I raise as many questions as I reveal musings, so this is more appropriate here in the Q&A thread than in the Astronomy thread.
Spoiler - many of my questions are rhetorical, answered later in this post! So please don't reel off an answer without first reading the post!
What would happen if we took all the matter in the universe (ok, "observable universe"), converted it to energy, added it to all the energy currently in existence, and expressed that in terms of one photon (the smallest EM unit)?
Yes, I know photons have an upper energy/temp limit... Just bear with me. Or, on second thought, let's cram it all into a volume with a dimension close to that of Planck length?
What would it's wavelength be? This may be answered by considering the relationship between an object's blackbody temperature and the wavelength of that object, as given by the Planckian locus.
Let's assume a standing wave. The minimal dimension for a standing wave, also called it's "fundamental mode," is 1/2 it's wavelength. However, in nature we find this exceedingly rare, due to the nature of a 1/2 standing wave (it lacks system stability). For our purposes, however, we're simply looking for something within the same order of magnitude of the wavelength. Thus, we'll set the wavelength equal to the Planck length.
We'll approach this from both ends - first, from the temperature, working towards a wavelength. Next, from a wavelength, to find the temperature.
First, some constants:
Planck time: 5.39 x 10-44 seconds
Planck length: 1.62 x 10-35 meters
Planck temp: 1.42 x 1032 K
In order to find a wavelength, we need to find a temperature. Fortunately, "The relationship between the temperature T of a black body, and wavelength λmax at which the intensity of the radiation it produces is at a maximum is" given by Wien's displacement law, which states:
T = b / λmax, where b is Wien's displacement constant (a constant of proportionality) which is equal to 2.898(51) × 10–3 m K
Thus, at a wavelength equal to Planck length, the temperature is 1.79E+32 K.
Similarly, at a temperature equal to Planck temp, the wavelength is 2.04E-35 m.
Thus, we've clearly established there's a very close, if not perfect, relationship between Plank length, Planck temperature, and the temperature at the earliest moments of the Big Bang.
I am curious as to the realtionship between the derived wavelength and Planck length...
The ratio is 1.26 (derived length divided by Planck length). The inverse is 0.794. Not really indicative of anything.
I do find it interesting, though, since we are talking about standing waves, that it's almost precisely the ratio between 1/4 and 1/5, and both of which are common overtones, which are produced when the energy in a vibrational system exceeds the fundamental (quantum) energy required to establish a vibration at it's fundamental resonance frequency.
Hmmm... Something that's 4/5ths of something else sounds a lot like the 1/3 and 2/3 charges of quarks...
Anyway, how does this compare to the temperature at the inception of the Big Bang?
From Timeline of the Big Bang, we have: "Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. This singularity signals the breakdown of general relativity. How closely we can extrapolate towards the singularity is debated—certainly not earlier than the Planck epoch.[/quote]
Also: "Approximately [1E-35] seconds into the expansion, a phase transition caused a cosmic inflation, during which the universe grew exponentially."
This phase transition was triggered by slow expansion which lowered the temperature enough to separate strong and electroweak forces. It was this separation that resulted in the rapid expansion known as the inflationary epoch.
The key is that prior to this phase transition, the strong and electroweak forces were combined.
So what were the conditions like before the phrase transition which resulted in rapid expansion?
I've searched for a temperature, but can't find any stated temp for a time earlier than 10-43 s, when it was "1032." However, I did a log-log graph of the known temps and the curve is fairly flat, and the Planck time unit is very close to the earliest time-temp pair.
So, the Planck Time temp (derived from graph extrapolation) is: 1E32.165, which comes to 1.46 x 1032 K. Bear in mind this is merely a wag on my part, an extrapolation of known data.
I find this amazingly close to, just 3.2% greater than, 1.416 785(71) × 1032 K, which is the Planck temperature!
And since no time-temp relationship is given (that I can find) which is hotter than 1E32 K, and Planck's temp is 1.42E32, I'll assume my graphical wag, while very close, isn't as accurate as that calculated for the Planck temp.
So - What happens at Planck temp?
Why does this appear to correspond to a primordial Planck length and time?
Why do Planck units continue to pop up when talking about issues such as the temp at which the unification of gravity and everything else is said to occur, ie, during the transition between the Augustinian era and the Planck epoch, when electromagnetism, weak nuclear, strong nuclear, and gravitation all have the same strength?
Why is Planck's length, mass, time, charge, and temperature directly, and so simply related to the gravitational constant?
The Early Universe:
Augustinian era - energies and temps had reached the Planck scale.
Planck epoch, grand unification epoch, electroweak epoch, etc., follow. Other websites list slightly different terms for their eras/epochs.
1. Inflation occurred after approximately 1.9 Million units of Planck time.
2. Plank energy, Ep = h/tP, where h is the reduced Planck's constant, and tP is Planck time. It's equal to 0.5433 MW-h, which is roughly the energy contained in a bolt of lightening, or about two weeks per-capita consumption of electricity in the US.
That's an awful lot of energy to be contained in a space so small... Perhaps there's an energy-mass equivalence in there relating to Planck densities (5.155 x 1096 kg/m3) contained in Planck volume (4.2221E-105 m3)?
Well, running the calculation, that does seem to equate to Planck mass (2.18E-8 kg).
In summary, this all seems to work. Modern values, tried, tested, and true in various laboratories, all seem to correlate very well with the known/expected conditions at the beginning of the universe, when the electronuclear force (GUT) and gravity were on a level playing field (Theory of Everything).
Thus, the key to TOE will probably be found in the relationship between these concepts. And, I even suspect that somewhere, someone may find there's a 4/5ths or 5/4ths relationship between something, somewhere...