Orion1

Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] (ref. 2, pg. 3) [math]\Omega_{\nu} = 1.258 \cdot 10^{-3}[/math] (ref. 2, pg. 3) [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] (ref. 2, pg. 3) Neutrino particle mass: [math]\boxed{m_{\nu} \neq 0}[/math] Photon particle mass: [math]\boxed{m_{\gamma} = 0}[/math] Hubble radius: (ref. 3) [math]R_h = \frac{c}{H_0}[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 4, pg. 11) [math]z = \left( \frac{T_{\gamma , t}}{T_{\gamma}} \right) - 1 = 1090.43[/math] Observable Universe total co-moving radius: (ref. 5) [math]R_u = R_h \left( \frac{\ln{(1 + z)}}{2} \right) = \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} = \frac{c \ln (1 + z)}{2 H_0} = 4.606 \cdot 10^{26} \; \text{m}[/math] [math]\boxed{R_u = \frac{c \ln (1 + z)}{2 H_0}}[/math] [math]\boxed{R_u = 4.606 \cdot 10^{26} \; \text{m}} \; \; \; (48.689 \cdot 10^{9} \; \text{ly})[/math] Friedmann equations critical mass: (ref. 6) [math]M_c = \frac{H_0^2 R_u^3}{2 G} = \frac{H_0^2}{2 G} \left[ \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} \right]^3 = \left( \frac{c^3}{16 G H_0} \right) [\ln (1 + z)]^3 = 3.794 \cdot 10^{54} \; \text{kg}[/math] [math]\boxed{M_c = \left( \frac{c^3}{16 G H_0} \right) [\ln (1 + z)]^3}[/math] [math]\boxed{M_c = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Fermi-Dirac and Bose-Einstein total neutrino distribution constant: (ref. 7) [math]C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5} = 0.236[/math] [math]\boxed{C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/math] Observable Universe total co-moving volume: [math]V_u = \frac{4 \pi R_u^3}{3} = \frac{4 \pi}{3} \left( \frac{c \ln (1 + z)}{2 H_0} \right)^3 = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 = 4.094 \cdot 10^{80} \; \text{m}^3[/math] [math]\boxed{V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3}[/math] [math]\boxed{V_u = 4.094 \cdot 10^{80} \; \text{m}^3} \; \; \; (1.154 \cdot 10^{32} \; \text{ly}^3)[/math] Observable Universe total mass: [math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Milky Way galaxy mass: (ref. 8, pg. 1) [math]M_{mw} = 1.26 \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] Observable Universe total galaxy number: (ref. 9) [math]N_g = \Omega_{b} \left( \frac{M_u}{M_{mw}} \right) = \frac{\pi \Omega_{b}}{6 M_{mw}} \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_g = \frac{\pi \Omega_{b}}{6 M_{mw}} \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_g = 7.496 \cdot 10^{10} \; \text{galaxies}}[/math] Stellar class number parameters: (ref. 12) [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] [math]\boxed{M_a = 1.183 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 10, pg. 20) [math]M_a = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] Stellar baryon density: (ref. 2, 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: (ref. 11) [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] [math]\boxed{N_s = 7.889 \cdot 10^{21} \; \text{stars}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Hubble radius: (ref. 3) https://en.wikipedia.org/wiki/Hubble_volume Planck 2013 results. XVI. Cosmological parameters: (ref. 4) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf UCLA Division of Astronomy and Astrophysics - Homogeneity and Isotropy: (ref. 5) http://www.astro.ucla.edu/~wright/cosmo_02.htm#DL Orion1 - Friedmann equations critical mass: (ref. 6) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915 Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 7) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Mass models of the Milky Way: (ref. 8) http://arxiv.org/pdf/1102.4340v1 Orion1 - Observable Universe total galaxy number: (ref. 9) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry874489 On The Mass Distribution Of Stars...: (ref. 10) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Orion1 - Observable Universe total observable stellar number: (ref. 11) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry871244 Wikipedia - Stellar classification - Harvard spectral classification: (ref. 12) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification

The calculation would not become 'irrelevant', the transitional epoch times would remain relatively the same, however the calculation would require an update to the latest data. For example, transitioning the calculation from RNA to DNA or the discovery of life fossils on other solar planets that are older than those on Earth. The Drake equation was not considered when the equation on post #1 was derived, however I am intrigued by the similarities. Extrapolating relevant variables from the Drake equation and using current data results from Wikipedia and setting the duration of the RNA world epoch between biogenic carbon in zircon at 4.1 Gy and microbial mat fossils found in 3.48 billion-year-old sandstone: [math]L_{RNA} = (4.1 \; \text{Gy} - 3.48 \; \text{Gy}) = 620 \; \text{My}[/math] Total number of planets currently experiencing the RNA world epoch: (ref. 5) [math]N_{RNA} = R_{\ast} \cdot f_p \cdot n_e \cdot f_{\ell} \cdot L_{RNA} = 1.736 \cdot 10^{9} \; \text{planets}[/math] [math]\boxed{N_{RNA} = 1.736 \cdot 10^{9} \; \text{planets}}[/math] Reference: Wikipedia - The earliest biological evidence for life on Earth (ref. 1) https://en.wikipedia.org/wiki/Abiogenesis#The_earliest_biological_evidence_for_life_on_Earth Bell, Elizabeth A.; Boehnike, Patrick; Harrison, T. Mark; et al. - Potentially biogenic carbon preserved in a 4.1 billion-year-old zircon: (ref. 2) http://www.pnas.org/content/early/2015/10/14/1517557112.full.pdf Wikipedia - Abiogenesis - support_and_difficulties: (ref. 3) https://en.wikipedia.org/wiki/RNA_world#Support_and_difficulties Wikipedia - RNA world - Support and difficulties: (ref. 4) https://en.wikipedia.org/wiki/RNA_world#Support_and_difficulties Wikipedia - Drake equation: (ref. 5) https://en.wikipedia.org/wiki/Drake_equation#Equation

According to stellar data, first generation (Population III) stars lasted less than 21.5 million years. (ref. 2) However, the first generation stellar epoch itself could have lasted up to 800 million years after the Big Bang (ref. 3), and placing the formation of the oldest second generation (Population II) star (SMSS J0313-6708) in the Milky Way galaxy within the first generation stellar epoch at 199 million years: [math]\boxed{dt_1 = (t_u - t_s) = 1.99 \cdot 10^{8} \; \text{y}}[/math] And what a spectacular stellar display that epoch would have been. Affirmative, very dynamic factors are involved, however nature's efficiency is intrinsically built into the chronological equation on post #1. Reference: Wikipedia - First Generation stars (Population III stars) (ref. 1) https://en.wikipedia.org/wiki/Stellar_population#Population_III_stars Wikipedia - Stellar evolution (ref. 2) https://en.wikipedia.org/wiki/Stellar_evolution Wikipedia - Cosmos Redshift 7: (ref. 3) https://en.wikipedia.org/wiki/Cosmos_Redshift_7

[math]t_u = 13.799 \cdot 10^{9} \; \text{y}[/math] - Universe age (ref. 1) [math]t_s = 13.6 \cdot 10^{9} \; \text{y}[/math] - oldest second generation star age in Milky Way galaxy (ref. 2) [math]t_{\odot} = 4.57 \cdot 10^{9} \; \text{y}[/math] - solar age (ref. 3) [math]t_E = 4.54 \cdot 10^{9} \; \text{y}[/math] - Earth age (ref. 4) [math]t_z = 4.4 \cdot 10^{9} \; \text{y}[/math] - oldest Zircon age (ref. 5) [math]t_l = 4.0 \cdot 10^{9} \; \text{y}[/math] - oldest fossilized RNA life age (ref. 6) key: [math]dt_1[/math] - minimum time required for second generation stars to form. [math]dt_2[/math] - minimum time required for inner planets to form. [math]dt_3[/math] - minimum time required for liquid water to form. [math]dt_4[/math] - minimum time required for RNA life to form. According to this basic equation, the amount of universal evolutionary time required to generate self-replicating RNA: [math]t_{RNA} = dt_1 + dt_2 + dt_3 + dt_4 = (t_u - t_s) + (t_{\odot} - t_E) + (t_E - t_z) + (t_z - t_l)[/math] [math]\boxed{t_{RNA} = (t_u - t_s) + (t_{\odot} - t_E) + (t_E - t_z) + (t_z - t_l)}[/math] [math]\boxed{t_{RNA} = 7.69 \cdot 10^{8} \; \text{y}}[/math] The first self-replicating RNA life could have been generated 769 million years after the Big Bang on the inner planets around second generation and third generation stars, which formed together at the same time. Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Universe: (ref. 1) https://en.wikipedia.org/wiki/Universe Wikipedia - SMSS J0313-6708: (ref. 2) https://en.wikipedia.org/wiki/SM0313 Wikipedia - Sun: (ref. 3) https://en.wikipedia.org/wiki/Sun Wikipedia - Earth: (ref. 4) https://en.wikipedia.org/wiki/Earth Wikipedia - Cryptic era: (ref. 5) https://en.wikipedia.org/wiki/Cryptic_era Wikipedia - Basin Groups: (ref. 6) https://en.wikipedia.org/wiki/Basin_Groups

Radial decrease rate between two bodies versus time: (ref. 1, pg. 20), (ref. 2) [math]\frac{dr}{dt} = - \frac{64 G^3 (m_1 m_2)(m_1 + m_2)}{5 c^5 r^3}[/math] Time decrease rate between two bodies versus radius: (ref. 1, pg. 20), (ref. 2) [math]\boxed{\frac{dt}{dr} = - \frac{5 c^5 r^3}{64 G^3 (m_1 m_2)(m_1 + m_2)}}[/math] Integration for total inspiral time from total radius [math]R[/math] to zero. [math]\tau = - \frac{5 c^5}{64 G^3 (m_1 m_2)(m_1 + m_2)} \int_R^0 r^3 dr = \frac{5 c^5 R^4}{256 G^3 (m_1 m_2)(m_1 + m_2)}[/math] Two-body total orbit inspiral lifetime: (ref. 3) [math]\boxed{\tau = \frac{5 c^5 R^4}{256 G^3 (m_1 m_2)(m_1 + m_2)}}[/math] Milky Way galaxy mass: (ref. 4, pg. 1) [math]M_{g} = 1.26 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math] [math]\boxed{M_{g} = 2.506 \cdot 10^{42} \; \text{kg}}[/math] Milky Way galaxy radius: (ref. 5), (ref. 6) [math]R_g = 8.515 \cdot 10^{20} \; \text{m} \; \; \; (90 \; \text{kly})[/math] Milky Way galaxy total orbit inspiral lifetime: [math]\boxed{\tau_g = \frac{5 c^5 R_g^4}{256 G^3 (M_g M_{\odot})(M_g + M_{\odot})}}[/math] [math]\boxed{\tau_g = 6.697 \cdot 10^{39} \; \text{s}} \; \; \; (2.122 \cdot 10^{32} \; \text{y})[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Orbital decay from gravitational radiation: (ref. 1) http://www.eftaylor.com/exploringblackholes/GravWaves150909v1.pdf Wikipedia - Orbital decay from gravitational radiation: (ref. 2) https://en.wikipedia.org/wiki/Gravitational_wave#Orbital_decay_from_gravitational_radiation Wikipedia - Orbital lifetime limits from gravitational radiation: (ref. 3) https://en.wikipedia.org/wiki/Gravitational_wave#Orbital_lifetime_limits_from_gravitational_radiation Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Space.com - Milky Way galaxy size: (ref. 5) http://www.space.com/29270-milky-way-size-larger-than-thought.html Wikipedia - Milky Way galaxy: (ref. 6) https://en.wikipedia.org/wiki/Milky_Way

Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] Dark matter scalar particle species total effective degeneracy number: (ref. 2) [math]\boxed{N_{\phi} = 1}[/math] Dark matter scalar particle radiation temperature is equivalent to neutrino cosmic background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Bose-Einstein total dark matter scalar particle distribution constant: (ref. 3) [math]C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5} = 3.640 \cdot 10^{-4}[/math] [math]\boxed{C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] Observable Universe total mass: [math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 5.485 \cdot 10^{54} \; \text{kg}}[/math] Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number: (ref. 3) [math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\phi} = T_{\nu}[/math] Observable Universe total dark matter scalar particle number: [math]N_{\phi t} = \Omega_{\phi} \left( \frac{M_u}{m_{\phi}} \right) = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{N_{\phi t} = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{N_{\phi t} = 2.232 \cdot 10^{93} \; \phi \; \text{particles}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Orion 1 - total effective degeneracy number: (ref. 2) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233 Orion1 - dark matter scalar particle composition: (ref. 3) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry887017 Dark matter - Wikipedia https://en.wikipedia.org/wiki/Dark_matter

Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] (ref. 2, pg. 3) [math]\Omega_{\nu} = 1.258 \cdot 10^{-3}[/math] (ref. 2, pg. 3) [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] (ref. 2, pg. 3) Neutrino particle mass: [math]\boxed{m_{\nu} \neq 0}[/math] Photon particle mass: [math]\boxed{m_{\gamma} = 0}[/math] Hubble radius: (ref. 3) [math]R_h = \frac{c}{H_0}[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 4, pg. 11) [math]z = \left( \frac{T_{\gamma , t}}{T_{\gamma}} \right) - 1 = 1090.43[/math] Observable Universe total co-moving radius: (ref. 5) [math]R_u = R_h \left( \frac{\ln{(1 + z)}}{2} \right) = \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} = \frac{c \ln (1 + z)}{2 H_0} = 4.606 \cdot 10^{26} \; \text{m}[/math] [math]\boxed{R_u = \frac{c \ln (1 + z)}{2 H_0}}[/math] [math]\boxed{R_u = 4.606 \cdot 10^{26} \; \text{m}} \; \; \; (48.689 \cdot 10^{9} \; \text{ly})[/math] Friedmann equations critical mass: (ref. 6) [math]M_c = \frac{H_0^2 R_u^3}{2 G} = \frac{H_0^2}{2 G} \left[ \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} \right]^3 = \left( \frac{c^3}{16 G H_0} \right) [\ln (1 + z)]^3 = 3.794 \cdot 10^{54} \; \text{kg}[/math] [math]\boxed{M_c = \left( \frac{c^3}{16 G H_0} \right) [\ln (1 + z)]^3}[/math] [math]\boxed{M_c = 3.794 \cdot 10^{54} \; \text{kg}}[/math] Fermi-Dirac and Bose-Einstein total neutrino distribution constant: (ref. 7) [math]C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5} = 0.236[/math] [math]\boxed{C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/math] Observable Universe total co-moving volume: [math]V_u = \frac{4 \pi R_u^3}{3} = \frac{4 \pi}{3} \left( \frac{c \ln (1 + z)}{2 H_0} \right)^3 = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 = 4.094 \cdot 10^{80} \; \text{m}^3[/math] [math]\boxed{V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3}[/math] [math]\boxed{V_u = 4.094 \cdot 10^{80} \; \text{m}^3} \; \; \; (1.154 \cdot 10^{32} \; \text{ly}^3)[/math] Observable Universe total mass: [math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math] [math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 5.485 \cdot 10^{54} \; \text{kg}}[/math] Milky Way galaxy mass: (ref. 8, pg. 1) [math]M_{mw} = 1.26 \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] Observable Universe total galaxy number: (ref. 9) [math]N_g = \Omega_{b} \left( \frac{M_u}{M_{mw}} \right) = \frac{\pi \Omega_{b}}{6 M_{mw}} \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_g = \frac{\pi \Omega_{b}}{6 M_{mw}} \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_g = 1.084 \cdot 10^{12} \; \text{galaxies}}[/math] Observable Universe average stellar mass: (ref. 10, pg. 20) [math]M_a = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]\boxed{M_a = 1.193 \cdot 10^{30} \; \text{kg}}[/math] Stellar baryon density: (ref. 2, 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: (ref. 11) [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] [math]\boxed{N_s = 1.131 \cdot 10^{22} \; \text{stars}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Hubble radius: (ref. 3) https://en.wikipedia.org/wiki/Hubble_volume Planck 2013 results. XVI. Cosmological parameters: (ref. 4) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf UCLA Division of Astronomy and Astrophysics - Homogeneity and Isotropy: (ref. 5) http://www.astro.ucla.edu/~wright/cosmo_02.htm#DL Orion1 - Friedmann equations critical mass: (ref. 6) http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915 Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 7) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Mass models of the Milky Way: (ref. 8) http://arxiv.org/pdf/1102.4340v1 Orion1 - Observable Universe total galaxy number: (ref. 9) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry874489 On The Mass Distribution Of Stars...: (ref. 10) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Orion1 - Observable Universe total observable stellar number: (ref. 11) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry871244

Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] Dark matter scalar particle species total effective degeneracy number: (ref. 2) [math]\boxed{N_{\phi} = 1}[/math] Dark matter scalar particle radiation temperature is equivalent to neutrino cosmic background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Dark matter scalar particle radiation energy density Bose-Einstein distribution: [math]\epsilon_{\phi} = \frac{4 \pi k_B^4 N_{\phi} T_{\phi}^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math] Dark matter quantum scalar particle Bose-Einstein distribution integral: [math]I_{\phi} = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math] Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number: [math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\phi} = T_{\nu}[/math] Dark matter scalar particle cosmic background radiation energy density Bose-Einstein distribution integration via substitution: [math]\epsilon_{\phi} = \alpha_{\phi} T_{\phi}^4 = \frac{4 \pi N_{\phi} (k_B T_{\phi})^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv = \frac{4 \pi N_{\phi} (k_B T_{\phi})^4}{( 2 \pi \hbar c )^3} \left( \frac{\pi^4}{C_{\phi}} \right) = \frac{\pi^2 N_{\phi} (k_B T_{\phi})^4}{2 C_{\phi} ( \hbar c )^3}[/math] Dark matter scalar particle cosmic background radiation energy density: [math]\boxed{\epsilon_{\phi} = \frac{\pi^2 N_{\phi} (k_B T_{\phi})^4}{2 C_{\phi} ( \hbar c )^3}}[/math] Dark matter scalar particle cosmic background radiation constant: [math]\boxed{\alpha_{\phi} = \frac{\pi^2 N_{\phi} k_B^4}{2 C_{\phi} ( \hbar c )^3}}[/math] Dark matter scalar particle cosmic background radiation composition integration via substitution: [math]\Omega_{\phi} = \frac{\epsilon_{\phi}}{\epsilon_c} = \frac{\alpha_{\phi} T_{\phi}^4}{\rho_c c^2} = \left( \frac{\pi^2 N_{\phi} (k_B T_{\phi})^4}{2 C_{\phi} ( \hbar c )^3} \right) \left( \frac{8 \pi G}{3 (c H_0)^2} \right) = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}[/math] Dark matter scalar particle cosmic background radiation composition: [math]\boxed{\Omega_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}}[/math] Bose-Einstein total dark matter scalar particle distribution constant: [math]C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5} = 3.640 \cdot 10^{-4}[/math] [math]\boxed{C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Orion 1 - total effective degeneracy number: (ref. 2) http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233 Dark matter - Wikipedia https://en.wikipedia.org/wiki/Dark_matter

Planck satellite cosmological parameters: (ref. 4, pg. 11) [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] (ref. 1, pg. 3) [math]\Omega_{\nu} = 1.258 \cdot 10^{-3}[/math] (ref. 1, pg. 3) [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] (ref. 1, pg. 3) Neutrino particle mass: [math]\boxed{m_{\nu} \neq 0}[/math] Photon particle mass: [math]\boxed{m_{\gamma} = 0}[/math] Universe total observable radius: [math]R_u = 4.408 \cdot 10^{26} \; \text{m} \; \; \; (46.6 \cdot 10^{9} \; \text{ly})[/math] Cosmic neutrino background radiation total neutrino composition: (ref. 1, pg. 3) [math]\Omega_{\nu} = \sum_{n = 1}^3 \Omega_{\nu} (n) = (\Omega_{\nu_{e}} + \Omega_{\nu_{\mu}} + \Omega_{\nu_{\tau}}) = 1.258 \cdot 10^{-3} = 10^{-2.9}[/math] [math]\boxed{\Omega_{\nu} = (\Omega_{\nu_{e}} + \Omega_{\nu_{\mu}} + \Omega_{\nu_{\tau}})}[/math] [math]\Omega_{\nu} = 1.258 \cdot 10^{-3}[/math] Cosmic Microwave Background Radiation photon composition: (ref. 2) [math]\boxed{\Omega_{\gamma} = \frac{8 \pi^3 G (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Cosmic Microwave Background Radiation photon composition: (ref. 1, pg. 3) [math]\boxed{\Omega_{\gamma} = 5.012 \cdot 10^{-5}} = 10^{-4.3}[/math] Cosmic neutrino background radiation and photon background radiation compositional ratio: [math]\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \frac{1.258 \cdot 10^{-3}}{5.012 \cdot 10^{-5}} = 25.118[/math] [math]\boxed{\frac{\Omega_{\nu}}{\Omega_{\gamma}} = 25.118}[/math] Fermi-Dirac and Bose-Einstein total neutrino distribution constant: (ref. 3) [math]C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5} = 0.236[/math] [math]\boxed{C_{\nu} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/math] Observable Universe total mass: [math]\boxed{M_u = \frac{4 \pi R_u^3}{3} \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 4.809 \cdot 10^{54} \; \text{kg}}[/math] Milky Way galaxy mass: (ref. 5, pg. 1) [math]M_{mw} = 1.26 \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] Observable Universe total galaxy number: (ref. 6) [math]\boxed{N_g = \frac{4 \pi R_u^3}{3 M_{mw}} \left( \frac{\Omega_{b}}{\Omega_{dm} + \Omega_{b}} \right) \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_g = 1.245 \cdot 10^{12} \; \text{galaxies}}[/math] Observable Universe average stellar mass: (ref. 7, pg. 20) [math]M_a = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]\boxed{M_a = 1.193 \cdot 10^{30} \; \text{kg}}[/math] Stellar baryon density: (ref. 1, 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: (ref. 8) [math]\boxed{N_s = \frac{4 \pi R_u^3}{3 M_{a}} \left( \frac{\Omega_{s}}{\Omega_{dm} + \Omega_{b}} \right) \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 = 1.300 \cdot 10^{22} \; \text{stars}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: The Cosmic Energy Inventory: (ref. 1) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Orion1 - Cosmic Microwave Background Radiation photon composition: (ref. 2) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882069 Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 3) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367 Planck 2013 results. XVI. Cosmological parameters: (ref. 4) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Mass models of the Milky Way: (ref. 5) http://arxiv.org/pdf/1102.4340v1 Orion1 - Observable Universe total galaxy number: (ref. 6) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry874489 On The Mass Distribution Of Stars...: (ref. 7) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Orion1 - Observable Universe total observable stellar number: (ref. 8) http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry871244

Correction to post #32: Neutrino primeval thermal remnant composition: (ref. 3, pg. 3) Electron neutrino particle rest mass: (ref. 6) ref. 6: Orion1 - Neutrino mass from Fermi-Dirac statistics...: http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233

Planck satellite cosmological parameters: (ref. 5, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\gamma} = 4.909 \cdot 10^{-5}[/math] (ref. 4) Massless photon mass: [math]\boxed{m_{\gamma} = 0}[/math] Standard Model photon species total effective degeneracy number: (ref. 1, pg. 41, eq. 197) [math]N_{\gamma} = 2[/math] Cosmic Microwave Background Radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Cosmic neutrino background radiation temperature at present time: (ref. 1, 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] Photon radiation energy density Bose-Einstein distribution: (ref. 1, pg. 42, eq. 204) [math]\epsilon_{\gamma} = \frac{4 \pi N_{\gamma} (k_B T_{\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[/math] Photon quantum particle Bose-Einstein distribution integration: [math]I_{\gamma} = \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = \frac{\pi^4}{15}[/math] Photon Bose-Einstein distribution integral constant: [math]\boxed{I_{\gamma} = \frac{\pi^4}{15}}[/math] Photon radiation energy density Bose-Einstein distribution integration via substitution: [math]\epsilon_{\gamma} = \alpha_{\gamma} T_{\gamma}^4 = \frac{4 \pi N_{\gamma} (k_B T_{\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{\pi^2 (k_B T_{\gamma})^4}{15 ( \hbar c )^3}[/math] Photon radiation energy density: [math]\boxed{\epsilon_{\gamma} = \frac{\pi^2 (k_B T_{\gamma})^4}{15 ( \hbar c )^3}}[/math] Photon radiation constant: [math]\boxed{\alpha_{\gamma} = \frac{\pi^2 k_B^4}{15 ( \hbar c )^3}}[/math] --- Standard Model neutrino species total effective degeneracy number: (ref. 2, pg. 16) [math]N_{\nu} = 3.046[/math] Neutrino radiation energy density Fermi-Dirac distribution: (ref. 1, pg. 44, eq. 221) [math]\epsilon_{\nu} = \frac{4 \pi k_B^4 N_{\nu} T_{\nu}^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math] Neutrino quantum particle Fermi-Dirac distribution integral: [math]I_{\nu} = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math] Solve for neutrino particle rest mass with highest relative maximum at critical mass number: [math]\frac{d}{dm} I_{\nu} (m_{\nu}) = 0[/math] Electron neutrino particle rest mass: [math]\boxed{m_{\nu_{e}} = 7.515 \cdot 10^{-40} \; \text{kg}}[/math] Fermi-Dirac and Bose-Einstein distribution Riemann sum limit numerical integration ratio: [math]\frac{I_{\nu_{e}}}{I_{\gamma}} = \frac{\int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv}{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega} = \frac{1}{3083.574}[/math] Fermi-Dirac and Bose-Einstein distribution Riemann sum limit numerical integration ratio: [math]\boxed{\frac{I_{\nu_{e}}}{I_{\gamma}} = \frac{1}{3083.574}}[/math] [math]I_{\nu_{e}} = \frac{I_{\gamma}}{3083.574} = \left( \frac{\pi^4}{15} \right) \frac{1}{3083.574} = \frac{\pi^4}{47221.1} = \frac{\pi^4}{C_{\nu_{e}}}[/math] [math]\boxed{C_{\nu_{e}} = 47221.1}[/math] Electron neutrino Fermi-Dirac distribution integral constant: [math]\boxed{I_{\nu_{e}} = \frac{\pi^4}{47221.1}} \; \; \; T_{\nu_{e}} = T_{\nu}[/math] Electron neutrino cosmic background radiation energy density Fermi-Dirac distribution integration via substitution:: [math]\epsilon_{\nu_{e}} = \alpha_{\nu_{e}} T_{\nu}^4 = \frac{4 \pi N_{\nu} (k_B T_{\nu})^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv = \frac{4 \pi N_{\nu} (k_B T_{\nu})^4}{( 2 \pi \hbar c )^3} \left( \frac{\pi^4}{C_{\nu_{e}}} \right) = \frac{\pi^2 N_{\nu} (k_B T_{\nu})^4}{2 C_{\nu_{e}} ( \hbar c )^3}[/math] Electron neutrino cosmic background radiation energy density: [math]\boxed{\epsilon_{\nu_{e}} = \frac{\pi^2 N_{\nu} (k_B T_{\nu})^4}{2 C_{\nu_{e}} ( \hbar c )^3}}[/math] Electron neutrino cosmic background radiation constant: [math]\boxed{\alpha_{\nu_{e}} = \frac{\pi^2 N_{\nu} k_B^4}{2 C_{\nu_{e}} ( \hbar c )^3}}[/math] Cosmic neutrino background radiation electron neutrino composition integration via substitution: [math]\Omega_{\nu_{e}} = \frac{\epsilon_{\nu_{e}}}{\epsilon_c} = \frac{\alpha_{\nu_{e}} T_{\nu}^4}{\rho_c c^2} = \left( \frac{\pi^2 N_{\nu} (k_B T_{\nu})^4}{2 C_{\nu_{e}} ( \hbar c )^3} \right) \left( \frac{8 \pi G}{3 (c H_0)^2} \right) = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 C_{\nu_{e}} H_0^2 \hbar^3 c^5} = 6.293 \cdot 10^{-9}[/math] Cosmic neutrino background radiation electron neutrino composition: [math]\boxed{\Omega_{\nu_{e}} = \frac{4 \pi^3 G N_{\nu} (k_B T_{\nu})^4}{3 C_{\nu_{e}} H_0^2 \hbar^3 c^5}}[/math] Cosmic neutrino background radiation electron neutrino composition: [math]\boxed{\Omega_{\nu_{e}} = 6.293 \cdot 10^{-9}}[/math] Neutrino primeval thermal remnant composition: (ref. 5, pg. 3) [math]\boxed{\Omega_{\nu} = 6.425 \cdot 10^{-9}} = 10^{-2.9}[/math] Observable Universe total mass: [math]\boxed{M_u = \frac{4 \pi R_u^3}{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_{e}}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math] [math]\boxed{M_u = 4.862 \cdot 10^{54} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: PHYS: 652 Cosmic Inventory I: Radiation: http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Relic neutrino decoupling including flavour oscillations: http://arxiv.org/pdf/hep-ph/0506164.pdf The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf Orion1 - Cosmic Microwave Background Radiation photon composition http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882069 Planck 2013 results. XVI. Cosmological parameters: http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

A new video animation of dwarf planet Ceres, based on images taken by NASA's Dawn spacecraft, provides dramatic flyover views of this heavily cratered, mysterious world. The images come from Dawn's first mapping orbit at Ceres, at an altitude of 8,400 mile (13,600 kilometers), as well as navigational images taken from 3,200 miles (5,100 kilometers) away. The images provided information for a three-dimensional terrain model. The vertical dimension has been exaggerated by a factor of two, and a star field has been added in the background. Any discussions and/or peer reviews about this specific topic thread? Fly over dwarf planet Ceres: https://www.youtube.com/embed/uSaLVAl-ObY Reference: https://en.wikipedia.org/wiki/Ceres_%28dwarf_planet%29

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{\gamma , t} = 0.15[/math] [math]\Omega_{\nu , t} = 0.10[/math] [math]\Omega_{\Lambda , t} \neq 0[/math] Photon radiation constant: (ref. 7, p. 42, eq. 205) [math]\alpha_{\gamma} = \frac{\pi^2 k_B^4}{15 (\hbar c)^3}[/math] Observable Universe cosmic background photon radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 2, pg. 11),(ref. 3) [math]z = \left( \frac{T_{\gamma , t}}{T_{\gamma}} \right) - 1 = 1090.43[/math] Observable Universe cosmic photon background radiation temperature at photon decoupling time: (ref. 3) [math]T_{\gamma , t} = T_{\gamma} (1 + z) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma , t} = 2974.67 \; \text{K}}[/math] Cosmic photon background radiation photon composition at current time: (ref. 4) [math]\boxed{\Omega_{\gamma} = \frac{8 \pi^3 G (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Solve for Hubble parameter [math]H_{\gamma , t}[/math] at photon decoupling time: [math]H_{\gamma , t} = \sqrt{\frac{8 \pi^3 G (k_B T_{\gamma , t})^4}{45 \Omega_{\gamma, t} \hbar^3 c^5}} = \frac{2 (k_B T_{\gamma , t})^2}{3} \sqrt{\frac{2 \pi^3 G}{5 \Omega_{\gamma, t} \hbar^3 c^5}} = 4.956 \cdot 10^{-14} \; \text{s}^{-1}[/math] Cosmic photon background radiation Hubble parameter at photon decoupling time: [math]\boxed{H_{\gamma , t} = \frac{2 (k_B T_{\gamma , t})^2}{3} \sqrt{\frac{2 \pi^3 G}{5 \Omega_{\gamma, t} \hbar^3 c^5}}}[/math] Cosmic photon background radiation Hubble parameter at photon decoupling time: [math]\boxed{H_{\gamma , t} = 4.956 \cdot 10^{-14} \; \text{s}^{-1}}[/math] Observable Universe cosmic photon background radiation photon decoupling time: [math]T_{u , \gamma} = \frac{1}{H_{\gamma , t}} = 2.017 \cdot 10^{13} \; \text{s} = 6.393 \cdot 10^{5} \; \text{years}[/math] Observable Universe cosmic photon background radiation photon decoupling time: [math]\boxed{T_{u , \gamma} = 6.393 \cdot 10^{5} \; \text{years}}[/math] --- Massless neutrino mass: [math]\boxed{m_{\nu} = 0}[/math] Standard Model neutrino species total effective degeneracy number: [math]N_{\nu} = 3.046[/math] Massless neutrino particle radiation constant: (ref. 5) [math]\boxed{\alpha_{\nu} = \frac{7 \pi^2 k_B^4 N_{\nu}}{120 \hbar^3 c^3}}[/math] Cosmic neutrino background radiation temperature at present time: [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] Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 6) [math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math] Photon background radiation temperature is equivalent to neutrino background radiation temperature at neutrino decoupling time: [math]\boxed{T_{\gamma , t} = T_{\nu , t}}[/math] Redshift parameter at neutrino decoupling time: [math]z = \left( \frac{T_{\nu , t}}{T_{\nu}} \right) - 1 = 5.140 \cdot 10^{9}[/math] Redshift parameter at neutrino decoupling time: [math]\boxed{z = 5.140 \cdot 10^{9}}[/math] Solve for Hubble parameter [math]H_{\nu , t}[/math] at massless neutrino decoupling time: [math]H_{\nu , t} = \sqrt{ \left( \frac{7}{8} \right) \frac{8 \pi^3 G N_{\nu} (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 \; \text{s}^{-1}[/math] Hubble parameter at massless neutrino decoupling time: [math]\boxed{H_{\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] Hubble parameter at massless neutrino decoupling time: [math]\boxed{H_{\nu , t} = 1.120 \; \text{s}^{-1}}[/math] Observable Universe neutrino background radiation massless neutrino decoupling time: [math]T_{u , \nu} = \frac{1}{H_{\nu , t}} = 0.893 \; \text{s}[/math] Observable Universe neutrino background radiation massless neutrino decoupling time: [math]\boxed{T_{u , \nu} = 0.893 \; \text{s}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite content of the Universe: http://map.gsfc.nasa.gov/media/080998/index.html Planck 2013 results. XVI. Cosmological parameters: http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Orion1 - Cosmology scale factor: http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry876504 Orion1 - Cosmic Microwave Background Radiation photon composition: http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882069 Orion1 - massless neutrino particle radiation constant: http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry875980 Wikipedia - neutrino decoupling: https://en.wikipedia.org/wiki/Neutrino_decoupling PHYS: 652 Cosmic Inventory I: Radiation: http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

Photon radiation constant: [math]\alpha_{\gamma} = \frac{\pi^2 k_B^4}{15 (\hbar c)^3}[/math] Friedmann equations critical mass density: [math]\rho_c = \frac{3 H_0^2}{8 \pi G}[/math] Friedmann equations critical energy density integration via substitution: [math]\epsilon_c = \rho_c c^2 = \frac{3 (c H_0)^2}{8 \pi G}[/math] Friedmann equations critical energy density: [math]\boxed{\epsilon_c = \frac{3 (c H_0)^2}{8 \pi G}}[/math] Cosmic Microwave Background Radiation photon composition integration via substitution: (ref. 2, pg. 43, eq. 208) [math]\Omega_{\gamma} = \frac{\epsilon_{\gamma}}{\epsilon_c} = \frac{\alpha_{\gamma} T_{\gamma}^4}{\rho_c c^2} = \left( \frac{\pi^2 (k_B T_{\gamma})^4}{15 (\hbar c)^3} \right) \left( \frac{8 \pi G}{3 (c H_0)^2} \right) = \frac{8 \pi^3 G (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5} = 4.909 \cdot 10^{-5}[/math] Cosmic Microwave Background Radiation photon composition: [math]\boxed{\Omega_{\gamma} = \frac{8 \pi^3 G (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5}}[/math] Cosmic Microwave Background Radiation photon composition: [math]\boxed{\Omega_{\gamma} = 4.909 \cdot 10^{-5}}[/math] Cosmic Microwave Background Radiation primeval thermal remnant composition: (ref. 1, pg. 3) [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5} = 10^{-4.3}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf PHYS: 652 Cosmic Inventory I: Radiation: http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

The specific formulation in this thread is sophisticated, however it is not very complex. A college or university student that has successfully completed a course on Calculus I and Physics II is qualified to solve all these equations. The mathematical skills acquired involve memorization of the semantic mathematical symbolic definitions and the ability to utilize these symbolic definitions to effectively solve mathematical equations. Absent such student academic qualifications definitely requires the use of math evaluation software and three months solid home study with a college or university level subject textbook. Equation (1) is semantically symbolizing the summation of an area under a sloped curve using the Riemann Sum formula, it is called an integral. Equation (3) is a function domain parameter that levels the slope of the slope formula to flat or horizontal and equivalent to zero. Equation (4) is semantically symbolizing the range location where the slope of the slope formula is flat or horizontal and equivalent to zero, it is called a derivative. Neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature: [math]\tag{0} \boxed{T_{\nu} = T_{\nu}}[/math] Use Riemann sum limit to solve [math]I_{\nu} (m_{\nu})[/math]: [math]\tag{1} I_{\nu} (m_{\nu}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math] [math]\tag{2} \boxed{I_{\nu} (m_{\nu}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv}[/math] [math]\tag{3} \Delta m_{\nu} = (m_{b} - m_{a})[/math] [math]\tag{4} \frac{d}{dm} I_{\nu} (m_{\nu}) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{d}{dx} f(x) = \lim_{\Delta m_{\nu} \rightarrow 0} \frac{I_{\nu}(m_{\nu} + \Delta m_{\nu}) - I_{\nu}(m_{\nu})}{\Delta m_{\nu}} = 0[/math] [math]\tag{5} \boxed{\frac{d}{dm} I_{\nu} (m_{\nu}) = \lim_{\Delta m_{\nu} \rightarrow 0} \frac{I_{\nu}(m_{\nu} + \Delta m_{\nu}) - I_{\nu}(m_{\nu})}{\Delta m_{\nu}} = 0}[/math] Solve for [math]m_{\nu}[/math]: Because there is no formal solution to the derivative, the next step is to plot the function [math]I_{\nu} (m_{\nu})[/math] and solve for the critical mass using the derivative. (ref. plot 1) The derivative is solved by slowly increasing the domain parameter [math]m_a[/math] and slowly decreasing domain parameter [math]m_b[/math], so that the difference in the parameters approach zero. Once a solution of three decimal places is achieved, the derivative is considered solved. (ref. plot 2) This Mathematica source code is nearly equivalent to the symbolic formulas expressed in equations 1 to 4. Mathematica 7.0.0 source code: t = AbsoluteTime[] (* Riemann Sum - MidPoint Rule *) MidPointRule[a0_, b0_, n0_] := Module[{a = a0, b = b0, c, \[CapitalDelta]X, k, n = n0, X}, \[CapitalDelta]X = (b - a)/n; Subscript[c, k_] = a + (k - 1/2)*\[CapitalDelta]X; Return[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(f[ \*SubscriptBox[\(c\), \(k\)]] \[CapitalDelta]X\)\)];]; (* End Module *) c = 2.99792458*10^8 Subscript[k, B] = 1.380648813* \!\(\*SuperscriptBox["10", RowBox[{"\[Minus]", "23"}]]\) Subscript[T, \[Gamma]] = 2.72548 Subscript[T, \[Nu]] = (4/11)^(1/3)*Subscript[T, \[Gamma]] Subscript[\[Epsilon], 1] = Subscript[k, B]*Subscript[T, \[Nu]] Subscript[\[Epsilon], 0] = m*c^2 Subscript[\[Epsilon], p][v_] := (m*c*v)/Sqrt[1 - (v/c)^2] Subscript[\[Epsilon], t][v_] := Sqrt[ Subscript[\[Epsilon], p][v]^2 + Subscript[\[Epsilon], 0]^2] f[v_] := Subscript[\[Epsilon], t][v]^3/( E^(Subscript[\[Epsilon], t][v]/Subscript[\[Epsilon], 1]) + 1) a = 0 b = c n = 100 F[m_] := MidPointRule[a, b, n] Subscript[m, a] = 1*10^-41 Subscript[m, b] = 3*10^-39 Plot[ F[m], {m, Subscript[m, a], Subscript[m, b]}, ImageSize -> {640, Automatic}] (* m=7.515*10^-40 *) Subscript[m, a] = 7.5150*10^-40 Subscript[m, b] = 7.5156*10^-40 Plot[ F[m], {m, Subscript[m, a], Subscript[m, b]}, ImageSize -> {640, Automatic}] Print["Evaluation Time: ", AbsoluteTime[] - t, " seconds."] Print["Maximum Memory Used: ", MaxMemoryUsed[], " bytes."] Evaluation Time: 4.9062500 seconds. Maximum Memory Used: 11784056 bytes. Answer: 5 seconds. According to Mathematica 7, the solution is... Fermi-Dirac fermionic particle rest mass: [math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}\; \; \; T_{\nu} = T_{\nu}[/math] Any discussions and/or peer reviews about this specific topic thread? plot01.bmp plot02.bmp Reference: mathfaculty.fullerton.edu - Riemann Sum: http://mathfaculty.fullerton.edu/mathews/n2003/riemannsum/RiemannSumMod/Links/RiemannSumMod_lnk_4.html Wikipedia - Function Limit: https://en.wikipedia.org/wiki/Limit_of_a_function

The best mathematical description is that the mass solution is a Fermi-Dirac fermionic particle rest mass with highest relative maximum at a critical mass number within the mass spectrum range of a Standard Model neutrino. This Fermi-Dirac fermionic particle rest mass is an interesting artifact of the mathematical properties of the Fermi-Dirac distribution equation. Consider for a moment that when a Standard Model electron neutrino attains a non-zero particle mass via some asymmetric Higgs mechanism, that the neutrino mass attains the Fermi-Dirac fermionic particle rest mass with highest relative maximum at a critical mass number by virtue of the particles fermionic quantum spin properties. Electron neutrino particle rest mass is equivalent to the Fermi-Dirac fermionic particle rest mass: [math]\boxed{m_{\nu_{e}} = m_{\nu_{fd}}}[/math] What are the possibilities that the Standard Model neutrino masses obey weak particle mass hierarchy? [math]\boxed{\frac{m_{e}}{m_{\mu}} = \frac{m_{\nu_{e}}}{m_{\nu_{\mu}}}}[/math] [math]\boxed{m_{\nu_{e}} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}} \; \; \; T_{\nu} = T_{\nu}[/math] [math]m_{\nu_{\mu}} \leq m_{\nu_{e}} \left( \frac{m_{\mu}}{m_{e}} \right) \leq 0.087 \; \frac{\text{eV}}{c^2}[/math] [math]\boxed{m_{\nu_{\mu}} \leq 0.087 \; \frac{\text{eV}}{c^2}}[/math] [math]\boxed{\frac{m_{e}}{m_{\tau}} = \frac{m_{\nu_{e}}}{m_{\nu_{\tau}}}}[/math] [math]m_{\nu_{\tau}} \leq m_{\nu_{e}} \left( \frac{m_{\tau}}{m_{e}} \right) \leq 1.465 \; \frac{\text{eV}}{c^2}[/math] [math]\boxed{m_{\nu_{\tau}} \leq 1.465 \; \frac{\text{eV}}{c^2}}[/math] The sum of these three neutrino flavors: [math]\boxed{\sum m_{\nu} \leq 1.553 \; \frac{\text{eV}}{c^2}}[/math] Active neutrino model with three degenerate neutrinos: (ref. 5) [math]\sum m_{\nu} \leq 0.320 \pm 0.081 \; \frac{\text{eV}}{c^2}[/math] The neutrino mass sum formula: [math]\sum m_{\nu} = \left( m_{\nu_{e}} + m_{\nu_{\mu}} + m_{\nu_{\tau}} \right)[/math] Integration via substitution and factor: [math]\sum m_{\nu} = m_{\nu_{e}} \left( 1 + \frac{m_{\mu}}{m_{e}} + \frac{m_{\tau}}{m_{e}} \right) = 0.320 \pm 0.081 \; \frac{\text{eV}}{c^2}[/math] Solve for [math]m_{\nu_{e}}[/math]. Active neutrino model degenerate electron neutrino rest mass: [math]\boxed{m_{\nu_{e}} = \sum m_{\nu} \left( \frac{m_{e}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 8.684 \cdot 10^{-5} \; \frac{\text{eV}}{c^2}[/math] Active neutrino model degenerate electron neutrino rest mass: [math]\boxed{m_{\nu_{e}} = 8.684 \cdot 10^{-5} \; \frac{\text{eV}}{c^2}}[/math] Active neutrino model degenerate electron neutrino rest mass: [math]\boxed{m_{\nu_{e}} = 1.548 \cdot 10^{-40} \; \text{kg}}[/math] Active neutrino model degenerate muon neutrino rest mass: [math]\boxed{m_{\nu_{\mu}} = \sum m_{\nu} \left( \frac{m_{\mu}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 0.018 \; \frac{\text{eV}}{c^2}[/math] Active neutrino model degenerate muon neutrino rest mass: [math]\boxed{m_{\nu_{\mu}} = 0.018 \; \frac{\text{eV}}{c^2}}[/math] Active neutrino model degenerate tau neutrino rest mass: [math]\boxed{m_{\nu_{\tau}} = \sum m_{\nu} \left( \frac{m_{\tau}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 0.302 \; \frac{\text{eV}}{c^2}[/math] Active neutrino model degenerate tau neutrino rest mass: [math]\boxed{m_{\nu_{\tau}} = 0.302 \; \frac{\text{eV}}{c^2}}[/math] Fermi-Dirac fermionic particle rest mass: [math]\boxed{m_{\nu_{fd}} = 7.515 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\nu} = T_{\nu}[/math] Fermi-Dirac fermionic particle rest mass: [math]\boxed{m_{\nu_{fd}} = 1.548 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\nu} = 0.4 \; \text{K}[/math] Active neutrino model degenerate electron neutrino rest mass: [math]\boxed{m_{\nu_{e}} = 1.548 \cdot 10^{-40} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: https://en.wikipedia.org/wiki/Neutrino https://en.wikipedia.org/wiki/Electron https://en.wikipedia.org/wiki/Muon https://en.wikipedia.org/wiki/Tau_(particle) http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.051303

Continued from post #1... Solve for neutrino particle rest mass with highest relative maximum at critical mass number: [math]\frac{d}{dm} I_{\nu} (m_{\nu}) = 0[/math] Neutrino particle rest mass: [math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}[/math] [math]\boxed{m_{\nu} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math] Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number: [math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}}[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 3.694 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math] [math]\boxed{\frac{m_{\phi}}{m_{\nu}} = \frac{7}{8}} \; \; \; T_{\phi} = T_{\nu}[/math] Any discussions and/or peer reviews about this specific topic thread?

[math]\begin{tabular}{l*{6}{c}r} bosons (b) = integer spin & Bose-Einstein statistics \\ fermions (f) = half-integer spin & Fermi-Dirac statistics \\ \end{tabular}[/math] [math][/math] [math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math] Symbolic identity key: [math]n_s[/math] - spin states total number [math]N_s[/math] - species total number [math]N_n[/math] - total effective degeneracy number If [math]n_s \geq N_s[/math] then [math]N_n = n_s[/math] If [math]n_s \leq N_s[/math] then [math]N_n = N_s[/math] CMBR temperature at present time: (ref. 5) [math]T_{\gamma} = 2.72548 \; \text{K}[/math] Cosmic neutrino background radiation temperature at present time: (ref. 1, 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] CMBR primeval thermal remnant composition: (ref. 2, pg. 3) [math]\Omega_{\gamma} = 10^{-4.3}[/math] Neutrino primeval thermal remnant composition: (ref. 2, pg. 3) [math]\Omega_{\nu} = 10^{-2.9}[/math] Standard Model photon species total effective degeneracy number: (ref. 1, pg. 41, eq. 197) [math]N_{\gamma} = 2[/math] Standard Model neutrino species total effective degeneracy number: (ref. 4, pg. 16) [math]N_{\nu} = 3.046[/math] Total photon energy: (ref. 3) [math]E_t (\omega) = \hbar \omega[/math] Total thermal energy: [math]E_1 (T_{\gamma}) = k_B T_{\gamma}[/math] Photon radiation energy density Bose-Einstein distribution: (ref. 1, pg. 42, eq. 204) [math]\rho_{\gamma} = \frac{4 \pi k_B^4 N_{\gamma} T_{\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[/math] Total neutrino rest mass energy: (ref. 3) [math]E_0 = m_0 c^2[/math] Total neutrino relativistic momentum energy: (ref. 3) [math]E_p (v) = p(v) c = (\gamma m_0 v) c = \frac{m_0 c v}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }}[/math] [math]\boxed{E_p (v) = \frac{m_0 c v}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }}}[/math] Total neutrino energy: (ref. 3) [math]E_t (v) = \sqrt{E_p(v)^2 + E_0^2}[/math] Neutrino radiation energy density Fermi-Dirac distribution: (ref. 1, pg. 44, eq. 221) [math]\rho_{\nu} = \frac{4 \pi k_B^4 N_{\nu} T_{\nu}^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math] Massless neutrino compositional ratio integration via substitution: [math]\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \frac{\rho_{\nu}}{\rho_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \frac{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\nu})}} + 1} d \omega}{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 C_1[/math] Massless integration constants: [math]\boxed{C_1 = 0.875 = \frac{7}{8}} \; \; \; \; \; \; T_{\nu} = T_{\gamma}[/math] [math]\boxed{C_1 = 0.227 = \frac{2}{9}} \; \; \; \; \; \; T_{\nu} = T_{\nu}[/math] Massless neutrino compositional ratio: [math]\boxed{\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 C_1}[/math] Mass neutrino compositional ratio: [math]\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \frac{\rho_{\nu}}{\rho_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \frac{\int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv}{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega}[/math] Neutrino Riemann sum limit numerical integration criterion: [math]I_{\nu} (m_{\nu}) = I_{\gamma}[/math] [math]I_{\nu} (m_{\nu}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv = \frac{\Omega_{\nu}}{\Omega_{\gamma}} \left( \frac{N_{\gamma}}{N_{\nu}} \right) \left( \frac{T_{\gamma}}{T_{\nu}} \right)^4 \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = I_{\gamma}[/math] Use Riemann sum limit to solve [math]I_{\nu} (m_{\nu})[/math]: [math]I_{\nu} (m_{\nu}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math] Riemann sum limit mass numerical integration probability: [math]P(m_{\nu}) = \frac{I_{\gamma}}{I_{\nu} (m_{\nu})}[/math] Solve for neutrino particle rest mass with highest Riemann sum limit rest mass numerical integration probability. Neutrino rest mass: [math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}[/math] Neutrino rest mass: [math]\boxed{m_{\nu} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math] --- Dark matter compositional fraction: (ref. 6, pg. 11) [math]\Omega_{dm} = 0.268[/math] Dark matter scalar particle temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] Dark matter scalar particle species total effective degeneracy number: [math]\boxed{N_{\phi} = 1}[/math] Dark matter scalar particle Riemann sum limit numerical integration criterion: [math]I_{\phi} (m_{\phi}) = I_{\gamma}[/math] [math]I_{\phi} (m_{\phi}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv = \frac{\Omega_{dm}}{\Omega_{\gamma}} \left( \frac{N_{\gamma}}{N_{\phi}} \right) \left( \frac{T_{\gamma}}{T_{\phi}} \right)^4 \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = I_{\gamma}[/math] Use Riemann sum limit to solve [math]I_{\phi} (m_{\phi})[/math]: [math]I_{\phi} (m_{\phi}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math] Riemann sum limit mass numerical integration probability: [math]P(m_{\phi}) = \frac{I_{\gamma}}{I_{\phi} (m_{\phi})}[/math] Solve for dark matter scalar particle rest mass with highest Riemann sum limit rest mass numerical integration probability. Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 1.818 \cdot 10^{-37} \; \text{kg}}[/math] Dark matter scalar particle rest mass: [math]\boxed{m_{\phi} = 0.102 \; \frac{\text{eV}}{c^2}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: PHYS: 652 Cosmic Inventory I: Radiation: http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf Hyperphysics: Momentum of Photon: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html#c4 Relic neutrino decoupling including flavour oscillations: http://arxiv.org/pdf/hep-ph/0506164.pdf Wikipedia - cosmic microwave background radiation: https://en.wikipedia.org/wiki/Cosmic_microwave_background Planck 2013 results. XVI. Cosmological parameters: http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - dark matter: https://en.wikipedia.org/wiki/Dark_matter Wikipedia - Riemann sum: https://en.wikipedia.org/wiki/Riemann_sum Wikipedia - Bose-Einstein statistics: https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics Wikipedia - Fermi-Dirac statistics: https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1) [math]\Omega_{dm} = 0.63[/math] [math]\Omega_{b} = 0.12[/math] [math]\Omega_{\gamma} = 0.15[/math] [math]\Omega_{\nu} = 0.10[/math] [math]\Omega_{\Lambda} \neq 0[/math] Planck satellite redshift parameter at photon decoupling time: (ref. 2, pg. 11) [math]z = 1090.43[/math] Cosmology scale factor: [math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math] Symbolic definition key: [math]R_u(t_0)[/math] - Observable Universe total radius at present time. [math]R_u(t)[/math] - Observable Universe total radius at past time. [math]a(t_0)[/math] - scale factor at present time. [math]a(t)[/math] - scale factor at past time. [math]T_t[/math] - cosmic background radiation temperature at past time. [math]T_0[/math] - cosmic background radiation temperature at present time. [math]z[/math] - redshift at photon decoupling time. Observable Universe total radius at photon decoupling time: [math]R_u(t) = \frac{R_u(t_0)}{1 + z} = 4.039 \cdot 10^{23} \; \text{m}[/math] [math]\boxed{R_u(t) = 4.039 \cdot 10^{23} \; \text{m}}[/math] Observable Universe photon cosmic background radiation temperature at photon decoupling time: [math]T_t = T_0(1 + z) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma , t} = 2974.67 \; \text{K}}[/math] Observable Universe neutrino cosmic background radiation temperature at photon decoupling time: [math]T_t = T_0(1 + z) = 2128.29 \; \text{K}[/math] [math]\boxed{T_{\nu , t} = 2128.29 \; \text{K}}[/math] Observable Universe total mass at photon decoupling time: [math]\boxed{M_u = \frac{4 \pi R_u^3}{3} \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\nu} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_t^2}{8 \pi G} \right]}[/math] Observable Universe total mass at photon decoupling time: [math]\boxed{M_u = 9.271 \cdot 10^{53} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: WMAP satellite content of the Universe: http://map.gsfc.nasa.gov/media/080998/index.html Planck 2013 results. XVI. Cosmological parameters: http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

Refined particle masses from 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] Dark matter scalar particle mass: [math]\boxed{m_{\phi} = 4.143 \cdot 10^{-35} \; \; \text{kg}}[/math] Dark matter scalar particle mass: [math]\boxed{m_{\phi} = 23.244 \; \; \frac{\text{eV}}{\text{c}^2}}[/math] Neutrino particle mass: [math]\boxed{m_{\nu} = 3.651 \cdot 10^{-38} \; \; \text{kg}}[/math] Neutrino particle mass: [math]\boxed{m_{\nu} = 0.020484 \; \; \frac{\text{eV}}{\text{c}^2}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Planck 2013 results. XVI. Cosmological parameters: http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf

Dark matter scalar particle mass: [math]\boxed{m_{\phi} = 4.309 \cdot 10^{-35} \; \; \text{kg}}[/math] Dark matter scalar particle mass: [math]\boxed{m_{\phi} = 24.171 \; \; \frac{\text{eV}}{\text{c}^2}}[/math] Neutrino primeval thermal remnant composition: [math]\Omega_{\nu} = 10^{-2.9}[/math] Photon radiation constant: [math]\alpha_{\gamma} = \frac{\pi^2 k_B^4}{15 \hbar^3 c^3}[/math] Standard Model neutrino species effective number: [math]N_{\nu} = 3.046[/math] Neutrino particle radiation constant is equivalent to the photon radiation constant times seven-eighths times the Standard Model neutrino species effective number: [math]\alpha_{\nu} = \alpha_{\gamma} \left( \frac{7}{8} \right) N_{\nu} = \left( \frac{7}{8} \right) \frac{\pi^2 k_B^4 N_{\nu}}{15 \hbar^3 c^3} = \frac{7 \pi^2 k_B^4 N_{\nu}}{120 \hbar^3 c^3}[/math] Neutrino particle radiation constant: [math]\boxed{\alpha_{\nu} = \frac{7 \pi^2 k_B^4 N_{\nu}}{120 \hbar^3 c^3}}[/math] Cosmic neutrino background radiation temperature at present time: [math]T_{\nu} = 1.95 \; \text{K}[/math] Neutrino particle radiation energy density: [math]\boxed{\epsilon_{\nu} = \alpha_{\nu} T_{\nu}^4}[/math] Average neutrino particle energy: [math]\boxed{E_{\nu} = \frac{3 k_B T_{\nu}}{2}}[/math] The total average neutrino particle radiation number density is equivalent to the neutrino particle radiation energy density divided by the average neutrino particle energy: [math]n_{\nu} = \frac{\epsilon_{\nu}}{E_{\nu}} = \frac{2 \epsilon_{\nu}}{3 k_B T_{\nu}} = \frac{2 \alpha_{\nu} T_{\nu}^3}{3 k_B} = \frac{7 \pi^2 N_{\nu}}{180} \left( \frac{k_B T_{\nu}}{\hbar c} \right)^3[/math] Total average neutrino particle radiation number density: [math]\boxed{n_{\nu} = \frac{7 \pi^2 N_{\nu}}{180} \left( \frac{k_B T_{\nu}}{\hbar c} \right)^3}[/math] The total Observable Universe neutrino fraction is equivalent to the neutrino compositional fraction divided by the total compositional mass fractions: [math]\boxed{\Omega_{\nu f} = \left( \frac{\Omega_{\nu}}{\Omega_{dm} + \Omega_{b}} \right)}[/math] Observable Universe total density derived from neutrino particles integration via substitution: [math]\rho_{u, \nu} = \frac{n_{\nu} m_{\nu}}{\Omega_{\nu f}} = n_{\nu} m_{\nu} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) = \frac{7 \pi^2 N_{\nu} m_{\nu}}{180} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) \left( \frac{k_B T_{\nu}}{\hbar c} \right)^3[/math] Observable Universe total density derived from neutrino particles: [math]\boxed{\rho_{u, \nu} = \frac{7 \pi^2 N_{\nu} m_{\nu}}{180} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) \left( \frac{k_B T_{\nu}}{\hbar c} \right)^3}[/math] Observable Universe total density derived from cosmic background radiation and dark energy: [math]\boxed{\rho_{u, \gamma , \Lambda} = \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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 density derived from neutrino particles is equivalent to the Observable Universe total density derived from cosmic background radiation and dark energy: [math]\boxed{\rho_{u, \nu} = \rho_{u, \gamma , \Lambda}}[/math] Integration via substitution: [math]\frac{7 \pi^2 N_{\nu} m_{\nu}}{180} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) \left( \frac{k_B T_{\nu}}{\hbar c} \right)^3 = \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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] Solve for neutrino particle mass: [math]m_{\nu}[/math] Neutrino particle mass: [math]\boxed{m_{\nu} = \frac{180}{7 \pi^2 N_{\nu}} \left( \frac{\Omega_{\nu}}{\Omega_{dm} + \Omega_{b}} \right) \left( \frac{\hbar c}{k_B T_{\nu}} \right)^3 \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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] Neutrino particle mass: [math]\boxed{m_{\nu} = 4.424 \cdot 10^{-38} \; \; \text{kg}}[/math] Neutrino particle mass: [math]\boxed{m_{\nu} = 0.0248206 \; \; \frac{\text{eV}}{\text{c}^2}}[/math] Calculated neutrino particle mass: (ref. 1) [math]\boxed{m_{\nu} \leq 0.320 \pm 0.081 \; \; \frac{\text{eV}}{\text{c}^2}}[/math] Do neutrinos follow Dirac or Majorana statistics? Any discussions and/or peer reviews about this specific topic thread? Reference: Neutrino mass - Wikipedia https://en.wikipedia.org/wiki/Neutrino#Mass Cosmic Neutrino Background Radiation - Wikipedia http://en.wikipedia.org/wiki/Cosmic_neutrino_background The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf http://star-www.dur.ac.uk/~csf/homepage/CosmicHistory_lectures/lecture_7-8_notes.pdf http://arxiv.org/pdf/astro-ph/0406095v2.pdf http://arxiv.org/pdf/hep-ph/0506164.pdf

Photon radiation constant: [math]\alpha_{\gamma} = \frac{\pi^2 k_B^4}{15 \hbar^3 c^3}[/math] Scalar particle radiation constant is equivalent to one-half the photon radiation constant: (ref. 1, pg. 363) [math]\alpha_{\phi} = \frac{\alpha_{\gamma}}{2} = \frac{\pi^2 k_B^4}{30 \hbar^3 c^3}[/math] Scalar particle radiation constant: (ref. 1, pg. 363) [math]\boxed{\alpha_{\phi} = \frac{\pi^2 k_B^4}{30 \hbar^3 c^3}}[/math] Scalar particle radiation temperature is equivalent to cosmic neutrino background radiation temperature: (ref. 1, pg. 363) [math]\boxed{T_{\phi} = T_{\nu}}[/math] Scalar particle radiation energy density: [math]\boxed{\epsilon_{\phi} = \alpha_{\phi} T_{\phi}^4}[/math] Average scalar particle energy: (ref. 2, pg. 4) [math]\boxed{E_{\phi} = \frac{3 k_B T_{\phi}}{2}}[/math] The total average scalar particle radiation number density is equivalent to the scalar particle radiation energy density divided by the average scalar particle energy: [math]n_{\phi} = \frac{\epsilon_{\phi}}{E_{\phi}} = \frac{2 \epsilon_{\phi}}{3 k_B T_{\phi}} = \frac{2 \alpha_{\phi} T_{\phi}^3}{3 k_B} = \frac{\pi^2}{45} \left( \frac{k_B T_{\phi}}{\hbar c} \right)^3[/math] Total average scalar particle radiation number density: [math]\boxed{n_{\phi} = \frac{\pi^2}{45} \left( \frac{k_B T_{\phi}}{\hbar c} \right)^3}[/math] The total Observable Universe dark matter fraction is equivalent to the dark matter compositional fraction divided by the total compositional mass fractions: [math]\boxed{\Omega_{dmf} = \left( \frac{\Omega_{dm}}{\Omega_{dm} + \Omega_{b}} \right)}[/math] Observable Universe total density derived from dark matter scalar particles integration via substitution: [math]\rho_{u, \phi} = \frac{n_{\phi} m_{\phi}}{\Omega_{dmf}} = n_{\phi} m_{\phi} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{dm}} \right) = \frac{\pi^2 m_{\phi}}{45} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{dm}} \right) \left( \frac{k_B T_{\phi}}{\hbar c} \right)^3[/math] Observable Universe total density derived from dark matter scalar particles: [math]\boxed{\rho_{u, \phi} = \frac{\pi^2 m_{\phi}}{45} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{dm}} \right) \left( \frac{k_B T_{\phi}}{\hbar c} \right)^3}[/math] Observable Universe total density derived from cosmic background radiation and dark energy: [math]\boxed{\rho_{u, \gamma , \Lambda} = \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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 density derived from dark matter scalar particles is equivalent to the Observable Universe total density derived from cosmic background radiation and dark energy: [math]\boxed{\rho_{u, \phi} = \rho_{u, \gamma , \Lambda}}[/math] Integration via substitution: [math]\frac{\pi^2 m_{\phi}}{45} \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{dm}} \right) \left( \frac{k_B T_{\phi}}{\hbar c} \right)^3 = \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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] Solve for dark matter scalar particle mass: [math]m_{\phi}[/math] Dark matter scalar particle mass: [math]\boxed{m_{\phi} = \frac{45}{\pi^2} \left( \frac{\Omega_{dm}}{\Omega_{dm} + \Omega_{b}} \right) \left( \frac{\hbar c}{k_B T_{\phi}} \right)^3 \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\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] Dark matter scalar particle mass: [math]\boxed{m_{\phi} = 4.309 \cdot 10^{-35} \; \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Neutrinos in Particle Physics, Astrophysics and Cosmology https://books.google.ca/books?id=mRGThuuYHgsC&pg=PA363#v=onepage&q&f=false PHYS 390 Lecture 13 - Simon Fraser University http://www.sfu.ca/~boal/390lecs/390lec13.pdf The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf Dark Matter - Wikipedia https://en.wikipedia.org/wiki/Dark_matter

Observable Universe total galaxy number: (ref. 4, 5, 6) [math]N_g = 1.7 \cdot 10^{11} \; \; \; \text{galaxies}[/math] Milky Way galaxy mass: (ref. 1, pg. 1) [math]M_{mw} = 1.26 \cdot 10^{12} \cdot M_{\odot}[/math] Observable Universe average galaxy mass: [math]M_a = f_{a} \cdot M_{mw}[/math] Dimensionless average galaxy mass fraction: [math]f_{a}[/math] CEI galaxy baryon density: (ref. 2, pg. 3) [math]\Omega_b = 0.045[/math] Galaxy compositional fraction is equivalent to baryon rest mass compositional fraction: [math]\boxed{\Omega_g = \Omega_b}[/math] The total Observable Universe galactic mass fraction is equivalent to the baryon rest mass fraction divided by the total compositional mass fractions. [math]\boxed{\Omega_{gf} = \left( \frac{\Omega_{b}}{\Omega_{dm} + \Omega_{b}} \right)}[/math] Observable Universe total mass from average galaxy mass: [math]M_{u,1} = \frac{N_g M_a}{\Omega_{gf}} = N_g M_a \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{b}} \right)[/math] Observable Universe total mass from cosmic background radiation: (ref. 3) [math]M_{u,2} = \frac{4 \pi R_u^3}{3} \left( \frac{\epsilon_r}{c^2 \Omega_{rf}} + \rho_{\Lambda} \right) = \frac{4 \pi R_u^3}{3} \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right][/math] Observable Universe total mass from average galaxy mass is equivalent to Observable Universe total mass from cosmic background radiation: [math]\boxed{M_{u,1} = M_{u,2}}[/math] Integration via substitution: [math]N_g M_a \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{b}} \right) = \frac{4 \pi R_u^3}{3} \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right][/math] Solve for average galaxy mass fraction: [math]f_{a}[/math]. Observable Universe average galaxy mass fraction: [math]\boxed{f_a = \frac{4 \pi R_u^3}{3 N_g M_{mw}} \left( \frac{\Omega_{b}}{\Omega_{dm} + \Omega_{b}} \right) \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right]}[/math] Observable Universe dimensionless average galaxy mass fraction: [math]\boxed{f_a = 0.962}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Mass models of the Milky Way: http://arxiv.org/pdf/1102.4340v1 The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf Observable Universe total mass from cosmic background radiation: http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry864237 A Map of the Universe: http://arxiv.org/pdf/astro-ph/0310571v2 Galaxy: https://en.wikipedia.org/wiki/Galaxy List of galaxies: https://en.wikipedia.org/wiki/List_of_galaxies

Observable Universe average stellar mass: (ref. 1, pg. 20) [math]M_a = 0.6 \cdot M_{\odot}[/math] CEI stellar Baryon density: (ref. 2, pg. 3) [math]\Omega_s = (\Omega_{ms} + \Omega_{wd} + \Omega_{ns}) = 0.00246[/math] Observable Universe total mass from average stellar mass: (ref. 3) [math]M_{u,1} = \frac{N_s M_a}{\Omega_{sf}} = N_s M_a \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{s}} \right)[/math] Observable Universe total mass from cosmic background radiation: (ref. 4) [math]M_{u,2} = \frac{4 \pi R_u^3}{3} \left( \frac{\epsilon_r}{c^2 \Omega_{rf}} + \rho_{\Lambda} \right) = \frac{4 \pi R_u^3}{3} \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right][/math] Observable Universe total mass from average stellar mass is equivalent to Observable Universe total mass from cosmic background radiation: [math]\boxed{M_{u,1} = M_{u,2}}[/math] Integration via substitution: [math]N_s M_a \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{s}} \right) = \frac{4 \pi R_u^3}{3} \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right][/math] Solve for total observable stellar number [math]N_s[/math]. Observable Universe total observable stellar number: [math]\boxed{N_s = \frac{4 \pi R_u^3}{3 \cdot 0.6 \cdot M_{\odot}} \left( \frac{\Omega_{s}}{\Omega_{dm} + \Omega_{b}} \right) \left[ \frac{\epsilon_r}{c^2} \left( \frac{\Omega_{dm} + \Omega_b}{\Omega_{\gamma} + \Omega_{\nu}} \right)+ \rho_{\Lambda} \right]}[/math] [math]\boxed{N_s = 1.877 \cdot 10^{22} \; \; \; \text{stars}}[/math] Hubble Space Telescope (HST) stellar survey Observable Universe total observable stellar number: [math]\boxed{N_s = 1.000 \cdot 10^{22} \; \; \; \text{stars}}[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: On The Mass Distribution Of Stars...: http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf The Cosmic Energy Inventory: http://arxiv.org/pdf/astro-ph/0406095v2.pdf Observable Universe total mass from stellar mass: http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry839572 Observable Universe total mass from cosmic background radiation: http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry864237

Observable Universe total mass integration via substitution: [math] M_u = \rho_u V_u = \frac{4 \pi R_u^3}{3} \left( \frac{\epsilon_r}{c^2 \Omega_{rf}} + \rho_{\Lambda} \right) = \frac{4 \pi R_u^3}{3} \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\nu} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right] [/math] Observable Universe total mass: [math] \boxed{M_u = \frac{4 \pi R_u^3}{3} \left[ \frac{\pi^2 k_B^4}{15 \hbar^3 c^5} \left( T_{\gamma}^4 + T_{\nu}^4 \left( \frac{7}{8} \right) N_{\nu} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]} [/math] [math] \boxed{M_u = 2.504 \cdot 10^{54} \; \text{kg}} [/math] Any discussions and/or peer reviews about this specific topic thread? Reference: http://www.scienceforums.net/topic/86694-observable-universe-mass/