Does expanding universe length units also expand?

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I have read an explanation about the expanding universe which represents our universe as a metal plate that tends to expand simultaneously in all directions while being heated. One can imagine that some stars and maybe galaxies as small metal balls welded to this plate and there is a bunch of such small balls that become further from each other while the plane is being heated.

My understanding

What I do not understand is whether the distances also grow being measured in the units of a system itself: If we also scratch a coordinate grid on this plane say let's make a grid where each cell is $1 \times 1 ~ cm$ before we start heating. It is clear that the distance measured in these very cells will remain the same no matter how the grid has expanded. But if we will measure with a ruler that was not heated we will see that the distances had changed.

The question

Finally the question: when physicists say that the universe expands do they also mean that the distances do not change as in the scratched grid metaphor?

Note that cosmic expansion doesn't cause objects to expand, separate, or anything of that sort. See for example benrg's answer to "Why does space expansion not expand matter?" and Andrew Steane's answer to "Is space expansion causing increasing distances between galaxies?". — Sten, Jul 03 '23 at 09:35

score 0 · Answer 1 · answered Jul 03 '23 at 10:02

We cannot measure anything if our units don’t stay constant. In your own analogy the metal plate is expanding regardless of the grid you draw. What you are describing is rescaling, i.e., the transformation to a new set of units so that the new size in the units is numerically the same as the old size in the old units.

Does expanding universe length units also expand?

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My understanding

The question

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