Neutrino mass from Fermi-Dirac statistics...

[Orion1]

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

[math][/math]

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

 

Symbolic identity key:

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

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

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

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

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

 

CMBR temperature at present time: (ref. 5)

[math]T_{\gamma} = 2.72548 \; \text{K}[/math]

 

Cosmic neutrino background radiation temperature at present time: (ref. 1, pg. 44, eq. 220)

[math]T_{\nu} = \left( \frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math]

 

[math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math]

 

CMBR primeval thermal remnant composition: (ref. 2, pg. 3)

[math]\Omega_{\gamma} = 10^{-4.3}[/math]

 

Neutrino primeval thermal remnant composition: (ref. 2, pg. 3)

[math]\Omega_{\nu} = 10^{-2.9}[/math]

 

Standard Model photon species total effective degeneracy number: (ref. 1, pg. 41, eq. 197)

[math]N_{\gamma} = 2[/math]

 

Standard Model neutrino species total effective degeneracy number: (ref. 4, pg. 16)

[math]N_{\nu} = 3.046[/math]

 

Total photon energy: (ref. 3)

[math]E_t (\omega) = \hbar \omega[/math]

 

Total thermal energy:

[math]E_1 (T_{\gamma}) = k_B T_{\gamma}[/math]

 

Photon radiation energy density Bose-Einstein distribution: (ref. 1, pg. 42, eq. 204)

[math]\rho_{\gamma} = \frac{4 \pi k_B^4 N_{\gamma} T_{\gamma}^4}{( 2 \pi \hbar c )^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega[/math]

 

Total neutrino rest mass energy: (ref. 3)

[math]E_0 = m_0 c^2[/math]

 

Total neutrino relativistic momentum energy: (ref. 3)

[math]E_p (v) = p(v) c = (\gamma m_0 v) c = \frac{m_0 c v}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }}[/math]

 

[math]\boxed{E_p (v) = \frac{m_0 c v}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }}}[/math]

 

Total neutrino energy: (ref. 3)

[math]E_t (v) = \sqrt{E_p(v)^2 + E_0^2}[/math]

 

Neutrino radiation energy density Fermi-Dirac distribution: (ref. 1, pg. 44, eq. 221)

[math]\rho_{\nu} = \frac{4 \pi k_B^4 N_{\nu} T_{\nu}^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math]

 

Massless neutrino compositional ratio integration via substitution:

[math]\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \frac{\rho_{\nu}}{\rho_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \frac{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\nu})}} + 1} d \omega}{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 C_1[/math]

 

Massless integration constants:

[math]\boxed{C_1 = 0.875 = \frac{7}{8}} \; \; \; \; \; \; T_{\nu} = T_{\gamma}[/math]

 

[math]\boxed{C_1 = 0.227 = \frac{2}{9}} \; \; \; \; \; \; T_{\nu} = T_{\nu}[/math]

 

Massless neutrino compositional ratio:

[math]\boxed{\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 C_1}[/math]

 

Mass neutrino compositional ratio:

[math]\frac{\Omega_{\nu}}{\Omega_{\gamma}} = \frac{\rho_{\nu}}{\rho_{\gamma}} = \left( \frac{N_{\nu}}{N_{\gamma}} \right) \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \frac{\int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv}{\int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega}[/math]

 

Neutrino Riemann sum limit numerical integration criterion:

[math]I_{\nu} (m_{\nu}) = I_{\gamma}[/math]

 

[math]I_{\nu} (m_{\nu}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv = \frac{\Omega_{\nu}}{\Omega_{\gamma}} \left( \frac{N_{\gamma}}{N_{\nu}} \right) \left( \frac{T_{\gamma}}{T_{\nu}} \right)^4 \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = I_{\gamma}[/math]

 

Use Riemann sum limit to solve [math]I_{\nu} (m_{\nu})[/math]:

[math]I_{\nu} (m_{\nu}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math]

 

Riemann sum limit mass numerical integration probability:

[math]P(m_{\nu}) = \frac{I_{\gamma}}{I_{\nu} (m_{\nu})}[/math]

 

Solve for neutrino particle rest mass with highest Riemann sum limit rest mass numerical integration probability.

 

Neutrino rest mass:

[math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}[/math]

 

Neutrino rest mass:

[math]\boxed{m_{\nu} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math]

 

---

 

Dark matter compositional fraction: (ref. 6, pg. 11)

[math]\Omega_{dm} = 0.268[/math]

 

Dark matter scalar particle temperature:

[math]\boxed{T_{\phi} = T_{\nu}}[/math]

 

Dark matter scalar particle species total effective degeneracy number:

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

 

Dark matter scalar particle Riemann sum limit numerical integration criterion:

[math]I_{\phi} (m_{\phi}) = I_{\gamma}[/math]

 

[math]I_{\phi} (m_{\phi}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv = \frac{\Omega_{dm}}{\Omega_{\gamma}} \left( \frac{N_{\gamma}}{N_{\phi}} \right) \left( \frac{T_{\gamma}}{T_{\phi}} \right)^4 \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_1 (T_{\gamma})}} - 1} d \omega = I_{\gamma}[/math]

 

Use Riemann sum limit to solve [math]I_{\phi} (m_{\phi})[/math]:

[math]I_{\phi} (m_{\phi}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math]

 

Riemann sum limit mass numerical integration probability:

[math]P(m_{\phi}) = \frac{I_{\gamma}}{I_{\phi} (m_{\phi})}[/math]

 

Solve for dark matter scalar particle rest mass with highest Riemann sum limit rest mass numerical integration probability.

 

Dark matter scalar particle rest mass:

[math]\boxed{m_{\phi} = 1.818 \cdot 10^{-37} \; \text{kg}}[/math]

 

Dark matter scalar particle rest mass:

[math]\boxed{m_{\phi} = 0.102 \; \frac{\text{eV}}{c^2}}[/math]

 

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

 

Reference:

PHYS: 652 Cosmic Inventory I: Radiation:

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

The Cosmic Energy Inventory:

http://arxiv.org/pdf/astro-ph/0406095v2.pdf

Hyperphysics: Momentum of Photon:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html#c4

Relic neutrino decoupling including flavour oscillations:

http://arxiv.org/pdf/hep-ph/0506164.pdf

Wikipedia - cosmic microwave background radiation:

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

Planck 2013 results. XVI. Cosmological parameters:

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

Wikipedia - dark matter:

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

Wikipedia - Riemann sum:

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

Wikipedia - Bose-Einstein statistics:

https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics

Wikipedia - Fermi-Dirac statistics:

https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

Edited August 8, 2015 by Orion1

[Orion1]

Continued from post #1...

 

Solve for neutrino particle rest mass with highest relative maximum at critical mass number:

[math]\frac{d}{dm} I_{\nu} (m_{\nu}) = 0[/math]

 

Neutrino particle rest mass:

[math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}[/math]

 

[math]\boxed{m_{\nu} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math]

 

Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number:

[math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math]

 

Dark matter scalar particle rest mass:

[math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}}[/math]

 

Dark matter scalar particle rest mass:

[math]\boxed{m_{\phi} = 3.694 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}}[/math]

 

[math]\boxed{\frac{m_{\phi}}{m_{\nu}} = \frac{7}{8}} \; \; \; T_{\phi} = T_{\nu}[/math]

 

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

 

 

 

Edited August 9, 2015 by Orion1

[swansont]

Which neutrino has that mass? There are three different ones.

[Orion1]

Which neutrino has that mass? There are three different ones.

 

The best mathematical description is that the mass solution is a Fermi-Dirac fermionic particle rest mass with highest relative maximum at a critical mass number within the mass spectrum range of a Standard Model neutrino.

 

This Fermi-Dirac fermionic particle rest mass is an interesting artifact of the mathematical properties of the Fermi-Dirac distribution equation.

 

Consider for a moment that when a Standard Model electron neutrino attains a non-zero particle mass via some asymmetric Higgs mechanism, that the neutrino mass attains the Fermi-Dirac fermionic particle rest mass with highest relative maximum at a critical mass number by virtue of the particles fermionic quantum spin properties.

 

Electron neutrino particle rest mass is equivalent to the Fermi-Dirac fermionic particle rest mass:

[math]\boxed{m_{\nu_{e}} = m_{\nu_{fd}}}[/math]

 

What are the possibilities that the Standard Model neutrino masses obey weak particle mass hierarchy?

 

[math]\boxed{\frac{m_{e}}{m_{\mu}} = \frac{m_{\nu_{e}}}{m_{\nu_{\mu}}}}[/math]

 

[math]\boxed{m_{\nu_{e}} = 4.215 \cdot 10^{-4} \; \frac{\text{eV}}{c^2}} \; \; \; T_{\nu} = T_{\nu}[/math]

 

[math]m_{\nu_{\mu}} \leq m_{\nu_{e}} \left( \frac{m_{\mu}}{m_{e}} \right) \leq 0.087 \; \frac{\text{eV}}{c^2}[/math]

 

[math]\boxed{m_{\nu_{\mu}} \leq 0.087 \; \frac{\text{eV}}{c^2}}[/math]

 

[math]\boxed{\frac{m_{e}}{m_{\tau}} = \frac{m_{\nu_{e}}}{m_{\nu_{\tau}}}}[/math]

 

[math]m_{\nu_{\tau}} \leq m_{\nu_{e}} \left( \frac{m_{\tau}}{m_{e}} \right) \leq 1.465 \; \frac{\text{eV}}{c^2}[/math]

 

[math]\boxed{m_{\nu_{\tau}} \leq 1.465 \; \frac{\text{eV}}{c^2}}[/math]

 

The sum of these three neutrino flavors:

[math]\boxed{\sum m_{\nu} \leq 1.553 \; \frac{\text{eV}}{c^2}}[/math]

 

Active neutrino model with three degenerate neutrinos: (ref. 5)

[math]\sum m_{\nu} \leq 0.320 \pm 0.081 \; \frac{\text{eV}}{c^2}[/math]

 

The neutrino mass sum formula:

[math]\sum m_{\nu} = \left( m_{\nu_{e}} + m_{\nu_{\mu}} + m_{\nu_{\tau}} \right)[/math]

 

Integration via substitution and factor:

[math]\sum m_{\nu} = m_{\nu_{e}} \left( 1 + \frac{m_{\mu}}{m_{e}} + \frac{m_{\tau}}{m_{e}} \right) = 0.320 \pm 0.081 \; \frac{\text{eV}}{c^2}[/math]

 

Solve for [math]m_{\nu_{e}}[/math].

 

Active neutrino model degenerate electron neutrino rest mass:

[math]\boxed{m_{\nu_{e}} = \sum m_{\nu} \left( \frac{m_{e}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 8.684 \cdot 10^{-5} \; \frac{\text{eV}}{c^2}[/math]

 

Active neutrino model degenerate electron neutrino rest mass:

[math]\boxed{m_{\nu_{e}} = 8.684 \cdot 10^{-5} \; \frac{\text{eV}}{c^2}}[/math]

 

Active neutrino model degenerate electron neutrino rest mass:

[math]\boxed{m_{\nu_{e}} = 1.548 \cdot 10^{-40} \; \text{kg}}[/math]

 

Active neutrino model degenerate muon neutrino rest mass:

[math]\boxed{m_{\nu_{\mu}} = \sum m_{\nu} \left( \frac{m_{\mu}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 0.018 \; \frac{\text{eV}}{c^2}[/math]

 

Active neutrino model degenerate muon neutrino rest mass:

[math]\boxed{m_{\nu_{\mu}} = 0.018 \; \frac{\text{eV}}{c^2}}[/math]

 

Active neutrino model degenerate tau neutrino rest mass:

[math]\boxed{m_{\nu_{\tau}} = \sum m_{\nu} \left( \frac{m_{\tau}}{m_{e} + m_{\mu} + m_{\tau}} \right)} = 0.302 \; \frac{\text{eV}}{c^2}[/math]

 

Active neutrino model degenerate tau neutrino rest mass:

[math]\boxed{m_{\nu_{\tau}} = 0.302 \; \frac{\text{eV}}{c^2}}[/math]

 

Fermi-Dirac fermionic particle rest mass:

[math]\boxed{m_{\nu_{fd}} = 7.515 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\nu} = T_{\nu}[/math]

 

Fermi-Dirac fermionic particle rest mass:

[math]\boxed{m_{\nu_{fd}} = 1.548 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\nu} = 0.4 \; \text{K}[/math]

 

Active neutrino model degenerate electron neutrino rest mass:

[math]\boxed{m_{\nu_{e}} = 1.548 \cdot 10^{-40} \; \text{kg}}[/math]

 

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

 

Reference:

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

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

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

https://en.wikipedia.org/wiki/Tau_(particle)

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.051303

Edited August 11, 2015 by Orion1

[Ant Sinclair]

Orion to those of Us here that don't have these math skills, it all looks so complex.

 

How long time-wise would it take to do these calculations?

Edited August 20, 2015 by Ant Sinclair

[Orion1]

Orion to those of Us here that don't have these math skills, it all looks so complex.

 

How long time-wise would it take to do these calculations?

The specific formulation in this thread is sophisticated, however it is not very complex.

 

A college or university student that has successfully completed a course on Calculus I and Physics II is qualified to solve all these equations. The mathematical skills acquired involve memorization of the semantic mathematical symbolic definitions and the ability to utilize these symbolic definitions to effectively solve mathematical equations. Absent such student academic qualifications definitely requires the use of math evaluation software and three months solid home study with a college or university level subject textbook.

 

Equation (1) is semantically symbolizing the summation of an area under a sloped curve using the Riemann Sum formula, it is called an integral.

 

Equation (3) is a function domain parameter that levels the slope of the slope formula to flat or horizontal and equivalent to zero.

 

Equation (4) is semantically symbolizing the range location where the slope of the slope formula is flat or horizontal and equivalent to zero, it is called a derivative.

 

Neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature:

[math]\tag{0} \boxed{T_{\nu} = T_{\nu}}[/math]

 

Use Riemann sum limit to solve [math]I_{\nu} (m_{\nu})[/math]:

[math]\tag{1} I_{\nu} (m_{\nu}) = \lim_{\parallel \Delta \parallel \rightarrow 0} \sum_{i = 1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv[/math]

 

[math]\tag{2} \boxed{I_{\nu} (m_{\nu}) = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\nu})}} + 1} dv}[/math]

 

[math]\tag{3} \Delta m_{\nu} = (m_{b} - m_{a})[/math]

 

[math]\tag{4} \frac{d}{dm} I_{\nu} (m_{\nu}) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{d}{dx} f(x) = \lim_{\Delta m_{\nu} \rightarrow 0} \frac{I_{\nu}(m_{\nu} + \Delta m_{\nu}) - I_{\nu}(m_{\nu})}{\Delta m_{\nu}} = 0[/math]

 

[math]\tag{5} \boxed{\frac{d}{dm} I_{\nu} (m_{\nu}) = \lim_{\Delta m_{\nu} \rightarrow 0} \frac{I_{\nu}(m_{\nu} + \Delta m_{\nu}) - I_{\nu}(m_{\nu})}{\Delta m_{\nu}} = 0}[/math]

 

Solve for [math]m_{\nu}[/math]:

 

Because there is no formal solution to the derivative, the next step is to plot the function [math]I_{\nu} (m_{\nu})[/math] and solve for the critical mass using the derivative. (ref. plot 1)

 

The derivative is solved by slowly increasing the domain parameter [math]m_a[/math] and slowly decreasing domain parameter [math]m_b[/math], so that the difference in the parameters approach zero. Once a solution of three decimal places is achieved, the derivative is considered solved. (ref. plot 2)

 

This Mathematica source code is nearly equivalent to the symbolic formulas expressed in equations 1 to 4.

 

Mathematica 7.0.0 source code:

t = AbsoluteTime[]
(* Riemann Sum - MidPoint Rule *)
MidPointRule[a0_, b0_, n0_] :=
  Module[{a = a0, b = b0, c, \[CapitalDelta]X, k, n = n0, X},
   \[CapitalDelta]X = (b - a)/n;
   Subscript[c, k_] = a + (k - 1/2)*\[CapitalDelta]X;
   Return[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(f[
\*SubscriptBox[\(c\), \(k\)]] \[CapitalDelta]X\)\)];];
(* End Module *)
c = 2.99792458*10^8
Subscript[k, B] = 1.380648813*
\!\(\*SuperscriptBox["10",
RowBox[{"\[Minus]", "23"}]]\)
Subscript[T, \[Gamma]] = 2.72548
Subscript[T, \[Nu]] = (4/11)^(1/3)*Subscript[T, \[Gamma]]
Subscript[\[Epsilon], 1] = Subscript[k, B]*Subscript[T, \[Nu]]
Subscript[\[Epsilon], 0] = m*c^2
Subscript[\[Epsilon], p][v_] := (m*c*v)/Sqrt[1 - (v/c)^2]
Subscript[\[Epsilon], t][v_] := Sqrt[
 Subscript[\[Epsilon], p][v]^2 + Subscript[\[Epsilon], 0]^2]
f[v_] := Subscript[\[Epsilon], t][v]^3/(
 E^(Subscript[\[Epsilon], t][v]/Subscript[\[Epsilon], 1]) + 1)
a = 0
b = c
n = 100
F[m_] := MidPointRule[a, b, n]
Subscript[m, a] = 1*10^-41
Subscript[m, b] = 3*10^-39
Plot[ F[m], {m, Subscript[m, a], Subscript[m, b]},
 ImageSize -> {640, Automatic}]
(* m=7.515*10^-40 *)
Subscript[m, a] = 7.5150*10^-40
Subscript[m, b] = 7.5156*10^-40
Plot[ F[m], {m, Subscript[m, a], Subscript[m, b]},
 ImageSize -> {640, Automatic}]
Print["Evaluation Time: ", AbsoluteTime[] - t, " seconds."]
Print["Maximum Memory Used: ", MaxMemoryUsed[], " bytes."]

Evaluation Time: 4.9062500 seconds.

Maximum Memory Used: 11784056 bytes.

 

 

How long time-wise would it take to do these calculations?

Answer: 5 seconds.

 

According to Mathematica 7, the solution is...

 

Fermi-Dirac fermionic particle rest mass:

[math]\boxed{m_{\nu} = 7.515 \cdot 10^{-40} \; \text{kg}}\; \; \; T_{\nu} = T_{\nu}[/math]

 

 

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

 

plot01.bmp

plot02.bmp

 

Reference:

mathfaculty.fullerton.edu - Riemann Sum:

http://mathfaculty.fullerton.edu/mathews/n2003/riemannsum/RiemannSumMod/Links/RiemannSumMod_lnk_4.html

Wikipedia - Function Limit:

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

Edited August 23, 2015 by Orion1

[Orion1]

Planck satellite cosmological parameters: (ref. 1, pg. 11)

[math]\Omega_{dm} = 0.268[/math]

 

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

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

 

Dark matter scalar particle species total effective degeneracy number: (ref. 2)

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

 

Dark matter scalar particle radiation temperature is equivalent to neutrino cosmic background radiation temperature:

[math]\boxed{T_{\phi} = T_{\nu}}[/math]

 

Dark matter scalar particle radiation energy density Bose-Einstein distribution:

[math]\epsilon_{\phi} = \frac{4 \pi k_B^4 N_{\phi} T_{\phi}^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math]

 

Dark matter quantum scalar particle Bose-Einstein distribution integral:

[math]I_{\phi} = \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv[/math]

 

Solve for dark matter scalar particle rest mass with highest relative maximum at critical mass number:

[math]\frac{d}{dm} I_{\phi} (m_{\phi}) = 0[/math]

 

Dark matter scalar particle rest mass:

[math]\boxed{m_{\phi} = 6.586 \cdot 10^{-40} \; \text{kg}} \; \; \; T_{\phi} = T_{\nu}[/math]

 

Dark matter scalar particle cosmic background radiation energy density Bose-Einstein distribution integration via substitution:

[math]\epsilon_{\phi} = \alpha_{\phi} T_{\phi}^4 = \frac{4 \pi N_{\phi} (k_B T_{\phi})^4}{( 2 \pi \hbar c )^3} \int_{0}^c \frac{E_t (v)^3}{e^{\frac{E_t (v)}{E_1 (T_{\phi})}} - 1} dv = \frac{4 \pi N_{\phi} (k_B T_{\phi})^4}{( 2 \pi \hbar c )^3} \left( \frac{\pi^4}{C_{\phi}} \right) = \frac{\pi^2 N_{\phi} (k_B T_{\phi})^4}{2 C_{\phi} ( \hbar c )^3}[/math]

 

Dark matter scalar particle cosmic background radiation energy density:

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

 

Dark matter scalar particle cosmic background radiation constant:

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

 

Dark matter scalar particle cosmic background radiation composition integration via substitution:

[math]\Omega_{\phi} = \frac{\epsilon_{\phi}}{\epsilon_c} = \frac{\alpha_{\phi} T_{\phi}^4}{\rho_c c^2} = \left( \frac{\pi^2 N_{\phi} (k_B T_{\phi})^4}{2 C_{\phi} ( \hbar c )^3} \right) \left( \frac{8 \pi G}{3 (c H_0)^2} \right) = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 C_{\phi} H_0^2 \hbar^3 c^5}[/math]

 

Dark matter scalar particle cosmic background radiation composition:

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

 

Bose-Einstein total dark matter scalar particle distribution constant:

[math]C_{\phi} = \frac{4 \pi^3 G N_{\phi} (k_B T_{\phi})^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5} = 3.640 \cdot 10^{-4}[/math]

 

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

 

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

 

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

 

Reference:

Planck 2013 results. XVI. Cosmological parameters: (ref. 1)

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

 

Orion 1 - total effective degeneracy number: (ref. 2)

http://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/#entry879233

 

Dark matter - Wikipedia

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

 

Edited October 5, 2015 by Orion1