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From Sabine Hossenfelder's recent video, New Evidence AGAINST Standard Cosmology:

enter image description here

And her source....

enter image description here

I don't get why a graph would show a quantity that is squared, then immediately 'square-rooted'.....

Also, the lowercase delta does stand for the Hubble constant difference or contrast, correct?

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    $\begingroup$ The plot she is showing averages over $\delta H$, not $H$, i.e. a fluctuation field. $\endgroup$ Sep 7 at 18:45
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The brackets refer to the average, so $\left< x^2 \right>^{1/2}$ is the root-mean-square (RMS) of $x$. That is the square root of the mean (or average) of the square of multiple $x$s.

The RMS average is useful when a quantity can be either negative or positive. For instance, a sine or cosine wave has an average of zero over one cycle, but its RMS average is proportional to its amplitude:

$$\left< A\sin x \right> = 0, \quad\quad \left< (A\sin x)^2 \right>^{1/2} = \frac{A}{\sqrt{2}}.$$

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    $\begingroup$ what further advantage does it have over the average of the absolute values? $\endgroup$
    – njzk2
    Sep 7 at 21:28
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    $\begingroup$ @njzk2 : You might benefit from reading about the RMS, learning, for instance, that it is related to the standard deviation of a (0-mean, or centered) sample. $\endgroup$ Sep 7 at 23:08

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