# What is the distance a comoving observer actually measures in a FRW universe, how do you define a standard ruler?

I am confused about distances in an expanding FRW universe. If we assume it's flat the metric takes the form $$ds^2 = -dt^2 + a(t)^2[d\chi^2 + \chi^2 d\Omega^2].$$ Say that a comoving observer (in rest w.r.t. the CMB) has some standard ruler (say a stick or something) to measure distances and measures the distance $$d_1$$ between to comoving objects. Then, a time later the physical distance between the objects has changed due to the expansion of the universe. But, since the standard ruler has grown in the same proportion the distance measured by the same observer will still be $$d_1$$.

To my understanding it is the physical distance ($$=$$scale factor times comoving distance) that an observer actually measures in an experiment. This means that the observer should not measure $$d_1$$ a time later since the physical distance has changed.

A possible way around this is to take the standard ruler to be $$\Delta t \cdot c$$ for some $$\Delta t$$, this way the ruler stays fixed in proper length. Q1: Is this the right way to define a standard ruler? And, since $$c$$ couple space and time, is it possible to think of it the other way around that a space scale defines a standard time scale?

My other question is; Q2: if we define the standard ruler to be a physical stick as before. Is the assumption that it grows in proper length as the universe expands correct? On the one hand, the whole universe expands so the stick should do it too, but on the other hand the stick is just held together by the forces of atoms and the physics should be unchanged by scaling the background.

Any suggestions of books with a more mathematical approach do distances in cosmology is greatly appreciated.

## 2 Answers

Firstly, your Q2: The 'standard rulers', cosmological or locally, do not expand with the universe. There is just progressively more of them that fit into the distance between objects at cosmological distances.

Your Q1: For the real universe, we have to a do a numerical integration of the first Friedman equation between two redshifts. If you want the details of how that is done, we can discuss it here, or by means of messaging on this forum.

A good reference is: http://arxiv.org/pdf/astro-ph/0402278v1.pdf

• Thank you, that reference helped a lot! Apr 10, 2020 at 13:37

Using the coordinates in your first equation, a standard ruler (being a local object) can have a length $$\delta s = a(t) \delta \chi$$. This is a fixed proper length, and it shows explicitly that the coordinate length $$\delta \chi$$ of the ruler appears to grows shorter in comoving coordinates. In other words, the expansion of the universe is relative to standard rulers.

It certainly makes more sense to think of the space scale as being defined by the time scale than the other way around, both because we use the speed of light to define the metre, and also because the structure of a physical ruler is determined by electromagnetic forces and consequently depends on the speed of light.