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I'm reading this post:

Definition of Fine-Tuning

and John Rennie's answer that we can calculate the probability that the cosmological constant has its observed value (the answer being around 1 in $10^{120}$.

I'm not a physicist. I'd like to understand what this probability means. When I think of probability, I think of a space of possibilities... for example a dice with 6 sides. The probability of getting a 3 is 1/6.

So what does it mean to say the probability of the cosmological constant being what it is, is 1 in $10^{120}$? My naive way to make sense of it is to say that there are multiple universes and around every 1 in $10^{120}$ of them has our particular cosmological constant. But I get the feeling this is the wrong way to think about probability in this situation.

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    $\begingroup$ It is the correct way. The number says how improbable it is to have the observed universe. That is why some people advocate the "anthropic principle" en.wikipedia.org/wiki/Anthropic_principle . $\endgroup$
    – anna v
    Commented Apr 27, 2021 at 8:34
  • $\begingroup$ @annav, so a physicist making this probability claim is implicitly saying there are multiple universes? $\endgroup$
    – Ameet Sharma
    Commented Apr 27, 2021 at 8:37
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    $\begingroup$ " and John Rennie's answer that we can calculate the probability that the cosmological constant has its observed value (the answer being around 1 in $10^{120}$ " . Is this number correct ? I find it hard to believe such a probability has been calculated. I would love to see some kind of citation, reference etc. $\endgroup$
    – silverrahul
    Commented Apr 27, 2021 at 8:45
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    $\begingroup$ @silverrhaul $10^{120}$ is famous as being the order of magnitude difference in the naive QFT prediction of $\rho_{\Lambda}$ vs the observed value. I assume this is where it comes from. $\endgroup$
    – Eletie
    Commented Apr 27, 2021 at 9:13
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    $\begingroup$ @silverrahul aapt.scitation.org/doi/10.1119/1.17850 $\endgroup$
    – anna v
    Commented Apr 27, 2021 at 9:14

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As John mentions on his comments, we should take this 'probability' with a grain of salt. We obviously can't seriously talk about probabilities when we have no other universes to compare to. (It's more like rolling a dice and getting '4' but not knowing how many sides the dice has, whether it's a fair dice/roll, etc). Calculating a probability in this case then necessarily relies on our assumptions about how the cosmological constant takes the value it does, as well as it's allowed values, and a whole bunch of other physics on the edge of our understanding (e.g. whether mechanisms do/don't exist that drive the CC to its observed value).

The probability mentioned here, the $1$ in $10^{120}$, is more akin to assuming the CC can randomly take any value, looking at its value and saying 'ah, the chance of having this value is 1 in whatever'. If I remember correctly, the $10^{120}$ figure is usually contrived by comparing the observed value of $\Lambda$ with the naive QFT vacuum prediction of $\Lambda$, which differ by around $10^{120}$ orders of magnitude. This is why I don't think there's any benefit to giving a probability to something like this. It also belittles the more serious theoretical problems with the cosmological constant, namely, it's radiative instability.

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  • $\begingroup$ Thanks. So is there any evidence of the cosmological constant changing within our universe? In that case maybe we could say something like the probability of the CC becoming the value it has after 14 billion is so and so? I'm trying to make sense of the probability talk I hear with regards to physics and fine tuning. $\endgroup$
    – Ameet Sharma
    Commented Apr 27, 2021 at 9:01
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    $\begingroup$ Nope, we have no evidence of it changing. When people talk about fine-tuning though they usually look at the (again, naive) values of the CC that seem compatible with something like structure formation (galaxies, etc). Then it might have a range of allowed values, and the probabilities come about by assuming the CC takes a random value. $\endgroup$
    – Eletie
    Commented Apr 27, 2021 at 9:11
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    $\begingroup$ I wouldn't get bogged down by the actual probability values, different analysis leads to wildly different numbers. The key message people are trying to make is that the observed $\Lambda$ is very very small but non-zero, whilst current QFT predictions are much larger. (I won't go into the details about why remedying this in QFT isn't as straightforward as it seems). Using numerical probabilities is just an attempt to encapsulate this point (and also hint that there should be some physical explanation, as opposed to being purely random/coincidence). $\endgroup$
    – Eletie
    Commented Apr 27, 2021 at 9:20
  • $\begingroup$ Thanks. One final question. I've heard physicists say our universe is very "unlikely". But is there any justification for this claim? For me, as far as I know this universe with this set of constants may very well be the only one that is possible (ie: maybe the probability of this universe is 1). Is there any reason to think other universes with other configurations of constants are possible? $\endgroup$
    – Ameet Sharma
    Commented Apr 27, 2021 at 9:26
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    $\begingroup$ @AmeetSharma you can probably have a long discussion about this question, I don't think I can answer it here. But in my opinion, we're not in a position to claim it is or isn't unlikely. There may be physical mechanisms making it the way it is, or there may not be. Anthropic reasoning might be answer, etc. We don't yet have a definitive answer. $\endgroup$
    – Eletie
    Commented Apr 27, 2021 at 9:46
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1x10 raised to 120 means you multiply 10 a hundred and 20 times. So the chance of one in that ginormous number means that the chances of that one event of happening is exponentially impossible. Our entire universe has 1x10 raised to the 90 particles, that is, subatomic particles.

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