Observable Universe mass...

Orion1 · July 14, 2015

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

Edited July 14, 2015 by Orion1

Mordred · July 14, 2015

Neutrinos as far as I know follow Fermi Dirac statistics. The majorana statistics application to neutrinos is still hypothetical( not conclusive enough atm). To the best of my knowledge.

In so far as a majarana neutrino would be a new type. As yet undiscovered.

Edited July 14, 2015 by Mordred

Orion1 · July 17, 2015

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

Edited July 17, 2015 by Orion1

Orion1 · July 18, 2015

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

Edited July 18, 2015 by Orion1

Orion1 · August 27, 2015

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

Orion1 · August 31, 2015

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

Edited August 31, 2015 by Orion1

Orion1 · September 12, 2015

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

Edited September 12, 2015 by Orion1

Orion1 · September 12, 2015

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

Edited September 13, 2015 by Orion1

Orion1 · October 4, 2015

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

Edited October 4, 2015 by Orion1

Orion1 · January 15, 2016

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

Edited January 15, 2016 by Orion1

Orion1 · January 23, 2016

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

Edited January 23, 2016 by Orion1

Orion1 · March 4, 2016

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

Edited March 4, 2016 by Orion1

Orion1 · March 5, 2016

Planck satellite cosmological parameters: (ref. 1, pg. 11)
[math]\Omega_{dm} = 0.268[/math]

Dark matter scalar particle composition is equivalent to dark matter composition:
[math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math]

Dark matter scalar particle species total effective degeneracy number: (ref. 2)
[math]\boxed{N_{\phi} = 1}[/math]

Dark matter scalar particle radiation temperature is equivalent to neutrino cosmic background radiation temperature:
[math]\boxed{T_{\phi} = T_{\nu}}[/math]

Bose-Einstein total dark matter scalar particle distribution constant: (ref. 3)
[math]C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5} = 3.640 \cdot 10^{-4}[/math]

[math]\boxed{C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math]

[math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math]

Observable Universe total mass:
[math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math]

[math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math]

[math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math]

Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number: (ref. 3)
[math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math]

Dark matter scalar particle rest mass:
[math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\phi} = T_{\nu}[/math]

Observable Universe total dark matter scalar particle number:
[math]N_{\phi} = \Omega_{\phi} \left( \frac{M_u}{m_{\phi}} \right) = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math]

[math]\boxed{N_{\phi} = \frac{\pi \Omega_{\phi}}{6 m_{\phi}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{\phi} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math]

[math]\boxed{N_{\phi} = 1.544 \cdot 10^{93} \; \phi \; \text{particles}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:
Planck 2013 results. XVI. Cosmological parameters: (ref. 1)
http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

Orion 1 - total effective degeneracy number: (ref. 2)
http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233

Orion1 - dark matter scalar particle composition: (ref. 3)
http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry887017

Dark matter - Wikipedia
https://en.wikipedia.org/wiki/Dark_matter

Edited March 5, 2016 by Orion1

Mordred · March 5, 2016

Are you basing your dark matter scalar particle composition on dark matter being a flavor of neutrinos? Ie neutralinos or sterile neutriinos ? Isn't that model dependant? Ie S0(10) MSSM

something along these lines?

http://arxiv.org/pdf/1306.4954v1.pdf

If you could can you post a better peer review detail on which specifics your following for the dark matter scalar particle composition.

Part of the reason I ask is your particle number density for DM is nearly the same as the number density most textbooks give for the number of elementary particles 10^90 particles.

Edited March 5, 2016 by Mordred

Orion1 · March 6, 2016

Are you basing your dark matter scalar particle composition on dark matter being a flavor of neutrinos?

Negative, a scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number [math]N_n[/math]. (ref. 1), (ref. 2)

[math]\begin{tabular}{l*{6}{c}r} bosons (b) = integer spin & Bose-Einstein statistics \\ fermions (f) = half-integer spin & Fermi-Dirac statistics \\ \end{tabular}[/math]

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math]

Symbolic identity key:

[math]n_s[/math] - spin states total number

[math]N_s[/math] - species total number

[math]N_n[/math] - total effective degeneracy number

If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math]

If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math]

Note that if each neutrino species has a corresponding supersymmetric sterile neutrino species, then [math]\boxed{N_n = N_s = 3}[/math].

Neutralinos and sterile neutrinos are hypothetical particles. (ref. 3), (ref. 4)

The mass spectrum range for neutralinos is 10 GeV to 1 TeV. (ref. 3)

The mass spectrum range for sterile neutrinos is 1 eV to 10^15 GeV. (ref. 4)

Isn't that model dependent? Ie S0(10) MSSM

Affirmative, theses particles are theoretically embedded into SO(10) MSSM models. (ref. 5), (ref. 6), (ref. 11)

If you could can you post a better peer review detail on which specifics your following for the dark matter scalar particle composition.

See PHYS: 652 Cosmic Inventory I: Radiation (ref. 9) and Formaggio - Relic Neutrinos - Institute for Nuclear Theory (ref. 10) and Orion1 - Dark matter scalar particle radiation energy density Bose-Einstein distribution (ref. 12) and Orion1 - Photon radiation energy density Bose-Einstein distribution (ref. 13) in Reference.

the number density most textbooks give for the number of elementary particles 10^90 particles.

Do you have a citation reference link for the equation derivation for that number?

Reference:

Wikipedia - Scalar boson (ref. 1)

https://en.wikipedia.org/wiki/Scalar_boson

Wikipedia - Fermion (ref. 2)

https://en.wikipedia.org/wiki/Fermion

Wikipedia - Neutralino - Origins in supersymmetric theories (ref. 3)

https://en.wikipedia.org/wiki/Neutralino#Origins_in_supersymmetric_theories

Wikipedia - Sterile neutrinos (ref. 4)

https://en.wikipedia.org/wiki/Sterile_neutrino

Wikipedia - Standard Model - Total particle count (ref. 5)

https://en.wikipedia.org/wiki/Standard_Model#Total_particle_count

Wikipedia - SO[10] (ref. 6)

https://en.wikipedia.org/wiki/SO%2810%29

CERN - Next decade of sterile neutrino studies (ref. 7)

http://arxiv.org/pdf/1306.4954v1.pdf

Wikipedia - Dark matter (ref. 8)

https://en.wikipedia.org/wiki/Dark_matter

PHYS: 652 Cosmic Inventory I: Radiation (ref. 9)

http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

Formaggio - Relic Neutrinos - Institute for Nuclear Theory (ref. 10)

http://www.int.washington.edu/talks/WorkShops/int_10_44W/People/Formaggio_J/Formaggio.pdf

WIkipedia - Minimal supersymmetric standard model - dark matter (ref. 11)

https://en.wikipedia.org/wiki/Minimal_Supersymmetric_Standard_Model#Dark_matter

Orion1 - Dark matter scalar particle radiation energy density Bose-Einstein distribution (ref. 12)

http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/?p=887017

Orion1 - Photon radiation energy density Bose-Einstein distribution (ref. 13)

http://www.scienceforums.net/topic/86694-observable-universe-mass/?p=884367

Edited March 6, 2016 by Orion1

Mordred · March 6, 2016

Yes I know the spin zero correlates to scalar particles. My question is why are you considering dark matter as a scalar quantity?

Ah didn't see the portion where your counting as a sterile neutrino.

Question answered

Except according to your references sterile neutrinos has a spin 1/2 statistic which is not scalar. Nor is it a boson but is a fermion

Which means you should be using the Fermi-Dirac statistics for dark matter not the Bose-Einstein statistics check spin statistic reference 4

Edited March 6, 2016 by Mordred

Orion1 · March 7, 2016

Yes I know the spin zero correlates to scalar particles. My question is why are you considering dark matter as a scalar quantity?

Sterile neutrinos and neutralinos are potential dark matter candidates. However, scalar particles are also dark matter candidates. (ref. 1)

Scalar particles are only the first particles on that quantum spin list to model as dark matter.

Reference:

Wikipedia - Scalar field dark matter (ref. 1)

https://en.wikipedia.org/wiki/Scalar_field_dark_matter

Edited March 7, 2016 by Orion1

Mordred · March 7, 2016

Yeah gotcha, I looked at the reference. The reference papers are modelling DM as being ultralight. I noted the references are back on 2000.

They are based partly on quintessence. Or at least the papers include quintessence.

If I'm not mistaken it's a counter argument for hot dark matter rather than cold dark matter. The ultralight mass would correspond to that.

Here is one of the reference papers for that wiki page

http://arxiv.org/abs/astro-ph/0004332

Judging from your math skills and knowledge. As well as attention to detail. I would hazard a guess that your working on a thesis paper.

Showing various modelling of DM is a good step. I'm assuming based on the above that you will probably include the fermionic sterile neutrinos idea into your paper

If I'm correct let me know and I'll go through my database of articles for reference materials

Orion1 · March 12, 2016

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)

[math]\Omega_{dm} = 0.63[/math]

[math]\Omega_{b} = 0.12[/math]

[math]\Omega_{\gamma} = 0.15[/math]

[math]\Omega_{\nu} = 0.10[/math]

[math]\Omega_{\Lambda} \neq 0[/math]

Planck satellite redshift parameter at photon decoupling time: (ref. 2, pg. 11)

[math]z = 1090.43[/math]

Cosmology scale factor:

[math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math]

Symbolic definition key:

[math]R_u(t_0)[/math] - Observable Universe total radius at present time.

[math]R_u(t)[/math] - Observable Universe total radius at past time.

[math]a(t_0)[/math] - scale factor at present time.

[math]a(t)[/math] - scale factor at past time.

[math]T_t[/math] - cosmic background radiation temperature at past time.

[math]T_0[/math] - cosmic background radiation temperature at present time.

[math]z[/math] - redshift at photon decoupling time.

Observable Universe total co-moving radius at present time: (ref. 3)

[math]\boxed{R_u(t_0) = \frac{c \ln (1 + z)}{2 H_0}}[/math]

Observable Universe total radius at photon decoupling time:

[math]R_u(t) = \frac{R_u(t_0)}{1 + z} = \frac{c \ln (1 + z)}{2 H_0 (1 + z)} = 4.220 \cdot 10^{23} \; \text{m}[/math]

[math]\boxed{R_u(t) = \frac{c \ln (1 + z)}{2 H_0 (1 + z)}}[/math]

[math]\boxed{R_u(t) = 4.220 \cdot 10^{23} \; \text{m}} \; \; \; 4.461 \cdot 10^{7} \; \text{ly}[/math]

Observable Universe photon cosmic background radiation temperature at photon decoupling time:

[math]T_{\gamma , t} = T_{\gamma} (1 + z) = 2974.67 \; \text{K}[/math]

[math]\boxed{T_{\gamma , t} = 2974.67 \; \text{K}}[/math]

Observable Universe neutrino cosmic background radiation temperature at photon decoupling time:

[math]T_{\nu , t} = T_{\nu} (1 + z) = 2123.22 \; \text{K}[/math]

[math]\boxed{T_{\nu , t} = 2123.22 \; \text{K}}[/math]

Observable Universe total mass at photon decoupling time:

[math]\boxed{M_u = \frac{4 \pi}{3} \left( \frac{c \ln (1 + z)}{2 H_0 (1 + z)} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math]

Observable Universe total mass at photon decoupling time:

[math]\boxed{M_u = 1.245 \cdot 10^{54} \; \text{kg}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:

WMAP satellite content of the Universe: (ref. 1)

http://map.gsfc.nasa.gov/media/080998/index.html

Planck 2013 results. XVI. Cosmological parameters: (ref. 2)

http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

UCLA Division of Astronomy and Astrophysics - Homogeneity and Isotropy: (ref. 3)

http://www.astro.ucla.edu/~wright/cosmo_02.htm#DL

Orion1 - Friedmann equations critical mass: (ref. 4)

http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915

Edited March 12, 2016 by Orion1

Orion1 · March 27, 2016

A scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and
sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number [math]N_n[/math]. (ref. 1), (ref. 2)

[math]\begin{tabular}{l*{6}{c}r} bosons (b) = integer spin & Bose-Einstein statistics \\ fermions (f) = half-integer spin & Fermi-Dirac statistics \\ \end{tabular}[/math]

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math]

Symbolic identity key:
[math]n_s[/math] - spin states total number
[math]N_s[/math] - species total number
[math]N_n[/math] - total effective degeneracy number
If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math]
If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math]

Note that if each neutrino species has a corresponding supersymmetric sterile neutrino species, then [math]\boxed{N_n = N_s = 3}[/math].

Neutralinos and sterile neutrinos are hypothetical particles. (ref. 3), (ref. 4)
The mass spectrum range for neutralinos is 10 GeV to 1 TeV. (ref. 3)
The mass spectrum range for sterile neutrinos is 1 eV to 10^15 GeV. (ref. 4)
These particles are theoretically embedded into SO(10) MSSM models. (ref. 5), (ref. 6), (ref. 11)

Observable Universe baryonic Hydrogen composition:

[math]\Omega_{\text{H}} = 0.75[/math]

Observable Universe baryonic Helium composition:

[math]\Omega_{\text{He}} = 0.25[/math]

Observable Universe total Hydrogen particle number:

[math]N_{\text{H}} = \Omega_b \Omega_{\text{H}} \left( \frac{M_u}{m_{\text{H}}} \right) = 8.417 \cdot 10^{79} \; \text{H particles}[/math]

[math]\boxed{N_{\text{H}} = \Omega_b \Omega_{\text{H}} \left( \frac{M_u}{m_{\text{H}}} \right)}[/math]

[math]\boxed{N_{\text{H}} = 8.417 \cdot 10^{79} \; \text{H particles}}[/math]

Observable Universe total Helium particle number:

[math]N_{\text{He}} = \Omega_b \Omega_{\text{He}} \left( \frac{M_u}{m_{\text{He}}} \right) = 7.065 \cdot 10^{78} \; \text{He particles}[/math]

[math]\boxed{N_{\text{He}} = \Omega_b \Omega_{\text{He}} \left( \frac{M_u}{m_{\text{He}}} \right)}[/math]

[math]\boxed{N_{\text{He}} = 7.065 \cdot 10^{78} \; \text{He particles}}[/math]

Observable Universe total proton particle number:

[math]N_{pt} = N_{\text{H}} + 2N_{\text{He}} = 9.830 \cdot 10^{79} \; \text{p particles}[/math]

[math]\boxed{N_{pt} = N_{\text{H}} + 2N_{\text{He}}}[/math]

[math]\boxed{N_{pt} = 9.830 \cdot 10^{79} \; \text{p particles}}[/math]

Observable Universe total neutron particle number:

[math]N_{nt} = 2N_{\text{He}} = 1.413 \cdot 10^{79} \; \text{n particles}[/math]

[math]\boxed{N_{nt} = 1.413 \cdot 10^{79} \; \text{n particles}}[/math]

Observable Universe total electron particle number:

[math]N_{\beta} = N_{\text{H}} + 2 N_{\text{He}} = 9.830 \cdot 10^{79} \; \beta \; \text{particles}[/math]

[math]\boxed{N_{\beta t} = N_{\text{H}} + 2 N_{\text{He}}}[/math]

[math]\boxed{N_{\beta t} = 9.830 \cdot 10^{79} \; \beta \; \text{particles}}[/math]

Observable Universe total up quark particle number:

[math]N_{u t} = 2 N_{\text{H}} + 6 N_{\text{He}} = 2.107 \cdot 10^{80} \; \text{u quark particles}[/math]

[math]\boxed{N_{u t} = 2.107 \cdot 10^{80} \; \text{u quark particles}}[/math]

Observable Universe total down quark particle number:

[math]N_{d t} = N_{\text{H}} + 6 N_{\text{He}} = 1.266 \cdot 10^{80} \; \text{d quark particles}[/math]

[math]\boxed{N_{d t} = 1.266 \cdot 10^{80} \; \text{d quark particles}}[/math]

Observable Universe total photon particle number: (ref. 13, eq. 7)

[math]N_{\gamma t} = n_{\gamma} V_u = \left( \frac{\zeta (3) N_{\gamma} (k_{B} T_{\gamma} )^3}{\pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\gamma}}{6 \pi} \left( \frac{k_B T_{\gamma} \ln (1 + z)}{\hbar H_0} \right)^3 = 1.682 \cdot 10^{89} \; \gamma \; \text{particles}[/math]

[math]\boxed{N_{\gamma t} = \frac{\zeta (3) N_{\gamma}}{6 \pi} \left( \frac{k_B T_{\gamma} \ln (1 + z)}{\hbar H_0} \right)^3}[/math]

[math]\boxed{N_{\gamma t} = 1.682 \cdot 10^{89} \; \gamma \; \text{particles}}[/math]

Observable Universe total neutrino particle number: (ref. 13, eq. 8)

[math]N_{\nu t} = n_{\nu} V_u = \left( \frac{3 \zeta (3) N_{\nu} (k_{B} T_{\nu} )^3}{4 \pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3 = 6.984 \cdot 10^{88} \; \nu \; \text{particles}[/math]

[math]\boxed{N_{\nu t} = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3}[/math]

[math]\boxed{N_{\nu t} = 6.984 \cdot 10^{88} \; \nu \; \text{particles}}[/math]

Observable Universe total dark matter scalar particle number: (ref. 13, eq. 7)

[math]N_{\phi t} = n_{\phi} V_u = \left( \frac{\zeta (3) N_{\phi} (k_{B} T_{\phi} )^3}{\pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\phi}}{6 \pi} \left( \frac{k_B T_{\phi} \ln (1 + z)}{\hbar H_0} \right)^3 = 3.057 \cdot 10^{88} \; \phi \; \text{particles}[/math]

[math]\boxed{N_{\phi t} = \frac{\zeta (3) N_{\phi}}{6 \pi} \left( \frac{k_B T_{\phi} \ln (1 + z)}{\hbar H_0} \right)^3}[/math]

[math]\boxed{N_{\phi t} = 3.057 \cdot 10^{88} \; \phi \; \text{particles}}[/math]

Observable Universe total dark matter sterile neutrino particle number: (ref. 13, eq. 8)

[math]N_{\nu t} = n_{\nu} V_u = \left( \frac{3 \zeta (3) N_{\nu} (k_{B} T_{\nu} )^3}{4 \pi^2 (\hbar c)^3} \right) \left[ \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_{0}} \right)^3 \right] = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3 = 6.984 \cdot 10^{88} \; \nu \; \text{particles}[/math]

[math]\boxed{N_{\nu t} = \frac{\zeta (3) N_{\nu}}{8 \pi} \left( \frac{k_B T_{\nu} \ln (1 + z)}{\hbar H_0} \right)^3}[/math]

[math]\boxed{N_{\nu t} = 6.984 \cdot 10^{88} \; \nu \; \text{particles}}[/math]

Observable Universe dark matter scalar particle total fundamental particle number:

[math]N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\phi t}) = 2.686 \cdot 10^{89} \; \text{particles}[/math]

[math]\boxed{N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\phi t})}[/math]

[math]\boxed{N_t = 2.686 \cdot 10^{89} \; \text{particles}}[/math]

Observable Universe dark matter sterile neutrino particle total fundamental particle number:

[math]N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\nu t}) = 3.078 \cdot 10^{89} \; \text{particles}[/math]

[math]\boxed{N_t = (N_{ut} + N_{dt} + N_{\beta t} + N_{\gamma t} + N_{\nu t} + N_{\nu t})}[/math]

[math]\boxed{N_t = 3.078 \cdot 10^{89} \; \text{particles}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Introduction to Cosmology: Lecture 6 - Thermal history of the Universe - Joao G. Rosa (ref. 13)

http://gravitation.web.ua.pt/sites/gravitation.web.ua.pt/files/Lecture_6.pdf

Edited March 27, 2016 by Orion1

Orion1 · April 4, 2016

Observable Universe electron neutrino mass:

[math]m_{\nu_e} = \frac{\Omega_{\nu} M_u}{N_{\nu t}} = 6.839 \cdot 10^{-38} \; \text{kg}[/math]

[math]\boxed{m_{\nu_e} = \frac{\Omega_{\nu} M_u}{N_{\nu t}}}[/math]

[math]\boxed{m_{\nu_e} = 6.839 \cdot 10^{-38} \; \text{kg}}[/math]

Observable Universe dark matter scalar particle mass:

[math]m_{\phi} = \frac{\Omega_{\phi} M_u}{N_{\phi t}} = 3.326 \cdot 10^{-35} \; \text{kg}[/math]

[math]\boxed{m_{\phi} = \frac{\Omega_{\phi} M_u}{N_{\phi t}}}[/math]

[math]\boxed{m_{\phi} = 3.326 \cdot 10^{-35} \; \text{kg}}[/math]

Observable Universe dark matter sterile neutrino mass:

[math]m_{\nu} = \frac{\Omega_{dm} M_u}{N_{\nu t}} = 1.456 \cdot 10^{-35} \; \text{kg}[/math]

[math]\boxed{m_{\nu} = \frac{\Omega_{dm} M_u}{N_{\nu t}}}[/math]

[math]\boxed{m_{\nu} = 1.456 \cdot 10^{-35} \; \text{kg}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:

Wikipedia - Cosmic neutrino background radiation: (ref. 1)
https://en.wikipedia.org/wiki/Cosmic_neutrino_background

Wikipedia - Scalar field dark matter: (ref. 2)
https://en.wikipedia.org/wiki/Scalar_field_dark_matter

Wikipedia - Sterile neutrino: (ref. 3)
https://en.wikipedia.org/wiki/Sterile_neutrino

Orion1 · July 11, 2016

A scalar particle is a boson with spin 0. A flavor of neutrinos such as neutralinos and
sterile neutrinos are fermions with spin 1/2. However, the resulting mathematical modelling of physical radiation properties due to the differences in quantum spin properties are intrinsically built into this model as total effective degeneracy number $256bed78c559f82e94a4589e0e8a15cc-1.png$ . (ref. 1), (ref. 2)

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math]

Symbolic identity key:
[math]n_s[/math] - spin states total number
[math]N_s[/math] - species total number
[math]N_n[/math] - total effective degeneracy number
If [math]n_s \geq N_s[/math] Then [math]N_n = n_s[/math]
If [math]n_s \leq N_s[/math] Then [math]N_n = N_s[/math]

Dark Energy scalar particle mass:

[math]\boxed{m_{\Lambda} = 0}[/math]

Dark Energy scalar particle mass total effective degeneracy number:

[math]\boxed{N_{\Lambda} = 1}[/math]

Planck satellite cosmological parameters: (ref. 3, pg. 11)

[math]\Omega_{\Lambda} = 0.6825[/math]

Planck satellite redshift parameter at photon decoupling time: (ref. 3, pg. 11)

[math]z = 1090.43[/math]

Dark Energy scalar particle radiation energy density Bose-Einstein distribution integration via substitution:

[math]\epsilon_{\Lambda , 1} = \alpha_{\Lambda} T_{\Lambda}^4 = \frac{4 \pi N_{\Lambda} (k_B T_{\Lambda})^4}{( 2 \pi \hbar c )^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\Lambda})}} - 1} d \omega = \frac{4 \pi N_{\Lambda} (k_B T_{\Lambda})^4}{( 2 \pi \hbar c )^3} \left( \frac{\pi^4}{15} \right) = \frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3}[/math]

Dark Energy scalar particle radiation energy density:

[math]\boxed{\epsilon_{\Lambda , 1} = \frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3}}[/math]

Dark Energy scalar particle radiation constant:

[math]\boxed{\alpha_{\Lambda} = \frac{\pi^2 k_B^4 N_{\Lambda}}{30 ( \hbar c )^3}}[/math]

Dark Energy scalar particle radiation composition energy density: (ref. 7)

[math]\epsilon_{\Lambda , 2} = \rho_{\Lambda} c^2 = \frac{3 \Omega_{\Lambda} \left(c H_{0} \right)^2}{8 \pi G} = 5.685 \cdot 10^{-10} \; \frac{\text{j}}{\text{m}^3}[/math]

Dark Energy scalar particle radiation composition energy density:

[math]\boxed{\epsilon_{\Lambda , 2} = \frac{3 \Omega_{\Lambda} \left(c H_{0} \right)^2}{8 \pi G}}[/math]

Dark Energy scalar particle radiation composition energy density:

[math]\boxed{\epsilon_{\Lambda , 2} = 5.685 \cdot 10^{-10} \; \frac{\text{j}}{\text{m}^3}}[/math]

Dark Energy scalar particle radiation energy density is equivalent to Dark Energy scalar radiation composition energy density:

[math]\boxed{\epsilon_{\Lambda , 1} = \epsilon_{\Lambda , 2}}[/math]

Dark Energy scalar particle radiation energy density is equivalent to Dark Energy scalar particle radiation composition energy density integration via substitution:

[math]\frac{\pi^2 N_{\Lambda} (k_B T_{\Lambda})^4}{30 ( \hbar c )^3} = \frac{3 \Omega_{\Lambda} \left(c H_0 \right)^2}{8 \pi G}[/math]

Solve for Dark Energy scalar particle radiation temperature: [math]T_{\Lambda}[/math]

[math]T_{\Lambda}^4 = \frac{30 ( \hbar c )^3}{\pi^2 N_{\Lambda} k_B^4} \left( \frac{3 \Omega_{\Lambda} \left(c H_0 \right)^2}{8 \pi G} \right) = \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4}[/math]

[math]T_{\Lambda}^4 = \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4}[/math]

Dark Energy scalar particle radiation temperature:

[math]\boxed{T_{\Lambda} = \left( \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_{B}^4} \right)^{1/4}}[/math]

Dark Energy scalar particle radiation temperature:

[math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math]

Observable Universe total Dark Energy scalar particle number: (ref. 5, eq. 7)

[math]\boxed{N_{\Lambda t} = \frac{\zeta (3) N_{\Lambda}}{6 \pi} \left( \frac{k_B T_{\Lambda} \ln (1 + z)}{\hbar H_0} \right)^3}[/math]

Observable Universe total Dark Energy scalar particle number: (ref. 5, eq. 7)

[math]\boxed{N_{\Lambda t} = 1.782 \cdot 10^{92} \; \Lambda \; \text{particles}}[/math]

Dark Energy scalar particle peak energy:

[math]E_{\Lambda} = k_B T_{\Lambda}[/math]

Dark Energy scalar particle peak frequency:

[math]E_{\Lambda} = 2 \pi \hbar f_{\Lambda} = k_B T_{\Lambda}[/math]

Dark Energy scalar particle peak frequency integration via substitution:

[math]f_{\Lambda} = \frac{k_B T_{\Lambda}}{2 \pi \hbar} = \frac{k_B}{2 \pi \hbar} \left( \frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3 k_B^4} \right)^{1/4} = \left( \frac{45 \Omega_{\Lambda} c^5}{64 \hbar G N_{\Lambda} \pi^7} \right)^{1/4}[/math]

Dark Energy scalar particle peak T-ray frequency: (ref. 6)

[math]\boxed{f_{\Lambda} = \left( \frac{45 \Omega_{\Lambda} c^5}{64 \hbar G N_{\Lambda} \pi^7} \right)^{1/4}}[/math]

Dark Energy scalar particle peak terahertz radiation T-ray frequency: (ref. 6)

[math]\boxed{f_{\Lambda} = 7.295 \cdot 10^{11} \; \text{Hz}}[/math]

[math]f_{\Lambda} = 7.295 \cdot 10^{11} \; \text{Hz} = 0.729 \; \text{THz}[/math]

Dark Energy scalar particle peak terahertz radiation T-ray frequency: (ref. 6)

[math]\boxed{f_{\Lambda} = 0.729 \; \text{THz}}[/math]

Dark Energy scalar particle peak energy:

[math]\boxed{E_{\Lambda} = 4.834 \cdot 10^{-22} \; \text{j}}[/math]

Dark Energy scalar particle peak energy:

[math]\boxed{E_{\Lambda} = 3.018 \; \frac{\text{mV}}{c^2}}[/math]

What are the possibilities for a terahertz antenna or a terahertz receiver tuned to the Dark Energy scalar particle peak terahertz frequency of being capable of detecting Dark Energy scalar particle terahertz T-ray radiation?

What are the possibilities of terahertz radiation T-ray astronomy?

Any discussions and/or peer reviews about this specific topic thread?

Reference:
Wikipedia - Scalar boson (ref. 1)
https://en.wikipedia...ki/Scalar_boson

Wikipedia - Fermion (ref. 2)
https://en.wikipedia.org/wiki/Fermion

Planck 2013 results. XVI. Cosmological parameters: (ref. 3)

http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

Wikipedia - Dark energy: (ref. 4)

https://en.wikipedia.org/wiki/Dark_energy

Introduction to Cosmology: Lecture 6 - Thermal history of the Universe - Joao G. Rosa (ref. 5)

http://gravitation.web.ua.pt/sites/gravitation.web.ua.pt/files/Lecture_6.pdf

Wikipedia - Terahertz radiation: (ref. 6)

https://en.wikipedia.org/wiki/Terahertz_radiation

Orion1 - Dark Energy mass density: (ref. 7)

http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850783

Edited July 11, 2016 by Orion1

Orion1 · April 1, 2017

Observable Universe Friedmann equations critical mass: (ref. 1)

[math]M_c = \frac{H_0^2 R_u^3}{2 G} = \frac{H_0^2}{2 G} \left[ \left( \frac{c}{H_0} \right) \frac{\ln (1 + z)}{2} \right]^3 = \frac{[c \ln (1 + z)]^3}{16 G H_0} = 3.794 \cdot 10^{54} \; \text{kg}[/math]

[math]\boxed{M_c = \frac{[c \ln (1 + z)]^3}{16 G H_0}}[/math]

[math]\boxed{M_c = 3.794 \cdot 10^{54} \; \text{kg}}[/math]

Observable Universe total mass: (ref. 2)

[math]M_u = \rho_u V_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math]

[math]\boxed{M_u = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math]

[math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math]

Observable Universe Friedmann equations critical mass is equivalent to Observable Universe total mass:

[math]\boxed{M_c = M_u}[/math]

Observable Universe Friedmann equations critical mass is equivalent to Observable Universe total mass integration via substitution:

[math]\frac{[c \ln (1 + z)]^3}{16 G H_0} = \frac{\pi}{6} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math]

[math]\frac{8 \pi G}{3 H_0^2} \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right] = 1[/math]

Observable Universe compositional Equation of State:

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \Omega_{\Lambda} = 1}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:

Orion1 - Friedmann equations critical mass: (ref. 1)

http://www.scienceforums.net/topic/86694-observable-universe-mass/#entry850915

Orion1 - Observable Universe total mass: (ref. 2)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry909462

Edited April 1, 2017 by Orion1

Orion1 · April 4, 2017

Photon particle mass:

[math]\boxed{m_{\gamma} = 0}[/math]

Neutrino particle mass:

[math]\boxed{m_{\nu} \neq 0}[/math]

Dark Energy scalar particle mass:

[math]\boxed{m_{\Lambda} = 0}[/math]

Dark Matter scalar particle mass:

[math]\boxed{m_{\phi} \neq 0}[/math]

Observable Universe Bose-Einstein photon composition: (ref. 1)

[math]\boxed{\Omega_{\gamma} = \frac{4 G N_{\gamma} \pi^3 (k_B T_{\gamma})^4}{45 H_0^2 \hbar^3 c^5}}[/math]

Observable Universe Fermi-Dirac neutrino composition: (ref. 2)

[math]\boxed{\Omega_{\nu} = \frac{4 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{3 C_{\nu} H_0^2 \hbar^3 c^5}}[/math]

Observable Universe Bose-Einstein Dark Energy composition: (ref. 3)

[math]\boxed{\Omega_{\Lambda} = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5}}[/math]

Observable Universe Bose-Einstein Dark Matter composition: (ref. 4)

[math]\boxed{\Omega_{\phi} = \frac{4 G N_{\phi} \pi^3 (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}}[/math]

Observable Universe compositional Equation of State:

[math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1[/math]

Observable Universe compositional Equation of State: (ref. 5)

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \Omega_{\Lambda} = 1}[/math]

Observable Universe compositional Equation of State integration via substitution:

[math]\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5} = \frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1[/math]

Observable Universe compositional Equation of State:

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math]

Observable Universe compositional Equation of State: (ref. 4)

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:

Orion1 - Cosmic Microwave Background Radiation photon composition: (ref. 1)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882069

Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 2)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry884367

Orion1 - Dark Energy: (ref. 3)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry931225

Orion1 - Dark Matter scalar particle composition: (ref. 4)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry909539

Orion1 - Observable Universe compositional Equation of State: (ref. 5)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry980016

Edited April 4, 2017 by Orion1

Orion1 · May 1, 2017

[math]\begin{tabular}{l*{6}{c}r} & identity & state & spin & ns & Ns & Nn \\ b & scalar & 0 & 0 & 1 & 1 & 1 \\ f & neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & sterile neutrino & +,- & 1/2 & 2 & 3 & 3 \\ f & neutralino & +,- & 1/2 & 2 & 4 & 4 \\ b & photon & +,- & 1 & 2 & 1 & 2 \\ b & graviton & +,- & 2 & 2 & 1 & 2 \\ \end{tabular}[/math]

Observable Universe composition Equation of State:

[math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1[/math]

Observable Universe photon and neutrino and scalar particle Dark Energy composition Equation of State:

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math]

Observable Universe scalar particle Dark Matter and scalar particle Dark Energy composition Equation of State:

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\phi}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math]

Observable Universe sterile neutrino Dark Matter and scalar particle Dark Energy composition Equation of State:

[math]\boxed{\frac{4 G \pi^3 k_B^4}{3 H_0^2 \hbar^3 c^5} \left[ \left( \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] = 1}[/math]

Observable Universe total energy integration via substitution:

[math]E_u = M_u c^2 = \frac{\pi^3 k_B^4}{12 (\hbar H_0)^3} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] \left( \ln (1 + z) \right)^3 = 3.410 \cdot 10^{71} \; \text{j}[/math]

Observable Universe total energy:

[math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 (\hbar H_0)^3} \left[ \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \right] \left( \ln (1 + z) \right)^3}[/math]

Observable Universe total energy:

[math]\boxed{E_u = 3.410 \cdot 10^{71} \; \text{j}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Edited May 1, 2017 by Orion1

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