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Orion1

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  1. \[\color{blue}{\text{Solar surface photon radiation temperature:} \; (\text{ref. 1})}\] \[T_{\odot} = 5772 \; \text{K}\] \[\color{blue}{\text{Bose-Einstein solar surface photon radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\odot}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\gamma} = 3.39332 \cdot 10^{14} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 339.332 \; \text{THz}}\] \[\color{blue}{\text{Calculated solar surface photon radiation spectral radiance peak frequency:} \; (\text{ref. 2})}\] \[f_{\gamma} = \frac{\left[W_{0}\left(-3e^{-3} \right) + 3 \right] k_B T_{\odot}}{h}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[T_{1} = 6000 \; \text{K} \; \; \; f_{\gamma} = 352.735 \; \text{THz}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] 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
  2. \[\color{blue}{\text{Toy model quantum particle properties chart:}}\] \[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\] \[\color{blue}{\text{Planck's law:}} \; (\color{blue}{\text{ref. 1}})\] \[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} \pm 1 \right)}\] \[\color{blue}{\text{A plus sign in the denominator is a Fermi-Dirac distribution, a minus sign in the denominator is a Bose-Einstein distribution.}}\] \[\color{blue}{\text{+ sign - Fermi-Dirac distribution}}\] \[\color{blue}{\text{- sign - Bose-Einstein distribution}}\] \[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 1}})\] \[114.366 \; \text{GHz} \; \; \; 126.915 \; \text{GHz} \; \; \; 160.229 \; \text{GHz}\] \[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 2}})\] \[2.058 \; \text{THz}\] \[\color{blue}{\text{In this toy model, some quantum particle radiation distributions are embedded within the photon radiation distribution.}}\] \[\color{blue}{\text{Would a quantum particle radiation distribution embedded within the photon radiation distribution induce cosine anisotropy in the photon radiation?}} \; (\color{blue}{\text{ref. 2}})\] \[\color{blue}{\text{If cosine anisotropy is a detectable amount, is it possible to map cosine anisotropy amounts verses frequency within the photon radiation distribution?}} \; (\color{blue}{\text{ref. 2}})\] \[\color{blue}{\text{Based upon the frequency multi-distribution plot, what frequency domain in the photon radiation distribution would you expect to detect cosine anisotropy?}}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] 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
  3. \[\color{blue}{\text{Symbolic identity key:}}\] \[\begin{array}{lcl} \text{s} \text{ - spin quantum number} \\ n_{s} \text{ - spin states total integer helicity number} \\ N_{s} \text{ - species total integer number} \\ N_{n} \text{ - total effective degeneracy number} \\ \end{array}\] \[n_{s} = 2 s + 1 \; \; \; \; \; \; s < 2 \; \; \; \; \; \; m = 0\] \[n_{s} = 2 \; \; \; \; \; \; s \geq 2 \; \; \; \; \; \; m = 0\] \[\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}\] \[\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}\] \[\color{blue}{\text{Toy model quantum particle properties chart:}}\] \[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\] \[\color{blue}{\text{For massless quantum particles, the transverse modes cannot exist due to Lorentz invariance.}}\] \[\color{blue}{\text{Only positive and negative helicity states remain. For massless scalar particles, only zero helicity states remain.}}\] \[\color{blue}{\text{The spin 1 photon is also restricted to its positive and negative helicity states, and has a total effective degeneracy number of 2.}}\] \[\color{blue}{\text{A massless graviton has only 2 helicity states, and has a total effective degeneracy number of 2.}}\] \[\color{blue}{\text{Dark energy quinton total effective degeneracy number:}}\] \[\boxed{N_{\Lambda} = 1}\] \[\color{blue}{\text{Planck satellite dark energy cosmological composition parameter:} \; (\text{ref. 1, pg. 11})}\] \[\Omega_{\Lambda} = 0.6825\] \[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation temperature:} \; (\text{ref. 2})}\] \[\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3} \right)^{1/4}}\] \[\boxed{T_{\Lambda} = 35.013 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\Lambda}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right]}{2 \pi} \left(\frac{45 \Omega_{\Lambda} H_0^2 c^5}{4 \hbar G N_{\Lambda} \pi^3} \right)^{1/4}}\] \[\boxed{f_{\Lambda} = 2.05837 \cdot 10^{12} \; \text{Hz}}\] \[\boxed{f_{\Lambda} = 2.058 \; \text{THz}}\] \[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\] \[T_{\gamma} = 2.72548 \; \text{K}\] \[\color{blue}{\text{Cosmic neutrino and sterile neutrino background radiation temperature at present time:} \; (\text{ref. 5})}\] \[T_{\nu} = \left(\frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}\] \[\boxed{T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Dark matter scalaron and sterile neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\] \[\boxed{T_{\phi} = T_{s \nu} = T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein dark matter cosmic scalaron background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\phi} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\phi}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\phi} = 1.14366 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\phi} = 114.366 \; \text{GHz}}\] \[\color{blue}{\text{Fermi-Dirac dark matter sterile neutrino background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{s \nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{s \nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{s \nu} = 126.915 \; \text{GHz}}\] \[\color{blue}{\text{Solve for cosmic neutrino background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Neutrino species total effective degeneracy number:}}\] \[N_{\nu} = 3.046\] \[\color{blue}{\text{Neutrino radiation energy density Fermi-Dirac distribution:}}\] \[\epsilon_{\nu} = \frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4}{\left(2 \pi \hbar c \right)^3} \int_{0}^c \frac{E_t \left(v \right)^3}{e^{\frac{E_t \left(v \right)}{E_1 \left(T_{\nu} \right)}} + 1} dv\] \[\color{blue}{\text{Solve Fermi-Dirac distribution first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{d \epsilon_{\nu}}{df_{\nu}} = \frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\nu} \right)}} + 1 \right)} \right) = 0\] \[\frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 \left(h f_{\nu} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\nu}}} + 1 \right)} \right) = \frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} - \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2} = 0\] \[\frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} = \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2}\] \[\frac{h f_{\nu}}{k_B T_{\nu}} = 3 \left(1 + \frac{1}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \right) \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = 3 + \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = \frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = 3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h} = 1.26915 \cdot 10^{11} \; \text{Hz} = 126.915 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Fermi-Dirac cosmic neutrino background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\] \[\boxed{f_{\nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\nu} = 126.915 \; \text{GHz}}\] \[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Photon species total effective degeneracy number:}}\] \[\boxed{N_{\gamma} = 2}\] \[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\] \[T_{\gamma} = 2.72548 \; \text{K}\] \[\color{blue}{\text{Planck's law:} \; (\text{ref. 6})}\] \[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)}\] \[\color{blue}{\text{Solve Bose-Einstein Planck's law first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = 0\] \[\color{blue}{\text{Planck's law first derivative with respect to frequency:}}\] \[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = \frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\] \[\frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\] \[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\] \[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\] \[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\] \[f_{\gamma} = 160.23 \; \text{GHz}\] \[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Photon radiation energy density Bose-Einstein distribution:}}\] \[\epsilon_{\gamma} = \frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4}{\left( 2 \pi \hbar c \right)^3} \int_{0}^\infty \frac{E_t \left(\omega \right)^3}{e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1} d \omega\] \[\color{blue}{\text{Solve Bose-Einstein distribution first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{d \epsilon_{\gamma}}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1 \right)} \right) = 0\] \[\frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 \left(h f_{\gamma} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\gamma}}} - 1 \right)} \right) = \frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\] \[\frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\] \[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\] \[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\] \[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\] \[f_{\gamma} = 160.23 \; \text{GHz}\] \[\color{blue}{\text{Cosmic graviton background radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\] \[\boxed{T_{G} = T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein cosmic graviton background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{G} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{G}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{G} = 1.14366 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{G} = 114.366 \; \text{GHz}}\] \[\color{blue}{\text{Are Fermilab quantum particle detectors capable of detecting cold dark matter quantum particles?} \; (\text{ref. 7})}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] 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
  4. y=∫f(x)dx Thanks joigus! \[ \color{blue}{\text{test}} \]
  5. @joigus, how did you center your equations? I cannot get \begin{center} command to work.
  6. [math]\begin{align} \text{text} \end{align}[/math] [math]\color{blue}{\text{text}}[/math] text [math]F = ma[/math]
  7. Stephen Hawking thought an asteroid impact posed the greatest threat to life on Earth. Thanks to <deleted> for sponsoring this video. Any Discussions About This Topic Thread?
  8. Those images appear to be Kaleidoscopic (having complex patterns of colors; multicolored, made up of a complex mix of elements; multifaceted.) (ref. 1) Reference: Wikipedia - Kaleidoscope: (ref. 1) https://en.wikipedia.org/wiki/Kaleidoscope
  9. Those appear to be Moiré patterns. (ref. 1) The interference effect can be produced when two or more zone plates overlap and are rotated, or when an opaque ruled pattern with transparent gaps is overlaid on another similar pattern and are rotated. Reference: Wikipedia - Moiré pattern - rotated patterns: (ref. 1) https://en.wikipedia.org/wiki/Moiré_pattern#Rotated_patterns https://upload.wikimedia.org/wikipedia/commons/9/97/Moire_Lines.svg
  10. The parameters are identical for modelling any real physical object. (ref. 1) Note that all the interference patterns that you have pictured are formally called Fresnel zone plates, and informally just called zone plates. The results are identical, with the exception that the original object may be physical and the other object may be a virtual hologram. The mathematics in (ref. 1) are capable of modelling any Fresnel zone plate, including point source holograms (ref. 2). Can a physical Fresnel lens project a virtual hologram?....Affirmative. (ref. 3) Reference: Wikipedia - Fresnel Zone Plate: (ref. 1) https://en.wikipedia.org/wiki/Zone_plate Wikipedia - Hologram - point sources: (ref. 2) https://en.wikipedia.org/wiki/Holography#Point_sources Physical Fresnel Lens Hologram Test: (ref. 3) https://www.youtube.com/watch?v=f8wSabvv1xs
  11. As of 2018, the Supreme Court had overruled more than 300 of its own cases. (ref. 1) Negative, there are no legal mandates to "balance the courts" by partisan boundaries. There are only three legal mandates, one mandate is an Act, one mandate is a congressional statute, and the other mandate is a constitutional article. By "even split", I presume you mean an even partisan (a strong supporter of a party, cause, or person; prejudiced in favor of a particular cause) boundary for the total number of justices. It is constitutionally possible, though improbable based upon the partisan affiliation of the nominating president. Although nominating Supreme Court justices based upon suspected partisan affiliation is in my opinion, constitutionally repugnant due to the "apolitical" constitutional design for judicial selection. In my opinion it would have the opposite effect of splitting the court down partisan lines and decide cases more on partisan politics than on case precedent merits. Only if the federal district courts or the Supreme Court ruled that the legislative acts that constructed the judiciary structure were illegal or unconstitutional (not in accordance with a political constitution, especially the United States Constitution, or with procedural rules). Affirmative, if the Supreme Court ruled that the addition of a new justice office was illegal by Act, statute or unconstitutional by established boundaries or with procedural rules. For example, a majority of congressional partisans wanted to "stack" the court with more than nine justices, without a presidential nomination and based upon suspected partisan affiliation for generating judicial bias for political gain regarding any existent de jure (by Right; according to law) legal precedent, in violation of the Judiciary Act of 1869 and Title 28 U.S.C. § 1 and the United States Constitution, Article II, Section 2, Clause 2. Constitutionally, the alteration of the judiciary branch of government requires a Act of Congress, which creates a law, or modifies an existing law, with a simple majority of both the House of Representatives, and the Senate. (> = 51%), (51 of 100). However, if the structural judiciary branch of government is already bounded within the constitution and therefore requires a modification or constitutional amendment, then to become an operative part of the United States Constitution, an amendment, whether proposed by Congress or a national constitutional convention, must be ratified by either: The legislatures of three-fourths (> = 75%) (at present 38) of the states; or State ratifying conventions in three-fourths (> = 75%) (at present 38) of the states. (ref. 6) Negative, the court has evolved to the nine members only by congressional Acts and statute, a constitutional power and boundary entrusted only to Congress. Article III of the United States Constitution sets neither the size of the Supreme Court nor any specific positions on it, though the existence of the office of the chief justice is tacitly acknowledged in Article I, Section 3, Clause 6. Instead, these powers are entrusted to Congress, which initially established a six-member Supreme Court composed of a chief justice and five associate justices through the Judiciary Act of 1789. The Judiciary Act of 1869 also called the Circuit Judges Act, and Title 28 U.S.C. § 1, returned the number of justices to nine, where it has since remained. (ref. 3, 4) Year - justice number, Congressional Act: 1789 - 6, Judiciary Act of 1789 1807 - 7, Seventh Circuit Act of 1807 1837 - 9, Eighth and Ninth Circuits Act of 1837 1863 - 10, Tenth Circuit Act of 1863 1866 - 9, Judicial Circuits Act of 1866 1869 - 9, Judiciary Act of 1869 1948 - 9, Title 28 U.S.C. § 1 2020 - 9, remained. U.S. Citizens do not have the Right to vote for a president, registered voters only have the constitutional Right to vote for an elector, under the United States Constitution Twelfth Amendment. (ref. 7) When people cast their vote, they are not actually voting for president, but for a group of people called electors. The number of electors each state gets is equal to its total number of Senators and Representatives in Congress. A total of 538 electors form the Electoral College. Each elector casts one vote following the general election. Only an elector has the constitutional Right to vote for a president. (ref. 7) In the United States, federal and state courts at all levels, both appellate and trial are able to review and declare the "constitutionality", or agreement with the Constitution, or unconstitutionality of legislation by a process of judicial interpretation that is relevant to any case properly within their jurisdiction. In American legal language, "judicial review" refers primarily to the adjudication of constitutionality of statutes, especially by the Supreme Court of the United States. This is commonly held to have been established in the case of Marbury v. Madison, which was argued before the Supreme Court in 1803. (ref. 8) Reference: Wikipedia - SCOTUS overruled decisions: (ref. 1) https://en.wikipedia.org/wiki/List_of_overruled_United_States_Supreme_Court_decisions Wikipedia - SCOTUS Nomination, confirmation, and appointment: (ref. 2) https://en.wikipedia.org/wiki/Supreme_Court_of_the_United_States#Nomination,_confirmation,_and_appointment Wikipedia - SCOTUS size: (ref. 3) https://en.wikipedia.org/wiki/Supreme_Court_of_the_United_States#Size_of_the_court Cornell University - 28 U.S.C. § 1: (ref. 4) https://www.law.cornell.edu/uscode/text/28/1 Wikipedia - Act of Congress: (ref. 5) https://en.wikipedia.org/wiki/Act_of_Congress Wikipedia - United States constitutional amendment - federal constitution: (ref. 6) https://en.wikipedia.org/wiki/Constitutional_amendment#Federal_constitution Wikipedia - United States Constitution - Twelfth Amendment: (ref. 7) https://en.wikipedia.org/wiki/Twelfth_Amendment_to_the_United_States_Constitution Wikipedia - United States - Judicial Review: (ref. 8) https://en.wikipedia.org/wiki/Judicial_review#Review_by_general_courts Wikipedia - United States - Marbury v. Madison (1803): (ref. 9) https://en.wikipedia.org/wiki/Marbury_v._Madison
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