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  1. Affirmative, revision complete. WMAP satellite cosmological composition parameters at photon decoupling time: (ref. 1) [math]\Omega_{\gamma,t} = 0.15[/math] [math]\;[/math] Photon species total effective degeneracy number: [math]\boxed{N_{\gamma} = 2}[/math] [math]\;[/math] Solar surface temperature: (ref. 2) [math]T_{\odot} = 5772 \; \text{K}[/math] [math]\;[/math] Total solar radius: (ref. 2) [math]R_{\odot} = 6.963 \cdot 10^{5} \; \text{km}[/math] [math]\;[/math] Earth orbital radius semi-major axis: (ref. 3) [math]r_{\oplus} = 1.496 \cdot 10^{8} \; \text{km}[/math] [math]\;[/math] Earth albedo reflectivity: (ref. 3) [math]\alpha_{\oplus} = 0.367[/math] [math]\;[/math] Earth surface temperature at present time: (ref. 4) [math]T_{\oplus} = T_{\odot} \left(\frac{\left(1 - \alpha_{\oplus} \right)^{\frac{1}{2}} R_{\odot}}{2 r_{\oplus}} \right)^{\frac{1}{2}} = 248.367 \; \text{K}[/math] [math]\boxed{T_{\oplus} = 248.367 \; \text{K}}[/math] [math]\;[/math] Isotropic cosmic photon background radiation temperature at present time: (ref. 5) [math]T_{\gamma} = 2.72548 \; \text{K}[/math] [math]\;[/math] Toy model universe age at isotropic cosmic photon background radiation time: (ref. 6) [math]\boxed{T_{u,\gamma} = \frac{3}{4\left(k_B T_{\gamma,t} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \pi^3}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Isotropic cosmic photon background radiation temperature at past time is equivalent to Earth surface temperature at present time: [math]\boxed{T_{\gamma,t} = T_{\oplus}}[/math] [math]\;[/math] Toy model universe age at isotropic cosmic photon background radiation time integration via substitution: [math]T_{u,\gamma} = \frac{3}{4\left(k_B T_{\gamma,t} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \pi^3}} = \frac{3}{2 \left(k_B T_{\odot} \right)^2} \left( \frac{r_{\oplus}}{R_{\odot}} \right) \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \left(1 - \alpha_{\oplus} \right) \pi^3}}[/math] [math]\;[/math] Toy model universe age at abiogenesis epoch time: [math]\boxed{T_{u,\gamma} = \frac{3 r_{\oplus}}{2 R_{\odot} \left(k_B T_{\odot} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \left(1 - \alpha_{\oplus} \right) \pi^3}}}[/math] [math]\boxed{T_{u,\gamma} = 1.447 \cdot 10^{15} \; \text{s}} \; \; \; \left(4.585 \cdot 10^{7} \; \text{years} \right)[/math] [math]\;[/math] If every planet around every second generation star and every third generation star were illuminated with a luminous isotropic cosmic photon background radiation source that is more cleaner and isotropic and more stable than the host star radiation source, could this cosmic photon background radiation have initiated a universe abiogenesis epoch? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - Sun Sol: (ref. 2) https://en.wikipedia.org/wiki/Sun Wikipedia - Earth: (ref. 3) https://en.wikipedia.org/wiki/Earth UVIC - The blackbody temperature of a planet: (ref. 4) http://www.astro.uvic.ca/~venn/A201/maths.7.planet_temperature.pdf Wikipedia - Cosmic microwave background radiation: (ref. 5) https://en.wikipedia.org/wiki/Cosmic_microwave_background Importance_of_precise_measurement Science Forums - Orion1 - Photon decoupling time: (ref. 6) https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1037262
  2. Solar surface temperature: (ref. 1) [math]T_{\odot} = 5772 \; \text{K}[/math] [math]\;[/math] Total solar radius: (ref. 1) [math]R_{\odot} = 6.963 \cdot 10^{5} \; \text{km}[/math] [math]\;[/math] Earth orbital radius semi-major axis: (ref. 2) [math]r_{\oplus} = 1.496 \cdot 10^{8} \; \text{km}[/math] [math]\;[/math] Earth albedo reflectivity: (ref. 2) [math]\alpha_{\oplus} = 0.367[/math] [math]\;[/math] Earth surface temperature at present time: (ref. 3) [math]T_{\oplus} = T_{\odot} \left(\frac{\left(1 - \alpha_{\oplus} \right)^{\frac{1}{2}} R_{\odot}}{2 r_{\odot}} \right)^{\frac{1}{2}} = 248.367 \; \text{K}[/math] [math]\boxed{T_{\oplus} = 248.367 \; \text{K}}[/math] [math]\;[/math] Cosmic photon background radiation temperature at present time: (ref. 4) [math]T_{\gamma,0} = 2.72548 \; \text{K}[/math] [math]\;[/math] Cosmology scale factor: [math]\boxed{\frac{a\left(t_0 \right)}{a\left(t \right)} = \frac{T_{\gamma,t}}{T_{\gamma,0}}}[/math] [math]\;[/math] Cosmology scale factor at present time: [math]\boxed{a\left(t_0 \right) = 1}[/math] [math]\;[/math] Cosmic photon background radiation temperature at past time is equivalent to Earth surface temperature at present time: [math]\boxed{T_{\gamma,t} = T_{\oplus}}[/math] [math]\;[/math] Universe abiogenesis epoch scale factor at past time: [math]a\left(t \right) = \frac{T_{\gamma,0}}{T_{\gamma,t}} = 0.011[/math] [math]\boxed{a\left(t \right) = \frac{T_{\gamma,0}}{T_{\gamma,t}}}[/math] [math]\boxed{a\left(t \right) = 0.011}[/math] [math]\;[/math] If every planet around every second generation star and every third generation star were illuminated with a luminous cosmic photon background radiation source that is cleaner and more stable than the host star radiation source, could this cosmic photon background radiation have initiated a universe abiogenesis epoch? [math]\;[/math] According to your universe model calculator, what is the universe age at this scale factor? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Sun Sol: (ref. 1) https://en.wikipedia.org/wiki/Sun Wikipedia - Earth: (ref. 2) https://en.wikipedia.org/wiki/Earth UVIC - The blackbody temperature of a planet: (ref. 3) http://www.astro.uvic.ca/~venn/A201/maths.7.planet_temperature.pdf Wikipedia - Cosmic microwave background radiation: (ref. 4) https://en.wikipedia.org/wiki/Cosmic_microwave_background#Importance_of_precise_measurement
  3. Planck satellite baryonic cosmological composition parameter: (ref. 1, pg. 11, ref. 2, pg. 3) [math]\Omega_{b} = 0.0495[/math] [math]\;[/math] Black holes cosmological composition parameter: {ref. 2, pg. 3) [math]\Omega_{bh} = 0.00007[/math] [math]\;[/math] Solar mass: (ref. 3) [math]M_{\odot} = 1.9885 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] Milky Way galaxy mass: (ref. 4, pg. 1) [math]M_{mw} = 1.260 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math] [math]\boxed{M_{mw} = 2.506 \cdot 10^{42} \; \text{kg}}[/math] [math]\;[/math] PSR J2215+5135 pulsar Tolman-Oppenheimer-Volkoff observational lower mass limit: (ref. 5) [math]\boxed{M_{bh} \geq 2.27 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 4.514 \cdot 10^{30} \; \text{kg}}[/math] [math]\;[/math] Stellar class O upper mass limit: (ref. 6) [math]\boxed{M_{bh} \geq 16 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 3.182 \cdot 10^{31} \; \text{kg}}[/math] [math]\;[/math] Toy model black holes per galaxy average number: [math]\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] Synthetic catalog black holes per galaxy average number: (ref. 7, pg. 1) [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Sun Sol: (ref. 3) https://en.wikipedia.org/wiki/Sun Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 5) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Stellar classification - Harvard spectral classification: (ref. 6) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification Synthetic catalog of black holes in the Milky Way: (ref. 7) https://arxiv.org/pdf/1908.08775.pdf
  4. Affirmative, I seem to have reached a reference citation impasse regarding the tensor field functions integration of [math]\Lambda^{\mu}_{\alpha}[/math] and [math]\Lambda^{\nu}_{\beta}[/math]. All of the references that have been cited do not demonstrate a functions integration past this initial point for a tensor field. Is this approach at least mathematically and symbolically correct to this point? Any citations or recommendations? Any discussions and/or peer reviews about this specific topic thread?
  5. Affirmative. Your nearest college or university book store should have Calculus I and Physics and Astrophysics books available for purchase. Or you could locate the same or similar books at your local city, college or university library available for checkout. If these options are not available due to remote locality, then online purchases may also be available.
  6. Toy model calculation versus observation comparison summary: [math]\begin{array}{l*{3}{c}r} \text{symbol} & \text{quantity} & \text{percent} \\ N_{g} & 2.330 \cdot 10^{12} \; \text{galaxies} & 116.5 \\ N_{g} & 2.000 \cdot 10^{12} \; \text{galaxies} & \\ N_{s} & 2.453 \cdot 10^{23} \; \text{stars} & 81.767 \\ N_{s} & 3.000 \cdot 10^{23} \; \text{stars} \\ N_p & 3.536 \cdot 10^{23} \; \text{planets} & 110.5 \\ N_p & 3.200 \cdot 10^{23} \; \text{planets} \\ \frac{N_s}{N_g} & 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} & 70.2 \\ \frac{N_s}{N_g} & 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} \\ \frac{N_p}{N_g} & 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}} & 94.875 \\ \frac{N_p}{N_g} & 1.600 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}} \\ \frac{N_p}{N_s} & 1.443 \; \frac{\text{planets}}{\text{star}} & 135.24 \\ \frac{N_p}{N_s} & 1.067 \; \frac{\text{planets}}{\text{star}} \\ \end{array}[/math] [math]\;[/math] Toy model particle properties chart: [math]\begin{array}{l*{9}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{decoupling time} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & >0 \; \text{s} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 0.0255 \; \text{s} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 0.0255 \; \text{s} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 0.148 \; \text{s} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 1.009 \cdot 10^{13} \; \text{s} & \left(3.197 \cdot 10^{5} \; \text{years} \right) \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & >0 \; \text{s} \\ \end{array}[/math] [math][/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Scienceforums - Orion1 - toy model revision: https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1089828 https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1091408
  7. Affirmative, in this toy model, sterile neutrinos have equivalent dark matter candidacy as scalar particles. [math]\;[/math] Planck satellite cosmological parameters at present time: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\;[/math] sterile neutrino composition is equivalent to dark matter composition at present time. [math]\boxed{\Omega_{s \nu} = \Omega_{dm}}[/math] [math]\;[/math] Fermi-Dirac total dark matter sterile neutrino distribution constant: [math]\boxed{C_{s \nu} = \frac{4 G N_{s \nu} \pi^3 \left(k_B T_{s \nu} \right)^4}{3 \Omega_{s \nu} H_0^2 \hbar^3 c^5}}[/math] [math]\;[/math] [math]\boxed{C_{s \nu} = 1.109 \cdot 10^{-3}}[/math] [math]\;[/math] Fermi-Dirac sterile neutrino density: [math]\boxed{\rho_{s \nu} = \frac{N_{s \nu} \pi^2 \left(k_B T_{s \nu} \right)^4}{2 C_{s \nu} \hbar^3 c^5}}[/math] [math]\;[/math] Fermi-Dirac sterile neutrino number density: (ref. 3, eq. 8) [math]n_{s \nu} = \left(\frac{3 \zeta \left(3 \right) N_{s \nu} \left(k_{B} T_{s \nu} \right)^3}{4 \pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino mass integration via substitution: [math]m_{s \nu} = \frac{\rho_{s \nu}}{n_{s \nu}} = \left(\frac{N_{s \nu} \pi^2 \left(k_B T_{s \nu} \right)^4}{2 C_{s \nu} \hbar^3 c^5} \right)\left(\frac{4 \pi^2 \left(\hbar c \right)^3}{3 \zeta \left(3 \right) N_{s \nu} \left(k_{B} T_{s \nu} \right)^3} \right) = \frac{2 \pi^{4} k_B T_{s \nu}}{3 C_{s \nu} c^{2} \zeta \left(3 \right)} = 1.456 \cdot 10^{-35} \; \text{kg}[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino mass: [math]\boxed{m_{s \nu} = \frac{2 \pi^{4} k_B T_{s \nu}}{3 C_{s \nu} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{s \nu} = 1.456 \cdot 10^{-35} \; \text{kg}}[/math] [math]\boxed{m_{s \nu} = 8.167 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] WMAP satellite cosmological parameters at photon decoupling time: (ref. 2) [math]\Omega_{dm,t} = 0.63[/math] [math]\;[/math] Sterile neutrino composition is equivalent to dark matter composition at photon decoupling time. [math]\boxed{\Omega_{s \nu,t} = \Omega_{dm,t}}[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino decoupling time: [math]\boxed{T_{u,s \nu} = \frac{}{2\left(k_B T_{s \nu,t} \right)^2} \sqrt{\frac{3 \Omega_{s \nu,t} C_{s \nu} \hbar^3 c^5}{G N_{s \nu} \pi^3}}} \; \; \; m_{s \nu} \neq 0[/math] [math]\;[/math] [math]\boxed{T_{u,s \nu} = 0.0255 \; \text{s}} \; \; \; m_{s \nu} \neq 0[/math] [math]\;[/math] Toy model dark matter particle candidates chart: [math]\begin{array}{l*{8}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{decoupling time} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 0.0255 \; \text{s} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 0.0255 \; \text{s} \\ \end{array}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf WMAP satellite cosmological parameters at photon decoupling time: (ref. 2) https://map.gsfc.nasa.gov/media/080998/index.html Introduction to Cosmology: Lecture 6 - Thermal history of the Universe: (ref. 3) http://gravitation.web.ua.pt/sites/default/files/migrated2016/Lecture_6.pdf Wikipedia - Scalar boson: (ref. 4) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Sterile neutrinos: (ref. 5) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Dark matter: (ref. 6) https://en.wikipedia.org/wiki/Dark_matter
  8. Toy model particle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math] [math]\;[/math] Supersymmetry particle and sparticle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ b & \text{Higgs} & 0 & 0 & 1 & 1 & 1 & 125.18 \; \text{GeV} & H^{0} \\ sf & \text{higgsino} & +,- & 1/2 & 2 & 1 & 2 & 1.1 \; \text{TeV} & \ddot{H} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & <1 \; \text{eV} - 10^{15} \; \text{GeV} & \nu \\ sf & \text{neutralino} & +,- & 1/2 & 2 & 4 & 4 & 100 \; \text{GeV} - 1 \; \text{TeV} & \ddot{N}^{0} \\ sf & \text{gravitino} & +,- & 3/2 & 2 & 1 & 2 & 1 \; \text{TeV} & \ddot{G} \\ \end{array}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter density: [math]\boxed{\rho_{\phi} = \frac{N_{\phi} \pi^2 \left(k_B T_{\phi} \right)^4}{2 C_{\phi} \hbar^3 c^5}}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter number density: (ref. 1, eq. 7) [math]n_{\phi} = \left(\frac{\zeta \left(3 \right) N_{\phi} \left(k_{B} T_{\phi} \right)^3}{\pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter mass: [math]m_{\phi} = \frac{\rho_{\phi}}{n_{\phi}} = \left(\frac{N_{\phi} \pi^2 \left(k_B T_{\phi} \right)^4}{2 C_{\phi} \hbar^3 c^5} \right)\left(\frac{\pi^2 \left(\hbar c \right)^3}{\zeta \left(3 \right) N_{\phi} \left(k_{B} T_{\phi} \right)^3} \right) = \frac{\pi^{4} k_B T_{\phi}}{2 C_{\phi} c^{2} \zeta \left(3 \right)} = 3.326 \cdot 10^{-35} \; \text{kg}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter mass: [math]\boxed{m_{\phi} = \frac{\pi^{4} k_B T_{\phi}}{2 C_{\phi} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{\phi} = 3.326 \cdot 10^{-35} \; \text{kg}}[/math] [math]\boxed{m_{\phi} = 18.658 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] Fermi-Dirac neutrino density: [math]\boxed{\rho_{\nu} = \frac{N_{\nu} \pi^2 \left(k_B T_{\nu} \right)^4}{2 C_{\nu} \hbar^3 c^5}}[/math] [math]\;[/math] Fermi-Dirac neutrino number density: (ref. 1, eq. 8) [math]n_{\nu} = \left(\frac{3 \zeta \left(3 \right) N_{\nu} \left(k_{B} T_{\nu} \right)^3}{4 \pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Fermi-Dirac neutrino mass: [math]m_{\nu} = \frac{\rho_{\nu}}{n_{\nu}} = \left(\frac{N_{\nu} \pi^2 \left(k_B T_{\nu} \right)^4}{2 C_{\nu} \hbar^3 c^5} \right)\left(\frac{4 \pi^2 \left(\hbar c \right)^3}{3 \zeta \left(3 \right) N_{\nu} \left(k_{B} T_{\nu} \right)^3} \right) = \frac{2 \pi^{4} k_B T_{\nu}}{3 C_{\nu} c^{2} \zeta \left(3 \right)} = 6.839 \cdot 10^{-38} \; \text{kg}[/math] [math]\;[/math] Fermi-Dirac neutrino mass: [math]\boxed{m_{\nu} = \frac{2 \pi^{4} k_B T_{\nu}}{3 C_{\nu} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{\nu} = 6.839 \cdot 10^{-38} \; \text{kg}}[/math] [math]\boxed{m_{\nu} = 0.038 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] Disqualifying dark matter particle candidates for this toy model based upon the predicted mass range, results in the remaining candidates chart. [math]\;[/math] Toy model dark matter particle remaining candidates chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & <1 \; \text{eV} - 10^{15} \; \text{GeV} & \nu \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ \end{array}[/math] [math]\;[/math] Would dark matter particle candidates that quantum mechanically decouple before/after neutrinos be a candidate qualifier/disqualifier? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Introduction to Cosmology: Lecture 6 - Thermal history of the Universe: (ref. 1) http://gravitation.web.ua.pt/sites/default/files/migrated2016/Lecture_6.pdf Wikipedia - Scalar boson: (ref. 2) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Sterile neutrinos: (ref. 3) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Dark matter: (ref. 4) https://en.wikipedia.org/wiki/Dark_matter
  9. Toy model particle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math] [math]\;[/math] Supersymmetry particle and sparticle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda^{0} \\ b & \text{higgson} & 0 & 0 & 1 & 1 & 1 & \neq 0 & h^{0} \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ b & \text{Higgs} & 0 & 0 & 1 & 1 & 1 & \neq 0 & H^{0} \\ sf & \text{higgsino} & +,- & 1/2 & 2 & 1 & 2 & \neq 0 & \ddot{H} \\ sf & \text{gravitino} & +,- & 3/2 & 2 & 1 & 2 & \neq 0 & \ddot{G} \\ \end{array}[/math] [math]\;[/math] A quintessence boson scalar particle is named a 'quinton' for speculative discussion, because no quintessence scalar boson particle is named in quintessence research. [math]\;[/math] A Higgs boson supersymmetry superpartner scalar sparticle boson is named a 'higgson', for speculative discussion. [math]\;[/math] Higgson sparticles generation via nuclear reaction is speculated to be: [math]H^{0} + H^{0} \rightarrow h^{0} + h^{0}[/math] [math]h^{0} + h^{0} \rightarrow H^{0} + H^{0}[/math] [math]\;[/math] Do these sparticle nuclear reactions violate any already known conservation laws? [math]\;[/math] Note that only a quinton and a higgson particle is all that is required to explain all observations in this toy model. [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Quintessence (physics): https://en.wikipedia.org/wiki/Quintessence_(physics) Wikipedia - Lightest Supersymmetric Particle: https://en.wikipedia.org/wiki/Lightest_Supersymmetric_Particle
  10. Affirmative. The quantum-mechanical model boundary conditions for the scalar particle decoupling time versus the [math]\Lambda[/math]CDM model scalar particle decoupling time and a non-zero neutrino mass particle decoupling time and a non-zero sterile neutrino mass particle decoupling time appears to favor a scalar particle for dark matter. [math]\;[/math] Scalar particle decoupling time: [math]T_{u,\phi} = \frac{1}{H_{\phi,t}} = \frac{}{2 \left(k_B T_{\phi,t} \right)^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}} = 0.0255 \; \text{s}[/math] [math]\boxed{T_{u,\phi} = \frac{}{2 \left(k_B T_{\phi,t} \right)^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\Lambda[/math]CDM universe model semi-emperical temperature-time scale factor: (ref. 2, pg. 18, eq. 2, ref. 3) [math]\left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = \frac{T_{u,\phi}}{T_{u}} = T_{u,\phi} H_0[/math] [math]T_{u,\phi} = \frac{}{H_0} \left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = 0.0166 \; \text{s}[/math] [math]\Lambda[/math]CDM universe model semi-emperical scalar particle decoupling time: [math]\boxed{T_{u,\phi} = \frac{}{H_0} \left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2}}[/math] [math]\boxed{T_{u,\phi} = 0.0166 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Neutrino decoupling time: [math]T_{u,\nu} = \frac{1}{H_{\nu,t}} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.148 \; \text{s}[/math] [math]\boxed{T_{u,\nu} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.148 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Sterile neutrino decoupling time: [math]\boxed{T_{u,\nu} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle decoupling time summary: [math]\boxed{T_{u,\phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\phi} = 0.0166 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.148 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Is it possible for the Higgs boson to have a related supersymmetric scalar particle boson with non-zero mass? (ref. 4) Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Cosmology: nucleosynthesis and inflation: (ref. 2) http://www.uio.no/studier/emner/matnat/astro/AST1100/h07/undervisningsmateriale/lecture25.pdf Wikipedia - Lambda-CDM_model parameters: (ref. 3) https://en.wikipedia.org/wiki/Lambda-CDM_model Wikipedia - Lightest Supersymmetric Particle: (ref. 4) https://en.wikipedia.org/wiki/Lightest_Supersymmetric_Particle
  11. Affirmative, revision complete. Derivation of neutrino mass from neutrino scattering: [math]\;[/math] [math]\theta[/math] - scattered neutrino angle [math]\phi[/math] - electron recoil angle [math]E_{\nu i}[/math] - initial neutrino total energy [math]E_{\nu f}[/math] - final neutrino total energy [math]E_{e}[/math] - electron total energy [math]E_{\nu}[/math] - neutrino total energy Scattered particles rebounding with relativistic momentum have total energy: [math]E_{e} = \gamma_{e} m_{e} c^{2} \; \; \; \; \; \; E_{\nu} = \gamma_{\nu} m_{\nu} c^{2}[/math] Where [math]m_{e}[/math] and [math]m_{\nu}[/math] are the particle rest masses. [math]\gamma_{e}[/math] and [math]\gamma_{\nu}[/math] are the Lorentz factors. (ref. 1) [math]\;[/math] Observational measurements of both particle total energy and velocity, it is possible to calculate the particle rest masses [math]m_{e}[/math] and [math]m_{\nu}[/math]: [math]\boxed{m_{e} = \frac{E_{e}}{\gamma_{e} c^{2}}} \; \; \; \; \; \; \boxed{m_{\nu} = \frac{E_{\nu}}{\gamma_{\nu} c^{2}}} \tag{0}[/math] [math]\;[/math] [math]E_{e} \sin \phi = E_{\nu f} \sin \theta \tag{1}[/math] [math]\;[/math] [math]E_{e} \cos \phi + E_{\nu f} \cos \theta = E_{\nu i} \tag{2}[/math] [math]\;[/math] Isolate [math]E_{e} \cos \phi[/math] from equation (2): [math]E_{e} \cos \phi = E_{\nu i} - E_{\nu f} \cos \theta \tag{3}[/math] [math]\;[/math] Divide equation (1) by equation (3) for an expression for [math]\tan \phi[/math]. [math]\;[/math] [math]\tan \phi = \frac{E_{\nu f} \sin \theta}{E_{\nu i} - E_{\nu f} \cos \theta} = \frac{\sin \theta}{\frac{E_{\nu i}}{E_{\nu f}} - \cos \theta} \tag{4}[/math] [math]\;[/math] Acquire a substitution for [math]\frac{E_{\nu i}}{E_{\nu f}}[/math] to eliminate [math]E_{\nu f}[/math]. Use the Compton equation, which can be rearranged to yield [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}}[/math] in terms of [math]\lambda_{\nu i}[/math] alone. [math]\;[/math] [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}} \tag{5}[/math] [math]\;[/math] [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}} = 1 + \frac{E_{\nu i}}{E_{e}} \left(1 - \cos \theta \right) = 1 + \frac{\gamma_{\nu} m_{\nu} c^2}{\gamma_{e} m_{e} c^2} \left(1 - \cos \theta \right) = 1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \left(1 - \cos \theta \right) \tag{6}[/math] [math]\;[/math] Substituting equation (6) into equation (4) and eliminate [math]E_{\nu i}[/math] and [math]E_{\nu f}[/math] in favor of [math]m_{\nu}[/math] alone. [math]\tan \phi = \frac{\sin \theta}{\frac{E_{\nu i}}{E_{\nu f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \left(1 - \cos \theta \right) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \right)\left(1 - \cos \theta \right)} \tag{7}[/math] [math]\;[/math] Utilizing a trigonometric identity produces the desired result, specifically: [math]\frac{1 - \cos \theta}{\sin \theta} = \tan \left(\frac{\theta}{2} \right) \tag{8}[/math] [math]\;[/math] Substituting this trigonometric identity into equation (7) results in: [math]\left(1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \right) \tan \phi = \cot \frac{\theta}{2} \tag{9}[/math] [math]\;[/math] Solve for neutrino rest mass [math]m_{\nu}[/math]: [math]\tan \phi + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \tan \phi = \cot \frac{\theta}{2} \tag{10}[/math] [math]\;[/math] [math]\frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/math] [math]\;[/math] Electron-neutrino scattering neutrino rest mass: [math]\boxed{m_{\nu} = \frac{\gamma_{e} m_{e} \cot \phi}{\gamma_{\nu}} \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{12}[/math] [math]\;[/math] Nuclear-neutrino scattering neutrino rest mass: [math]\boxed{m_{\nu} = \frac{\gamma_{n} m_{n} \cot \phi}{\gamma_{\nu}} \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{13}[/math] [math]m_{n}[/math] - nuclear rest mass [math]\;[/math] Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(\gamma_{e} m_{e} + \gamma_{\nu} m_{\nu} \right) \tan \phi}{\gamma_{e} m_{e}} \right)} \tag{14}[/math] [math]\;[/math] Neutrino interaction electron recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{\gamma_{e} m_{e} \cot \frac{\theta}{2}}{\gamma_{e} m_{e} + \gamma_{\nu} m_{\nu}} \right)} \tag{15}[/math] [math]\;[/math] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(\gamma_{n} m_{n} + \gamma_{\nu} m_{\nu} \right) \tan \phi}{\gamma_{n} m_{n}} \right)} \tag{16}[/math] [math]\;[/math] Neutrino interaction nuclear recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{\gamma_{n} m_{n} \cot \frac{\theta}{2}}{\gamma_{n} m_{n} + \gamma_{\nu} m_{\nu}} \right)} \tag{17}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Lorentz factor: (ref. 1) https://en.wikipedia.org/wiki/Lorentz_factor Wikipedia - Compton scattering - Derivation of the scattering formula: https://en.wikipedia.org/wiki/Compton_scattering#Derivation_of_the_scattering_formula Physics 253 - Compton Scattering - Patrick LeClair http://pleclair.ua.edu//PH253/Notes/compton.pdf Orion1 - Neutrino mass from Fermi-Dirac statistics...: https://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/ Science News - Neutrinos seen scattering off an atom’s nucleus for the first time: https://www.sciencenews.org/article/neutrinos-seen-scattering-atoms-nucleus-first-time
  12. De Broglie relativistic momentum: (ref. 1) [math]p = \frac{\hbar}{\overline{\lambda}} = \gamma m_0 v[/math] [math]\;[/math] Relativistic energy-momentum relation and relativistic mass particle total energy identity: [math]\boxed{E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] [math]\boxed{E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(\frac{\hbar c}{\overline{\lambda}} \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] Is this equation an identity for a relativistic mass particle? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Matter wave: (ref. 1) https://en.wikipedia.org/wiki/Matter_wave
  13. Relativistic energy-momentum relation total energy: (ref. 1) [math]E_{t}^{2} = \left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2} [/math] [math]E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2}}[/math] [math]\;[/math] Relativistic mass particle total energy: (ref. 2) [math]E_{t} = \gamma m_0 c^{2}[/math] [math]\;[/math] Relativistic energy-momentum relation and relativistic mass particle total energy identity: [math]\boxed{E_{t} = \sqrt{\left(m_{0}c^{2} \right)^{2} + \left(pc \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] Is this equation an identity for a relativistic mass particle? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Energy-momentum relation: (ref. 1) https://en.wikipedia.org/wiki/Energy–momentum_relation#Special_relativity Wikipedia - Energy-momentum relation - heuristic approach for massive particles: (ref. 2) https://en.wikipedia.org/wiki/Energy–momentum_relation#Heuristic_approach_for_massive_particle
  14. Derivation of neutrino mass from neutrino scattering: [math]\;[/math] [math]\theta[/math] - scattered neutrino angle [math]\phi[/math] - electron recoil angle [math]p_{i}[/math] - initial neutrino momentum [math]p_{f}[/math] - final neutrino momentum [math]p_{e}[/math] - electron momentum [math]\;[/math] Scattered particles rebounding with relativistic momentum have total energy: [math]E_{e} = \gamma m_{0e} c^{2} = m_{e} c^{2} \; \; \; \; \; \; E_{\nu} = \gamma m_{0\nu} c^{2} = m_{\nu} c^{2}[/math] Where [math]m_{e}[/math] and [math]m_{\nu}[/math] are the relativistic particle masses. [math]\;[/math] Observational measurements of both particle total energy and velocity, it is possible to calculate the rest particle masses [math]m_{0e}[/math] and [math]m_{0\nu}[/math]: [math]\boxed{m_{0e} = \frac{E_{e}}{\gamma c^{2}}} \; \; \; \; \; \; \boxed{m_{0\nu} = \frac{E_{\nu}}{\gamma c^{2}}} \tag{0}[/math] [math]\;[/math] [math]p_{e} \sin \phi = p_{f} \sin \theta \tag{1}[/math] [math]\;[/math] [math]p_{e} \cos \phi + p_{f} \cos \theta = p_{i} \tag{2}[/math] [math]\;[/math] Isolate [math]p_{e} \cos \phi[/math] from equation [math](2)[/math]: [math]p_{e} \cos \phi = p_{i} - p_{f} \cos \theta \tag{3}[/math] [math]\;[/math] Divide equation [math](1)[/math] by equation [math](3)[/math] for an expression for [math](3)[/math] for an expression for [math]\tan \phi[/math]: [math]\;[/math] [math]\tan \phi = \frac{p_{f} \sin \theta}{p_{i} - p_{f} \cos \theta} = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} \tag{4}[/math] [math]\;[/math] Acquire a substitution for [math]\frac{p_{i}}{p_{f}}[/math] to eliminate [math]p_{f}[/math]. Use the Compton equation, which can be rearranged to yield [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}}[/math] in terms of [math]\lambda_{i}[/math] alone. [math]\;[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} \tag{5}[/math] [math]\;[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} = 1 + \frac{E_{\nu}}{E_{e}} \left(1 - \cos \theta \right) = 1 + \frac{m_{\nu} c^2}{m_{e} c^2} \left(1 - \cos \theta \right) = 1 + \frac{m_{\nu}}{m_{e}} \left(1 - \cos \theta \right) \tag{6}[/math] [math]\;[/math] Substituting equation [math](6)[/math] into equation [math](4)[/math], and eliminate [math]p_{i}[/math] and [math]p_{f}[/math] in favor of [math]m_{\nu}[/math] alone. [math]\;[/math] [math]\tan \phi = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{m_{\nu}}{m_{e}} \left(1 - \cos \theta \right) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{m_{\nu}}{m_{e}} \right)\left(1 - \cos \theta \right)} \tag{7}[/math] [math]\;[/math] Utilizing a trigonometric identity produces the desired result, specifically: [math]\frac{1 - \cos \theta}{\sin \theta} = \tan \left(\frac{\theta}{2} \right) \tag{8}[/math] [math]\;[/math] Substituting this trigonometric identity into equation [math](7)[/math] results in: [math]\left(1 + \frac{m_{\nu}}{m_{e}} \right) \tan \phi = \cot \frac{\theta}{2} \tag{9}[/math] [math]\;[/math] Solve for neutrino mass [math]m_{\nu}[/math]: [math]\tan \phi + \frac{m_{\nu}}{m_{e}} \tan \phi = \cot \frac{\theta}{2} \tag{10}[/math] [math]\;[/math] [math]\frac{m_{\nu}}{m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/math] [math]\;[/math] Electron-neutrino scattering neutrino mass: [math]\boxed{m_{\nu} = m_{e} \cot \phi \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{12}[/math] [math]\;[/math] Nuclear-neutrino scattering neutrino mass: [math]\boxed{m_{\nu} = m_{n} \cot \phi \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{13}[/math] [math]\;[/math] [math]m_{n}[/math] - nuclear mass [math]\;[/math] Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(m_{e} + m_{\nu} \right) \tan \phi}{m_{e}} \right)} \tag{14}[/math] [math]\;[/math] Neutrino interaction electron recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{m_{e} \cot \frac{\theta}{2}}{m_{e} + m_{\nu}} \right)} \tag{15}[/math] [math]\;[/math] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(m_{n} + m_{\nu} \right) \tan \phi}{m_{n}} \right)} \tag{16}[/math] [math]\;[/math] Neutrino interaction nuclear recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{m_{n} \cot \frac{\theta}{2}}{m_{n} + m_{\nu}} \right)} \tag{17}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Compton scattering - Derivation of the scattering formula: https://en.wikipedia.org/wiki/Compton_scattering#Derivation_of_the_scattering_formula Physics 253 - Compton Scattering - Patrick LeClair http://pleclair.ua.edu//PH253/Notes/compton.pdf Orion1 - Neutrino mass from Fermi-Dirac statistics...: https://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/ Science News - Neutrinos seen scattering off an atom’s nucleus for the first time: https://www.sciencenews.org/article/neutrinos-seen-scattering-atoms-nucleus-first-time
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