If the energy of $1 \, \mathrm{bit}$ of information is $k_{\small{\text{B}}}T \ln{2}$, then the energy of that bit increases with the temperature of the system. When I try to calculate how much energy 1 bit of information would have had around the Planck time, when the temperature of the universe was in the ballpark of ${10}^{32}\,\mathrm{K}$ and the diameter was around ${10}^{-33}\,\mathrm{cm}$, then the energy of that $1\,\mathrm{bit}$ of information is suspiciously close to the Bekenstein bound. In fact, the result I got was $1.3719436998375747 ,$ which I think means that the energy of $1\,\mathrm{bit}$ of information at planck time was just a bit less than would have been required to collapse the universe into a black hole before it had even begun (perhaps even before inflation could save it from said fate).

Is it just coincidence that the number I happen to get is so close to the Bekenstein bound? Was the Information (or entropy or negentropy or whatever you want to call it) contained in the big bang really THAT low? I mean, I knew it had to be low, but 1 bit seems to be cutting things a bit fine, doesn't it?

  • $\begingroup$ Wait, I may have been reading that wrong. My result of 1.3719436998375747 might be just inside the threshold to create a black hole, not outside... Does that mean that there could have been 0 bits of information encoded in the big bang? Instead of 1 bit? Either way, a rounding error or slight error in my inputs (from the UCLA website) could easily be enough to push it either way. And I still find it suspicious that the number is so close to the Berkenstein Bound. $\endgroup$ – Thor Oct 13 '18 at 6:16
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    $\begingroup$ I suspect there is some circularity to your calculation: the Planck time is effectively defined as (the ball park) time when the energy density is such that quantum gravity kicks in and the Berkenstein Bound is reached. $\endgroup$ – Bruce Greetham Oct 13 '18 at 9:22
  • $\begingroup$ Lol! You might well be right... Still though, any idea how many bits of information would have come with the Big Bang? If the universe had a finite size and a finite volume and a finite temperature, then we should be able to calculate the energy (on that note, I can't believe I didn't do that last night...). What is that figure and how does it compare to the 1 bit = 956992961.692908J I got in my calculation. $\endgroup$ – Thor Oct 13 '18 at 15:31
  • $\begingroup$ I think you can only study information-of-universe questions from the other end starting with the known observable universe. Check out physics.stackexchange.com/questions/35920/… $\endgroup$ – Bruce Greetham Oct 13 '18 at 17:00
  • $\begingroup$ Thanks Bruce, I'll definitely check out that thread. I still think there's something to my question though... Ultimately, all the information that exists in today's universe evolved from the informational content of the universe at the time of the big bang. And, following that, there's the question of how much information will exist at the final moment of the universe? The temperature will be MUCH lower, but will the temperature ever get low enough that once again only 1 bit of information will exist? What does that total information content curve look like? $\endgroup$ – Thor Oct 13 '18 at 17:07

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