\[\color{blue}{\text{Symbolic identity key:}}\]
\[\begin{array}{lcl} \text{s} \text{ - spin quantum number} \\ n_{s} \text{ - spin states total integer helicity number} \\ N_{s} \text{ - species total integer number} \\ N_{n} \text{ - total effective degeneracy number} \\ \end{array}\]
\[n_{s} = 2 s + 1 \; \; \; \; \; \; s < 2 \; \; \; \; \; \; m = 0\]
\[n_{s} = 2 \; \; \; \; \; \; s \geq 2 \; \; \; \; \; \; m = 0\]
\[\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}\]
\[\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}\]
\[\color{blue}{\text{Toy model quantum particle properties chart:}}\]
\[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\]
\[\color{blue}{\text{For massless quantum particles, the transverse modes cannot exist due to Lorentz invariance.}}\]
\[\color{blue}{\text{Only positive and negative helicity states remain. For massless scalar particles, only zero helicity states remain.}}\]
\[\color{blue}{\text{The spin 1 photon is also restricted to its positive and negative helicity states, and has a total effective degeneracy number of 2.}}\]
\[\color{blue}{\text{A massless graviton has only 2 helicity states, and has a total effective degeneracy number of 2.}}\]
\[\color{blue}{\text{Dark energy quinton total effective degeneracy number:}}\]
\[\boxed{N_{\Lambda} = 1}\]
\[\color{blue}{\text{Planck satellite dark energy cosmological composition parameter:} \; (\text{ref. 1, pg. 11})}\]
\[\Omega_{\Lambda} = 0.6825\]
\[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation temperature:} \; (\text{ref. 2})}\]
\[\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3} \right)^{1/4}}\]
\[\boxed{T_{\Lambda} = 35.013 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\Lambda}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right]}{2 \pi} \left(\frac{45 \Omega_{\Lambda} H_0^2 c^5}{4 \hbar G N_{\Lambda} \pi^3} \right)^{1/4}}\]
\[\boxed{f_{\Lambda} =  2.05837 \cdot 10^{12} \; \text{Hz}}\]
\[\boxed{f_{\Lambda} =  2.058 \; \text{THz}}\]
\[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\]
\[T_{\gamma} = 2.72548 \; \text{K}\]
\[\color{blue}{\text{Cosmic neutrino and sterile neutrino background radiation temperature at present time:} \; (\text{ref. 5})}\]
\[T_{\nu} = \left(\frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}\]
\[\boxed{T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Dark matter scalaron and sterile neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\]
\[\boxed{T_{\phi} = T_{s \nu} = T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein dark matter cosmic scalaron background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\phi} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\phi}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\phi} = 1.14366 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\phi} = 114.366 \; \text{GHz}}\]
\[\color{blue}{\text{Fermi-Dirac dark matter sterile neutrino background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{s \nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{s \nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{s \nu} = 126.915 \; \text{GHz}}\]
\[\color{blue}{\text{Solve for cosmic neutrino background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Neutrino species total effective degeneracy number:}}\]
\[N_{\nu} = 3.046\]
\[\color{blue}{\text{Neutrino radiation energy density Fermi-Dirac distribution:}}\]
\[\epsilon_{\nu} = \frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4}{\left(2 \pi \hbar c \right)^3} \int_{0}^c \frac{E_t \left(v \right)^3}{e^{\frac{E_t \left(v \right)}{E_1 \left(T_{\nu} \right)}} + 1} dv\]
\[\color{blue}{\text{Solve Fermi-Dirac distribution first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{d \epsilon_{\nu}}{df_{\nu}} = \frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\nu} \right)}} + 1 \right)} \right) = 0\]
\[\frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 \left(h f_{\nu} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\nu}}} + 1  \right)} \right) = \frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} - \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2} = 0\]
\[\frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} = \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2}\]
\[\frac{h f_{\nu}}{k_B T_{\nu}} = 3 \left(1 + \frac{1}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \right) \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = 3 + \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = \frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = 3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h} = 1.26915 \cdot 10^{11} \; \text{Hz} =  126.915 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Fermi-Dirac cosmic neutrino background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\]
\[\boxed{f_{\nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\nu} = 126.915 \; \text{GHz}}\]
\[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Photon species total effective degeneracy number:}}\]
\[\boxed{N_{\gamma} = 2}\]
\[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\]
\[T_{\gamma} = 2.72548 \; \text{K}\]
\[\color{blue}{\text{Planck's law:} \; (\text{ref. 6})}\]
\[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)}\]
\[\color{blue}{\text{Solve Bose-Einstein Planck's law first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = 0\]
\[\color{blue}{\text{Planck's law first derivative with respect to frequency:}}\]
\[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = \frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\]
\[\frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\]
\[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\]
\[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\]
\[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\]
\[f_{\gamma} = 160.23 \; \text{GHz}\]
\[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\color{blue}{\text{Photon radiation energy density Bose-Einstein distribution:}}\]
\[\epsilon_{\gamma} = \frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4}{\left( 2 \pi \hbar c  \right)^3} \int_{0}^\infty \frac{E_t \left(\omega \right)^3}{e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1} d \omega\]
\[\color{blue}{\text{Solve Bose-Einstein distribution first derivative x-axis zero intercept with respect to frequency:}}\]
\[\frac{d \epsilon_{\gamma}}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1  \right)} \right) = 0\]
\[\frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 \left(h f_{\gamma} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\gamma}}} - 1  \right)} \right) = \frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\]
\[\frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\]
\[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\]
\[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\]
\[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\]
\[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\]
\[f_{\gamma} = 160.23 \; \text{GHz}\]
\[\color{blue}{\text{Cosmic graviton background radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\]
\[\boxed{T_{G} = T_{\nu} = 1.945 \; \text{K}}\]
\[\color{blue}{\text{Bose-Einstein cosmic graviton background radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{G} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{G}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{G} = 1.14366 \cdot 10^{11} \; \text{Hz}}\]
\[\boxed{f_{G} = 114.366 \; \text{GHz}}\]
\[\color{blue}{\text{Are Fermilab quantum particle detectors capable of detecting cold dark matter quantum particles?} \; (\text{ref. 7})}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Planck 2013 results. XVI. Cosmological parameters: (ref. 1)
http://www.ymambrini.com/My_World/Articles_files/Planck_2013_results_16.pdf
Science Forums - Dark energy quinton radiation temperature - Orion1: (ref. 2)
https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=931225
Wikipedia - Lambert W function: (ref. 3)
https://en.wikipedia.org/wiki/Lambert_W_function
Wikipedia - Cosmic microwave background - Importance of precise measurement: (ref. 4)
https://en.wikipedia.org/wiki/Cosmic_microwave_background#Importance_of_precise_measurement
Wikipedia - Cosmic neutrino background - Derivation of the CvB temperature: (ref. 5)
https://en.wikipedia.org/wiki/Cosmic_neutrino_background#Derivation_of_the_CνB_temperature
Wikipedia - Planck's law: (ref. 6)
https://en.wikipedia.org/wiki/Planck's_law
Fermilab - How scientists at Fermilab search for dark matter particles: (ref. 7)
https://bit.ly/3AvGTo9
Science Forums - Toy model calculation versus observation comparison summary - Orion1: (ref. 8 )
https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1112539

 

Edited July 7, 20214 yr by Orion1
source code correction...

...

Edited July 11, 20214 yr by Orion1
source code correction...

\[\color{blue}{\text{Toy model quantum particle properties chart:}}\]
\[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\]
\[\color{blue}{\text{Planck's law:}} \; (\color{blue}{\text{ref. 1}})\]
\[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} \pm 1 \right)}\]
\[\color{blue}{\text{A plus sign in the denominator is a Fermi-Dirac distribution, a minus sign in the denominator is a Bose-Einstein distribution.}}\]
\[\color{blue}{\text{+ sign - Fermi-Dirac distribution}}\]
\[\color{blue}{\text{- sign - Bose-Einstein distribution}}\]
\[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 1}})\]
\[114.366 \; \text{GHz} \; \; \; 126.915 \; \text{GHz} \; \; \; 160.229 \; \text{GHz}\]
\[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 2}})\]
\[2.058 \; \text{THz}\]
\[\color{blue}{\text{In this toy model, some quantum particle radiation distributions are embedded within the photon radiation distribution.}}\]
\[\color{blue}{\text{Would a quantum particle radiation distribution embedded within the photon radiation distribution induce cosine anisotropy in the photon radiation?}} \; (\color{blue}{\text{ref. 2}})\]
\[\color{blue}{\text{If cosine anisotropy is a detectable amount, is it possible to map cosine anisotropy amounts verses frequency within the photon radiation distribution?}} \; (\color{blue}{\text{ref. 2}})\]
\[\color{blue}{\text{Based upon the frequency multi-distribution plot, what frequency domain in the photon radiation distribution would you expect to detect cosine anisotropy?}}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Wikipedia - Planck's law: (ref. 1)
https://en.wikipedia.org/wiki/Planck's_law
Wikipedia - Anisotropy - Physics: (ref. 2)
https://en.wikipedia.org/wiki/Anisotropy#Physics

Edited July 11, 20214 yr by Orion1
source code correction...

\[\color{blue}{\text{Solar surface photon radiation temperature:} \; (\text{ref. 1})}\]
\[T_{\odot} = 5772 \; \text{K}\]
\[\color{blue}{\text{Bose-Einstein solar surface photon radiation spectral radiance peak frequency:}}\]
\[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\odot}}{h}}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[\boxed{f_{\gamma} = 3.39332 \cdot 10^{14} \; \text{Hz}}\]
\[\boxed{f_{\gamma} = 339.332 \; \text{THz}}\]
\[\color{blue}{\text{Calculated solar surface photon radiation spectral radiance peak frequency:} \; (\text{ref. 2})}\]
\[f_{\gamma} = \frac{\left[W_{0}\left(-3e^{-3} \right) + 3 \right] k_B T_{\odot}}{h}\]
\[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\]
\[T_{1} = 6000 \; \text{K} \; \; \; f_{\gamma} = 352.735 \; \text{THz}\]
\[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\]
Reference:
Wikipedia - Sun Sol: (ref. 1)
https://en.wikipedia.org/wiki/Sun
Wikipedia - Wien's displacement law: (ref. 2)
https://en.wikipedia.org/wiki/Wien's_displacement_law#Parameterization_by_frequency
Wikipedia - Lambert W function: (ref. 3)
https://en.wikipedia.org/wiki/Lambert_W_function

Edited August 31, 20214 yr by Orion1
source code correction...

Would any students or professors be interested in preserving these published exterior solutions in their academic library archive to survive the author's and server's longevity and for the future study of these exterior solutions by students or professors?

Any discussions and/or peer reviews about this specific topic thread?

"You will do well to expand your horizons." - Fortune Cookie

3 hours ago, Orion1 said:

Would any students or professors be interested in preserving these published exterior solutions in their academic library archive to survive the author's and server's longevity and for the future study of these exterior solutions by students or professors?

Any discussions and/or peer reviews about this specific topic thread?

"You will do well to expand your horizons." - Fortune Cookie

What for?

3 hours ago, Orion1 said:

Would any students or professors be interested in preserving these published exterior solutions in their academic library archive to survive the author's and server's longevity and for the future study of these exterior solutions by students or professors?

Archiving is really your job. As would be any submission to journals.