀, is also linked to the Big Bang and is felt to represent a measure of the expansion rate of the universe [17] - [22] . Though frequently quoted as a velocity, its actual unit is per second or Hz. The H₀ does not represent a fixed constant, but rather an evolving one that is only measured presently as a transient value. These two cosmology constants along CMB are defined by units of time either seconds or Hz so they can be directly inter-related and mathematically scaled as dimensionless ratios. These constants are presently not understood to represent an integrated system directly linking them into a coherent mathematical or physical system where one value can be used to derive another like many other physical systems, for example, the Bohr radius, a₀.

The structure of this paper follows a logical progression. An overview of the Harmonic Neutron Hypothesis, HNH, is introduced including the concepts of a dimensionless ratio fundamental constant system, Buckingham Pi theory, a single natural unit system related to the neutron, and integer based power laws. The fundamental constants are converted to frequency equivalents then dimensionless ratios. These dimensionless constants are converted to harmonic partial fraction exponents of the frequency equivalent of the neutron. A review of the universal nature of Planck time within physics and how it is derived within the HNH are demonstrated. From the Planck time power law the Hubble constant is derived based on its partial harmonic fraction. The Hubble constant is evaluated as a β velocity ratio. The kinetic energy associated with the Hubble constant β is derived. The assignment of the harmonic fraction related to CMB is logically analyzed. Utilizing the identical kinetic energy pattern related to the Hubble constant β the peak spectral radiance of CMB is derived. CMB is transformed to a Kelvin temperature using the Wien’s displacement factor. Mathematical methods to transform Planck time to the Hubble constant or to CMB are shown. The derived values are compared to the experimental known values. A discussion of the insights into the fundamental constants as a classic harmonic system is made. Finally the conclusions are summarized.

There are many fundamental constants and calculations utilizing these constants. For clarity they will be listed here. They include: neutron, n⁰, ν refers to the frequency equivalent of a constant, electron, e-, Bohr radius, a₀, Rydberg constant, R, speed of light, c, Planck’s constant, h, Planck’s time, t_P, Planck time squared, , Hubble constant, H₀, Higgs boson, H⁰, cosmic microwave background peak spectral radiance, CMB, β, a ratio of a velocity divided by c, α, the fine structure constant, the β related to the Hubble constant, a qf is an integer fraction exponent, and δ is the difference between a fundamental constant’s exponent and its quantum fraction.

The known values are labeled with a subscript k, and the derived values with a subscript d. There are multiple possible integer “n’s” described in this model and they are differentiated. Integer exponents, i.e., of ν_F are described as “n_ie”. Partial or harmonic integer fractional exponents, ife, “n” values are described as “n_ife”, whereas the “n” of non-exponent linear entities of a consecutive integer series are described as “n_cis”. For example black body radiation energy equals n_cis(νh).

All of the data for the fundamental constants were obtained from http://physics.nist.gov/cuu/Constants/ and www.wikipedia.org. The NIST site http://physics.nist.gov/cuu/Constants/energy.html has an online physical unit converter that can be used for these types of calculations so the values used in the model are all standard unit conversions.

The floating point (the number of accurate digits) is based upon known experimental atomic data, of approximately 5 × 10⁻⁸. Other constants have lower know accurate digits. All of the fundamental constants are converted to frequency equivalents, ν. Equation (1) demonstrates the frequency equivalent conversion of the neutron as an example. c is the speed of light, h is Planck’s constant, is the mass of the neutron. Though not intuitively obvious in this type of a single physical unit system, ν_F, is an invariant coupling constant independent of whatever physical unit system is initially chosen.

Conversation of the other constants has been previously published. Masses are converted by multiplying by c² (speed of light squared) then dividing by h (Planck’s constant). Distances are converted by dividing the wavelength into c. Energies in Joules are converted to Hz by dividing by h. The eV value for the neutron is 939.565378(21) × 10⁶. Its frequency in Hz is converted to eV by multiplying by the constant, 4.13566750(21) × 10^?15eV/Hz. The eV was converted to frequency by multiplying by the constant 2.41798930 (13) × 10¹⁴ Hz/eV.

This model has two parallel domains both describing identical physical values. One domain is the frequency

equivalent of any physical value. The other domain is the exponent of the base which when raised to

that exponent equals the frequency equivalent of that specific value, Equation (2). Table 1 and Table 2 lists the standard unit quantum fractions, δs, and exponents, exp of the constants evaluated in this paper. The exponent,

exp, of a fundamental constant is the ratio of the log_e of the frequency equivalent, vs, divided by the, Equation (2). Here, equals 53.780055612(22).

The minus the quantum fraction, qf, or partial harmonic fraction equals the δ, and the Y-axis value when plotted, y, Equation (3). A quantum fraction, qf, is a possible integer fractional exponent, not solely a

harmonic or partial fraction in some settings, for example composite constants like. The frequency equivalent of a constant, v, is calculated by raising to the exp in Equation (4). When the harmonic fraction

is negative the entity has a mass less than that of the neutron, example, the electron, −1/7. When the harmonic fraction is positive the entity has a mass more than that of the n⁰, example, H⁰, 1/11.

or

Each physical entity is plotted on a log log plane in a power law format. The X-axis is related to the known or derived qf or partial fraction minus 1, Figures 1-3. This centers the neutron at the (0, 0) central point. Each

integer division of the X-axis is an integer exponent change of. scales the X-axis. The

Y-axis is δ, or y. The Y-axis is scaled by the properties of hydrogen and the composite slopes and Y-intercepts of awk, bwk, and bem. Their derivation and origins are described in References [23] [27] [28] , Table 1 and Table 2.

The classic t_P equation when transformed to a frequency equivalent system relates the gravitational binding energy, GBE, as a frequency related to the product of Planck time squared, no ħ; and the frequency equivalents of one mass, the other mass, and the distance separating them, Equation (5). equals the ratio of the frequency equivalent of the GBE divided by the product of the frequency equivalents of one mass, the other mass, and the distance separating them, Equation (6).

Planck time squared in the frequency domain is equivalent to the Newtonian gravitational constant, G. The sum of harmonic fractions defining is related to the gravitational binding energy of the electron in hydrogen include: −1 ? 1 − (6/7) − (4/5), or −128/35. These are the fractions respectively associated with the proton, −1, the gravitational binding energy of the electron in hydrogen, −1, the e⁻, −6/7, and the a₀, −4/5.

The HNH has previously derived, h not ħ, Figure 2, Table 1 and Table 2 [25] . The line defining the point is related to a composite sum of awk, bwk, and bem just a t_P is composite of three different important fundamental constants constants. The δ and exponent calculations are shown in Equations (7) and (8). The δ_d, −1.3736509 (1) ×10⁻², exp_d, −3.670879366 (1). The actual is derived in Equation (9). The known experimental value is 1.82611(11) × 10⁻⁸⁶ s². The derived value is 1.82617124(09) × 10⁻⁸⁶ s². The derived standard ħ Planck time is 5.3911418(3) × 10⁻⁴⁴ s. The known experimental value is 5.39106(32) × 10⁻⁴⁴ s.

It was assumed that H_0d was associated with the -line since it is related to gravity, and a cosmic constant, Figure 2 and Figure 3 [28] . H₀ represents a reciprocal Lorentz factor, a red shift factor. Physically associated entities fall on a common power law line as seen with the quarks [26] [29] [31] . The H₀ point falls on the line. H₀ is associated with the partial fraction of -3/4. The partial harmonic fraction 3/4 is associated with the kinetic energy lost in the beta decay process [24] . On the power law the H₀ point is on the line at the x axis position −3/4 − 1 or −7/4. The previous derivation will not be repeated, but the methods are show in Equations (10) and (11), Table 1 and Table 2. The δ_d, −5.20211263(26) × 10⁻³, exp_d, −7.75520211(04) × 10⁻¹, and actual H₀ are derived in Equation (12). The derived value is 2.29726666(11) × 10⁻¹⁸ s⁻¹. This is within the known experimental range of approximately 2.3 × 10⁻¹⁸ s⁻¹. The WMAP data combined with other cosmological data generates the best H₀ estimate of 70.4 ± 1.4 (km/sec)/Mpc. The derived value is 70.886246(4) (km/sec)/Mpc.

In the HNH the ratio values of the points from (−1, 0) to (−2, 0) on the power law plane are equivalent to velocity ratios of v divided by c, ν/c, or a classic β values utilized in Lorentz factor, red shift factor, time dilation factor of cosmology and high energy physics, Figure 2 and Figure 3. The X-axis has multiple different interpretations depending on the physical setting [27] [28] . In the HNH when the velocity/c equals 1 the β is 1. This is equivalent to the point (−1, 0). This is also the point for h and a frequency of 1 Hz. All other velocities are frac-

tional β values. The β values range from at (−2, 0) to 1 at (−1, 0).

In the HNH the ratio values of the points from (−1, 0) to (−3, 0) on the power law are related to velocity squared ratios of v² divided by c², or β² values, Figure 2 and Figure 3. These β² range from at (-3, 0) to at (−1, 0).

The original conceptual construct of the fine structure constant, α, was related to the velocity ratio, β, of v/c that would transform the energy equivalent of an electron going the speed of light, the annihilation energy, to the ionization energy of the electron. The α represents a specific β. This is a transformation of matter into either electromagnetic energy to kinetic energy. This velocity equals α × c in standard units. The energy equivalent of the mass of an electron times (cα)² divided by 2 is the ionization energy of hydrogen. An electron is not physically possible to have a velocity of c, but these types of “virtual” transformations are valid concepts in standard physics, and the HNH. This type of α virtual transformation is used multiple times in the derivation of CMB. All of the transformation on the power law plane are virtual, but remain mathematically, physically, and conceptually valid as well.

On the power law plane a β represents a negative value vector. Its square is equal to two times that vector’s distance. Any standard kinetic energy calculation can be made on the power law plane by starting at any mass

equivalent then moving down or to the left twice the distance of the distance then down or to the left the vector related to.

It is logical to associate CMB with the partial harmonic fraction of 1/2. It has an even numbered denominator which is associated with kinetic properties [24] . It represents the lowest possible energy state of the positive partial fractions. CMB is logically associated as the minimum base kinetic energy state, and the smallest prime number, 2.

It is logical to analyze H₀ as associated with a specific β_H0d identical to any kinetic process. This represents the vector along the line starting from the virtual mass equivalent of the point, 0.8509858 (42) Hz at X-axis value of −1. It extends along the line to an X-axis value of −1 − 3/4, or −7/4, the point. Here equals 1.175107647. The is associated with a of , or, , or 2.69953563(14) × 10⁻¹⁸, Figure 3, Equation (13), Table 1 and Table 2. The exponent of equals, −7.5220175 × 10⁻¹. The δ_d of equals, −2.2017483 ×10⁻³. equals 7.28749189(37) × 10⁻³⁶.

In standard experimental setting this would be unmeasurable, but in the HNH these minute values are valid and essential. This equals 6.887032 × 10⁻¹⁰ m/s in standard units. equals 4.743121 × 10⁻¹⁹ m²/s² in standard units.

The kinetic energy, KE, frequency equivalent of the H_0d system, , is related to the energy equivalent of a virtual mass of, 8.509858 (42) × 10⁻¹ Hz times, 7.28749189 (37) × 10⁻³⁶ divided by the product of 2 and^.. This equals 3.10077661 (24) × 10⁻³⁶ Hz, Figure 3, Equation (14) and Equation

(15), Table 1 and Table 2. This represents the kinetic energy of a virtual mass equivalent slightly less than a frequency of 1 with a. Here equals −1.288855455 × 10⁻². The exponent equals plus, -1.50740386(08) - 1.28885545 × 10⁻², or −1.52029241 (08). The partial fraction equals −3/2. The δ_d on the line equals , −7.40386098(37) × 10⁻³. The δ_d related to the actual energy equals plus, or −2.02924155 (10) × 10⁻². This is plotted on the 2D power law at the point ((−3/2 − 1), −2.02924155 (10) × 10⁻².

The kinetic energy associated with is related to a system identical to α and independent of any specific mass. The total ratio change from a frequency of 1 to frequency is a constant, 3.10077661(24) × 10⁻³⁶.

This transition should scale CMB as well since they are both related to gravity, are kinetic, and scaled by H₀, and t_P. This transition spans an X-axis range of −3/2. For this to define a constant at harmonic fraction of 1/2,

and X-axis location of −1/2, the original mass must start at X-axis value of 1. The mass must be

matter in a black hole. This value has also been shown to be associated with all of the quarks so this point is of

fundamental significance, and not a new observation in the HNH [26] [27] [31] . equals 2.13456013 (11) × 10³² eV, or 3.80519890 (19) × 10⁻⁴ kg. The product of 3.10077661(24) × 10⁻³⁶ and equals the

derived CMB, , 160.041737(06) × 10⁹ Hz, Equation (16), Table 1 and Table 2. The reported experimental peak spectral radiance value is approximately 160 × 10⁹ Hz. The derived exponent equals 4.79707584 (24) × 10⁻¹, and the δ_d is -2.02924155(10) × 10⁻², Equation (16). The known experimental exponent equals approximately 4.7972 × 10⁻¹, and the δ_k is -2.027 × 10⁻². , Equations (17) and (18) is the derivation from β_H0d.

The ν_max for the Wien’s displacement law is 58.7267923 × 10⁹ Hz per K. [32] The divided by this Wien’s constant is the temperature in Kelvin. This value is from a communication by Markus Nielbock [32] . The derived temperature is 2.72519 K, Equation (19). The known range is 2.72548 ± 0.00057 K.

The is derived from either H_0d or CMB_d utilizing, integer fractions, and bwk, awk, and bem, Equation (20).

The CMB_d is derived from either H_0d or utilizing, integer fractions, and bwk, awk, and bem, Equation (21).

The H_0d is derived from either or utilizing, integer fractions, and bwk, awk, and bem, Equation (22).