The previous thread got shut down, and for good reason. It was likely to go in circles. I wanted to give the ideas a fair shot on these forums in a sense, despite not being anything more than perhaps mathematical possibilities. So, I tried to go through the material and made it to the end of the first page before my brain was fried:
Okay let’s begin.
The derivation of the first set of equations is odd. If the beginning is odd, then the rest of it would be hopeless.
- Equation 1: okay, this is the Stefan Boltzmann law. Nothing too odd here.
- Equation 2: what is this equation? What is this scale factor, τ? Initially it seems that you are saying that the surface area of our universe expands as a function of τ. What does tau mean exactly? You have defined things backwards from how τ is usually used as a characteristic time scale, because you defined such characteristic timescale already to be the planck time. If you look at your equation, what you are saying is that given an initial surface area, the surface area increases linearly depending on how many planck time units have passed. This equation is just wrong. By definition characteristic timescales do not change as a function of time.
- Equation 3: given that equation 2 was so odd, this one is also destined to fail. However, you did do the correct substitution of the Planck length for the Planck time.
- Equation 4: seems like a lot of work to just derive the equation that only describes the observable universe. If the speed of light has indeed remained constant, then yes, the observable universe has a radius grows at the speed of light in all directions. The surface area of such a sphere is given by exactly what you wrote: A=4piR^2=4pi(ct)^2.
All of these equations don’t really make sense in light of cosmological models for the scale factor of the universe. Why go through the trouble since it’s already pretty well established that the scale factor depends on the density of radiation vs. matter vs. dark energy. For example this graph is a very common graph in cosmology:
In different eras, you can derive the scale factor of the universe, like the radiation dominated era scales the radius of the universe like R~t^(1/2), the matter dominated R~t^(2/3), as outlined here: https://en.wikipedia.org/wiki/Scale_factor_(cosmology).
This is not going well, but I did skim the second page. You mention the so-called flatness problem that is “unsettling” with no “scientifically satisfying answer.” That’s not true at all. There are many good solutions, with the leading one being inflation which has produced a good 12,000+ papers since its inception by more than 9,000 different cosmologists. Anyone can casually read the wikipedia entry: https://en.wikipedia.org/wiki/Flatness_problem. I am personally pulling for cosmic inflation which we have some evidence for, but it’s not quite as nice as we’d like just yet.
I can’t do any more