Over what scale is curvature measurable

[newjerseyrunner]

Imagine I have three space probes that I send out radially. They have a superluminal way to determine each other's relative position to each other instantaneously. If each one measures the relative position of the other two and comes up with an angle for them, how far away would they have to be from each other before the sum of those angles became noticeably not 180 degrees?

 

[Bandersnatch]

If you're asking hypothetically - then it depends on how curved the space is, and how good are your measuring instruments.

If you're asking about our actual universe - then it's impossible to answer, since we don't know how, if at all, curved it is.

 

[Orodruin]

They have a superluminal way to determine each other's relative position to each other instantaneously.

No they don't.

If each one measures the relative position of the other two and comes up with an angle for them, how far away would they have to be from each other before the sum of those angles became noticeably not 180 degrees?

Since you are referring to spatial curvature, this depends on how you separate time from space and therefore on exactly how your super magical fictitious superluminal communication works. Space can be curved in a flat space-time and vice versa.

 

[bcrowell]

If you're asking about our actual universe - then it's impossible to answer, since we don't know how, if at all, curved it is.

We do know quite accurately how curved it is. For example, we know in great detail about the curvature of spacetime within our own solar system. You may have had in mind the average spatial curvature of the entire universe; we know that very accurately too, and it happens to be close to zero.

 

[Bandersnatch]

@bcrowell: yes, I meant the spatial curvature of the universe as a whole. The post is in cosmology, so I assumed that's what was meant.
I know it's close to zero, that's the whole point. Consider what the OP is asking. He basically wants to know how accurate measuring equipment (i.e., how large a triangle) is required to detect spatial curvature of the universe. At least that's how I read it.

In this question there's an unspoken assumption that we know that the universe definitely has some non-zero global spatial curvature.
After all, if it has none, then it's impossible to measure it, no matter the accuracy of equipment.
And if it has some, but small enough to fall within the error bars of current measurements consistent with flat universe, then it is impossible to answer how accurate the equipment needs to be to detect it, since we don't know how close to zero the value lies.edit: now that I read my response in post #2 again, I see that I could be taken to mean that the global curvature is completely unconstrained by measurements. I admit it was sloppy.