Thread: Friedmann Equation View Single Post
 Current Universe radius: $$\boxed{r_u(t_0) = 4.4 \cdot 10^{26} \; \text{m}}$$ Friedmann scale factor: $$\boxed{a(t) = \frac{r_u(t)}{r_u(t_0)}}$$ $$a(t_u) = \frac{r_u(t_u)}{r_u(t_0)} = \frac{\hbar}{m_u c r_u(t_0)}$$ $$\boxed{a(t_u) = \frac{\hbar}{m_u c r_u(t_0)}}$$ Quantum Compton-Friedmann equation: $$H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - \frac{kc^2}{a(t_u)^2} + \frac{\Lambda(t_u) c^2}{3}$$ Integration by substitution: $$H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - kc^2 \left( \frac{m_u c r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda(t_u) c^2}{3} = \frac{2 G c^3 m_u^4}{\hbar^3} - k \left( \frac{m_u c^2 r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda c^2}{3}$$ Quantum Compton-Friedmann equation: $$\boxed{H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{2 G c^3 m_u^4}{\hbar^3} - k \left( \frac{m_u c^2 r_u(t_0)}{\hbar} \right)^2 + \frac{\Lambda c^2}{3}}$$ $$\Lambda = \frac{1}{dt^2} = 10^{-35} \; \text{s}^{-2}$$ Reference: Scale_factor_(universe) - Wikipedia Universe - Wikipedia