Orion1

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\nu,t} = 0.10[/math] Observable Universe cosmology scale factor: [math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math] Observable Universe total neutrino co-moving radius at present time: [math]\boxed{R_{\nu} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right]}[/math] [math]\boxed{R_{\nu} = 1.472 \cdot 10^{27} \; \text{m}} \; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math] Observable Universe total neutrino co-moving radius at past time integration via substitution: [math]R_{\nu,t} = R_{\nu} \left( \frac{T_{\nu}}{T_{\nu,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) = 2.864 \cdot 10^{17} \; \text{m} \; \; \; (30.277 \; \text{ly})[/math] Observable Universe total neutrino co-moving radius at past time: [math]\boxed{R_{\nu,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right)}[/math] [math]\boxed{R_{\nu,t} = 2.864 \cdot 10^{17} \; \text{m}} \; \; \; (30.277 \; \text{ly})[/math] Neutrino decoupling time: [math]T_{u,\nu} = \frac{1}{H_{\nu,t}} = \frac{}{2 (k_B T_{\nu,t})^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 (k_B T_{\nu,t})^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] Observable Universe expansion rate at neutrino decoupling time integration via substitution: [math]\frac{dr}{dt} = \frac{R_{\nu,t}}{T_{u,\nu}} = \left[ \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) \right] 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 1.933 \cdot 10^{18} \; \frac{\text{m}}{\text{s}} \; \; \; \left( 204.337 \; \frac{\text{ly}}{\text{s}} \right)[/math] Observable Universe expansion rate at neutrino decoupling time: [math]\boxed{\frac{dr}{dt} = \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} \left( \frac{c}{H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) (k_B T_{\nu,t})^2}[/math] [math]\boxed{\frac{dr}{dt} = 1.933 \cdot 10^{18} \; \frac{\text{m}}{\text{s}}} \; \; \; \left( 204.337 \; \frac{\text{ly}}{\text{s}} \right)[/math] Did the universe have a co-moving radius of 30 light-years at neutrino decoupling time? Did the universe inflate and expand at a rate of 204 light-years per second at neutrino decoupling time? Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - Inflation cosmology: https://en.wikipedia.org/wiki/Inflation_(cosmology) Wikipedia - Metric space expansion cosmology: https://en.wikipedia.org/wiki/Metric_expansion_of_space

[math]\boxed{\frac{ds}{dt} = 1 \cdot 10^{2} \; \frac{\text{m}}{\text{s}}} \; \; \; (100 \; \frac{\text{m}}{\text{s}})[/math]

Observable Universe cosmology scale factor: [math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math] Symbolic definition key: [math]R_u(t_0)[/math] - Observable Universe total radius at present time. [math]R_u(t)[/math] - Observable Universe total radius at past time. [math]a(t_0)[/math] - scale factor at present time. [math]a(t)[/math] - scale factor at past time. [math]T_t[/math] - cosmic background radiation temperature at past time. [math]T_0[/math] - cosmic background radiation temperature at present time. [math]z[/math] - cosmic background radiation redshift parameter at decoupling time. Hubble radius: [math]R_{H} = \frac{c}{H_{0}}[/math] Cosmic photon background radiation temperature at present time: (ref. 1) [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Cosmic photon background radiation redshift parameter at photon decoupling time: (ref. 2, pg. 11) [math]z_{\gamma} = \left( \frac{T_{\gamma,t}}{T_{\gamma}} \right) - 1 = 1090.43[/math] Cosmic photon background radiation temperature at photon decoupling time: (ref. 1) [math]T_{\gamma,t} = T_{\gamma} (1 + z_{\gamma}) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math] Observable Universe total photon co-moving radius integration via substitution: [math]R_{\gamma} = \frac{R_{H} \ln [1 + z_{\gamma}]}{2} = \frac{1}{2} \left( \frac{c}{H_{0}} \right) \ln [1 + z_{\gamma}] = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\gamma,t}}{T_{\gamma}} \right] = 4.606 \cdot 10^{26} \; \text{m}[/math] Observable Universe total photon co-moving radius: [math]\boxed{R_{\gamma} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\gamma,t}}{T_{\gamma}} \right]}[/math] [math]\boxed{R_{\gamma} = 4.606 \cdot 10^{26} \; \text{m}}[/math][math]\; \; \; (48.689 \cdot 10^{9} \; \text{ly})[/math] Cosmic neutrino background radiation temperature at present time: (ref. 3, pg. 44, eq. 220), (ref. 4) [math]T_{\nu} = \left( \frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.94535 \; \text{K}[/math] [math]\boxed{T_{\nu} = 1.94535 \; \text{K}}[/math] Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 5) [math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe total neutrino co-moving radius: [math]\boxed{R_{\nu} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right]}[/math] [math]\boxed{R_{\nu} = 1.472 \cdot 10^{27} \; \text{m}}[/math][math]\; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math] Observable Universe total observable stellar number: [math]\boxed{N_s = \frac{\pi \Omega_{s}}{6 M_{a}} \left( \frac{c}{H_0} \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] Observable Universe total observable stellar number: [math]\boxed{N_s = 2.575 \cdot 10^{23} \; \text{stars}}[/math] Wikipedia observable universe total observable stellar number: (ref. 6) [math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math] Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 7) [math]N_s = 1 \cdot 10^{24} \; \text{stars}[/math] After neutrinos decoupled from heavy baryonic matter when the universe was one second old, is it possible for the cosmic neutrino background radiation to have inflated to a distance of 156 billion light years? Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Cosmic microwave background radiation: (ref. 1) https://en.wikipedia.org/wiki/Cosmic_microwave_background Planck 2013 results. XVI. Cosmological parameters: (ref. 2) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf PHYS: 652 Cosmic Inventory I: Radiation: (ref. 3) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Wikipedia - Cosmic neutrino background radiation: (ref. 4) https://en.wikipedia.org/wiki/Cosmic_neutrino_background Wikipedia - Neutrino decoupling: (ref. 5) https://en.wikipedia.org/wiki/Neutrino_decoupling Wikipedia - Observable universe total observable stellar number: (ref. 6) https://en.wikipedia.org/wiki/Star#Distribution Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 7) https://www.space.com/26078-how-many-stars-are-there.html

It appears that I have reached the limit of my available energy budget. Any recommendations on equation refinement? You were correct. I have attempted to calculate the actual average stellar mass from the equations and table references listed on this post. Any recommendations on equation refinement? You were correct. Your Wikipedia reference for estimates based on critical density, states the observable universe total mass at [math]1.46 \cdot 10^{53} \; \text{kg}[/math]. I was able to further refine these calculations down to a factor of only 26, instead of ~100. Observable Universe total mass: (ref. 1) [math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math] --- Stellar class number parameters: (ref. 2) [math]n_c = 1 \rightarrow 7[/math] key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M [math]\Omega_n[/math] - main sequence stars stellar class fraction [math]N_s[/math] - total observable stellar number [math]M_n[/math] - Main-sequence mass Observable Universe average stellar mass: [math]M_a = \frac{1}{N_s} \sum_{n_c = 1}^{7} (\Omega_n N_s M_n) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math] [math]\boxed{M_a = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math] [math]\boxed{M_a = (0.595 \rightarrow 0.769) \cdot M_{\odot}}[/math] Observable Universe average stellar mass lower bound limit: [math]\boxed{M_a = 1.183 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 3, pg. 20) [math]M_a = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] Observable Universe stellar baryon density: (ref. 4, pg. 3) [math]\Omega_s = (\Omega_{ms} + \Omega_{wd} + \Omega_{ns}) = 2.460 \cdot 10^{-3}[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] Observable Universe total observable stellar number: [math]N_s = \Omega_{s} \left( \frac{M_u}{M_{a}} \right) = \frac{\pi \Omega_{s}}{6 M_{a}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{N_s = \frac{\pi \Omega_{s}}{6 M_{a}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] Observable Universe total observable stellar number: [math]\boxed{N_s = 7.885 \cdot 10^{21} \; \text{stars}}[/math] Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 5) [math]N_s = 1 \cdot 10^{24} \; \text{stars}[/math] Wikipedia - observable universe total observable stellar number: (ref. 6) [math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - Observable Universe total mass: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=909462 Wikipedia - Stellar classification - Harvard spectral classification: (ref. 2) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification On The Mass Distribution Of Stars...: (ref. 3) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf The Cosmic Energy Inventory: (ref. 4) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 5) https://www.space.com/26078-how-many-stars-are-there.html Wikipedia - observable universe stellar number: (ref. 6) https://en.wikipedia.org/wiki/Star#Distribution

Photon species total effective degeneracy number: [math]\boxed{N_{\gamma} = 2}[/math] Planck's law: (ref. 1) [math]\boxed{I_{\gamma}(\nu,T_{\gamma}) = \frac{N_{\gamma} h \nu^3}{c^2 (e^{\frac{E_t}{E_{\gamma}}} - 1)}}[/math] Radiant emmittance integration via substitution: (ref. 2) [math]j^* = \int d\Omega \int_0^\infty I_{\gamma}(\nu,T_{\gamma}) \; d\nu[/math] [math]\int d\Omega = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d\theta[/math] [math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \int_0^\infty I_{\gamma}(\nu,T_{\gamma}) \; d\nu[/math] [math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \int_0^\infty \frac{\nu^3}{e^{\frac{E_t}{E_{\gamma}}} - 1} \; d\nu[/math] Differential calculus theorem: [math]\boxed{\frac{\int_a^b f(u)^n \; du}{\int_a^b f(v)^n \; dv} = \left( \frac{du}{dv} \right)^{n+1}}[/math] [math]\int_0^\infty \frac{\nu^3}{e^{\frac{E_t}{E_{\gamma}}} - 1} \; d\nu = \left( \frac{d\nu}{du} \right)^4 \int_0^\infty \frac{u^3}{e^u - 1} \; du[/math] [math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \left( \frac{d\nu}{du} \right)^4 \int_0^\infty \frac{u^3}{e^u - 1} \; du[/math] [math]\frac{d\nu}{du} = \frac{E_{\gamma}}{h}[/math] [math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \left( \frac{E_{\gamma}}{h} \right)^4 \int_{0}^\infty \frac{E_t (\nu)^3}{e^{\frac{E_t (\nu)}{E_{\gamma} (T_{\gamma})}} - 1} d \nu[/math] [math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 h^3} \int_{0}^\infty \frac{E_t (\nu)^3}{e^{\frac{E_t(\nu)}{E_{\gamma} (T_{\gamma})}} - 1} d \nu[/math] [math]j^{*} = \int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d \theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 (2 \pi \hbar)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_{\gamma} (T_{\gamma})}} - 1} d \omega = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}[/math] Radiant emmittance Stefan-Boltzmann law: (ref. 3) [math]\boxed{j^{*} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}}[/math] Is there a formal name or formal method name for this differential calculus theorem? [math]\boxed{\frac{\int_a^b f(u)^n \; du}{\int_a^b f(v)^n \; dv} = \left( \frac{du}{dv} \right)^{n+1}}[/math] 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 - Stefan–Boltzmann law - derivation from Planck's law: (ref. 2) https://en.wikipedia.org/wiki/Stefan–Boltzmann_law#Derivation_from_Planck.27s_law Wikipedia - Stefan-Boltzmann law: (ref. 3) https://en.wikipedia.org/wiki/Stefan–Boltzmann_law

Photon mass: [math]\boxed{m_{\gamma} = 0}[/math] Photon species total effective degeneracy number: [math]\boxed{N_{\gamma} = 2}[/math] Photon radiation energy radiant emmittance Bose-Einstein distribution integration via substitution: (ref. 1) [math]j^{*} = \sigma_{\gamma} T_{\gamma}^4 = \int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d \theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 (2 \pi \hbar)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = \frac{\pi N_{\gamma} (k_B T_{\gamma})^4}{c^2 (2 \pi \hbar)^3} \left( \frac{\pi^4}{15} \right) = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}[/math] Radiant emmittance Stefan-Boltzmann constant: (ref. 2) [math]\boxed{\sigma_{\gamma} = \frac{N_{\gamma} \pi^2 k_B^4}{120 c^2 \hbar^3}}[/math] Radiant emmittance Stefan-Boltzmann law: (ref. 2) [math]\boxed{j^{*} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}}[/math] Photon radiation energy density Bose-Einstein distribution integration via substitution: (ref. 3, pg. 43, eq. 204-206) [math]\epsilon_{\gamma} = \alpha_{\gamma} T_{\gamma}^4 = \int_{0}^{2 \pi} \int_{0}^{\pi} \sin \theta \; d \theta \; d \phi \; \frac{N_{\gamma} E_{\gamma}^4}{(2 \pi \hbar c)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = \frac{4 \pi N_{\gamma} (k_B T_{\gamma})^4}{(2 \pi \hbar c)^3} \left( \frac{\pi^4}{15} \right) = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{30 (\hbar c)^3}[/math] Photon radiation constant: [math]\boxed{\alpha_{\gamma} = \frac{N_{\gamma} \pi^2 k_B^4}{30 (\hbar c)^3}}[/math] Photon radiation energy density: [math]\boxed{\epsilon_{\gamma} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{30 (\hbar c)^3}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Stefan–Boltzmann law - derivation from Planck's law: (ref. 1) https://en.wikipedia.org/wiki/Stefan–Boltzmann_law#Derivation_from_Planck.27s_law Wikipedia - Stefan-Boltzmann law: (ref. 2) https://en.wikipedia.org/wiki/Stefan–Boltzmann_law PHYS: 652 Cosmic Inventory I: Radiation: (ref. 3) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

Symbolic identity key: [math]n_{s}[/math] - spin states total number [math]N_{s}[/math] - species total number [math]N_{n}[/math] - total effective degeneracy number if [math]n_{s} \geq N_{s}[/math] then [math]N_{n} = n_{s}[/math] if [math]n_{s} \leq N_{s}[/math] then [math]N_{n} = N_{s}[/math] [math]\begin{array}{l*{6}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 \\ f & \text{neutralino} & +,- & 1/2 & 2 & 4 & 4 \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 \\ \end{array}[/math] Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1) [math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: (ref. 2) [math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 1) [math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] Massless neutrino interaction rate at massless neutrino decoupling time: (ref. 3) [math]\boxed{\Gamma_{\nu,t} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} = 0[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725 Orion1 - Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725 Orion1 - Massless neutrino interaction rate at massless neutrino decoupling time: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1004223

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\nu , t} = 0.10[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2) [math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math] Neutrino mass: [math]\boxed{m_{\nu} = 0}[/math] Fermi-Dirac massless neutrino composition: (ref. 3) [math]\boxed{\Omega_{\nu} = \frac{7 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Solve for massless neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at massless neutrino decoupling time: [math]\Gamma_{\nu,t} = \sqrt{\frac{7 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{45 \Omega_{\nu,t} \hbar^3 c^5}} = \frac{(k_B T_{\nu,t})^2}{3} \sqrt{\frac{7 \pi^3 G N_{\nu}}{5 \Omega_{\nu,t} \hbar^3 c^5}} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math] Massless neutrino interaction rate at massless neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = \frac{(k_B T_{\nu,t})^2}{3} \sqrt{\frac{7 \pi^3 G N_{\nu}}{5 \Omega_{\nu,t} \hbar^3 c^5}}}[/math] Massless neutrino interaction rate at massless neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) https://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - neutrino decoupling: (ref. 2) https://en.wikipedia.org/wiki/Neutrino_decoupling Hubble parameter at massless neutrino decoupling time: (ref. 3)

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1) [math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}}[/math] Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: (ref. 2) [math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math] Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 1) [math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725 Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Observable Universe Dark Matter scalar particle decoupling time: (ref. 1) [math]\boxed{T_{u, \phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] Observable Universe cosmic neutrino background radiation neutrino decoupling time: (ref. 2) [math]\boxed{T_{u , \nu} = 0.148 \; \text{s}}\; \; \; m_{\nu} \neq 0[/math] Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 3) [math]\boxed{T_{u, \nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] Observable Universe cosmic neutrino background radiation massless neutrino decoupling time: (ref. 4) [math]\boxed{T_{u , \nu} = 0.893 \; \text{s}} \; \; \; m_{\nu} = 0[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - Observable Universe Dark Matter scalar particle decoupling time: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999388 Orion1 - Observable Universe neutrino background radiation neutrino decoupling time: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999033 Orion1 - Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725 Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\nu , t} = 0.10[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2) [math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math] Solve for Observable Universe cosmic neutrino background radiation neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at neutrino decoupling time: (ref. 3) [math]\Gamma_{\nu,t} = \sqrt{\frac{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 6.749 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math] Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}}}[/math] Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) https://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - neutrino decoupling: (ref. 2) https://en.wikipedia.org/wiki/Neutrino_decoupling Fermi-Dirac total neutrino distribution constant: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry886860

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{dm , t} = 0.63[/math] [math]\Omega_{\gamma , t} = 0.15[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Observable Universe Dark Matter sterile neutrino composition is equivalent to Dark Matter composition at photon decoupling time: [math]\boxed{\Omega_{\nu , t} = \Omega_{dm , t}}[/math] Observable Universe Dark Matter sterile neutrino mass: [math]\boxed{m_{\nu} \neq 0}[/math] Observable Universe Dark Matter sterile neutrino decoupling temperature is equivalent to cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2) [math]\boxed{T_{\nu, t} = T_{\nu , t}} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe Fermi-Dirac Dark Matter sterile neutrino composition: (ref. 3) [math]\boxed{\Omega_{\nu} = \frac{4 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{3 C_{\nu} H_0^2 \hbar^3 c^5}}[/math] Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 4) [math]\boxed{T_{u , \nu} = \frac{1}{H_{\nu , t}}}[/math] Solve for Observable Universe Dark Matter sterile neutrino decoupling time [math]T_{u , \nu}[/math]: [math]T_{u , \nu} = \frac{1}{H_{\nu, t}} = \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.372 \; \text{s}[/math] Observable Universe Dark Matter sterile neutrino decoupling time: [math]\boxed{T_{u , \nu} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}}}[/math] Observable Universe Dark Matter sterile neutrino decoupling time: [math]\boxed{T_{u, \nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] --- Particle interaction rate is equivalent to Hubble Parameter at particle decoupling time: (ref. 5), (ref .6) [math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math] Solve for Observable Universe photon interaction rate [math]\Gamma_{\gamma,t}[/math] at photon decoupling time: [math]\Gamma_{\gamma,t} = \sqrt{\frac{4 G N_{\gamma} \pi^3 (k_B T_{\gamma,t})^4}{45 \Omega_{\gamma,t} \hbar^3 c^5}} = \frac{2 (k_B T_{\gamma,t})^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} = 4.957 \cdot 10^{-14} \; \frac{ \gamma \; \text{particles}}{\text{s}}[/math] Observable Universe photon interaction rate at photon decoupling time: [math]\boxed{\Gamma_{\gamma,t} = \frac{2 (k_B T_{\gamma,t})^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}}}[/math] Observable Universe photon interaction rate at photon decoupling time: [math]\boxed{\Gamma_{\gamma,t} = 4.957 \cdot 10^{-14} \; \frac{ \gamma \; \text{particles}}{\text{s}}}[/math] Solve for Observable Universe Dark Matter scalar particle interaction rate [math]\Gamma_{\phi,t}[/math] at scalar particle decoupling time: [math]\Gamma_{\phi,t} = \sqrt{\frac{4 G N_{\phi} \pi^3 (k_B T_{\phi,t})^4}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 2 (k_B T_{\phi,t})^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 39.232 \; \frac{ \phi \; \text{particles}}{\text{s}}[/math] Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: [math]\boxed{\Gamma_{\phi,t} = 2 (k_B T_{\phi,t})^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}}}[/math] Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: [math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}}[/math] Solve for Observable Universe Dark Matter sterile neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at sterile neutrino decoupling time: [math]\Gamma_{\nu,t} = \sqrt{\frac{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2.689 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math] Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}}}[/math] Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: [math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) https://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - neutrino decoupling: (ref. 2) https://en.wikipedia.org/wiki/Neutrino_decoupling Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631 Wikipedia - Derivation of decoupling time: (ref. 5) https://en.wikipedia.org/wiki/Neutrino_decoupling#Derivation_of_decoupling_time Wikipedia - Cross section (physics): (ref. 6) https://en.wikipedia.org/wiki/Cross_section_(physics)

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{dm , t} = 0.63[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Observable Universe Dark Matter scalar particle composition is equivalent to Dark Matter composition at photon decoupling time: [math]\boxed{\Omega_{\phi , t} = \Omega_{dm , t}}[/math] Observable Universe Dark Matter scalar particle mass: [math]\boxed{m_{\phi} \neq 0}[/math] Observable Universe Dark Matter scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2) [math]\boxed{T_{\phi , t} = T_{\nu , t}} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe Bose-Einstein Dark Matter scalar particle composition: (ref. 3) [math]\boxed{\Omega_{\phi} = \frac{4 G N_{\phi} \pi^3 (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}}[/math] Observable Universe Dark Matter scalar particle decoupling time: (ref. 4) [math]\boxed{T_{u , \phi} = \frac{1}{H_{\phi , t}}}[/math] Solve for Observable Universe Dark Matter scalar particle decoupling time [math]T_{u , \phi}[/math]: [math]T_{u , \phi} = \frac{1}{H_{\phi, t}} = \sqrt{\frac{3 \Omega_{\phi,t} C_{\phi} \hbar^3 c^5}{4 G N_{\phi} \pi^3 (k_B T_{\phi,t})^4}} = \frac{}{2 (k_B T_{\phi,t})^2} \sqrt{\frac{3 \Omega_{\phi,t} C_{\phi} \hbar^3 c^5}{G N_{\phi} \pi^3}} = 0.0255 \; \text{s}[/math] Observable Universe Dark Matter scalar particle decoupling time: [math]\boxed{T_{u , \phi} = \frac{}{2 (k_B T_{\phi,t})^2} \sqrt{\frac{3 \Omega_{\phi,t} C_{\phi} \hbar^3 c^5}{G N_{\phi} \pi^3}}}[/math] Observable Universe Dark Matter scalar particle decoupling time: [math]\boxed{T_{u, \phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) https://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - neutrino decoupling: (ref. 2) https://en.wikipedia.org/wiki/Neutrino_decoupling Orion1 - Dark Matter scalar particle composition: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry909539 Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\nu , t} = 0.10[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Neutrino mass: [math]\boxed{m_{\nu} \neq 0}[/math] Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2) [math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe Fermi-Dirac neutrino composition: (ref. 3),(ref. 4) [math]\boxed{\Omega_{\nu} = \frac{4 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{3 C_{\nu} H_0^2 \hbar^3 c^5}}[/math] Observable Universe neutrino background radiation neutrino decoupling time: (ref. 5) [math]\boxed{T_{u , \nu} = \frac{1}{H_{\nu , t}}}[/math] Solve for Observable Universe neutrino decoupling time [math]T_{u , \nu}[/math]: [math]T_{u , \nu} = \frac{1}{H_{\nu , t}} = \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.148 \; \text{s}[/math] Observable Universe neutrino background radiation neutrino decoupling time: [math]\boxed{T_{u , \nu} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}}}[/math] Observable Universe neutrino background radiation neutrino decoupling time: [math]\boxed{T_{u , \nu} = 0.148 \; \text{s}}\; \; \; m_{\nu} \neq 0[/math] Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 5) [math]\boxed{T_{u , \nu} = 0.893 \; \text{s}} \; \; \; m_{\nu} = 0[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631 Wikipedia - neutrino decoupling: (ref. 2) https://en.wikipedia.org/wiki/Neutrino_decoupling Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Orion1 - Fermi-Dirac and Bose-Einstein total neutrino distribution constant: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry886860 Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 5) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Observable Universe Cosmological Constant: (ref. 1), (ref. 2) [math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] Observable Universe Bose-Einstein scalar particle Dark Energy composition: (ref. 3) [math]\boxed{\Omega_{\Lambda} = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Observable Universe scalar particle Dark Energy Cosmological Constant integration via substitution: [math]\Lambda_s = \frac{3 H_0^2}{c^2} \left( \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5} \right) = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] Observable Universe scalar particle Dark Energy Cosmological Constant: [math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{15 \hbar^3 c^7}}[/math] Observable Universe scalar particle Dark Energy Cosmological Constant: [math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Cosmological Constant: (ref. 1) http://en.wikipedia.org/wiki/Hubble%27s_law#Matter-dominated_universe_.28with_a_cosmological_constant.29 Orion1 - Cosmological Constant: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850783 Orion1 - Dark Energy: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry931225

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Observable Universe composition Equation of State: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1[/math] Observable Universe photon and neutrino and scalar particle Dark Energy composition Equation of State: [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math] Observable Universe scalar particle Dark Matter and scalar particle Dark Energy composition Equation of State: [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math] Observable Universe sterile neutrino Dark Matter and scalar particle Dark Energy composition Equation of State: [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math] Observable Universe total energy integration via substitution: [math]E_u = M_u c^2 = \frac{\pi^3 k_B^4}{12 (\hbar H_0)^3} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] \left( \ln (1 + z) \right)^3 = 3.410 \cdot 10^{71} \; \text{j}[/math] Observable Universe total energy: [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 (\hbar H_0)^3} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] \left( \ln (1 + z) \right)^3}[/math] Observable Universe total energy: [math]\boxed{E_u = 3.410 \cdot 10^{71} \; \text{j}}[/math] Any discussions and/or peer reviews about this specific topic thread?

Photon particle mass: [math]\boxed{m_{\gamma} = 0}[/math] Neutrino particle mass: [math]\boxed{m_{\nu} \neq 0}[/math] Dark Energy scalar particle mass: [math]\boxed{m_{\Lambda} = 0}[/math] Dark Matter scalar particle mass: [math]\boxed{m_{\phi} \neq 0}[/math] Observable Universe Bose-Einstein photon composition: (ref. 1) [math]\boxed{\Omega_{\gamma} = \frac{4 G N_{\gamma} \pi^3 (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Observable Universe Fermi-Dirac neutrino composition: (ref. 2) [math]\boxed{\Omega_{\nu} = \frac{4 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{3 C_{\nu} H_0^2 \hbar^3 c^5}}[/math] Observable Universe Bose-Einstein Dark Energy composition: (ref. 3) [math]\boxed{\Omega_{\Lambda} = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Observable Universe Bose-Einstein Dark Matter composition: (ref. 4) [math]\boxed{\Omega_{\phi} = \frac{4 G N_{\phi} \pi^3 (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}}[/math] Observable Universe compositional Equation of State: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1[/math] Observable Universe compositional Equation of State: (ref. 5) [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \Omega_{\Lambda} = 1}[/math] Observable Universe compositional Equation of State integration via substitution: [math]\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5} = \frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1[/math] Observable Universe compositional Equation of State: [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math] Observable Universe compositional Equation of State: (ref. 4) [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - Cosmic Microwave Background Radiation photon composition: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882069 Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Orion1 - Dark Energy: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry931225 Orion1 - Dark Matter scalar particle composition: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry909539 Orion1 - Observable Universe compositional Equation of State: (ref. 5) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry980016

Observable Universe Friedmann equations critical mass: (ref. 1) [math]M_c = \frac{H_0^2 R_u^3}{2 G} = \frac{H_0^2}{2 G} \left[ \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} \right]^3 = \frac{[c \ln (1 + z)]^3}{16 G H_0} = 3.794 \cdot 10^{54} \; \text{kg}[/math] [math]\boxed{M_c = \frac{[c \ln (1 + z)]^3}{16 G H_0}}[/math] [math]\boxed{M_c = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Observable Universe total mass: (ref. 2) [math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Observable Universe Friedmann equations critical mass is equivalent to Observable Universe total mass: [math]\boxed{M_c = M_u}[/math] Observable Universe Friedmann equations critical mass is equivalent to Observable Universe total mass integration via substitution: [math]\frac{[c \ln (1 + z)]^3}{16 G H_0} = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\frac{8 \pi G}{3 H_0^2} \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right] = 1[/math] Observable Universe compositional Equation of State: [math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \Omega_{\Lambda} = 1}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orion1 - Friedmann equations critical mass: (ref. 1) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915 Orion1 - Observable Universe total mass: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry909462

A scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number . (ref. 1), (ref. 2) [math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Symbolic identity key: [math]n_s[/math] - spin states total number [math]N_s[/math] - species total number [math]N_n[/math] - total effective degeneracy number If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math] If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math] Dark Energy scalar particle mass: [math]\boxed{m_{\Lambda} = 0}[/math] Dark Energy scalar particle mass total effective degeneracy number: [math]\boxed{N_{\Lambda} = 1}[/math] Planck satellite cosmological parameters: (ref. 3, pg. 11) [math]\Omega_{\Lambda} = 0.6825[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 3, pg. 11) [math]z = 1090.43[/math] Dark Energy scalar particle radiation energy density Bose-Einstein distribution integration via substitution: [math]\epsilon_{\Lambda , 1} = \alpha_{\Lambda} T_{\Lambda}^4 = \frac{4 \pi N_{\Lambda} (k_B T_{\Lambda})^4}{( 2 \pi \hbar c )^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\Lambda})}} - 1} d \omega = \frac{4 \pi N_{\Lambda} (k_B T_{\Lambda})^4}{( 2 \pi \hbar c )^3} \left( \frac{\pi^4}{15} \right) = \frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3}[/math] Dark Energy scalar particle radiation energy density: [math]\boxed{\epsilon_{\Lambda , 1} = \frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3}}[/math] Dark Energy scalar particle radiation constant: [math]\boxed{\alpha_{\Lambda} = \frac{\pi^2 k_B^4 N_{\Lambda}}{30 ( \hbar c )^3}}[/math] Dark Energy scalar particle radiation composition energy density: (ref. 7) [math]\epsilon_{\Lambda , 2} = \rho_{\Lambda} c^2 = \frac{3 \Omega_{\Lambda} \left(c H_{0} \right)^2}{8 \pi G} = 5.685 \cdot 10^{-10} \; \frac{\text{j}}{\text{m}^3}[/math] Dark Energy scalar particle radiation composition energy density: [math]\boxed{\epsilon_{\Lambda , 2} = \frac{3 \Omega_{\Lambda} \left(c H_{0} \right)^2}{8 \pi G}}[/math] Dark Energy scalar particle radiation composition energy density: [math]\boxed{\epsilon_{\Lambda , 2} = 5.685 \cdot 10^{-10} \; \frac{\text{j}}{\text{m}^3}}[/math] Dark Energy scalar particle radiation energy density is equivalent to Dark Energy scalar radiation composition energy density: [math]\boxed{\epsilon_{\Lambda , 1} = \epsilon_{\Lambda , 2}}[/math] Dark Energy scalar particle radiation energy density is equivalent to Dark Energy scalar particle radiation composition energy density integration via substitution: [math]\frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3} = \frac{3 \Omega_{\Lambda} \left(c H_0 \right)^2}{8 \pi G}[/math] Solve for Dark Energy scalar particle radiation temperature: [math]T_{\Lambda}[/math] [math]T_{\Lambda}^4 = \frac{30 ( \hbar c )^3}{\pi^2 N_{\Lambda} k_B^4} \left( \frac{3 \Omega_{\Lambda} \left(c H_0 \right)^2}{8 \pi G} \right) = \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4}[/math] [math]T_{\Lambda}^4 = \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4}[/math] Dark Energy scalar particle radiation temperature: [math]\boxed{T_{\Lambda} = \left( \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4} \right)^{1/4}}[/math] Dark Energy scalar particle radiation temperature: [math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math] Observable Universe total Dark Energy scalar particle number: (ref. 5, eq. 7) [math]\boxed{N_{\Lambda t} = \frac{\zeta (3) N_{\Lambda}}{6 \pi} \left( \frac{k_B T_{\Lambda} \ln (1 + z)}{\hbar H_0} \right)^3}[/math] Observable Universe total Dark Energy scalar particle number: (ref. 5, eq. 7) [math]\boxed{N_{\Lambda t} = 1.782 \cdot 10^{92} \; \Lambda \; \text{particles}}[/math] Dark Energy scalar particle peak energy: [math]E_{\Lambda} = k_B T_{\Lambda}[/math] Dark Energy scalar particle peak frequency: [math]E_{\Lambda} = 2 \pi \hbar f_{\Lambda} = k_B T_{\Lambda}[/math] Dark Energy scalar particle peak frequency integration via substitution: [math]f_{\Lambda} = \frac{k_B T_{\Lambda}}{2 \pi \hbar} = \frac{k_B}{2 \pi \hbar} \left( \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_B^4} \right)^{1/4} = \left( \frac{45 \Omega_{\Lambda} c^5}{64 \hbar G N_{\Lambda} \pi^7} \right)^{1/4}[/math] Dark Energy scalar particle peak T-ray frequency: (ref. 6) [math]\boxed{f_{\Lambda} = \left( \frac{45 \Omega_{\Lambda} c^5}{64 \hbar G N_{\Lambda} \pi^7} \right)^{1/4}}[/math] Dark Energy scalar particle peak terahertz radiation T-ray frequency: (ref. 6) [math]\boxed{f_{\Lambda} = 7.295 \cdot 10^{11} \; \text{Hz}}[/math] [math]f_{\Lambda} = 7.295 \cdot 10^{11} \; \text{Hz} = 0.729 \; \text{THz}[/math] Dark Energy scalar particle peak terahertz radiation T-ray frequency: (ref. 6) [math]\boxed{f_{\Lambda} = 0.729 \; \text{THz}}[/math] Dark Energy scalar particle peak energy: [math]\boxed{E_{\Lambda} = 4.834 \cdot 10^{-22} \; \text{j}}[/math] Dark Energy scalar particle peak energy: [math]\boxed{E_{\Lambda} = 3.018 \; \frac{\text{mV}}{c^2}}[/math] What are the possibilities for a terahertz antenna or a terahertz receiver tuned to the Dark Energy scalar particle peak terahertz frequency of being capable of detecting Dark Energy scalar particle terahertz T-ray radiation? What are the possibilities of terahertz radiation T-ray astronomy? Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Scalar boson (ref. 1) https://en.wikipedia...ki/Scalar_boson Wikipedia - Fermion (ref. 2) https://en.wikipedia.org/wiki/Fermion Planck 2013 results. XVI. Cosmological parameters: (ref. 3) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - Dark energy: (ref. 4) https://en.wikipedia.org/wiki/Dark_energy Introduction to Cosmology: Lecture 6 - Thermal history of the Universe - Joao G. Rosa (ref. 5) http://gravitation.web.ua.pt/sites/gravitation.web.ua.pt/files/Lecture_6.pdf Wikipedia - Terahertz radiation: (ref. 6) https://en.wikipedia.org/wiki/Terahertz_radiation Orion1 - Dark Energy mass density: (ref. 7) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850783

Observable Universe electron neutrino mass: [math]m_{\nu_e} = \frac{\Omega_{\nu} M_u}{N_{\nu t}} = 6.839 \cdot 10^{-38} \; \text{kg}[/math] [math]\boxed{m_{\nu_e} = \frac{\Omega_{\nu} M_u}{N_{\nu t}}}[/math] [math]\boxed{m_{\nu_e} = 6.839 \cdot 10^{-38} \; \text{kg}}[/math] Observable Universe dark matter scalar particle mass: [math]m_{\phi} = \frac{\Omega_{\phi} M_u}{N_{\phi t}} = 3.326 \cdot 10^{-35} \; \text{kg}[/math] [math]\boxed{m_{\phi} = \frac{\Omega_{\phi} M_u}{N_{\phi t}}}[/math] [math]\boxed{m_{\phi} = 3.326 \cdot 10^{-35} \; \text{kg}}[/math] Observable Universe dark matter sterile neutrino mass: [math]m_{\nu} = \frac{\Omega_{dm} M_u}{N_{\nu t}} = 1.456 \cdot 10^{-35} \; \text{kg}[/math] [math]\boxed{m_{\nu} = \frac{\Omega_{dm} M_u}{N_{\nu t}}}[/math] [math]\boxed{m_{\nu} = 1.456 \cdot 10^{-35} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Cosmic neutrino background radiation: (ref. 1) https://en.wikipedia.org/wiki/Cosmic_neutrino_background Wikipedia - Scalar field dark matter: (ref. 2) https://en.wikipedia.org/wiki/Scalar_field_dark_matter Wikipedia - Sterile neutrino: (ref. 3) https://en.wikipedia.org/wiki/Sterile_neutrino

A scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number [math]N_n[/math]. (ref. 1), (ref. 2) [math]\begin{tabular}{l*{6}{c}r} bosons (b) = integer spin & Bose-Einstein statistics \\ fermions (f) = half-integer spin & Fermi-Dirac statistics \\ \end{tabular}[/math] [math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Symbolic identity key: [math]n_s[/math] - spin states total number [math]N_s[/math] - species total number [math]N_n[/math] - total effective degeneracy number If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math] If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math] Note that if each neutrino species has a corresponding supersymmetric sterile neutrino species, then [math]\boxed{N_n = N_s = 3}[/math]. Neutralinos and sterile neutrinos are hypothetical particles. (ref. 3), (ref. 4) The mass spectrum range for neutralinos is 10 GeV to 1 TeV. (ref. 3) The mass spectrum range for sterile neutrinos is 1 eV to 10^15 GeV. (ref. 4) These particles are theoretically embedded into SO(10) MSSM models. (ref. 5), (ref. 6), (ref. 11) Observable Universe baryonic Hydrogen composition: [math]\Omega_{\text{H}} = 0.75[/math] Observable Universe baryonic Helium composition: [math]\Omega_{\text{He}} = 0.25[/math] Observable Universe total Hydrogen particle number: [math]N_{\text{H}} = \Omega_b \Omega_{\text{H}} \left( \frac{M_u}{m_{\text{H}}} \right) = 8.417 \cdot 10^{79} \; \text{H particles}[/math] [math]\boxed{N_{\text{H}} = \Omega_b \Omega_{\text{H}} \left( \frac{M_u}{m_{\text{H}}} \right)}[/math] [math]\boxed{N_{\text{H}} = 8.417 \cdot 10^{79} \; \text{H particles}}[/math] Observable Universe total Helium particle number: [math]N_{\text{He}} = \Omega_b \Omega_{\text{He}} \left( \frac{M_u}{m_{\text{He}}} \right) = 7.065 \cdot 10^{78} \; \text{He particles}[/math] [math]\boxed{N_{\text{He}} = \Omega_b \Omega_{\text{He}} \left( \frac{M_u}{m_{\text{He}}} \right)}[/math] [math]\boxed{N_{\text{He}} = 7.065 \cdot 10^{78} \; \text{He particles}}[/math] Observable Universe total proton particle number: [math]N_{pt} = N_{\text{H}} + 2N_{\text{He}} = 9.830 \cdot 10^{79} \; \text{p particles}[/math] [math]\boxed{N_{pt} = N_{\text{H}} + 2N_{\text{He}}}[/math] [math]\boxed{N_{pt} = 9.830 \cdot 10^{79} \; \text{p particles}}[/math] Observable Universe total neutron particle number: [math]N_{nt} = 2N_{\text{He}} = 1.413 \cdot 10^{79} \; \text{n particles}[/math] [math]\boxed{N_{nt} = 1.413 \cdot 10^{79} \; \text{n particles}}[/math] Observable Universe total electron particle number: [math]N_{\beta} = N_{\text{H}} + 2 N_{\text{He}} = 9.830 \cdot 10^{79} \; \beta \; \text{particles}[/math] [math]\boxed{N_{\beta t} = N_{\text{H}} + 2 N_{\text{He}}}[/math] [math]\boxed{N_{\beta t} = 9.830 \cdot 10^{79} \; \beta \; \text{particles}}[/math] Observable Universe total up quark particle number: [math]N_{u t} = 2 N_{\text{H}} + 6 N_{\text{He}} = 2.107 \cdot 10^{80} \; \text{u quark particles}[/math] [math]\boxed{N_{u t} = 2.107 \cdot 10^{80} \; \text{u quark particles}}[/math] Observable Universe total down quark particle number: [math]N_{d t} = N_{\text{H}} + 6 N_{\text{He}} = 1.266 \cdot 10^{80} \; \text{d quark particles}[/math] [math]\boxed{N_{d t} = 1.266 \cdot 10^{80} \; \text{d quark particles}}[/math] Observable Universe total photon particle number: (ref. 13, eq. 7) [math]N_{\gamma t} = n_{\gamma} V_u = \left( \frac{\zeta (3) N_{\gamma} (k_{B} T_{\gamma} )^3}{\pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\gamma}}{6 \pi} \left( \frac{k_B T_{\gamma} \ln (1 + z)}{\hbar H_0} \right)^3 = 1.682 \cdot 10^{89} \; \gamma \; \text{particles}[/math] [math]\boxed{N_{\gamma t} = \frac{\zeta (3) N_{\gamma}}{6 \pi} \left( \frac{k_B T_{\gamma} \ln (1 + z)}{\hbar H_0} \right)^3}[/math] [math]\boxed{N_{\gamma t} = 1.682 \cdot 10^{89} \; \gamma \; \text{particles}}[/math] Observable Universe total neutrino particle number: (ref. 13, eq. 8) [math]N_{\nu t} = n_{\nu} V_u = \left( \frac{3 \zeta (3) N_{\nu} (k_{B} T_{\nu} )^3}{4 \pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3 = 6.984 \cdot 10^{88} \; \nu \; \text{particles}[/math] [math]\boxed{N_{\nu t} = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3}[/math] [math]\boxed{N_{\nu t} = 6.984 \cdot 10^{88} \; \nu \; \text{particles}}[/math] Observable Universe total dark matter scalar particle number: (ref. 13, eq. 7) [math]N_{\phi t} = n_{\phi} V_u = \left( \frac{\zeta (3) N_{\phi} (k_{B} T_{\phi} )^3}{\pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\phi}}{6 \pi} \left( \frac{k_B T_{\phi} \ln (1 + z)}{\hbar H_0} \right)^3 = 3.057 \cdot 10^{88} \; \phi \; \text{particles}[/math] [math]\boxed{N_{\phi t} = \frac{\zeta (3) N_{\phi}}{6 \pi} \left( \frac{k_B T_{\phi} \ln (1 + z)}{\hbar H_0} \right)^3}[/math] [math]\boxed{N_{\phi t} = 3.057 \cdot 10^{88} \; \phi \; \text{particles}}[/math] Observable Universe total dark matter sterile neutrino particle number: (ref. 13, eq. 8) [math]N_{\nu t} = n_{\nu} V_u = \left( \frac{3 \zeta (3) N_{\nu} (k_{B} T_{\nu} )^3}{4 \pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3 = 6.984 \cdot 10^{88} \; \nu \; \text{particles}[/math] [math]\boxed{N_{\nu t} = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3}[/math] [math]\boxed{N_{\nu t} = 6.984 \cdot 10^{88} \; \nu \; \text{particles}}[/math] Observable Universe dark matter scalar particle total fundamental particle number: [math]N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\phi t}) = 2.686 \cdot 10^{89} \; \text{particles}[/math] [math]\boxed{N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\phi t})}[/math] [math]\boxed{N_t = 2.686 \cdot 10^{89} \; \text{particles}}[/math] Observable Universe dark matter sterile neutrino particle total fundamental particle number: [math]N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\nu t}) = 3.078 \cdot 10^{89} \; \text{particles}[/math] [math]\boxed{N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\nu t})}[/math] [math]\boxed{N_t = 3.078 \cdot 10^{89} \; \text{particles}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Scalar boson (ref. 1) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Fermion (ref. 2) https://en.wikipedia.org/wiki/Fermion Wikipedia - Neutralino - Origins in supersymmetric theories (ref. 3) https://en.wikipedia.org/wiki/Neutralino#Origins_in_supersymmetric_theories Wikipedia - Sterile neutrinos (ref. 4) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Standard Model - Total particle count (ref. 5) https://en.wikipedia.org/wiki/Standard_Model#Total_particle_count Wikipedia - SO[10] (ref. 6) https://en.wikipedia.org/wiki/SO%2810%29 CERN - Next decade of sterile neutrino studies (ref. 7) http://arxiv.org/pdf/1306.4954v1.pdf Wikipedia - Dark matter (ref. 8) https://en.wikipedia.org/wiki/Dark_matter PHYS: 652 Cosmic Inventory I: Radiation (ref. 9) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Formaggio - Relic Neutrinos - Institute for Nuclear Theory (ref. 10) http://www.int.washington.edu/talks/WorkShops/int_10_44W/People/Formaggio_J/Formaggio.pdf WIkipedia - Minimal supersymmetric standard model - dark matter (ref. 11) https://en.wikipedia.org/wiki/Minimal_Supersymmetric_Standard_Model#Dark_matter Orion1 - Dark matter scalar particle radiation energy density Bose-Einstein distribution (ref. 12) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/?p=887017 Introduction to Cosmology: Lecture 6 - Thermal history of the Universe - Joao G. Rosa (ref. 13) http://gravitation.web.ua.pt/sites/gravitation.web.ua.pt/files/Lecture_6.pdf

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{dm} = 0.63[/math] [math]\Omega_{b} = 0.12[/math] [math]\Omega_{\gamma} = 0.15[/math] [math]\Omega_{\nu} = 0.10[/math] [math]\Omega_{\Lambda} \neq 0[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 2, pg. 11) [math]z = 1090.43[/math] Cosmology scale factor: [math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math] Symbolic definition key: [math]R_u(t_0)[/math] - Observable Universe total radius at present time. [math]R_u(t)[/math] - Observable Universe total radius at past time. [math]a(t_0)[/math] - scale factor at present time. [math]a(t)[/math] - scale factor at past time. [math]T_t[/math] - cosmic background radiation temperature at past time. [math]T_0[/math] - cosmic background radiation temperature at present time. [math]z[/math] - redshift at photon decoupling time. Observable Universe total co-moving radius at present time: (ref. 3) [math]\boxed{R_u(t_0) = \frac{c \ln (1 + z)}{2 H_0}}[/math] Observable Universe total radius at photon decoupling time: [math]R_u(t) = \frac{R_u(t_0)}{1 + z} = \frac{c \ln (1 + z)}{2 H_0 (1 + z)} = 4.220 \cdot 10^{23} \; \text{m}[/math] [math]\boxed{R_u(t) = \frac{c \ln (1 + z)}{2 H_0 (1 + z)}}[/math] [math]\boxed{R_u(t) = 4.220 \cdot 10^{23} \; \text{m}} \; \; \; 4.461 \cdot 10^{7} \; \text{ly}[/math] Observable Universe photon cosmic background radiation temperature at photon decoupling time: [math]T_{\gamma , t} = T_{\gamma} (1 + z) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma , t} = 2974.67 \; \text{K}}[/math] Observable Universe neutrino cosmic background radiation temperature at photon decoupling time: [math]T_{\nu , t} = T_{\nu} (1 + z) = 2123.22 \; \text{K}[/math] [math]\boxed{T_{\nu , t} = 2123.22 \; \text{K}}[/math] Observable Universe total mass at photon decoupling time: [math]\boxed{M_u = \frac{4 \pi}{3} \left( \frac{c \ln (1 + z)}{2 H_0 (1 + z)} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] Observable Universe total mass at photon decoupling time: [math]\boxed{M_u = 1.245 \cdot 10^{54} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Planck 2013 results. XVI. Cosmological parameters: (ref. 2) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf UCLA Division of Astronomy and Astrophysics - Homogeneity and Isotropy: (ref. 3) http://www.astro.ucla.edu/~wright/cosmo_02.htm#DL Orion1 - Friedmann equations critical mass: (ref. 4) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915

Sterile neutrinos and neutralinos are potential dark matter candidates. However, scalar particles are also dark matter candidates. (ref. 1) Scalar particles are only the first particles on that quantum spin list to model as dark matter. Reference: Wikipedia - Scalar field dark matter (ref. 1) https://en.wikipedia.org/wiki/Scalar_field_dark_matter

Negative, a scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number [math]N_n[/math]. (ref. 1), (ref. 2) [math]\begin{tabular}{l*{6}{c}r} bosons (b) = integer spin & Bose-Einstein statistics \\ fermions (f) = half-integer spin & Fermi-Dirac statistics \\ \end{tabular}[/math] [math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Symbolic identity key: [math]n_s[/math] - spin states total number [math]N_s[/math] - species total number [math]N_n[/math] - total effective degeneracy number If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math] If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math] Note that if each neutrino species has a corresponding supersymmetric sterile neutrino species, then [math]\boxed{N_n = N_s = 3}[/math]. Neutralinos and sterile neutrinos are hypothetical particles. (ref. 3), (ref. 4) The mass spectrum range for neutralinos is 10 GeV to 1 TeV. (ref. 3) The mass spectrum range for sterile neutrinos is 1 eV to 10^15 GeV. (ref. 4) Affirmative, theses particles are theoretically embedded into SO(10) MSSM models. (ref. 5), (ref. 6), (ref. 11) See PHYS: 652 Cosmic Inventory I: Radiation (ref. 9) and Formaggio - Relic Neutrinos - Institute for Nuclear Theory (ref. 10) and Orion1 - Dark matter scalar particle radiation energy density Bose-Einstein distribution (ref. 12) and Orion1 - Photon radiation energy density Bose-Einstein distribution (ref. 13) in Reference. Do you have a citation reference link for the equation derivation for that number? Reference: Wikipedia - Scalar boson (ref. 1) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Fermion (ref. 2) https://en.wikipedia.org/wiki/Fermion Wikipedia - Neutralino - Origins in supersymmetric theories (ref. 3) https://en.wikipedia.org/wiki/Neutralino#Origins_in_supersymmetric_theories Wikipedia - Sterile neutrinos (ref. 4) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Standard Model - Total particle count (ref. 5) https://en.wikipedia.org/wiki/Standard_Model#Total_particle_count Wikipedia - SO[10] (ref. 6) https://en.wikipedia.org/wiki/SO%2810%29 CERN - Next decade of sterile neutrino studies (ref. 7) http://arxiv.org/pdf/1306.4954v1.pdf Wikipedia - Dark matter (ref. 8) https://en.wikipedia.org/wiki/Dark_matter PHYS: 652 Cosmic Inventory I: Radiation (ref. 9) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Formaggio - Relic Neutrinos - Institute for Nuclear Theory (ref. 10) http://www.int.washington.edu/talks/WorkShops/int_10_44W/People/Formaggio_J/Formaggio.pdf WIkipedia - Minimal supersymmetric standard model - dark matter (ref. 11) https://en.wikipedia.org/wiki/Minimal_Supersymmetric_Standard_Model#Dark_matter Orion1 - Dark matter scalar particle radiation energy density Bose-Einstein distribution (ref. 12) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/?p=887017 Orion1 - Photon radiation energy density Bose-Einstein distribution (ref. 13) http://www.scienceforums.net/topic/86694-observable-universe-mass/?p=884367

Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] Dark matter scalar particle species total effective degeneracy number: (ref. 2) [math]\boxed{N_{\phi} = 1}[/math] Dark matter scalar particle radiation temperature is equivalent to neutrino cosmic background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Bose-Einstein total dark matter scalar particle distribution constant: (ref. 3) [math]C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5} = 3.640 \cdot 10^{-4}[/math] [math]\boxed{C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] Observable Universe total mass: [math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number: (ref. 3) [math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\phi} = T_{\nu}[/math] Observable Universe total dark matter scalar particle number: [math]N_{\phi} = \Omega_{\phi} \left( \frac{M_u}{m_{\phi}} \right) = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{N_{\phi} = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{N_{\phi} = 1.544 \cdot 10^{93} \; \phi \; \text{particles}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Orion 1 - total effective degeneracy number: (ref. 2) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233 Orion1 - dark matter scalar particle composition: (ref. 3) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry887017 Dark matter - Wikipedia https://en.wikipedia.org/wiki/Dark_matter