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Orion1

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  1. Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/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 [math]\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}[/math] [math]\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}[/math] [math]\;[/math] [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] Total stellar class number: (ref. 2) [math]n_c = 7[/math] key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M [math]\Omega_f[/math] - main sequence stars stellar class fraction [math]N_s[/math] - total observable stellar number [math]M_s[/math] - Main-sequence mass [math]\;[/math] Observable Universe average stellar mass: [math]M_{as} = \frac{1}{N_s} \sum_{n = 1}^{n_c} \Omega_f\left(n\right) N_s M_s\left(n\right) = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right) = 0.219 \cdot M_{\odot} \rightarrow 0.595 \cdot M_{\odot}[/math] [math]\boxed{M_{as} = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)} \; \; \; n_c = 7[/math] [math]\boxed{M_{as} = \left(0.219 \rightarrow 0.595\right) \cdot M_{\odot}}[/math] Observable Universe average stellar mass upper bound limit: [math]\boxed{M_{as} = 1.184 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 3, pg. 20) [math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]M_{as} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] Observable Universe stellar baryon composition: (ref. 4, pg. 3) [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] [math]---[/math] Milky Way galaxy mass: (ref. 5, 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] Redshift parameter at photon decoupling time: [math]z = 1090.43[/math] [math]\;[/math] Cosmic photon background radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] [math]\;[/math] Cosmic photon background radiation temperature at photon decoupling time: [math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math] [math]\;[/math] Cosmic neutrino background radiation temperature at present time: (ref. 6, pg. 44, eq. 220) [math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math] [math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math] [math]\;[/math] Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 7) [math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math] [math]\;[/math] Observable Universe dark matter scalar particle temperature is equivalent to cosmic neutrino background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] [math]\;[/math] Observable Universe dark matter scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation decoupling temperature: [math]\boxed{T_{\phi,t} = T_{\nu,t}}[/math] [math]\;[/math] Cosmic scalar particle dark energy background radiation temperature: [math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math] [math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math] [math]\;[/math] Observable Universe Cosmological Constant: [math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] [math]\;[/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 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] [math]\;[/math] Observable Universe scalar particle dark energy Cosmological Constant: [math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7}}[/math] [math]\;[/math] Observable Universe scalar particle dark energy Cosmological Constant: [math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math] [math]\;[/math] Dark matter density: [math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math] Baryonic density: [math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math] Dark energy density: [math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math] [math]\;[/math] Bose-Einstein total dark matter scalar particle distribution constant: [math]\boxed{C_{\phi} = \frac{4 G N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] [math]\;[/math] Fermi-Dirac total neutrino distribution constant: [math]\boxed{C_{\nu} = \frac{4 G N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/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 and Fermi-Dirac baryonic photon and neutrino matter density: [math]\boxed{\rho_{b} = \frac{N_{\gamma} \pi^2 \left(k_B T_{\gamma}\right)^4}{30 \hbar^3 c^5} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) + \frac{N_{\nu} \pi^2 \left(k_B T_{\nu}\right)^4}{2 C_{\nu} \hbar^3 c^5} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right)}[/math] [math]\;[/math] Bose-Einstein scalar particle dark energy density: [math]\boxed{\rho_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{30 \hbar^3 c^5}}[/math] [math]\;[/math] Observable Universe total scalar particle dark matter and photon and neutrino co-moving volumes: [math]V_{\phi} = \frac{4 \pi R_{\phi}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]\right)^3[/math] [math]V_{\gamma} = \frac{4 \pi R_{\gamma}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\right)^3[/math] [math]V_{\nu} = \frac{4 \pi R_{\nu}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]\right)^3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy and neutrino total co-moving volumes are equivalent: [math]\boxed{V_{\phi} = V_{\Lambda} = V_{\nu}}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter total mass: [math]\boxed{\rho_{\phi} V_{\phi} = \frac{N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{12 C_{\phi} c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] [math]\;[/math] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter relative composition total mass: [math]\boxed{\rho_{b} V_{b} = \frac{N_{\gamma} \pi^3 \left(k_B T_{\gamma}\right)^4}{180 c^2 \left(\hbar H_0\right)^3} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{12 C_{\nu} c^2 \left(\hbar H_0\right)^3} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] [math]\;[/math] Bose-Einstein scalar particle dark energy total mass: [math]\boxed{\rho_{\Lambda} V_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{180 c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] [math]\;[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] [math]\;[/math] Observable Universe total critical mass: [math]M_{c} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{dm} V_{dm} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_c = \frac{c^3}{16 G H_0} \left(\left(\Omega_{dm} + \Omega_{\Lambda}\right) \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \Omega_b \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3\right)}[/math] [math]\boxed{M_c = 1.179 \cdot 10^{56} \; \text{kg}}[/math] [math]\;[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{\phi} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] [math]\;[/math] Observable Universe scalar particle dark matter and scalar particle dark energy composition total mass: [math]M_{u} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{\phi} V_{\phi} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\;[/math] [math]\boxed{M_u = \frac{\pi^3 k_B^4}{12 c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\boxed{M_u = 1.179 \cdot 10^{56} \; \text{kg}}[/math] [math]\;[/math] Observable Universe total energy: [math]E_u = M_u c^2[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] Observable Universe total stellar number: [math]\boxed{N_s = \frac{\Omega_s \pi^3 k_B^4}{12 M_{as} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\;[/math] Observable Universe total stellar number: [math]\boxed{N_s = 2.453 \cdot 10^{23} \; \text{stars}}[/math] [math]\;[/math] Wikipedia observable universe total stellar number: (ref. 8) [math]N_s = 3.000 \cdot 10^{23} \; \text{stars}[/math] [math]\;[/math] Observable Universe total galaxy number: [math]\boxed{N_g = \frac{\Omega_b \pi^3 k_B^4}{12 M_{mw} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\;[/math] Observable Universe total galaxy number: [math]\boxed{N_g = 2.330 \cdot 10^{12} \; \text{galaxies}}[/math] [math]\;[/math] Wikipedia observable universe total galaxy number: (ref. 9) [math]N_g = 2.000 \cdot 10^{12} \; \text{galaxies}[/math] [math]\;[/math] Observable Universe stars per galaxy average number: [math]\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\boxed{\frac{N_s}{N_g} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}}[/math] [math]\;[/math] Wikipedia stars per galaxy average number: (ref. 8, ref. 9) [math]\frac{N_s}{N_g} = 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total stellar number: (ref. 10) [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] Observable Universe planetary composition: (ref. 4, pg. 3) [math]\Omega_p = 1 \cdot 10^{-6}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 10, ref. 11) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] [math]\;[/math] Observable Universe average planetary mass: [math]\boxed{M_{ap} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{N_g}{N_p}\right)}[/math] [math]\boxed{M_{ap} = \left(52.974 \rightarrow 84.758\right) \cdot M_{\oplus}}[/math] [math]\;[/math] Solar star system average planetary mass: (ref. 12) [math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{M_{ap} = 55.855 \cdot M_{\oplus}}[/math] [math]\;[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{\frac{N_p}{N_g} = 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}}}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 10, ref. 11) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] [math]\;[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\;[/math] Observable Universe stars per galaxy average number: [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\;[/math] Observable Universe average stellar mass: [math]\boxed{M_{as} = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)} \; \; \; n_c = 7[/math] [math]\;[/math] Observable Universe planets per star average number based upon solar star system integration via substitution: [math]\frac{N_p}{N_s} = \left(\frac{N_p}{N_g}\right)\left(\frac{N_g}{N_s}\right) = \left[\frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}\right]\left(\frac{\Omega_b M_{as}}{\Omega_s M_{mw}}\right)[/math] [math]\frac{N_p}{N_s} = \left[\frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}\right]\left(\frac{\Omega_b}{\Omega_s M_{mw}}\right)\left(\sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)\right) = \frac{\Omega_p}{\Omega_s} \left(\sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)\right)\left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}[/math] [math]\;[/math] Observable Universe planets per star average number based upon solar star system: [math]\boxed{\frac{N_p}{N_s} = \frac{\Omega_p}{\Omega_s} \left(\sum_{n = 1}^{n_c} \Omega_{f}\left(n\right) M_{s}\left(n\right)\right)\left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_{p}\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_c = 7, n_p = 8[/math] [math]\;[/math] [math]\boxed{\frac{N_p}{N_s} = 1.443 \; \frac{\text{planets}}{\text{star}}}[/math] [math]\;[/math] Wikipedia planets per star average number: [math]\boxed{\frac{N_p}{N_s} = \left(0.667 \rightarrow 1.067\right) \; \frac{\text{planets}}{\text{star}}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Is there anything else that you want to see quantified based upon this model for observational comparison? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - Stellar classification - Harvard spectral classification: (ref. 2) https://en.wikipedia.org/wiki/Stellar_classification Harvard_spectral_classification (insert number symbol) 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 Mass models of the Milky Way: (ref. 5) http://arxiv.org/pdf/1102.4340v1 PHYS: 652 Cosmic Inventory I: Radiation: (ref. 6) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Wikipedia - neutrino decoupling: (ref. 7) https://en.wikipedia.org/wiki/Neutrino_decoupling Wikipedia - Observable universe total stellar number: (ref. 8) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Galaxy: (ref. 9) https://en.wikipedia.org/wiki/Galaxy Wikipedia - Milky Way Galaxy: (ref. 10) https://en.wikipedia.org/wiki/Milky_Way Space.com - 160 Billion Alien Planets May Exist in Our Milky Way Galaxy: (ref. 11) https://www.space.com/14200-160-billion-alien-planets-milky-galaxy.html Wikipedia - Planetary mass: (ref. 12) https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris (source code errors corrected on 01-18-2019)
  2. Planck satellite cosmological parameters: (ref. 1) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] [math]\;[/math] Planck mass: (ref. 2) [math]m_{P} = \sqrt{\frac{\hbar c}{G}}[/math] [math]\;[/math] Planck temperature (ref. 3) [math]T_{P} = \frac{E_{P}}{k_{B}} = \frac{m_{P} c^{2}}{k_{B}} = \left(\sqrt{\frac{\hbar c}{G}}\right) \frac{c^2}{k_{B}} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}} = 1.417 \cdot 10^{32} \; \text{K}[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\boxed{T_{P} = 1.417 \cdot 10^{32} \; \text{K}}[/math] [math]\;[/math] Planck radius: (ref. 4) [math]r_P = \sqrt{\frac{\hbar G}{c^{3}}}[/math] [math]\;[/math] Planck volume: (ref. 5) [math]V_{P} = \frac{4 \pi R_{P}^{3}}{3} = \frac{4 \pi}{3} \left(\sqrt{\frac{\hbar G}{c^{3}}}\right)^{3} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}} = 1.768 \cdot 10^{-104} \; \text{m}^3[/math] [math]\;[/math] [math]\boxed{V_{P} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}}}[/math] [math]\boxed{V_{P} = 1.768 \cdot 10^{-104} \; \text{m}^3}[/math] [math]\;[/math] Observable Universe total energy at present time: [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] All particle peak radiation temperatures are equivalent to Planck temperature at Planck time: [math]\boxed{T_{\phi,t_P} = T_{\gamma,t_P} = T_{\nu,t_P} = T_P}[/math] Dark energy scalar particle peak radiation temperature is non-zero at Planck time: [math]\boxed{T_{\Lambda,t_P} \neq 0}[/math] [math]\;[/math] Present time radial metric decoupling temperature redshift parameter global quantizations remain the same: [math]\boxed{\ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right] \; \; \; \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \; \; \; \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]}[/math] [math]\;[/math] Observable Universe total energy at Planck time integration via substitution: [math]E_{u}\left(T_P\right) = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}\right)^{4} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] Observable Universe total energy at Planck time: [math]\boxed{E_{u}\left(T_P\right) = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_{u}\left(T_P\right) = 8.439 \cdot 10^{199} \; \text{j}}[/math] [math]\;[/math] Observable Universe total mass at Planck time: [math]M_{u}\left(T_P\right) = \frac{E_{u}\left(T_P\right)}{c^{2}} = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\boxed{M_{u}\left(T_P\right) = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_{u}\left(T_P\right) = 9.390 \cdot 10^{182} \; \text{kg}}[/math] [math]\;[/math] Observable Universe total energy density at Planck time integration via substitution: [math]\epsilon_{u}\left(T_{P}\right) = \frac{E_{u}\left(T_{P}\right)}{V_{P}} = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] Observable Universe total energy density at Planck time: [math]\boxed{\epsilon_{u}\left(T_{P}\right) = \frac{\pi^{2}}{16 H_{0}^{3}} \sqrt{\frac{c^{29}}{\hbar^{5} G^{7}}} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\;[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = 4.772 \cdot 10^{303} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck energy density integration via substitution: [math]\epsilon_{P} = \frac{E_{P}}{V_{P}} = \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)\left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) = \frac{3 c^{7}}{4 \pi \hbar G^{2}} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\;[/math] Planck energy density: (ref. 5) [math]\boxed{\epsilon_{P} = \frac{3 c^{7}}{4 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{P} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck vacuum energy density at Planck time: (ref. 6) [math]\epsilon_{vac} = \frac{\epsilon_{P}}{2} = \frac{3 c^{7}}{8 \pi \hbar G^{2}} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\boxed{\epsilon_{vac} = \frac{3 c^{7}}{8 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{vac} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck time: (ref. 7) [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] [math]\;[/math] Particle interaction rate is equivalent to Hubble Parameter at Planck interaction time: [math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math] [math]\Gamma_{P} = H_{P} = \frac{}{t_{P}} = \sqrt{\frac{c^{5}}{\hbar G}} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}[/math] Planck particle interaction rate at Planck interaction time: [math]\boxed{\Gamma_{P} = \sqrt{\frac{c^{5}}{\hbar G}}}[/math] [math]\;[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\;[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 2 \left(k_B T_{\phi,t_P}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Dark matter scalar particle interaction rate at scalar particle Planck time integration via substitution: [math]\Gamma_{\phi,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = \frac{2 \hbar c^{5}}{G} \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4 \left(k_B T_{\gamma,t_P}\right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Photon interaction rate at photon Planck time integration via substitution: [math]\Gamma_{\gamma,t_P} = \frac{4}{3} \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}[/math] [math]\;[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 2 \left(k_B T_{\nu,t_P}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time integration via substitution: [math]\Gamma_{\nu,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle interaction rate summary: Planck particle interaction rate at Planck interaction time: [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Planck particle strong decoupling time: (ref. 7) [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] [math]\;[/math] Dark matter scalar particle strong decoupling time: [math]\boxed{t_{\phi,t} = \frac{}{2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}{N_{\phi} \pi^3 c^{5}}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{t_{\phi,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Photon particle strong decoupling time: [math]\boxed{t_{\gamma,t} = \frac{3}{4} \sqrt{\frac{5 \Omega_{\gamma,t_P} \hbar G}{N_{\gamma} \pi^3 c^{5}}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{t_{\gamma,t} = 8.128 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Cosmic neutrino background radiation particle strong decoupling time: [math]\boxed{t_{\nu,t} = \frac{}{2} \sqrt{ \frac{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}{N_{\nu} \pi^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{t_{\nu,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle strong decoupling time summary: Planck particle strong decoupling time: [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] Dark matter scalar particle strong decoupling time: [math]\boxed{t_{\phi,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] Photon particle strong decoupling time: [math]\boxed{t_{\gamma,t} = 8.128 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\gamma} = 0[/math] Cosmic neutrino background radiation particle strong decoupling time: [math]\boxed{t_{\nu,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Is the Planck energy density the maximum energy density limit in the universe? [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 Wikipedia - Planck mass: (ref. 2) https://en.wikipedia.org/wiki/Planck_mass Wikipedia - Planck temperature: (ref. 3) https://en.wikipedia.org/wiki/Planck_temperature Wikipedia - Planck radius: (ref. 4) https://en.wikipedia.org/wiki/Planck_length Wikipedia - Planck volume: (ref. 5) https://en.wikipedia.org/wiki/Planck_units#Derived_units Wikipedia - Vacuum_energy: (ref. 6) https://en.wikipedia.org/wiki/Vacuum_energy Wikipedia - Planck time: (ref. 7) https://en.wikipedia.org/wiki/Planck_time (source code errors corrected on 12-30-2018)
  3. [math]\text{Planck satellite cosmological parameters:} \; \left(\text{ref. 1}\right)[/math] [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle composition is equivalent to dark matter composition:}[/math] [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] [math]\;[/math] [math]\text{Planck mass:} \; \left(\text{ref. 2}\right)[/math] [math]m_{P} = \sqrt{\frac{\hbar c}{G}}[/math] [math]\;[/math] [math]\text{Planck temperature} \; \left(\text{ref. 3}\right)[/math] [math]T_{P} = \frac{E_{P}}{k_{B}} = \frac{m_{P} c^{2}}{k_{B}} = \left(\sqrt{\frac{\hbar c}{G}}\right) \frac{c^2}{k_{B}} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}} = 1.417 \cdot 10^{32} \; \text{K}[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\boxed{T_{P} = 1.417 \cdot 10^{32} \; \text{K}}[/math] [math]\;[/math] [math]\text{Planck radius:} \; \left(\text{ref. 4}\right)[/math] [math]r_P = \sqrt{\frac{\hbar G}{c^{3}}}[/math] [math]\;[/math] [math]\text{Planck volume:} \; \left(\text{ref. 5}\right)[/math] [math]V_{P} = \frac{4 \pi R_{P}^{3}}{3} = \frac{4 \pi}{3} \left(\sqrt{\frac{\hbar G}{c^{3}}}\right)^{3} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}} = 1.768 \cdot 10^{-104} \; \text{m}^3[/math] [math]\;[/math] [math]\boxed{V_{P} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}}}[/math] [math]\boxed{V_{P} = 1.768 \cdot 10^{-104} \; \text{m}^3}[/math] [math]\;[/math] [math]\text{Observable Universe total energy at present time:}[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] [math]\text{All particle peak radiation temperatures are equivalent to Planck temperature at Planck time:}[/math] [math]\boxed{T_{\phi,t_P} = T_{\gamma,t_P} = T_{\nu,t_P} = T_P}[/math] [math]\text{Dark energy scalar particle peak radiation temperature is non-zero at Planck time:}[/math] [math]\boxed{T_{\Lambda,t_P} \neq 0}[/math] [math]\;[/math] [math]\text{Present time radial metric decoupling temperature redshift parameter global quantizations remain the same:}[/math] [math]\boxed{\ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right] \; \; \; \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \; \; \; \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]}[/math] [math]\;[/math] [math]\text{Observable Universe total energy at Planck time integration via substitution:}[/math] [math]E_{u}\left(T_P\right) = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}\right)^{4} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] [math]\text{Observable Universe total energy at Planck time:}[/math] [math]\boxed{E_{u}\left(T_P\right) = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_{u}\left(T_P\right) = 8.439 \cdot 10^{199} \; \text{j}}[/math] [math]\;[/math] [math]\text{Observable Universe total mass at Planck time:}[/math] [math]M_{u}\left(T_P\right) = \frac{E_{u}\left(T_P\right)}{c^{2}} = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\boxed{M_{u}\left(T_P\right) = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_{u}\left(T_P\right) = 9.390 \cdot 10^{182} \; \text{kg}}[/math] [math]\;[/math] [math]\text{Observable Universe total energy density at Planck time integration via substitution:}[/math] [math]\epsilon_{u}\left(T_{P}\right) = \frac{E_{u}\left(T_{P}\right)}{V_{P}} = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] [math]\text{Observable Universe total energy density at Planck time:}[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = \frac{\pi^{2}}{16 H_{0}^{3}} \sqrt{\frac{c^{29}}{\hbar^{5} G^{7}}} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\;[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = 4.772 \cdot 10^{303} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Planck energy density integration via substitution:}[/math] [math]\epsilon_{P} = \frac{E_{P}}{V_{P}} = \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)\left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) = \frac{3 c^{7}}{4 \pi \hbar G^{2}} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\;[/math] [math]\text{Planck energy density:} \; \left(\text{ref. 5}\right)[/math] [math]\boxed{\epsilon_{P} = \frac{3 c^{7}}{4 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{P} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Vacuum energy density at Planck time:} \; \left(\text{ref. 6}\right)[/math] [math]\epsilon_{vac} = \frac{\epsilon_{P}}{2} = \frac{3 c^{7}}{8 \pi \hbar G^{2}} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\boxed{\epsilon_{vac} = \frac{3 c^{7}}{8 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{vac} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Planck time:} \; \left(\text{ref. 7}\right)[/math] [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\;[/math] [math]\text{Particle interaction rate is equivalent to Hubble Parameter at Planck interaction time:}[/math] [math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math] [math]\Gamma_{P} = H_{P} = \frac{}{t_{P}} = \sqrt{\frac{c^{5}}{\hbar G}} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}[/math] [math]\text{Planck particle interaction rate at Planck interaction time:}[/math] [math]\boxed{\Gamma_{P} = \sqrt{\frac{c^{5}}{\hbar G}}}[/math] [math]\;[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 2 \left(k_B T_{\phi,t_P}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time integration via substitution:}[/math] [math]\Gamma_{\phi,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = \frac{2 \hbar c^{5}}{G} \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4 \left(k_B T_{\gamma,t_P}\right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time integration via substitution:}[/math] [math]\Gamma_{\gamma,t_P} = \frac{4}{3} \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 2 \left(k_B T_{\nu,t_P}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time integration via substitution:}[/math] [math]\Gamma_{\nu,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Particle interaction rate summary:}[/math] [math]\text{Planck particle interaction rate at Planck interaction time:}[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Is the Planck energy density the maximum energy density limit in the universe?}[/math] [math]\;[/math] [math]\text{Any discussions and/or peer reviews about this specific topic thread?}[/math] [math]\;[/math] [math]\text{Reference:}[/math] Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - Planck mass: (ref. 2) https://en.wikipedia.org/wiki/Planck_mass Wikipedia - Planck temperature: (ref. 3) https://en.wikipedia.org/wiki/Planck_temperature Wikipedia - Planck radius: (ref. 4) https://en.wikipedia.org/wiki/Planck_length Wikipedia - Planck volume: (ref. 5) https://en.wikipedia.org/wiki/Planck_units#Derived_units Wikipedia - Vacuum_energy: (ref. 6) https://en.wikipedia.org/wiki/Vacuum_energy Wikipedia - Planck time: (ref. 7)
  4. Orion1

    Latex test

    \[ F(x) = \int_{0}^{\infty} f(x) dx \] \[ F(x) = \int_{0}^{\infty} f(x) dx \]
  5. This is true for the Compton equation, where a mass-less particle is scattering from a mass particle. However, in equation [math](6)[/math], the energy terms for [math]E[/math] represent the total energy of a mass particle, which includes its rest mass plus kinetic energy. I should have been more clear about that equation description and will include the total energy description in the next revision, hence peer review. If I understand this inelastic scattering correctly, when a lighter particle scatters from a heavier particle, as total kinetic energy is increased, then more kinetic energy is absorbed by the recoiling heavier particle and less kinetic energy is carried away by the lighter particle. So, the limit of equations [math](12)[/math] and [math](13)[/math] as kinetic energy approaches infinity, should be the rest mass of the neutrino. [math]\lim_{E_k \to \infty} m_{\nu} = m_{\nu,0}[/math]
  6. Orion1

    Latex test

    This is some centered text
  7. Derivation of neutrino mass from neutrino scattering: [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]p_{e} \sin \phi = p_{f} \sin \theta \tag{1}[/math] [math]p_{e} \cos \phi + p_{f} \cos \theta = p_{i} \tag{2}[/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] Divide equation [math](1)[/math] by equation [math](3)[/math] for an expression for [math]\tan \phi[/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] 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, noting that [math]p = \frac{E}{c}[/math]. [math]\lambda_{f} - \lambda_{i} = \frac{h}{m_{e} c} (1 - \cos \theta) \tag{5}[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} = 1 + \frac{E_{\nu}}{E_{e}} (1 - \cos \theta) = 1 + \frac{m_{\nu} c^2}{m_{e} c^2} (1 - \cos \theta) = 1 + \frac{m_{\nu}}{m_{e}} (1 - \cos \theta) \tag{6}[/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]\tan \phi = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{m_{\nu}}{m_{e}} (1 - \cos \theta) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{m_{\nu}}{m_{e}} \right)(1 - \cos \theta)} \tag{7}[/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] 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] 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]\frac{m_{\nu}}{m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/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] 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]m_{n}[/math] - nuclear mass Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{(m_{e} + m_{\nu}) \tan \phi}{m_{e}} \right)} \tag{14}[/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] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{(m_{n} + m_{\nu}) \tan \phi}{m_{n}} \right)} \tag{16}[/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] Any discussions and/or peer reviews about this specific topic thread? 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
  8. Orion1

    Latex test

    [math]F(x) = \int_{0}^{\infty} f(x) dx[/math] [math]\boxed{F(x) = \int_{0}^{\infty} f(x) dx} \tag{1}[/math] [math]\text{Equation} (1) \tag{}[/math].
  9. Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/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 [math]\text{if} \; n_{s} \geq N_{s} \; \text{then} \; N_{n} = n_{s}[/math] [math]\text{if} \; n_{s} \leq N_{s} \; \text{then} \; N_{n} = N_{s}[/math] [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] 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_{as} = \frac{1}{N_s} \sum_{n_c = 1}^{7} \left(\Omega_n N_s M_n\right) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math] [math]\boxed{M_{as} = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math] [math]\boxed{M_{as} = \left(0.595 \rightarrow 0.769 \right) \cdot M_{\odot}}[/math] Observable Universe average stellar mass lower bound limit: [math]\boxed{M_{as} = 1.183 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 3, pg. 20) [math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]M_{as} = 1.193 \cdot 10^{30} \; \text{kg}[/math] Observable Universe stellar baryon composition: (ref. 4, pg. 3) [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] --- Milky Way galaxy mass: (ref. 5, 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] --- Redshift parameter at photon decoupling time: [math]z = 1090.43[/math] Cosmic photon background radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Cosmic photon background radiation temperature at photon decoupling time: [math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math] Cosmic neutrino background radiation temperature at present time: (ref. 6, pg. 44, eq. 220) [math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math] [math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math] Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 7) [math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe dark matter scalar particle temperature is equivalent to cosmic neutrino background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Cosmic scalar particle dark energy background radiation temperature: [math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math] [math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math] Observable Universe Cosmological Constant: [math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/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 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^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 \left(k_B T_{\Lambda}\right)^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] Dark matter density: [math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math] Baryonic density: [math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math] Dark energy density: [math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] Bose-Einstein total dark matter scalar particle distribution constant: [math]\boxed{C_{\phi} = \frac{4 G N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] Fermi-Dirac total neutrino distribution constant: [math]\boxed{C_{\nu} = \frac{4 G N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/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] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter density: [math]\boxed{\rho_{b} = \frac{N_{\gamma} \pi^2 \left(k_B T_{\gamma}\right)^4}{30 \hbar^3 c^5} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) + \frac{N_{\nu} \pi^2 \left(k_B T_{\nu}\right)^4}{2 C_{\nu} \hbar^3 c^5} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right)}[/math] Bose-Einstein scalar particle dark energy density: [math]\boxed{\rho_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{30 \hbar^3 c^5}}[/math] Observable Universe total photon and neutrino co-moving volumes: [math]V_{\gamma} = \frac{4 \pi R_{\gamma}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\right)^3[/math] [math]V_{\nu} = \frac{4 \pi R_{\nu}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]\right)^3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy and neutrino total co-moving volumes are equivalent: [math]\boxed{V_{\phi} = V_{\Lambda} = V_{\nu}}[/math] Bose-Einstein scalar particle dark matter total mass: [math]\boxed{\rho_{\phi} V_{\phi} = \frac{N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{12 C_{\phi} c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter relative composition total mass: [math]\boxed{\rho_{b} V_{b} = \frac{N_{\gamma} \pi^3 \left(k_B T_{\gamma}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{12 C_{\nu} \left(\hbar H_0\right)^3 c^2} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] Bose-Einstein scalar particle dark energy total mass: [math]\boxed{\rho_{\Lambda} V_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] Observable Universe total critical mass: [math]M_{c} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{dm} V_{dm} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_c = \frac{c^3}{16 G H_0} \left(\left(\Omega_{dm} + \Omega_{\Lambda}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \Omega_b \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3\right)}[/math] [math]\boxed{M_c = 1.179 \cdot 10^{56} \; \text{kg}}[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{\phi} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy composition total mass: [math]M_{u} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{\phi} V_{\phi} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_u = \frac{\pi^3 k_B^4}{12 c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_u = 1.179 \cdot 10^{56} \; \text{kg}}[/math] Observable Universe total energy: [math]E_u = M_u c^2[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] Observable Universe total observable stellar number: [math]\boxed{N_s = \frac{\Omega_s \pi^3 k_B^4}{12 M_{as} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] Observable Universe total stellar number: [math]\boxed{N_s = 2.453 \cdot 10^{23} \; \text{stars}}[/math] Wikipedia observable universe total stellar number: (ref. 8) [math]N_s = 3.000 \cdot 10^{23} \; \text{stars}[/math] Observable Universe total galaxy number: [math]\boxed{N_g = \frac{\Omega_b \pi^3 k_B^4}{12 M_{mw} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] Observable Universe total galaxy number: [math]\boxed{N_g = 2.330 \cdot 10^{12} \; \text{galaxies}}[/math] Wikipedia observable universe total galaxy number: (ref. 9) [math]N_g = 2.000 \cdot 10^{12} \; \text{galaxies}[/math] Observable Universe stars per galaxy average number: [math]\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\boxed{\frac{N_s}{N_g} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}}[/math] Wikipedia stars per galaxy average number: (ref. 8, ref. 9) [math]\frac{N_s}{N_g} = 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] Wikipedia Milky Way galaxy total stellar number: (ref. 10) [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] Observable Universe planetary composition: (ref. 4, pg. 3) [math]\Omega_p = 1 \cdot 10^{-6}[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 10, ref. 11) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] Observable Universe average planetary mass: [math]\boxed{M_{ap} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{N_g}{N_p}\right)}[/math] [math]\boxed{M_{ap} = \left(52.974 \rightarrow 84.758\right) \cdot M_{\oplus}}[/math] Solar star system average planetary mass: (ref. 12) [math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_n} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{M_{ap} = 55.855 \cdot M_{\oplus}}[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_n\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{\frac{N_p}{N_g} = 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Is there anything else that you want to see quantified based upon this model for observational comparison? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf 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 Mass models of the Milky Way: (ref. 5) http://arxiv.org/pdf/1102.4340v1 PHYS: 652 Cosmic Inventory I: Radiation: (ref. 6) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Wikipedia - neutrino decoupling: (ref. 7) https://en.wikipedia.org/wiki/Neutrino_decoupling Wikipedia - Observable universe total observable stellar number: (ref. 8) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Galaxy: (ref. 9) https://en.wikipedia.org/wiki/Galaxy Wikipedia - Milky Way Galaxy: (ref. 10) https://en.wikipedia.org/wiki/Milky_Way Space.com - 160 Billion Alien Planets May Exist in Our Milky Way Galaxy: (ref. 11) https://www.space.com/14200-160-billion-alien-planets-milky-galaxy.html Wikipedia - Planetary mass: (ref. 12) https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris (source code errors corrected 05-19-2018)
  10. Orion1

    Latex test

    [math]F(x) = \int_{0}^{\infty} f(x) dx[/math] [math]\boxed{F(x) = \int_{0}^{\infty} f(x) dx}[/math] My pizza is arriving squashed also, no tip for the delivery driver! The boxed code appears to override whatever is causing it, but I do not want to box every equation, only my solutions.
  11. Photon species total effective degeneracy number: [math]\boxed{N_{\gamma} = 2}[/math] Photon radiation energy radiant emmittance Bose-Einstein distribution integration via substitution: [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_{\gamma} (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: [math]\boxed{\sigma_{\gamma} = \frac{N_{\gamma} \pi^2 k_B^4}{120 c^2 \hbar^3}}[/math] Radiant emmittance Stefan-Boltzmann law: [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: [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_{\gamma} (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] Planck's law: (ref. 1) [math]\boxed{I_{\gamma}(\nu,T_{\gamma}) = \frac{N_{\gamma} h \nu^3}{c^2 \left(e^{\frac{E_t}{E_{\gamma}}} - 1\right)}}[/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] --- The Stefan-Boltzmann law was first theorized in 1879, Planck's law was first theorized in 1914, I attempted to trace the derivation of the Stefan-Boltzmann law and Planck's law to determine the mathematical and theoretical origin of the number "2" in the numerator of Planck's law, however there was no formal derivation formulas published in their original papers. According to Planck, the specific intensity[math]\; K \;[/math]of a monochromatic plane polarized ray of frequency[math]\; \nu \;[/math]is: (ref. 4, pg. 168, eq. 274, 276) [math]K_{\nu}(\nu,T) = \frac{h \nu^3}{c^2} \frac{1}{e^{\frac{h \nu}{k_B T}} - 1}[/math] [math]E_{\lambda}(\lambda,T) = \frac{h c^2}{\lambda^5} \frac{1}{e^{\frac{h c}{\lambda k_B T}} - 1}[/math] Wikipedia Planck's laws: (ref. 1, ref. 5, pg. 22, eq. 1.51, 1.52) [math]B_{\nu}(\nu,T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{\frac{h \nu}{k_B T}} - 1}[/math] [math]B_{\lambda}(\lambda,T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{\frac{h c}{\lambda k_B T}} - 1}[/math] Note that the number "2" in the numerator was not published in Planck's original paper. These equations in Planck's original paper describe a scalar particle. According to the derivation above, the number "2" in the numerator corresponds to the photon species total effective degeneracy number[math]\; N_{\gamma}[/math]. According to Rybicki and Lightman, photons have two independent polarizations (two states per wave vector)[math]\; k[/math], corresponding to the density of states (the number of states per solid angle per volume per frequency): (ref. 5, pg. 20, eq 1.47) [math]\rho_{s} = \frac{2 \nu^2}{c^3}[/math] Is this the theoretical definition for the total effective degeneracy number[math]\; N_{\gamma} \;[/math]for photon radiation? [math]\boxed{\rho_{s} = \frac{N_{\gamma} \nu^2}{c^3}}[/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 Planck, Maxwell (1914) - The Theory of Heat Radiation: (ref. 4) https://archive.org/stream/theoryofheatradi00planrich?ref=ol#page/168/mode/2up Rybicki and Lightman (1979): (ref. 5) http://www.bartol.udel.edu/~owocki/phys633/RadProc-RybLightman.pdf
  12. 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 [math]\text{if} \; n_{s} \geq N_{s} \; \text{then} \; N_{n} = n_{s}[/math] [math]\text{if} \; n_{s} \leq N_{s} \; \text{then} \; N_{n} = N_{s}[/math] [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] Stellar class number parameters: (ref. 1) [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_{as} = \frac{1}{N_s} \sum_{n_c = 1}^{7} \left(\Omega_n N_s M_n\right) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math] [math]\boxed{M_{as} = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math] [math]\boxed{M_{as} = \left(0.595 \rightarrow 0.769 \right) \cdot M_{\odot}}[/math] Observable Universe average stellar mass lower bound limit: [math]\boxed{M_{as} = 1.183 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 2, pg. 20) [math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] Observable Universe stellar baryon density: (ref. 3, pg. 3) [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/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] --- Redshift parameter at photon decoupling time: [math]z = 1090.43[/math] Cosmic photon background radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Cosmic photon background radiation temperature at photon decoupling time: [math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math] Cosmic neutrino background radiation temperature at present time: (ref. 5, pg. 44, eq. 220) [math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math] [math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math] Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 6) [math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math] Observable Universe dark matter scalar particle temperature is equivalent to cosmic neutrino background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Cosmic scalar particle dark energy background radiation temperature: [math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math] [math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math] Observable Universe Cosmological Constant: [math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/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 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^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 \left(k_B T_{\Lambda}\right)^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] Dark matter density: [math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math] Baryonic density: [math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math] Dark energy density: [math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math] Bose-Einstein total dark matter scalar particle distribution constant: [math]\boxed{C_{\phi} = \frac{4 G N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] Fermi-Dirac total neutrino distribution constant: [math]\boxed{C_{\nu} = \frac{4 G N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/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] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter density: [math]\boxed{\rho_{b} = \frac{N_{\gamma} \pi^2 \left(k_B T_{\gamma}\right)^4}{30 \hbar^3 c^5} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) + \frac{N_{\nu} \pi^2 \left(k_B T_{\nu}\right)^4}{2 C_{\nu} \hbar^3 c^5} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right)}[/math] Bose-Einstein scalar particle dark energy density: [math]\boxed{\rho_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{30 \hbar^3 c^5}}[/math] Observable Universe total photon and neutrino co-moving volumes: [math]V_{\gamma} = \frac{4 \pi R_{\gamma}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\right)^3[/math] [math]V_{\nu} = \frac{4 \pi R_{\nu}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]\right)^3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy and neutrino total co-moving volumes are equivalent: [math]\boxed{V_{\phi} = V_{\Lambda} = V_{\nu}}[/math] Bose-Einstein scalar particle dark matter total mass: [math]\boxed{\rho_{\phi} V_{\phi} = \frac{N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{12 C_{\phi} c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter total mass: [math]\boxed{\rho_{b} V_{b} = \frac{N_{\gamma} \pi^3 \left(k_B T_{\gamma}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{12 C_{\nu} \hbar^3 c^2} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] Bose-Einstein scalar particle dark energy total mass: [math]\boxed{\rho_{\Lambda} V_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] Observable Universe total critical mass: [math]M_{c} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{dm} V_{dm} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_c = \frac{c^3}{16 G H_0} \left(\left(\Omega_{dm} + \Omega_{\Lambda}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \Omega_b \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3\right)}[/math] [math]\boxed{M_c = 1.179 \cdot 10^{56} \; \text{kg}}[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{\phi} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy total mass: [math]M_{u} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{\phi} V_{\phi} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_u = \frac{\pi^3 k_B^4}{12 c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_u = 1.179 \cdot 10^{56} \; \text{kg}}[/math] Observable Universe total energy: [math]E_u = M_u c^2[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] Observable Universe total observable stellar number: [math]\boxed{N_s = \frac{\Omega_s \pi^3 k_B^4}{12 M_{as} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] Observable Universe total stellar number: [math]\boxed{N_s = 2.453 \cdot 10^{23} \; \text{stars}}[/math] Wikipedia observable universe total stellar number: (ref. 7) [math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math] Observable Universe total galaxy number: [math]\boxed{N_g = \frac{\Omega_b \pi^3 k_B^4}{12 M_{mw} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] Observable Universe total galaxy number: [math]\boxed{N_g = 2.330 \cdot 10^{12} \; \text{galaxies}}[/math] Wikipedia observable universe total galaxy number: (ref. 8) [math]N_g = 2.000 \cdot 10^{12} \; \text{galaxies}[/math] Observable Universe stars per galaxy average number: [math]\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\boxed{\frac{N_s}{N_g} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}}[/math] Wikipedia Milky Way galaxy total stellar number: (ref. 9) [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] Observable Universe planetary composition: (ref. 3, pg. 3) [math]\Omega_p = 1 \cdot 10^{-6}[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 9, ref. 10) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] Observable Universe average planetary mass: [math]\boxed{M_{ap} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{N_g}{N_p}\right)}[/math] [math]\boxed{M_{ap} = \left(52.974 \rightarrow 84.758\right) \cdot M_{\oplus}}[/math] Solar star system average planetary mass: (ref. 11) [math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_n} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{M_{ap} = 55.855 \cdot M_{\oplus}}[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_n\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{\frac{N_p}{N_g} = 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}}}[/math] Any discussions and/or peer reviews about this specific topic thread? Is there anything else that you want to see quantified based upon this model for observational comparison? Reference: Wikipedia - Stellar classification - Harvard spectral classification: (ref. 1) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification On The Mass Distribution Of Stars...: (ref. 2) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf The Cosmic Energy Inventory: (ref. 3) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 PHYS: 652 Cosmic Inventory I: Radiation: (ref. 5) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Wikipedia - neutrino decoupling: (ref. 6) https://en.wikipedia.org/wiki/Neutrino_decoupling Wikipedia - Observable universe total observable stellar number: (ref. 7) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Galaxy: (ref. 8) https://en.wikipedia.org/wiki/Galaxy Wikipedia - Milky Way Galaxy: (ref. 9) https://en.wikipedia.org/wiki/Milky_Way Space.com - 160 Billion Alien Planets May Exist in Our Milky Way Galaxy: (ref. 10) https://www.space.com/14200-160-billion-alien-planets-milky-galaxy.html Wikipedia - Planetary mass: (ref. 11) https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris
  13. WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{dm,t} = 0.63[/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] Scalar particle temperature is equivalent to cosmic neutrino background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation temperature at neutrino decoupling time: [math]\boxed{T_{\phi,t} = T_{\nu,t}}[/math] Observable Universe total scalar particle co-moving radius at present time: [math]\boxed{R_{\phi} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right]}[/math] [math]\boxed{R_{\phi} = 1.472 \cdot 10^{27} \; \text{m}} \; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math] Observable Universe total scalar particle co-moving radius at past time integration via substitution: [math]R_{\phi,t} = R_{\phi} \left( \frac{T_{\phi}}{T_{\phi,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) = 2.864 \cdot 10^{17} \; \text{m} \; \; \; (30.277 \; \text{ly})[/math] Observable Universe total scalar particle co-moving radius at past time: [math]\boxed{R_{\phi,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right)}[/math] [math]\boxed{R_{\phi,t} = 2.864 \cdot 10^{17} \; \text{m}} \; \; \; (30.277 \; \text{ly})[/math] Bose-Einstein scalar particle decoupling time: [math]T_{u,\phi} = \frac{1}{H_{\phi,t}} = \frac{}{2 (k_B T_{\phi,t})^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 (k_B T_{\phi,t})^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]\Lambda[/math]CDM universe model semiemperical 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 semiemperical 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}}[/math] Observable Universe expansion rate at scalar particle decoupling time integration via substitution: [math]\frac{dr}{dt} = \frac{R_{\phi,t}}{T_{u,\phi}} = \left[ \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) \right] 2 (k_B T_{\phi,t})^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 1.124 \cdot 10^{19} \; \frac{\text{m}}{\text{s}} \; \; \; (1187.8 \; \frac{\text{ly}}{\text{s}})[/math] Observable Universe expansion rate at scalar particle decoupling time: [math]\boxed{\frac{dr}{dt} = \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} \left( \frac{c}{H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) (k_B T_{\phi,t})^2}[/math] [math]\boxed{\frac{dr}{dt} = 1.124 \cdot 10^{19} \; \frac{\text{m}}{\text{s}}} \; \; \; (1187.8 \; \frac{\text{ly}}{\text{s}})[/math] Is it possible for dark matter cosmic scalar particle background radiation to have inflated to a distance of 156 billion light-years? Did the universe have a co-moving radius of 30 light-years at scalar particle decoupling time? Did the universe inflate and expand at a rate of 1188 light-years per second at scalar particle decoupling time? Is dark matter composed of scalar particles? 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 Cosmology: nucleosynthesis and inflation: (ref. 2) http://www.uio.no/studier/emner/matnat/astro/AST1100/h07/undervisningsmateriale/lecture25.pdf Wikipedia - Lambda-CDM model: (ref. 3) https://en.wikipedia.org/wiki/Lambda-CDM_model Wikipedia - Dark matter: https://en.wikipedia.org/wiki/Dark_matter Wikipedia - Inflation cosmology: https://en.wikipedia.org/wiki/Inflation_(cosmology) Wikipedia - Metric space expansion cosmology: https://en.wikipedia.org/wiki/Metric_expansion_of_space
  14. WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\gamma,t} = 0.15[/math] Observable Universe cosmology scale factor: [math]\boxed{\frac{R_u \left(t_0 \right)}{R_u \left(t \right)} = \frac{a \left(t_0 \right)}{a \left(t \right)} = \frac{T_t}{T_0} = 1 + z}[/math] Observable Universe total photon co-moving radius at present time: [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}} \; \; \; \left( 48.689 \cdot 10^{9} \; \text{ly} \right)[/math] Observable Universe total photon co-moving radius at past time integration via substitution: [math]R_{\gamma,t} = R_{\gamma} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right) = 4.220 \cdot 10^{23} \; \text{m} \; \; \; \left( 44.610 \; \cdot 10^{6} \; \text{ly} \right)[/math] Observable Universe total photon co-moving radius at past time: [math]\boxed{R_{\gamma,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)}[/math] [math]\boxed{R_{\gamma,t} = 4.220 \cdot 10^{23} \; \text{m}} \; \; \; \left( 44.610 \; \cdot 10^{6} \; \text{ly} \right)[/math] Photon decoupling time: [math]\boxed{T_{u,\gamma} = \frac{3}{ \left(2 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]\boxed{T_{u,\gamma} = 1.009 \cdot 10^{13} \; \text{s}} \; \; \; \left( 3.197 \cdot 10^{5} \; \text{years} \right) \; \; \; m_{\gamma} = 0[/math] [math]\Lambda[/math]CDM universe model semiemperical temperature-time scale factor: (ref. 2, pg. 19, eq. 1, ref. 3) [math]\left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}} = \frac{T_{u,\gamma}}{T_{u}} = T_{u,\gamma} H_0[/math] [math]T_{u,\gamma} = \frac{}{H_0} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}} = 1.218 \cdot 10^{13} \; \text{s} \; \; \; \left( 3.861 \cdot 10^{5} \; \text{years} \right)[/math] [math]\Lambda[/math]CDM universe model semiemperical photon decoupling time: [math]\boxed{T_{u,\gamma} = \frac{}{H_0} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}}}[/math] [math]\boxed{T_{u,\gamma} = 1.218 \cdot 10^{13} \; \text{s}} \; \; \; \left(3.861 \cdot 10^{5} \; \text{years}\right)[/math] Observable Universe expansion rate at photon decoupling time integration via substitution: [math]\frac{dr}{dt} = \frac{R_{\gamma,t}}{T_{u,\gamma}} = \left[ \left(\frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)\right] \frac{ \left(2 k_B T_{\gamma,t} \right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} = 4.184 \cdot 10^{10} \; \frac{\text{m}}{\text{s}} \; \; \; \left( 139.557 \cdot c \; \; \; 4.422 \cdot 10^{-6} \; \frac{\text{ly}}{\text{s}} \right)[/math] Observable Universe expansion rate at photon decoupling time: [math]\boxed{\frac{dr}{dt} = \frac{2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} \left(\frac{c}{H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\left(\frac{T_{\gamma}}{T_{\gamma,t}}\right)\left(k_B T_{\gamma,t}\right)^2}[/math] [math]\boxed{\frac{dr}{dt} = 4.184 \cdot 10^{10} \; \frac{\text{m}}{\text{s}}} \; \; \; \left( 139.557 \cdot c \; \; \; 4.422 \cdot 10^{-6} \; \frac{\text{ly}}{\text{s}} \right)[/math] Does the universe have a photon co-moving radius of 49 billion light-years at present time? Did the universe have a co-moving radius of 45 million light-years at photon decoupling time? Did the universe inflate and expand at a rate of 140 c or 4 micro light-years per second at photon 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 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#Parameters Wikipedia - Inflation cosmology: https://en.wikipedia.org/wiki/Inflation_(cosmology) Wikipedia - Metric space expansion cosmology: https://en.wikipedia.org/wiki/Metric_expansion_of_space
  15. 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
  16. [math]\boxed{\frac{ds}{dt} = 1 \cdot 10^{2} \; \frac{\text{m}}{\text{s}}} \; \; \; (100 \; \frac{\text{m}}{\text{s}})[/math]
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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)
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