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Orion1
#7
Jan4-09, 09:45 AM
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P: 989

Current Universe radius:
[tex]\boxed{r_u(t_0) = 4.4 \cdot 10^{26} \; \text{m}}[/tex]

Friedmann scale factor:
[tex]\boxed{a(t) = \frac{r_u(t)}{r_u(t_0)}}[/tex]

[tex]a(t_u) = \frac{r_u(t_u)}{r_u(t_0)} = \frac{\hbar}{m_u c r_u(t_0)}[/tex]

[tex]\boxed{a(t_u) = \frac{\hbar}{m_u c r_u(t_0)}}[/tex]

Quantum Compton-Friedmann equation:
[tex]H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - \frac{kc^2}{a(t_u)^2} + \frac{\Lambda(t_u) c^2}{3}[/tex]

Integration by substitution:
[tex]H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - kc^2 \left( \frac{m_u c r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda(t_u) c^2}{3} = \frac{2 G c^3 m_u^4}{\hbar^3} - k \left( \frac{m_u c^2 r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda c^2}{3}[/tex]

Quantum Compton-Friedmann equation:
[tex]\boxed{H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - k \left( \frac{m_u c^2 r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda c^2}{3}}[/tex]

[tex]\Lambda = \frac{1}{dt^2} = 10^{-35} \; \text{s}^{-2}[/tex]

Reference:
Scale_factor_(universe) - Wikipedia
Universe - Wikipedia