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What is the currently accepted estimated range of the amount of energy of the Big Bang event?

In joules at some estimated size, so a temperature may be calculated.

For context, I wonder if the temperature of the Big Bang would make its heat radiation wavelength shorter than the Planck length.

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Let's start by making some points clear:

1. We don't know what the Big Bang was.

Rather, we know that the Universe is expanding. If you extrapolate backwards, you'd expect the Universe to be denser and denser. More specifically, we talk about this as a change in the scale factor $a$, and this gets smaller and smaller as we look further back in time. According to general relativity (our modern theory of gravity), 13.8 billion years ago, $a$ should have been $0$; however, you can't have a metric with $a = 0$.

Thus, we know that general relativity is necessarily incomplete. It breaks down at the conditions of the early universe, so we currently have no physical model to explain that time. Rather, we know that the early universe expanded, and the Big Bang is the time that perplexes cosmologists. Some theories, like quantum gravity, have emerged in an effort to explain the Big Bang; however, we currently have little understanding of what it actually was.

So no, we can't tell you what the energy output of the event was, since we don't know what actually happened.

2. The temperature of the early Universe was high

Our theories break down at the Planck epoch of the Universe. The Planck epoch was the earliest epoch of the Universe and lasted until $10^{-42}$ seconds after the Big Bang — that's 200 Planck times, which are the shortest meaningful measurement of time.

During this epoch, the entire Universe was at $1.417×10^{32} \; \mathrm{K}$, which is the Planck temperature. This is the hottest possible temperature; an object at this temperature will emit photons with wavelengths of a Planck length (you can read more about this in my answer here). The point is that there is no meaningful distance smaller than a Planck length, so the Universe couldn't be hotter than the Planck temperature.

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    $\begingroup$ +1 for answering the question, but don't forget that theories break down, and they are only theories. We don't actually know that the temperature of the very early universe was high. $\endgroup$
    – John Duffield
    Mar 18, 2017 at 19:20
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    $\begingroup$ @JohnDuffield The WMAP observations provide strong evidence that the temperature of the early universe was high, ruling out a cold Big Bang. See Komatsu et al. (2010). $\endgroup$
    – Sir Cumference
    Mar 31, 2017 at 17:36
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    $\begingroup$ @JohnDuffield Well, we don't really know what the Big Bang was, so I'm a bit confused as to how you derived that. $\endgroup$
    – Sir Cumference
    Apr 1, 2017 at 18:22
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    $\begingroup$ @SirCumference total size of the universe would be something like the mass of the matter plus the energy, though I am sure there are better ways to say that. I did not suggest you would equate as you've stated, but I think it's reasonable to expect a larger universe to have a larger energy big bang, no? I still think you need a 3rd uncertainty item for the size of the universe, of which we have no upper limit. $\endgroup$
    – uhoh
    Jan 11, 2020 at 10:09
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    $\begingroup$ @uhoh If we're talking about total energy of the universe, then yes. But the energy density of our universe at the Big Bang rises to infinity for any size of the universe. $\endgroup$
    – Sir Cumference
    Jan 11, 2020 at 10:47

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