Undergrad About fundamental constants and vacuum

[IamNobody]

Does it not raise question to be able to build two parameters with the same dimension and order of magnitude of two fundamental constants from five parameters (other fundamental constants and measured properties)?

$$\sqrt{10\frac{\left(\varepsilon_0 e^{-2}\right)^3\left(k_B T\right)^4}{\rho_c}} \sim 3\times 10^8~m.s^{-1}$$
$$21.7\frac{\sqrt{\rho_c}}{\left(\varepsilon_0 e^{-2}\right)^{5/2}\left(k_B T\right)^2} \sim 6.6\times 10^{-34}~kg.m^2.s^{-1}$$
with
Vacuum permittivity: $\varepsilon_0\sim 8.854\times 10^{-12}~m^{-3}.kg^{-1}.s^4.A^2$
Elementary charge: $e\sim 1.602\times 10^{-19}~A.s$
Boltzmann constant: $k_B\sim 1.381\times 10^{-23}~kg.m^2.s^{-2}.K^{-1}$
Cosmic Microwave Background (CMB) temperature: $T\sim 2.73~K$
Critical density: $\rho_c=\frac{3H^2}{8\pi G}\sim 9.2\times 10^{-27}~kg.m^{-3}$
where $G\sim 6.674 \times 10^{-11}~m^3.kg^{-1}.s^{-2}$ is the gravitational constant and $H\sim 70~km.s^{-1}.Mpc^{-1}$ is the Hubble constant (actually between $65$ and $75~km.s^{-1}.Mpc^{-1}$ according to the measurements).
Of course, the speed of light is $c\sim 3\times 10^8~m.s^{-1}$ and the Planck constant is $h\sim 6.626\times 10^{-34}~kg.m^2.s^{-1}$.

Besides, one can see both relations as two dimensionless numbers approximately equal to $\sqrt{10}$ and $21.7$ built from assumed independent parameters.

 

[fresh_42]

Any connection between quantities which are dependent on arbitrarily defined units is per construction arbitrary, hence numerology which cannot hint to any deeper insights.

Even worse, you can solve any equation system $f_k(\alpha_1,\ldots) = c_k$ with given values $c_k$ and arbitrary parameters $\alpha_j$ where the solutions are any possible arithmetic expressions $f_i$. Existence of such a solution is trivial, and uniqueness unachievable. This alone indicates the lack of meaning.

I'm afraid we will not debate such kind of conspiracies on PF.