Cosmic Coincidences

The critical density, as we have seen, is determined by introducing the Hubble parameter, $H$, defined from the velocity distance relation, $v=HR$. Rather than being a 'receding speed', $v$, and $R$ the distance the galaxy and the observer, the former appears as a revolving speed, so that $H$ is an angular speed of rotation, and the latter the radius measuring the distance between the galaxy and the cosmic center. Although the latter interpretation of Hubble's constant has not been open to direct observations, it lends itself to a series of numerical coincidences.

First, and foremost, is the determination of the critical density of the universe from---of all places---the Schwarzschild radius,

$$\mathcal{R}=\frac{2GM}{c^2}=\frac{c}{\omega}.$$

If the angular speed is identified with Hubble's constant, and the density is introduced as $\rho=3M/4\pi\mathcal{R}^3$, we immediately obtain the critical density,

$$\rho_c=\frac{3H^2}{8\pi G},$$

which is enough to raise eyebrows.

Second, if the angular velocity is $H\mathcal{R}$, the centripetal acceleration is

$$a=H^2\mathcal{R}.$$

Again using the critical density, the acceleration is found to be

$$a=Hc,$$

which is numerical $6.48\times 10^{-10}$ m/sec. This is the same order of magnitude as the MOND acceleration, $a_0$, in 

$$v\simeq\sqrt[4]{2GMa_0},$$

where $a_0\simeq (1.2\pm.03)\times 10^{-10}$ m/sec. Milgrom determined this value by fitting flat  rotation curves of galaxies obeying the Tully-Fisher relation,

$$v\propto L^{1/4},$$

where the luminosity $L$ is proportional to the mass $M$ of the spiral galaxy.

Finally, if we use Milgrom's relation, 

$$v=\sqrt[4]{2GMa_0},$$

introduce $v=H\mathcal{R}$, and the critical density, we find

$$a_0=H^2\mathcal{R}.$$

Alternatively, if we know $a_0$, from fitting, and $H$ from the results of the 2015 Planck survey, we find $\mathcal{R}\sim 10^{26}$ m/sec, which is the order of the radius of the universe!

This is truly remarkable in view of the fact that we are dealing with very high speed relativistic phenomena, and, yet, $a_0$, is about a 100 billion times smaller than the acceleration due to gravity at the earth's surface!

Universal constants, like Planck's constant, Boltzmann's constant, Newton's constant, charge, and the speed of light are often used as benchmarks for determining the limits of applicability of a theory. Consider the 'classical' radius of an electron, $r_c=e^2/mc^2$. It contains the charge and the speed of light so we know that it is electrodynamic, ultra-relativistic, and 'classical'. If we divide this radius by the fine structure constant $\alpha=e^2/\hbar c$, we get the Compton wavelength, $\hbar/mc$, which is still relativistic, but non-classical since it contains Planck's constant. Another division by $\alpha$ results in the 'classical' Bohr radius, $\hbar^2/me^2$, which is quantum and non-relativistic since $c$ has disappeared.  The same procedure applies to gravity by substituting $e\rightarrow \sqrt{G}M$.

In reference to the above 'coincidences', the critical mass is classical and non-relativistic, although the ultra-relativistic Schwarzschild radius was used to derived it. Moreover, Hubble's law is classical but the centrifugal acceleration used to derive Milgrom's constant, $a_0$ is ultra-relativistic. The big question is: How can the critical density be non-relativistic when all that depends on it is ultra-relativistic?

All we can conclude from this is that if there is any physics in this, which is extremely doubtful, it is very well hidden.