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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

1 Answer 1

In 1988, Weinberg used an equivalent relation as an historical starting point for the weak anthropic principle, in his "The cosmological constant problem". I quote, page 7:

An example is provided by what I think is the first use of anthropic arguments in modern physics, by Dicke (1961), in response to a problem posed by Dirac (1937). In effect, Dirac had noted that a combination of fundamental constants with the dimensions of a time turns out to be roughly of the order of the present age of the universe:

$\hbar/Gcm_{\pi}^3 = 4.5\times10^{10}~yr.~~~~~~~~~(5.1)$

[There are various other ways of writing this relation, such as replacing $m_{\pi}$ with various combinations of particle masses and introducing powers of $e^2/\hbar c$. Dirac's original "large-number" coincidence is equivalent to using Eq. (5.1) as a formula for the age of the universe, with $m_{\pi}$ replaced by $(137m_pm_e^2)^{1/3} =183~MeV$. In fact, there are so many different possibilities that one may doubt whether there is any coincidence that needs explaining.] Dirac reasoned that if this connection were a real one, then, since the age of the universe increases (linearly) with time, some of the constants on the left side of (5.1) must change with time. He guessed that it is $G$ that changes, like $1/t$. In response to Dirac, Dicke pointed out that the question of the age of the universe could only arise when the conditions are right for the existence of life. Specifically, the universe must be old enough so that some stars will have completed their time on the main sequence to produce the heavy elements necessary for life, and it must be young enough so that some stars would still be providing energy through nuclear reactions. Both the upper and lower bounds on the ages of the universe at which life can exist turn out to be roughly (very roughly) given by just the quantity (5.1). Hence there is no need to suppose that any of the fundamental constants vary with time to account for the rough agreement of the quantity (5.1) with the present age of the universe

I don't know any other references from Weinberg to that question, but if this is the one, the name of Weinberg seems to be there only to give some additional credibility to the formula, even if

  • Weinberg seemed not to be terribly convinced (my emphasis) by the coincidence(s).
  • Weinberg made a boo-boo in his counting of the $\hbar$s in his equation (5.1).

So, maybe the association of the name of Weinberg with the formula is a bit unjustified, and that's why the link is not so easy to find.

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