Observable Universe mass...

Orion1 · May 13, 2017

Observable Universe Cosmological Constant: (ref. 1), (ref. 2)

[math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

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

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

Observable Universe scalar particle Dark Energy Cosmological Constant integration via substitution:

[math]\Lambda_s = \frac{3 H_0^2}{c^2} \left( \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{45 H_0^2 \hbar^3 c^5} \right) = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

Observable Universe scalar particle Dark Energy Cosmological Constant:

[math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 (k_B T_{\Lambda})^4}{15 \hbar^3 c^7}}[/math]

Observable Universe scalar particle Dark Energy Cosmological Constant:

[math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math]

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

Reference:

Wikipedia - Cosmological Constant: (ref. 1)

http://en.wikipedia.org/wiki/Hubble%27s_law#Matter-dominated_universe_.28with_a_cosmological_constant.29

Orion1 - Cosmological Constant: (ref. 2)

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

Orion1 - Dark Energy: (ref. 3)

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

Edited May 13, 2017 by Orion1

Orion1 · June 28, 2017

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

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

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

Neutrino mass:

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

Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2)

[math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math]

Observable Universe Fermi-Dirac neutrino composition: (ref. 3),(ref. 4)

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

Observable Universe neutrino background radiation neutrino decoupling time: (ref. 5)

[math]\boxed{T_{u , \nu} = \frac{1}{H_{\nu , t}}}[/math]

Solve for Observable Universe neutrino decoupling time [math]T_{u , \nu}[/math]:

[math]T_{u , \nu} = \frac{1}{H_{\nu , t}} = \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.148 \; \text{s}[/math]

Observable Universe neutrino background radiation neutrino decoupling time:

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

Observable Universe neutrino background radiation neutrino decoupling time:

[math]\boxed{T_{u , \nu} = 0.148 \; \text{s}}\; \; \; m_{\nu} \neq 0[/math]

Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 5)

[math]\boxed{T_{u , \nu} = 0.893 \; \text{s}} \; \; \; m_{\nu} = 0[/math]

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

Reference:

Orion1 - WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Wikipedia - neutrino decoupling: (ref. 2)

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

Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 3)

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

Orion1 - Fermi-Dirac and Bose-Einstein total neutrino distribution constant: (ref. 4)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry886860

Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 5)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Edited June 28, 2017 by Orion1

Orion1 · July 1, 2017

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

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

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

Observable Universe Dark Matter scalar particle composition is equivalent to Dark Matter composition at photon decoupling time:

[math]\boxed{\Omega_{\phi , t} = \Omega_{dm , t}}[/math]

Observable Universe Dark Matter scalar particle mass:

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

Observable Universe Dark Matter scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2)

[math]\boxed{T_{\phi , t} = T_{\nu , t}} = 1 \cdot 10^{10} \; \text{K}[/math]

Observable Universe Bose-Einstein Dark Matter scalar particle composition: (ref. 3)

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

Observable Universe Dark Matter scalar particle decoupling time: (ref. 4)

[math]\boxed{T_{u , \phi} = \frac{1}{H_{\phi , t}}}[/math]

Solve for Observable Universe Dark Matter scalar particle decoupling time [math]T_{u , \phi}[/math]:

[math]T_{u , \phi} = \frac{1}{H_{\phi, t}} = \sqrt{\frac{3 \Omega_{\phi,t} C_{\phi} \hbar^3 c^5}{4 G N_{\phi} \pi^3 (k_B T_{\phi,t})^4}} = \frac{}{2 (k_B T_{\phi,t})^2} \sqrt{\frac{3 \Omega_{\phi,t} C_{\phi} \hbar^3 c^5}{G N_{\phi} \pi^3}} = 0.0255 \; \text{s}[/math]

Observable Universe Dark Matter scalar particle decoupling time:

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

Observable Universe Dark Matter scalar particle decoupling time:

[math]\boxed{T_{u, \phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math]

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

Reference:

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

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

Wikipedia - neutrino decoupling: (ref. 2)

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

Orion1 - Dark Matter scalar particle composition: (ref. 3)

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

Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Edited July 1, 2017 by Orion1

Mordred · July 1, 2017

Clarification on DM decoupling time. Your still modelling DM as zero spin in your use of the Bose-Einsten statistics correct?

edit spin zero, p=0 also you are still setting [latex] T_o=T_v[/latex] correct?

edit:Doh forget last question I see the statement.

I would still like some clarity on the decoupling time bssed on neutrino background temp. You may have posted the details already but I would like to see that shown as per the proofs of the equations you used...not stating its inaccurate but not familiar with this particular proof.

If you happen to have a reference correlating relevant boundaries on DM to allow the scalar treatment of DM for the usage you have that would be great.

(I have no objections so far but just looking for clarity ) particularly since the treatments I am familiar with usually has DM freezing out before neutrinos. Hence the details I need.

example the interaction cross section for DM and neutrinos in your examination.

Hot DM would make sense to decouple close to the same as neutrinos however this would not be the case with Cold DM with regards to Hubble expansion rates.

Edited July 1, 2017 by Mordred

Mordred · July 1, 2017

edit never mind I just answered my own question. Then reread the thread and confirmed. lol I had already asked about this but forgot about it. The numbers make sense with the DM variation you provided earlier.

Which if I follow correctly is an LSP variation under SO(10) MSSM.

If thats true here is a reference for your LSP scattering amplitudes.

"Thermal decoupling of WIMPs from first principles"

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-ph/0612238&ved=0ahUKEwjC7JOew-jUAhVDyoMKHbPYAqkQFggfMAE&usg=AFQjCNFHeuJybwvTjGpiDghXhtFBkBtyiw

PS its arxiv on phone atm

Edited July 1, 2017 by Mordred

Orion1 · July 3, 2017

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
[math]\Omega_{dm , t} = 0.63[/math]

[math]\Omega_{\gamma , t} = 0.15[/math]
[math]\Omega_{\Lambda , t} \neq 0[/math]

Observable Universe Dark Matter sterile neutrino composition is equivalent to Dark Matter composition at photon decoupling time:
[math]\boxed{\Omega_{\nu , t} = \Omega_{dm , t}}[/math]

Observable Universe Dark Matter sterile neutrino mass:
[math]\boxed{m_{\nu} \neq 0}[/math]

Observable Universe Dark Matter sterile neutrino decoupling temperature is equivalent to cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2)
[math]\boxed{T_{\nu, t} = T_{\nu , t}} = 1 \cdot 10^{10} \; \text{K}[/math]

Observable Universe Fermi-Dirac Dark Matter sterile neutrino composition: (ref. 3)
[math]\boxed{\Omega_{\nu} = \frac{4 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{3 C_{\nu} H_0^2 \hbar^3 c^5}}[/math]

Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 4)
[math]\boxed{T_{u , \nu} = \frac{1}{H_{\nu , t}}}[/math]

Solve for Observable Universe Dark Matter sterile neutrino decoupling time [math]T_{u , \nu}[/math]:
[math]T_{u , \nu} = \frac{1}{H_{\nu, t}} = \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.372 \; \text{s}[/math]

Observable Universe Dark Matter sterile neutrino decoupling time:
[math]\boxed{T_{u , \nu} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 \Omega_{\nu,t} C_{\nu} \hbar^3 c^5}{G N_{\nu} \pi^3}}}[/math]

Observable Universe Dark Matter sterile neutrino decoupling time:
[math]\boxed{T_{u, \nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math]

---

Particle interaction rate is equivalent to Hubble Parameter at particle decoupling time: (ref. 5), (ref .6)

[math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math]

Solve for Observable Universe photon interaction rate [math]\Gamma_{\gamma,t}[/math] at photon decoupling time:

[math]\Gamma_{\gamma,t} = \sqrt{\frac{4 G N_{\gamma} \pi^3 (k_B T_{\gamma,t})^4}{45 \Omega_{\gamma,t} \hbar^3 c^5}} = \frac{2 (k_B T_{\gamma,t})^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} = 4.957 \cdot 10^{-14} \; \frac{ \gamma \; \text{particles}}{\text{s}}[/math]

Observable Universe photon interaction rate at photon decoupling time:

[math]\boxed{\Gamma_{\gamma,t} = \frac{2 (k_B T_{\gamma,t})^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}}}[/math]

Observable Universe photon interaction rate at photon decoupling time:

[math]\boxed{\Gamma_{\gamma,t} = 4.957 \cdot 10^{-14} \; \frac{ \gamma \; \text{particles}}{\text{s}}}[/math]

Solve for Observable Universe Dark Matter scalar particle interaction rate [math]\Gamma_{\phi,t}[/math] at scalar particle decoupling time:

[math]\Gamma_{\phi,t} = \sqrt{\frac{4 G N_{\phi} \pi^3 (k_B T_{\phi,t})^4}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 2 (k_B T_{\phi,t})^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 39.232 \; \frac{ \phi \; \text{particles}}{\text{s}}[/math]

Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time:

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

Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time:

[math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}}[/math]

Solve for Observable Universe Dark Matter sterile neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at sterile neutrino decoupling time:

[math]\Gamma_{\nu,t} = \sqrt{\frac{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2.689 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math]

Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time:

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

Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time:

[math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math]

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

Reference:
WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
https://map.gsfc.nasa.gov/media/080998/index.html

Wikipedia - neutrino decoupling: (ref. 2)
https://en.wikipedia.org/wiki/Neutrino_decoupling

Orion1 - Cosmic neutrino background radiation electron neutrino composition: (ref. 3)

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

Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4)
http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Wikipedia - Derivation of decoupling time: (ref. 5)

https://en.wikipedia.org/wiki/Neutrino_decoupling#Derivation_of_decoupling_time

Wikipedia - Cross section (physics): (ref. 6)

https://en.wikipedia.org/wiki/Cross_section_(physics)

Edited July 3, 2017 by Orion1

Orion1 · July 11, 2017

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

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

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

Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2)

[math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math]

Solve for Observable Universe cosmic neutrino background radiation neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at neutrino decoupling time: (ref. 3)
[math]\Gamma_{\nu,t} = \sqrt{\frac{4 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 6.749 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math]

Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time:
[math]\boxed{\Gamma_{\nu,t} = 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}}}[/math]

Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time:
[math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math]

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

Reference:
WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
https://map.gsfc.nasa.gov/media/080998/index.html

Wikipedia - neutrino decoupling: (ref. 2)
https://en.wikipedia.org/wiki/Neutrino_decoupling

Fermi-Dirac total neutrino distribution constant: (ref. 3)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry886860

Edited July 12, 2017 by Orion1

Orion1 · July 13, 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 Dark Matter scalar particle decoupling time: (ref. 1)

[math]\boxed{T_{u, \phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math]

Observable Universe cosmic neutrino background radiation neutrino decoupling time: (ref. 2)

[math]\boxed{T_{u , \nu} = 0.148 \; \text{s}}\; \; \; m_{\nu} \neq 0[/math]

Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 3)

[math]\boxed{T_{u, \nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math]

Observable Universe cosmic neutrino background radiation massless neutrino decoupling time: (ref. 4)

[math]\boxed{T_{u , \nu} = 0.893 \; \text{s}} \; \; \; m_{\nu} = 0[/math]

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

Reference:

Orion1 - Observable Universe Dark Matter scalar particle decoupling time: (ref. 1)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999388

Orion1 - Observable Universe neutrino background radiation neutrino decoupling time: (ref. 2)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999033

Orion1 - Observable Universe Dark Matter sterile neutrino decoupling time: (ref. 3)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

Orion1 - Observable Universe neutrino background radiation massless neutrino decoupling time: (ref. 4)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-2#entry882631

Orion1 · July 21, 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 Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1)

[math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}}[/math]

Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: (ref. 2)

[math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math]

Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 1)

[math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}}[/math]

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

Reference:

Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 2)

http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

Orion1 · July 30, 2017

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

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

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

Observable Universe cosmic neutrino background radiation neutrino decoupling temperature: (ref. 2)

[math]T_{\nu , t} = 1 \cdot 10^{10} \; \text{K}[/math]

Neutrino mass:

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

Fermi-Dirac massless neutrino composition: (ref. 3)

[math]\boxed{\Omega_{\nu} = \frac{7 G N_{\nu} \pi^3 (k_B T_{\nu})^4}{45 H_0^2 \hbar^3 c^5}}[/math]

Solve for massless neutrino interaction rate [math]\Gamma_{\nu,t}[/math] at massless neutrino decoupling time:

[math]\Gamma_{\nu,t} = \sqrt{\frac{7 G N_{\nu} \pi^3 (k_B T_{\nu,t})^4}{45 \Omega_{\nu,t} \hbar^3 c^5}} = \frac{(k_B T_{\nu,t})^2}{3} \sqrt{\frac{7 \pi^3 G N_{\nu}}{5 \Omega_{\nu,t} \hbar^3 c^5}} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}[/math]

Massless neutrino interaction rate at massless neutrino decoupling time:

[math]\boxed{\Gamma_{\nu,t} = \frac{(k_B T_{\nu,t})^2}{3} \sqrt{\frac{7 \pi^3 G N_{\nu}}{5 \Omega_{\nu,t} \hbar^3 c^5}}}[/math]

Massless neutrino interaction rate at massless neutrino decoupling time:

[math]\boxed{\Gamma_{\nu,t} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}}[/math]

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

Reference:

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

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

Wikipedia - neutrino decoupling: (ref. 2)
https://en.wikipedia.org/wiki/Neutrino_decoupling

Hubble parameter at massless neutrino decoupling time: (ref. 3)

Edited July 30, 2017 by Orion1
sourcecode repair

Orion1 · August 9, 2017

Symbolic identity key:
[math]n_{s}[/math] - spin states total number
[math]N_{s}[/math] - species total number
[math]N_{n}[/math] - total effective degeneracy number
if [math]n_{s} \geq N_{s}[/math] then [math]N_{n} = n_{s}[/math]
if [math]n_{s} \leq N_{s}[/math] then [math]N_{n} = N_{s}[/math]

[math]\begin{array}{l*{6}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 \\ f & \text{neutralino} & +,- & 1/2 & 2 & 4 & 4 \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 \\ \end{array}[/math]

Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1)
[math]\boxed{\Gamma_{\phi,t} = 39.232 \; \frac{\phi \; \text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math]

Observable Universe cosmic neutrino background radiation neutrino interaction rate at neutrino decoupling time: (ref. 2)
[math]\boxed{\Gamma_{\nu,t} = 6.749 \; \frac{\nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math]

Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 1)
[math]\boxed{\Gamma_{\nu,t} = 2.689 \; \frac{\nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math]

Massless neutrino interaction rate at massless neutrino decoupling time: (ref. 3)
[math]\boxed{\Gamma_{\nu,t} = 1.120 \; \frac{ \nu \; \text{particles}}{\text{s}}} \; \; \; m_{\nu} = 0[/math]

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

Reference:
Orion1 - Observable Universe Dark Matter scalar particle interaction rate at scalar particle decoupling time: (ref. 1)
http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

Orion1 - Observable Universe Dark Matter sterile neutrino interaction rate at sterile neutrino decoupling time: (ref. 2)
http://www.scienceforums.net/topic/86694-observable-universe-mass/page-3#entry999725

Orion1 - Massless neutrino interaction rate at massless neutrino decoupling time: (ref. 3)
http://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1004223

Edited August 9, 2017 by Orion1
sourcecode repair

Orion1 · December 5, 2017

On 11/29/2014 at 2:26 AM, Mordred said:

You could add up the energy budget to check your numbers

It appears that I have reached the limit of my available energy budget. Any recommendations on equation refinement?

On 11/29/2014 at 5:11 AM, Sensei said:

You're making assumption that every star has the same mass as Sun and/or that Sun's mass is average star mass across entire observable Universe.

You were correct. I have attempted to calculate the actual average stellar mass from the equations and table references listed on this post. Any recommendations on equation refinement?

On 11/29/2014 at 6:46 AM, Strange said:

Your figure seems to be about two orders of magnitude larger than other estimates:

http://en.wikipedia.org/wiki/Observable_universe#Mass_of_ordinary_matter

(I haven't gone through the "Extrapolation from number of stars" to work out where the difference might come from. Overestimating average star mass, would only seem to account for a factor of 2.)

You were correct. Your Wikipedia reference for estimates based on critical density, states the observable universe total mass at [math]1.46 \cdot 10^{53} \; \text{kg}[/math]. I was able to further refine these calculations down to a factor of only 26, instead of ~100.

Observable Universe total mass: (ref. 1)
[math]\boxed{M_u = 3.794 \cdot 10^{54} \; \text{kg}}[/math]

---

Stellar class number parameters: (ref. 2)
[math]n_c = 1 \rightarrow 7[/math]

key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M

[math]\Omega_n[/math] - main sequence stars stellar class fraction
[math]N_s[/math] - total observable stellar number
[math]M_n[/math] - Main-sequence mass

Observable Universe average stellar mass:
[math]M_a = \frac{1}{N_s} \sum_{n_c = 1}^{7} (\Omega_n N_s M_n) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math]

[math]\boxed{M_a = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math]

[math]\boxed{M_a = (0.595 \rightarrow 0.769) \cdot M_{\odot}}[/math]

Observable Universe average stellar mass lower bound limit:
[math]\boxed{M_a = 1.183 \cdot 10^{30} \; \text{kg}}[/math]

Observable Universe average stellar mass: (ref. 3, pg. 20)
[math]M_a = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math]

Observable Universe stellar baryon density: (ref. 4, pg. 3)
[math]\Omega_s = (\Omega_{ms} + \Omega_{wd} + \Omega_{ns}) = 2.460 \cdot 10^{-3}[/math]

[math]\Omega_s = 2.460 \cdot 10^{-3}[/math]

Observable Universe total observable stellar number:
[math]N_s = \Omega_{s} \left( \frac{M_u}{M_{a}} \right) = \frac{\pi \Omega_{s}}{6 M_{a}} \left( \frac{c \ln (1 + z)}{H_0} \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right][/math]

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

Observable Universe total observable stellar number:
[math]\boxed{N_s = 7.885 \cdot 10^{21} \; \text{stars}}[/math]

Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 5)
[math]N_s = 1 \cdot 10^{24} \; \text{stars}[/math]

Wikipedia - observable universe total observable stellar number: (ref. 6)
[math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math]

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

Reference:
Orion1 - Observable Universe total mass: (ref. 1)
http://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=909462

Wikipedia - Stellar classification - Harvard spectral classification: (ref. 2)
https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification

On The Mass Distribution Of Stars...: (ref. 3)
http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf

The Cosmic Energy Inventory: (ref. 4)
http://arxiv.org/pdf/astro-ph/0406095v2.pdf

Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 5)
https://www.space.com/26078-how-many-stars-are-there.html

Wikipedia - observable universe stellar number: (ref. 6)
https://en.wikipedia.org/wiki/Star#Distribution

Edited December 5, 2017 by Orion1

Orion1 · January 9, 2018

Observable Universe cosmology scale factor:
[math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math]

Symbolic definition key:
[math]R_u(t_0)[/math] - Observable Universe total radius at present time.
[math]R_u(t)[/math] - Observable Universe total radius at past time.
[math]a(t_0)[/math] - scale factor at present time.
[math]a(t)[/math] - scale factor at past time.
[math]T_t[/math] - cosmic background radiation temperature at past time.
[math]T_0[/math] - cosmic background radiation temperature at present time.
[math]z[/math] - cosmic background radiation redshift parameter at decoupling time.

Hubble radius:
[math]R_{H} = \frac{c}{H_{0}}[/math]

Cosmic photon background radiation temperature at present time: (ref. 1)
[math]T_{\gamma} = 2.72548 \; \text{K}[/math]

Cosmic photon background radiation redshift parameter at photon decoupling time: (ref. 2, pg. 11)
[math]z_{\gamma} = \left( \frac{T_{\gamma,t}}{T_{\gamma}} \right) - 1 = 1090.43[/math]

Cosmic photon background radiation temperature at photon decoupling time: (ref. 1)
[math]T_{\gamma,t} = T_{\gamma} (1 + z_{\gamma}) = 2974.67 \; \text{K}[/math]
[math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math]

Observable Universe total photon co-moving radius integration via substitution:
[math]R_{\gamma} = \frac{R_{H} \ln [1 + z_{\gamma}]}{2} = \frac{1}{2} \left( \frac{c}{H_{0}} \right) \ln [1 + z_{\gamma}] = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\gamma,t}}{T_{\gamma}} \right] = 4.606 \cdot 10^{26} \; \text{m}[/math]

Observable Universe total photon co-moving radius:
[math]\boxed{R_{\gamma} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\gamma,t}}{T_{\gamma}} \right]}[/math]
[math]\boxed{R_{\gamma} = 4.606 \cdot 10^{26} \; \text{m}}[/math][math]\; \; \; (48.689 \cdot 10^{9} \; \text{ly})[/math]

Cosmic neutrino background radiation temperature at present time: (ref. 3, pg. 44, eq. 220), (ref. 4)
[math]T_{\nu} = \left( \frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.94535 \; \text{K}[/math]
[math]\boxed{T_{\nu} = 1.94535 \; \text{K}}[/math]

Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 5)
[math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math]

Observable Universe total neutrino co-moving radius:
[math]\boxed{R_{\nu} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right]}[/math]
[math]\boxed{R_{\nu} = 1.472 \cdot 10^{27} \; \text{m}}[/math][math]\; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math]

Observable Universe total observable stellar number:
[math]\boxed{N_s = \frac{\pi \Omega_{s}}{6 M_{a}} \left( \frac{c}{H_0} \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \right)^3 \left[ \frac{\pi^2 k_B^4}{2 \hbar^3 c^5} \left( \frac{N_{\gamma} T_{\gamma}^4}{15} + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \right) \left( \frac{\Omega_{dm} + \Omega_{b}}{\Omega_{\gamma} + \Omega_{\nu}} \right) + \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G} \right]}[/math]

Observable Universe total observable stellar number:
[math]\boxed{N_s = 2.575 \cdot 10^{23} \; \text{stars}}[/math]

Wikipedia observable universe total observable stellar number: (ref. 6)
[math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math]

Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 7)
[math]N_s = 1 \cdot 10^{24} \; \text{stars}[/math]

After neutrinos decoupled from heavy baryonic matter when the universe was one second old, is it possible for the cosmic neutrino background radiation to have inflated to a distance of 156 billion light years?

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

Reference:
Wikipedia - Cosmic microwave background radiation: (ref. 1)
https://en.wikipedia.org/wiki/Cosmic_microwave_background

Planck 2013 results. XVI. Cosmological parameters: (ref. 2)
http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf

PHYS: 652 Cosmic Inventory I: Radiation: (ref. 3)
http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

Wikipedia - Cosmic neutrino background radiation: (ref. 4)
https://en.wikipedia.org/wiki/Cosmic_neutrino_background

Wikipedia - Neutrino decoupling: (ref. 5)
https://en.wikipedia.org/wiki/Neutrino_decoupling

Wikipedia - Observable universe total observable stellar number: (ref. 6)
https://en.wikipedia.org/wiki/Star#Distribution

Hubble Space Telescope Kornreich stellar survey total observable stellar number: (ref. 7)
https://www.space.com/26078-how-many-stars-are-there.html

Edited January 9, 2018 by Orion1
source code correction

Orion1 · January 20, 2018

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
[math]\Omega_{\nu,t} = 0.10[/math]

Observable Universe cosmology scale factor:
[math]\boxed{\frac{R_u(t_0)}{R_u(t)} = \frac{a(t_0)}{a(t)} = \frac{T_t}{T_0} = 1 + z}[/math]

Observable Universe total neutrino co-moving radius at present time:
[math]\boxed{R_{\nu} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right]}[/math]
[math]\boxed{R_{\nu} = 1.472 \cdot 10^{27} \; \text{m}} \; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math]

Observable Universe total neutrino co-moving radius at past time integration via substitution:
[math]R_{\nu,t} = R_{\nu} \left( \frac{T_{\nu}}{T_{\nu,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) = 2.864 \cdot 10^{17} \; \text{m} \; \; \; (30.277 \; \text{ly})[/math]

Observable Universe total neutrino co-moving radius at past time:
[math]\boxed{R_{\nu,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right)}[/math]
[math]\boxed{R_{\nu,t} = 2.864 \cdot 10^{17} \; \text{m}} \; \; \; (30.277 \; \text{ly})[/math]

Neutrino decoupling time:
[math]T_{u,\nu} = \frac{1}{H_{\nu,t}} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.148 \; \text{s}[/math]
[math]\boxed{T_{u,\nu} = \frac{}{2 (k_B T_{\nu,t})^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}}} \; \; \; m_{\nu} \neq 0[/math]
[math]\boxed{T_{u,\nu} = 0.148 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math]

Observable Universe expansion rate at neutrino decoupling time integration via substitution:
[math]\frac{dr}{dt} = \frac{R_{\nu,t}}{T_{u,\nu}} = \left[ \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) \right] 2 (k_B T_{\nu,t})^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} = 1.933 \cdot 10^{18} \; \frac{\text{m}}{\text{s}} \; \; \; \left( 204.337 \; \frac{\text{ly}}{\text{s}} \right)[/math]

Observable Universe expansion rate at neutrino decoupling time:
[math]\boxed{\frac{dr}{dt} = \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}} \left( \frac{c}{H_{0}} \right) \ln \left[ \frac{T_{\nu,t}}{T_{\nu}} \right] \left( \frac{T_{\nu}}{T_{\nu,t}} \right) (k_B T_{\nu,t})^2}[/math]

[math]\boxed{\frac{dr}{dt} = 1.933 \cdot 10^{18} \; \frac{\text{m}}{\text{s}}} \; \; \; \left( 204.337 \; \frac{\text{ly}}{\text{s}} \right)[/math]

Did the universe have a co-moving radius of 30 light-years at neutrino decoupling time?
Did the universe inflate and expand at a rate of 204 light-years per second at neutrino decoupling time?
Any discussions and/or peer reviews about this specific topic thread?

Reference:
WMAP satellite content of the Universe: (ref. 1)
http://map.gsfc.nasa.gov/media/080998/index.html

Wikipedia - Inflation cosmology:
https://en.wikipedia.org/wiki/Inflation_(cosmology)

Wikipedia - Metric space expansion cosmology:
https://en.wikipedia.org/wiki/Metric_expansion_of_space

Edited January 20, 2018 by Orion1
source code correction

Orion1 · February 9, 2018

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
[math]\Omega_{\gamma,t} = 0.15[/math]

Observable Universe cosmology scale factor:
[math]\boxed{\frac{R_u \left(t_0 \right)}{R_u \left(t \right)} = \frac{a \left(t_0 \right)}{a \left(t \right)} = \frac{T_t}{T_0} = 1 + z}[/math]

Observable Universe total photon co-moving radius at present time:
[math]\boxed{R_{\gamma} = \left( \frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]}[/math]
[math]\boxed{R_{\gamma} = 4.606 \cdot 10^{26} \; \text{m}} \; \; \; \left( 48.689 \cdot 10^{9} \; \text{ly} \right)[/math]

Observable Universe total photon co-moving radius at past time integration via substitution:
[math]R_{\gamma,t} = R_{\gamma} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right) = 4.220 \cdot 10^{23} \; \text{m} \; \; \; \left( 44.610 \; \cdot 10^{6} \; \text{ly} \right)[/math]

Observable Universe total photon co-moving radius at past time:
[math]\boxed{R_{\gamma,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)}[/math]
[math]\boxed{R_{\gamma,t} = 4.220 \cdot 10^{23} \; \text{m}} \; \; \; \left( 44.610 \; \cdot 10^{6} \; \text{ly} \right)[/math]

Photon decoupling time:
[math]\boxed{T_{u,\gamma} = \frac{3}{ \left(2 k_B T_{\gamma,t} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \pi^3}}} \; \; \; m_{\gamma} = 0[/math]
[math]\boxed{T_{u,\gamma} = 1.009 \cdot 10^{13} \; \text{s}} \; \; \; \left( 3.197 \cdot 10^{5} \; \text{years} \right) \; \; \; m_{\gamma} = 0[/math]

[math]\Lambda[/math]CDM universe model semiemperical temperature-time scale factor: (ref. 2, pg. 19, eq. 1, ref. 3)
[math]\left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}} = \frac{T_{u,\gamma}}{T_{u}} = T_{u,\gamma} H_0[/math]
[math]T_{u,\gamma} = \frac{}{H_0} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}} = 1.218 \cdot 10^{13} \; \text{s} \; \; \; \left( 3.861 \cdot 10^{5} \; \text{years} \right)[/math]

[math]\Lambda[/math]CDM universe model semiemperical photon decoupling time:
[math]\boxed{T_{u,\gamma} = \frac{}{H_0} \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)^{\frac{3}{2}}}[/math]
[math]\boxed{T_{u,\gamma} = 1.218 \cdot 10^{13} \; \text{s}} \; \; \; \left(3.861 \cdot 10^{5} \; \text{years}\right)[/math]

Observable Universe expansion rate at photon decoupling time integration via substitution:
[math]\frac{dr}{dt} = \frac{R_{\gamma,t}}{T_{u,\gamma}} = \left[ \left(\frac{c}{2 H_{0}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \left( \frac{T_{\gamma}}{T_{\gamma,t}} \right)\right] \frac{ \left(2 k_B T_{\gamma,t} \right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} = 4.184 \cdot 10^{10} \; \frac{\text{m}}{\text{s}} \; \; \; \left( 139.557 \cdot c \; \; \; 4.422 \cdot 10^{-6} \; \frac{\text{ly}}{\text{s}} \right)[/math]

Observable Universe expansion rate at photon decoupling time:
[math]\boxed{\frac{dr}{dt} = \frac{2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t} \hbar^3 c^5}} \left(\frac{c}{H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\left(\frac{T_{\gamma}}{T_{\gamma,t}}\right)\left(k_B T_{\gamma,t}\right)^2}[/math]

[math]\boxed{\frac{dr}{dt} = 4.184 \cdot 10^{10} \; \frac{\text{m}}{\text{s}}} \; \; \; \left( 139.557 \cdot c \; \; \; 4.422 \cdot 10^{-6} \; \frac{\text{ly}}{\text{s}} \right)[/math]

Does the universe have a photon co-moving radius of 49 billion light-years at present time?
Did the universe have a co-moving radius of 45 million light-years at photon decoupling time?
Did the universe inflate and expand at a rate of 140 c or 4 micro light-years per second at photon decoupling time?
Any discussions and/or peer reviews about this specific topic thread?

Reference:
WMAP satellite content of the Universe: (ref. 1)
http://map.gsfc.nasa.gov/media/080998/index.html

Cosmology: nucleosynthesis and inflation: (ref. 2)
http://www.uio.no/studier/emner/matnat/astro/AST1100/h07/undervisningsmateriale/lecture25.pdf

Wikipedia - Lambda-CDM_model parameters: (ref. 3)
https://en.wikipedia.org/wiki/Lambda-CDM_model#Parameters

Wikipedia - Inflation cosmology:
https://en.wikipedia.org/wiki/Inflation_(cosmology)

Wikipedia - Metric space expansion cosmology:
https://en.wikipedia.org/wiki/Metric_expansion_of_space

Edited February 9, 2018 by Orion1
source code correction

Orion1 · February 27, 2018

WMAP satellite cosmological parameters at photon decoupling time: (ref. 1)
[math]\Omega_{dm,t} = 0.63[/math]

Observable Universe dark matter scalar particle composition is equivalent to dark matter composition at photon decoupling time:
[math]\boxed{\Omega_{\phi,t} = \Omega_{dm,t}}[/math]

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

Scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation temperature at neutrino decoupling time:
[math]\boxed{T_{\phi,t} = T_{\nu,t}}[/math]

Observable Universe total scalar particle co-moving radius at present time:
[math]\boxed{R_{\phi} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right]}[/math]
[math]\boxed{R_{\phi} = 1.472 \cdot 10^{27} \; \text{m}} \; \; \; (155.635 \cdot 10^{9} \; \text{ly})[/math]

Observable Universe total scalar particle co-moving radius at past time integration via substitution:
[math]R_{\phi,t} = R_{\phi} \left( \frac{T_{\phi}}{T_{\phi,t}} \right) = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) = 2.864 \cdot 10^{17} \; \text{m} \; \; \; (30.277 \; \text{ly})[/math]

Observable Universe total scalar particle co-moving radius at past time:
[math]\boxed{R_{\phi,t} = \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right)}[/math]
[math]\boxed{R_{\phi,t} = 2.864 \cdot 10^{17} \; \text{m}} \; \; \; (30.277 \; \text{ly})[/math]

Bose-Einstein scalar particle decoupling time:
[math]T_{u,\phi} = \frac{1}{H_{\phi,t}} = \frac{}{2 (k_B T_{\phi,t})^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}} = 0.0255 \; \text{s}[/math]
[math]\boxed{T_{u,\phi} = \frac{}{2 (k_B T_{\phi,t})^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}}} \; \; \; m_{\phi} \neq 0[/math]
[math]\boxed{T_{u,\phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math]

[math]\Lambda[/math]CDM universe model semiemperical temperature-time scale factor: (ref. 2, pg. 18, eq. 2), (ref. 3)
[math]\left( \frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = \frac{T_{u,\phi}}{T_{u}} = T_{u,\phi} H_0[/math]
[math]T_{u,\phi} = \frac{}{H_0} \left( \frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = 0.0166 \; \text{s}[/math]

[math]\Lambda[/math]CDM universe model semiemperical scalar particle decoupling time:
[math]\boxed{T_{u,\phi} = \frac{}{H_0} \left( \frac{T_{\phi}}{T_{\phi,t}} \right)^{2}}[/math]
[math]\boxed{T_{u,\phi} = 0.0166 \; \text{s}}[/math]

Observable Universe expansion rate at scalar particle decoupling time integration via substitution:
[math]\frac{dr}{dt} = \frac{R_{\phi,t}}{T_{u,\phi}} = \left[ \left( \frac{c}{2 H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) \right] 2 (k_B T_{\phi,t})^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} = 1.124 \cdot 10^{19} \; \frac{\text{m}}{\text{s}} \; \; \; (1187.8 \; \frac{\text{ly}}{\text{s}})[/math]

Observable Universe expansion rate at scalar particle decoupling time:
[math]\boxed{\frac{dr}{dt} = \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}} \left( \frac{c}{H_{0}} \right) \ln \left[ \frac{T_{\phi,t}}{T_{\phi}} \right] \left( \frac{T_{\phi}}{T_{\phi,t}} \right) (k_B T_{\phi,t})^2}[/math]

[math]\boxed{\frac{dr}{dt} = 1.124 \cdot 10^{19} \; \frac{\text{m}}{\text{s}}} \; \; \; (1187.8 \; \frac{\text{ly}}{\text{s}})[/math]

Is it possible for dark matter cosmic scalar particle background radiation to have inflated to a distance of 156 billion light-years?
Did the universe have a co-moving radius of 30 light-years at scalar particle decoupling time?
Did the universe inflate and expand at a rate of 1188 light-years per second at scalar particle decoupling time?
Is dark matter composed of scalar particles?
Any discussions and/or peer reviews about this specific topic thread?

Reference:
WMAP satellite content of the Universe: (ref. 1)
http://map.gsfc.nasa.gov/media/080998/index.html

Cosmology: nucleosynthesis and inflation: (ref. 2)
http://www.uio.no/studier/emner/matnat/astro/AST1100/h07/undervisningsmateriale/lecture25.pdf

Wikipedia - Lambda-CDM model: (ref. 3)
https://en.wikipedia.org/wiki/Lambda-CDM_model

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

Wikipedia - Inflation cosmology:
https://en.wikipedia.org/wiki/Inflation_(cosmology)

Wikipedia - Metric space expansion cosmology:
https://en.wikipedia.org/wiki/Metric_expansion_of_space

Edited February 27, 2018 by Orion1
source code correction

Orion1 · May 3, 2018

Symbolic identity key:
[math]n_{s}[/math] - spin states total number
[math]N_{s}[/math] - species total number
[math]N_{n}[/math] - total effective degeneracy number
[math]\text{if} \; n_{s} \geq N_{s} \; \text{then} \; N_{n} = n_{s}[/math]
[math]\text{if} \; n_{s} \leq N_{s} \; \text{then} \; N_{n} = N_{s}[/math]

[math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math]

Stellar class number parameters: (ref. 1)
[math]n_c = 1 \rightarrow 7[/math]
key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M
[math]\Omega_n[/math] - main sequence stars stellar class fraction
[math]N_s[/math] - total observable stellar number
[math]M_n[/math] - Main-sequence mass

Observable Universe average stellar mass:
[math]M_{as} = \frac{1}{N_s} \sum_{n_c = 1}^{7} \left(\Omega_n N_s M_n\right) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math]
[math]\boxed{M_{as} = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math]
[math]\boxed{M_{as} = \left(0.595 \rightarrow 0.769 \right) \cdot M_{\odot}}[/math]

Observable Universe average stellar mass lower bound limit:
[math]\boxed{M_{as} = 1.183 \cdot 10^{30} \; \text{kg}}[/math]

Observable Universe average stellar mass: (ref. 2, pg. 20)
[math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math]

Observable Universe stellar baryon density: (ref. 3, pg. 3)
[math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math]
[math]\Omega_s = 2.460 \cdot 10^{-3}[/math]
---

Milky Way galaxy mass: (ref. 4, pg. 1)
[math]M_{mw} = 1.260 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math]
[math]\boxed{M_{mw} = 2.506 \cdot 10^{42} \; \text{kg}}[/math]

---
Redshift parameter at photon decoupling time:
[math]z = 1090.43[/math]

Cosmic photon background radiation temperature at present time:
[math]T_{\gamma} = 2.72548 \; \text{K}[/math]

Cosmic photon background radiation temperature at photon decoupling time:
[math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math]
[math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math]

Cosmic neutrino background radiation temperature at present time: (ref. 5, pg. 44, eq. 220)
[math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math]
[math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math]

Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 6)
[math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math]

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

Cosmic scalar particle dark energy background radiation temperature:
[math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math]
[math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math]

Observable Universe Cosmological Constant:
[math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

Observable Universe scalar particle dark energy Cosmological Constant integration via substitution:
[math]\Lambda_s = \frac{3 H_0^2}{c^2} \left(\frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

Observable Universe scalar particle dark energy Cosmological Constant:
[math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7}}[/math]

Observable Universe scalar particle dark energy Cosmological Constant:
[math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math]

Dark matter density:
[math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math]
Baryonic density:
[math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math]
Dark energy density:
[math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math]

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

Fermi-Dirac total neutrino distribution constant:
[math]\boxed{C_{\nu} = \frac{4 G N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math]
[math]\boxed{C_{\nu} = 0.236}[/math]

Bose-Einstein scalar particle dark matter density:
[math]\boxed{\rho_{\phi} = \frac{N_{\phi} \pi^2 \left(k_B T_{\phi}\right)^4}{2 C_{\phi} \hbar^3 c^5}}[/math]

Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter density:
[math]\boxed{\rho_{b} = \frac{N_{\gamma} \pi^2 \left(k_B T_{\gamma}\right)^4}{30 \hbar^3 c^5} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) + \frac{N_{\nu} \pi^2 \left(k_B T_{\nu}\right)^4}{2 C_{\nu} \hbar^3 c^5} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right)}[/math]

Bose-Einstein scalar particle dark energy density:
[math]\boxed{\rho_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{30 \hbar^3 c^5}}[/math]

Observable Universe total photon and neutrino co-moving volumes:
[math]V_{\gamma} = \frac{4 \pi R_{\gamma}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\right)^3[/math]

[math]V_{\nu} = \frac{4 \pi R_{\nu}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]\right)^3[/math]

Observable Universe scalar particle dark matter and scalar particle dark energy and neutrino total co-moving volumes are equivalent:
[math]\boxed{V_{\phi} = V_{\Lambda} = V_{\nu}}[/math]

Bose-Einstein scalar particle dark matter total mass:
[math]\boxed{\rho_{\phi} V_{\phi} = \frac{N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{12 C_{\phi} c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math]

Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter total mass:
[math]\boxed{\rho_{b} V_{b} = \frac{N_{\gamma} \pi^3 \left(k_B T_{\gamma}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{12 C_{\nu} \hbar^3 c^2} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math]

Bose-Einstein scalar particle dark energy total mass:
[math]\boxed{\rho_{\Lambda} V_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{180 \left(\hbar H_0\right)^3 c^2} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math]

Observable Universe compositional Equation of State and compositional number:
[math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math]

Observable Universe total critical mass:
[math]M_{c} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{dm} V_{dm} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math]
[math]\boxed{M_c = \frac{c^3}{16 G H_0} \left(\left(\Omega_{dm} + \Omega_{\Lambda}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \Omega_b \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3\right)}[/math]
[math]\boxed{M_c = 1.179 \cdot 10^{56} \; \text{kg}}[/math]

Observable Universe compositional Equation of State and compositional number:
[math]\Omega_{\phi} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math]

Observable Universe scalar particle dark matter and scalar particle dark energy total mass:
[math]M_{u} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{\phi} V_{\phi} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math]

[math]\boxed{M_u = \frac{\pi^3 k_B^4}{12 c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math]

[math]\boxed{M_u = 1.179 \cdot 10^{56} \; \text{kg}}[/math]

Observable Universe total energy:
[math]E_u = M_u c^2[/math]
[math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math]
[math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math]

Observable Universe total observable stellar number:
[math]\boxed{N_s = \frac{\Omega_s \pi^3 k_B^4}{12 M_{as} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math]

Observable Universe total stellar number:
[math]\boxed{N_s = 2.453 \cdot 10^{23} \; \text{stars}}[/math]

Wikipedia observable universe total stellar number: (ref. 7)
[math]N_s = 3 \cdot 10^{23} \; \text{stars}[/math]

Observable Universe total galaxy number:
[math]\boxed{N_g = \frac{\Omega_b \pi^3 k_B^4}{12 M_{mw} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math]

Observable Universe total galaxy number:
[math]\boxed{N_g = 2.330 \cdot 10^{12} \; \text{galaxies}}[/math]

Wikipedia observable universe total galaxy number: (ref. 8)
[math]N_g = 2.000 \cdot 10^{12} \; \text{galaxies}[/math]

Observable Universe stars per galaxy average number:
[math]\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math]
[math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math]
[math]\boxed{\frac{N_s}{N_g} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}}[/math]

Wikipedia Milky Way galaxy total stellar number: (ref. 9)
[math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math]

Observable Universe planetary composition: (ref. 3, pg. 3)
[math]\Omega_p = 1 \cdot 10^{-6}[/math]

Wikipedia Milky Way galaxy total planetary number: (ref. 9, ref. 10)
[math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math]

Observable Universe average planetary mass:
[math]\boxed{M_{ap} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{N_g}{N_p}\right)}[/math]
[math]\boxed{M_{ap} = \left(52.974 \rightarrow 84.758\right) \cdot M_{\oplus}}[/math]

Solar star system average planetary mass: (ref. 11)
[math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_n} \; \; \; \; \; \; n_p = 8[/math]
[math]\boxed{M_{ap} = 55.855 \cdot M_{\oplus}}[/math]

Milky Way galaxy total planetary number based upon solar star system:
[math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_n\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math]
[math]\boxed{\frac{N_p}{N_g} = 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}}}[/math]

Any discussions and/or peer reviews about this specific topic thread?
Is there anything else that you want to see quantified based upon this model for observational comparison?

Reference:
Wikipedia - Stellar classification - Harvard spectral classification: (ref. 1)
https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification

On The Mass Distribution Of Stars...: (ref. 2)
http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf

The Cosmic Energy Inventory: (ref. 3)
http://arxiv.org/pdf/astro-ph/0406095v2.pdf

Mass models of the Milky Way: (ref. 4)
http://arxiv.org/pdf/1102.4340v1

PHYS: 652 Cosmic Inventory I: Radiation: (ref. 5)
http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf

Wikipedia - neutrino decoupling: (ref. 6)
https://en.wikipedia.org/wiki/Neutrino_decoupling

Wikipedia - Observable universe total observable stellar number: (ref. 7)
https://en.wikipedia.org/wiki/Star#Distribution

Wikipedia - Galaxy: (ref. 8)
https://en.wikipedia.org/wiki/Galaxy

Wikipedia - Milky Way Galaxy: (ref. 9)
https://en.wikipedia.org/wiki/Milky_Way

Space.com - 160 Billion Alien Planets May Exist in Our Milky Way Galaxy: (ref. 10)
https://www.space.com/14200-160-billion-alien-planets-milky-galaxy.html

Wikipedia - Planetary mass: (ref. 11)
https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris

Edited May 3, 2018 by Orion1

Orion1 · May 19, 2018

Planck satellite cosmological parameters: (ref. 1, pg. 11)
[math]\Omega_{dm} = 0.268[/math]
[math]\Omega_{b} = 0.0495[/math]
[math]\Omega_{\Lambda} = 0.6825[/math]

Symbolic identity key:
[math]n_{s}[/math] - spin states total number
[math]N_{s}[/math] - species total number
[math]N_{n}[/math] - total effective degeneracy number
[math]\text{if} \; n_{s} \geq N_{s} \; \text{then} \; N_{n} = n_{s}[/math]
[math]\text{if} \; n_{s} \leq N_{s} \; \text{then} \; N_{n} = N_{s}[/math]

[math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math]

Stellar class number parameters: (ref. 2)
[math]n_c = 1 \rightarrow 7[/math]
key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M
[math]\Omega_n[/math] - main sequence stars stellar class fraction
[math]N_s[/math] - total observable stellar number
[math]M_n[/math] - Main-sequence mass

Observable Universe average stellar mass:
[math]M_{as} = \frac{1}{N_s} \sum_{n_c = 1}^{7} \left(\Omega_n N_s M_n\right) = \sum_{n_c = 1}^{7} \Omega_n M_n = 0.595 \cdot M_{\odot} \rightarrow 0.769 \cdot M_{\odot}[/math]

[math]\boxed{M_{as} = \sum_{n_c = 1}^{7} \Omega_n M_n}[/math]
[math]\boxed{M_{as} = \left(0.595 \rightarrow 0.769 \right) \cdot M_{\odot}}[/math]

Observable Universe average stellar mass lower bound limit:
[math]\boxed{M_{as} = 1.183 \cdot 10^{30} \; \text{kg}}[/math]

Observable Universe average stellar mass: (ref. 3, pg. 20)
[math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math]

[math]M_{as} = 1.193 \cdot 10^{30} \; \text{kg}[/math]

Observable Universe stellar baryon composition: (ref. 4, pg. 3)
[math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math]
[math]\Omega_s = 2.460 \cdot 10^{-3}[/math]
---

Milky Way galaxy mass: (ref. 5, pg. 1)
[math]M_{mw} = 1.260 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math]
[math]\boxed{M_{mw} = 2.506 \cdot 10^{42} \; \text{kg}}[/math]

---
Redshift parameter at photon decoupling time:
[math]z = 1090.43[/math]

Cosmic photon background radiation temperature at present time:
[math]T_{\gamma} = 2.72548 \; \text{K}[/math]

Cosmic photon background radiation temperature at photon decoupling time:
[math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math]
[math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math]

Cosmic neutrino background radiation temperature at present time: (ref. 6, pg. 44, eq. 220)
[math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math]
[math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math]

Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 7)
[math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math]

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

Cosmic scalar particle dark energy background radiation temperature:
[math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math]
[math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math]

Observable Universe Cosmological Constant:
[math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

Observable Universe scalar particle dark energy Cosmological Constant integration via substitution:
[math]\Lambda_s = \frac{3 H_0^2}{c^2} \left(\frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math]

Observable Universe scalar particle dark energy Cosmological Constant:
[math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7}}[/math]

Observable Universe scalar particle dark energy Cosmological Constant:
[math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math]

Dark matter density:
[math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math]

Baryonic density:
[math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math]

Dark energy density:
[math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math]

Dark matter scalar particle composition is equivalent to dark matter composition:

[math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math]