Arthur Eddington discovered conspicuous coincidences, amongst the ratios, of various physical & cosmological constants, for our cosmos (Wolff. Exploring the Physics of the Unknown Universe, pp. 173-4), including:

 

[math]\frac{time \, since \, the \, Big \, Bang}{time \, for \, light \, to \, cross \, a \, hydrogen \, atom} \approx 10^{36}[/math]

 

[math]\frac{electrical \, repulsion \, between \, two \, protons}{gravitational \, attraction \, between \, two \, protons} \approx 10^{36}[/math]

[~

10

⁴²

for electrons]

 

[math]\frac{mass \, of \, visible \, cosmos}{mass \, of \, proton} \approx 10^{79}[/math]

[~

10

⁸²

for electrons]

The last ratio, is conspicuously close, to the square of either of the previous pair (Pirani & Roche. Introducing the Universe, 154-155) -- and, hence, to their product. Now, noting that R_H = c T_H, w.h.t.:

 

[math]\frac{R_H}{a_B} \approx \frac{e^2 / 4 \pi \epsilon_0}{G m_p^2}[/math]

 

[math]\therefore \frac{G m_p^2}{a_B} \approx \frac{e^2}{4 \pi \epsilon_0 \; R_H}[/math]

This says, the the GPE of two protons, at an atom's width apart, is comparable, to the EPE of two protons, at a Hubble Radius (!). And, from the product, w.h.t.:

 

[math]\frac{R_H}{a_B} \times \frac{e^2 / 4 \pi \epsilon_0}{G m_p^2} \approx \frac{M_V}{m_p}[/math]

 

[math]\therefore \frac{m_p \times e^2}{4 \pi \epsilon_0 \; a_B} \approx \frac{m_p \times G M_V m_p}{R_H}[/math]

 

[math]\therefore \frac{G M_V m_p}{R_H} \approx \frac{e^2}{4 \pi \epsilon_0 \; a_B}[/math]

This says, quasi-classically, the GPE binding protons to the visible cosmos, at the macroscopic scale, is closely comparable, to the EPE pushing protons apart, at the atomic, microscopic scale. Thus, there is seemingly some sort of "balance", between the forces of contraction (gravity), and those of expansion (electro-static repulsion). Doesn't this seem reasonable?

 

This says, quasi-classically, the GPE binding protons to the visible cosmos, at the macroscopic scale, is closely comparable, to the EPE pushing protons apart, at the atomic, microscopic scale. Thus, there is seemingly some sort of "balance", between the forces of contraction (gravity), and those of expansion (electro-static repulsion). Doesn't this seem reasonable?

 

Except, as far as we can tell, the universe is electrically neutral. So it seems more like a coincidence than a balance.

[math]\therefore \frac{G M_V m_p}{R_H} \approx \frac{e^2}{4 \pi \epsilon_0 \; a_B}[/math]

The LHS contains two terms, which both depend upon time (M_V, R_H). But, there ratio is (approximately) independent of time:

 

[math]M_V \approx \frac{4 \pi}{3}R_H^3 \rho_{crit}[/math]

 

[math]\rho_{crit} \equiv \frac{H^2}{8 \pi G} = \frac{c^2}{8 \pi G R_H^2}[/math]

And so:

 

[math]\frac{M_V}{R_H} \propto \frac{R_H^3}{R_H \times R_H^2} \propto 1[/math]

 

[math]\therefore \frac{G M_V m_p}{R_H} \approx \frac{e^2}{4 \pi \epsilon_0 \; a_B} \; \; \; \forall t[/math]

Such seemingly suggests, that (1) the cosmos is close to critical; (2) constants are constants. Both such suggestions seem rather reasonable, yes?

In 1973, Edward Tryon calculated, that the sum of matter's Rest-Mass Energy & Gravitational Potential Energy is E₀ + E_g ~ 0. This can be seen, from the above equations, by replacing the Bohr Radius w/ the classical electron radius (a_B --> r₀), which one can do b/c the correspondences, while quite close comparatively, are still rather rough:

 

[math]\frac{G M_V m}{R_H} \approx \frac{e^2}{4 \pi \epsilon_0 \, r_0} \equiv m c^2[/math]

And, thus,

 

[math]R_H \approx \frac{G M_V}{c^2} \approx R_S[/math]

To wit, the Hubble Radius (R_H), of the visible cosmos, is quite close to the Schwarschild Radius (R_S), of all the mass making up the visible cosmos. This again seemingly suggests, that the cosmos is quite close to "critical" & "closed" (M.Wolff. Schrodinger's Universe, pp. 136-137).