The first sentence above is, for a physicist, the key result. It says that, for the most common way one could modify gravity to incorporate a new force-mediator, the new interaction has to turn on at distances shorter than 39 microns.

At distances longer than that, these measurements show that gravity acts, and in a way that roughly follows the inverse-square law. (at distances of ~100 microns it very-much follows the ISL, and at distances longer than that, it really follows the ISL).

The easy way to think about this result: we are on the cusp of showing that gravity acts over distances shorter than the width of a human hair. Only the finest human hairs are smaller in diameter.

For many years, my goal in life (see HN handle) was to push this limit below 10-microns. Turns out that is really hard to do :).

Thanks.

> ... and at distances longer than [100 microns], it really follows the ISL

But at cosmic distances, there's also evidence that gravity doesn't follow ISL, right?

Are there any implications of this work for hypothesized dark matter and dark energy?

You are probably thinking of galaxy rotation curves which do not agree with theoretical predictions that assume only visible matter exists.

Gravity not following the ISL at large scales would indeed be a possible explanation for this. However, most alternative theories of gravity clash with other observations, such as the motion of binary pulsars or gravitational waves, whereas (unmodified) General Relativity is in excellent agreement. This is one of the reasons why the existence of dark matter is considered to be a much better explanation for the deviation of the rotation curves.

Yes, I was.

But I know so little physics that the distinction between dark matter and extra terms in gravitation is unclear.

It's pretty simple. You can use the known laws of gravitation and the visible distribution of matter in a galaxy to predict how fast stars at the edge of the galaxy should move. But when we do this we find that the stars are moving much faster than expected. So EITHER the formula for gravity is wrong OR there is extra matter there which we can't see, i.e. dark matter. Dark matter is currently the preferred explanation because it also explains a bunch of other, unrelated, phonenomena, but it has as yet never been directly detected or measured.