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Purpose: The cosmic microwave background radiation, CMB, is fundamental to observational cosmology, and is believed to be a remnant from the Big Bang. The CMB, Planck time,
t_{P}, and the Hubble constant,
H
_{0}, are important cosmologic constants. The goal is to accurately derive and demonstrate the inter-relationships of the CMB peak spectral radiance frequency,
t_{P}, and
H
_{0} from neutron and hydrogen quantum data only. Methods: The harmonic neutron hypothesis, HNH, evaluates physical phenomena within a finite consecutive integer and exponential power law harmonic fraction series that are scaled by a fundamental frequency of the neutron as the exponent base. The CMB and the
H
_{0} are derived from a previously published method used to derive
t_{P}. Their associated integer exponents are respectively +1/2, −3/4, and −128/35. Results: Precise mathematical relationships of these three constants are demonstrated. All of the derived values are within their known observational values. The derived and known values are:
ν_{CMB}, 160.041737 (06) × 10
^{9} Hz, ~160 × 10
^{9} Hz; 2.72519
K, 2.72548 ± 0.00057
K,
H
_{0} 2.29726666 (11) × 10
^{−18} s
^{−1}, ~2.3 × 10
^{−18} s
^{−1}; and
t_{P} 5.3911418 (3) × 10
^{−44} s, 5.39106 (32) × 10
^{−44} s. Conclusion: The cosmic fundamental constants
t_{P},
H
_{0}, and
CMB are mathematically inter-related constants all defined by gravity. They are also directly derivable from the quantum properties of the neutron and hydrogen within a harmonic power law.

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 [

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) [

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 [_{0}, is also linked to the Big Bang and is felt to represent a measure of the expansion rate of the universe [_{0} 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, a_{0}.

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.

The goal of this work is to derive and predict the peak spectral radiance, ν_{max}, ν_{CMB} and the associated temperature using the Wien’s displacement factor, t_{P} and H_{0} from the subatomic data of the frequency equivalents ν of the neutron (n^{0}), electron (e^{−}), Bohr radius a_{0}, the Rydberg constant R, and integers. The methods of the HNH will be utilized [_{P}, H_{0}, Higgs boson, H^{0}, the quarks, and the energy lost in neutron beta decay from the same limited subatomic data and integer exponent fractions, _{0}, t_{P}, and the concept of fine structure β constant, α will be demonstrated.

physical constant (derived_{d} and known_{k}) | value |
---|---|

2.2718590 (01) × 10^{23} | |

53.780055612 (22) | |

bwk: y intercept, weak force, wk line | 3.51638329 (18) × 10^{−3} |

awk: slope, weak force, wk line | 3.00036428 (15) × 10^{−3} |

bem: y intercept, electromagnetic, em line | −3.45168347 (17) × 10^{−3} |

aem: slope, electromagnetic, em line | −3.45168347 (17) × 10^{−3} |

_{ } | 1.82617124 (09) × 10^{−86} s^{2} |

1.82611 (11) ×10^{−86} s^{2} | |

t_{Pd}, with ħ_{ } | 5.3911418 (3) × 10^{−44}s |

t_{Pk}, with ħ_{ } | 5.39106 (32) × 10^{−44} s. |

H_{0d} | 2.29726666 (11) × 10^{−18} s^{−1} |

H_{0k} | ~2.3 × 10^{−18} s^{−1} |

CMB_{d}, ν_{max} | 160.041737 (06) × 10^{9} Hz |

CMB_{k}, ν_{max} | ~160 × 10^{9} Hz |

derived CMB in Temperature | 2.72519 K |

known CMB in Temperature | 2.72548 ± 0.00057 K |

The following is a limited review and explanation of the HNH. The details have been described in multiple previous publications, and will not be repeated [

The whole system is scaled by a single natural unit fundamental frequency, ν_{f}. In this model the annihilation

frequency of the neutron, n^{0}, times by 1 second, 2.2718590(01) × 10^{23} Hz × s,

fundamental constant that is invariant by definition when evaluated within a single unit coupling constant system similar to Planck units. These integers and fractions define a global harmonic system that encompasses the geometry of a circle, and therefore all of the associated properties of waves including π, frequency, and phase. All harmonic systems are power laws and therefore exponential by definition, though most commonly viewed from the linear perspective. The pure number properties such as even, odd, prime, composites, squares, sums, square roots, etcetera dominate and explain the organization of the model and secondarily all physical systems [

The HNH is a single natural unit harmonic system spanning different physical domains. Each physical state, distance, time, matter are defined with an integer scaled by the fundamental frequency, ν_{F} in the linear domain.

Each force/physical domain unit is defined by an integer exponent of

the unit for elemental gravity. This is related to two times the gravitational binding energy of the electron in hy-

drogen, 2 × 2.9001 × 10^{−24} Hz. The reciprocal of ^{−24} so they are nearly identical. This

is equivalent to the ionization energy of hydrogen, but in the gravitational domain. The exponent 0 is associated with 1 Hz which is related to the Hz equivalent of Planck’s constant, h. Each electromagnetic wave is in integer Hz, the unit for electromagnetic energy. Exponent 1 is related to the neutron, and the unit for atoms. The exponent 2 is related to the next logical consecutive integer of the series. This is related to a mass equivalent of

^{46} Hz or 3.8052 × 10^{−4} kg. This mass

must be a mass within a black hole state. The Compton wavelength of this dense mass is smaller than it Schwarzschild radius. Therefore, the exponent consecutive series corresponds to the units for gravity, electromagnetic, atomic, and black hole domains.

The frequency equivalent of other observable physical constants and particles, such as the Bohr radius, a_{0}, the Rydberg constant, R, Higg’s boson, H^{0}, the electron, e^{−}, leptons, bosons, quarks, mesons are also associated with an integer, n, in the exponential domain. This n defines a partial fraction, 1 ± (1/n), that is an exponent of the

The physical constants also follow power law properties within a harmonic system with the four natural unit integer values that scale the system, namely the n^{0}, e^{−}, a_{0}, and R. Only two of the hydrogen natural units need to be known since the other can be derived from other two. Therefore there are three essential natural units.

^{−}, a_{0}, and R scale the global system. The model has predictive pow-

er beyond standard methods by analyzing the system as an integrated harmonic spectrum rather than individual, isolated physical values. Harmonic systems have an inherent predictive power where if the ν_{F} is known then all of the possible harmonics can be defined or vice versa. This mathematical property accounts for the possibility of deriving cosmologic constants from non-cosmologic physical data (subatomic data in particular).

The model utilizes the dimensional analysis methods similar to that of Rayleigh and Buckingham’s Pi Theorem. The Buckingham Pi Theorem states that physical laws are independent of the form of the physical units. Therefore, acceptable laws of physics are homogeneous in all dimensions.

All of the physical phenomena are evaluated as frequency equivalents, and secondarily as dimensionless coupling constant ratios. The system is physical unit-independent. The fundamental frequency, ν_{F}, ratio we use as a dimensionless number, is that scaled by the annihilation frequency of the neutron. Most other units are 1, and drop out. All entities are defined by exponents of ν_{F} for the constants or integer values for the individual states.

All predictions and derivation are from two (2) finite integer sets. The first set includes four natural units based on known atomic quantities as frequency equivalents. Included in this set is the neutron, n^{0}, and comprises ν_{F}. A partition of this set includes the frequency equivalents of the_{R}. From this initial data composite universal slopes and Y-intercepts of two lines have been reported,

The unit system of the HNH is simplified with the units for Planck’s constant, electric charge, time, the distance light travels in one unit of time, and the speed of light all equaling 1. All entities are defined as in terms of ratios of two linear states, or exponents of ν_{F}. Energy, mass, frequency, temperature are all scaled equally. Dis-

tance is defined as 1 divided by frequency, 1/ν. The speed of light equals

There are many fundamental constants and calculations utilizing these constants. For clarity they will be listed here. They include: neutron, n^{0}, ν refers to the frequency equivalent of a constant, electron, e-, Bohr radius, a_{0}, Rydberg constant, R, speed of light, c, Planck’s constant, h, Planck’s time, t_{P}, Planck time squared, _{0}, Higgs boson, H^{0}, cosmic microwave background peak spectral radiance, CMB, β, a ratio of a velocity divided by c, α, the fine structure constant,

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, _{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 c^{2} (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) × 10^{6}. Its frequency in Hz is converted to eV by multiplying by the constant, 4.13566750(21) × 10^{?15}eV/Hz. The eV was converted to frequency by multiplying by the constant 2.41798930 (13) × 10^{14} 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

that exponent equals the frequency equivalent of that specific value, Equation (2).

exp, of a fundamental constant is the ratio of the log_{e} of the frequency equivalent, vs, divided by the

The

harmonic or partial fraction in some settings, for example composite constants like

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 n^{0}, example, H^{0}, 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

Physical constant | qf | δ | exponent |
---|---|---|---|

−128/35 | −1.3736509 (1) ×10^{−2} | −3.670879366 (1) | |

H_{0d} | −3/4 | −5.20211263 (26) × 10^{−3} | −7.75520211 (04) × 10^{−1} |

β_{H}_{0d} | −3/4 | −2.2017483 ×10^{−3} | −7.5220175 × 10^{−1} |

−3/2 | −2.02924155 (10) × 10^{−2} | −1.52029241 (08) | |

v_{CMBd} | 1/2 | −2.02924155 (10) × 10^{−2} | 4.79707584 (24) × 10^{−1} |

integer division of the X-axis is an integer exponent change of

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 [

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,

Planck time squared in the frequency domain is equivalent to the Newtonian gravitational constant, G. The sum of harmonic fractions defining ^{−}, −6/7, and the a_{0}, −4/5.

The HNH has previously derived_{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 ^{−86} s^{2}. The derived value is 1.82617124(09) × 10^{−86} s^{2}. 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 _{0} represents a reciprocal Lorentz factor, a red shift factor. Physically associated entities fall on a common power law line as seen with the quarks [_{0} point falls on the _{0} 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 [_{0} point is on the _{d}, −5.20211263(26) × 10^{−3}, exp_{d}, −7.75520211(04) × 10^{−1}, and actual H_{0} 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 H_{0} 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,

tional β values. The β values range from

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 v^{2} divided by c^{2}, or β^{2} values, ^{2} range from

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

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 [

It is logical to analyze H_{0} as associated with a specific β_{H}_{0d} identical to any kinetic process. This ^{−18}, ^{−1}. The δ_{d} of ^{−3}. ^{−36}.

In standard experimental setting this ^{−10} m/s in standard units. ^{−19} m^{2}/s^{2} in standard units.

The kinetic energy, KE, frequency equivalent of the H_{0d} system, ^{−1} Hz times^{−36} divided by the product of 2 and^{.}. This equals 3.10077661 (24) × 10^{−36} Hz,

(15), ^{−2}. The ^{−2}, or −1.52029241 (08). The partial fraction equals −3/2. The δ_{d} on the ^{−3}. The δ_{d} related to the actual energy equals ^{−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 ^{−36}.

This transition should scale CMB as well since they are both related to gravity, are kinetic, and scaled by H_{0}, 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

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 [^{32} eV, or 3.80519890 (19) × 10^{−4} kg. The product of 3.10077661(24) × 10^{−36} and

derived CMB, ^{9} Hz, Equation (16), ^{9} 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 × 10^{9} Hz per K. [

The _{0d} or CMB_{d} utilizing

The CMB_{d} is derived from either H_{0d} or

The H_{0d} is derived from either

A robust model that scales from quantum to cosmic physical constants does not exist. The SM and ST fail. An important unique aspect of the harmonic neutron hypothesis is that it does resolve some of these limitations by analyzing the scaling relationships between the physical constants as a classic unified harmonic power law system. Derivation of an accurate CMB, and directly relating CMB to previously published derived values of H_{0} and t_{P} supports the validity of the model. There is no experimental astronomy data utilized. The initial data utilized for all of the calculations are subatomic data. This accounts for the high resolution result. It is remarkable that three of the most important constants related to cosmology are all related to a single δ-line that defines gravity and Planck time.

In 2009 it was proposed that the fundamental constants were related to four physical values, n^{0}, e^{−}, a_{0}, and R; and integers. Composite values uniting the subatomic properties of hydrogen including: awk, bwk, and bem, were published. Only these values have been used for all of the subsequent physical constant derivations. The composite slopes and Y-intercepts utilized on power law plots in the interval have been shown to accurately predict and derive some of the most important physical constants. Utilization ofthis highly limited physical datain conjunction with fixed integer fractions makes it very unlikely that all of these derivations are coincidence. The predicted H^{0} was 125.120961 GeV/c^{2} in 2014, and recent data reports a mass of 125.09 ± 0.21(stat.) ± 0.11(syst.) GeV/c^{2} [

The HNH is based on pure integer mathematical properties. It has recently been shown that π/2 can be derived from the orbital energy levels seen in the hydrogen atom [

A high accuracy CMB is an important physical constant for physics. The experimental data are moderately precise [

In the HNH t_{P}, CMB, H^{0}, and H_{0} are analogous to the other free space constants, permittivity and permeability, and represent natural inherent properties of space. Conceptually they need not be the result of some other previous physical event such as the Big Bang. Both CMB and H_{0} are viewed within the standard model of astronomy as “transient” values that are measured today, but will evolve over many years. This is not the case within the HNH. There will never be any actual physical proof that these values are changing because of their immense time scale.

The HNH model is purely quantum and it is inappropriate, and conceptually inaccurate to try, and interpret physical phenomena from a classic perspective including all of the cosmology or quantum constants. There are many probabilistic properties of cosmic phenomena. The location of short gamma foci are randomly distributed. The dimension of galaxies and rotation rates of stars in galaxies are distributed over a range, but there is a general pattern.

This derivation demonstrates that H_{0} has a triple physical context. H_{0} is typically viewed as a measure of the expansion of the universe. In this model as a rotation rate of the visible universe, a cosmic “year”. H_{0} can be viewed as associated with a specific β. The kinetic energy of that β^{2} follows the classic definition of the product of the mass times β^{2} divided by 2. H_{0} also represents a red shift factor that is associated with a reciprocal of Lorentz factor. It is hypothesized that this Lorentz factor should be related to the maximum cosmic ray energy. The harmonic fraction is logically related to 3/2, 1 + (1/2). CMB is the smallest cosmic ray and related to a harmonic fractions of 1/2, 1 − (1/2). Therefore its harmonic fraction partner 3/2, 1 + (1/2) should be the largest cosmic ray. This is analogous to the relationship of the quarks. Both of these systems have the same harmonic fraction, but just inverse signs.

physical entity. In this case CMB represents the kinetic energy of this mass with a β related to H_{0}. This is combining gravity related to G and the

This is a unique unification of three of the most important cosmology observations into one unified system all based on a common line that defines Planck time. t_{P} is a natural composite by definition. It is the only physical constant that spans all domains. The values of t_{P}, H_{0}, and ν_{CMB} in the existing SM are not directly mathematically inter-related variables. They are in part conceptually related, but not in a format that helps scale their relative values. This paper shows that they are all closely related, and all are related to points on the

The harmonic neutron hypothesis accurately scales quantum to cosmologic physical phenomena. This is a property of power laws. This method allows for derivation of physical constants independent of any specific associated typical experimental data. This is a classic property of all harmonic systems so this is not a new concept. The fundamental constants viewed as a unified harmonic power law system are a new concept. This is not the standard perspective of the Standard Model, but all of the foundations of the HNH are within the SM. The SM also fails to unite quantum and cosmic phenomena. The fundamental cosmologic constants CMB, H_{0}, and t_{P} can all be derived and accurately predicted from subatomic data only. The predictions exceed the known experimental precision so they could be of experimental significance. These constants are also mathematically and conceptually closely inter-related. This is a new observation uniting cosmology within a harmonic system. These constants are all based on Planck time. Therefore all the observable cosmic phenomena including black holes are directly related to the single unifying force of gravity. It is assumed that the maximum cosmic ray energy will represent another harmonic fraction scaled by gravity as well.

I would like to thank Richard White MD, and Richard Vento for their support and help.

Donald William Chakeres,Vola Andrianarijaona, (2016) The Derivation of the Cosmic Microwave Background Radiation Peak Spectral Radiance, Planck Time, and the Hubble Constant from the Neutron and Hydrogen. Journal of Modern Physics,07,573-586. doi: 10.4236/jmp.2016.76060