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The numerical coincidence that triggered Dirac to postulate his Large Number Hypothesis can be summarized by expressing the proton-electron gravitational angular momentum in units $\hbar c$:

$$\frac{G m_p m_e}{\hbar c} = 10^{-41.49}$$

and the Hubble parameter $H_0$ (a measure for the inverse of the lifetime of the universe) in the Compton frequency of the proton $m_p c^2/\hbar$:

$$\frac{2 \hbar H_0}{m_p c^2} = 10^{-41.51}$$

Different guises of the near equality:

$$G c m_p^2 m_e \approx 2 \hbar^2 H_0$$

equating the scale of the universe to subatomic scales are often referred to as the Eddington-Weinberg relation. Why? I can see how Eddington's name got attached, as Dirac did build on his work. But why Weinberg? Did he investigate this cosmic coincidence? Any references?

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I don't know about this particular result, but Weinberg's certainly done a bunch of work on the Anthropic principle, including one of the successes of it--a guess of the order of magnitude of the cosmological constant before it was measured. –  Jerry Schirmer Aug 19 '13 at 17:18
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