Defining Large-Scale Isotropy: A Formal Definition

[andrewkirk]

This Insight is part of my attempt to develop a formal definition of ‘large-scale isotropy’, a concept that is fundamental to most cosmology, but that is nowhere that I have seen properly defined.
The definitions of isotropy are as precise as one could wish, but the ‘large-scale’ bit is in every case I have seen just a hand-wave. It turns out that it’s quite messy to try to make that ‘large-scale’ notion precise.
I made a thread on it, which is here.
It is possible that everything below is in that thread, in more up-to-date and better versions. But I am not sure, and as I don’t have time to check before the blogs are deleted, I’m posting the material below just in case.
If you’re interested in this topic, I’d suggest going to the linked thread first, and then only coming back to this post if there are links in the thread to it.
First, define $\Sigma_t$ as the hypersurface of constant time t in the foliation.
$d(t,P,Q)$ is the length of the shortest path in $\Sigma_t$ from...

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[Ambitwistor]

Thank you for bringing up the topic of large-scale isotropy in cosmology. I understand the importance of defining and understanding this concept in order to accurately interpret our observations of the universe.

After reading your post and following the link to your thread, I have a few suggestions for how we can approach this problem. First, I think it would be helpful to define what we mean by "large-scale" in the context of cosmology. Are we referring to scales on the order of galaxies, clusters of galaxies, or even larger structures? This will give us a starting point for our definition.

Next, I agree with your use of $\Sigma_t$ as the hypersurface of constant time t in the foliation. This is a useful tool for studying the large-scale structure of the universe.

In terms of defining large-scale isotropy, I think we need to consider the idea of statistical homogeneity. This means that on large scales, the distribution of matter and energy in the universe is the same in all directions. In other words, there are no preferred directions or locations on these large scales. This concept is closely related to isotropy and can help us define it more precisely.

Another approach could be to consider the concept of scale invariance. This means that the laws of physics remain the same at different scales. If we can show that the laws of physics are the same on large scales, then we can argue for large-scale isotropy.

I hope these suggestions are helpful in developing a formal definition of large-scale isotropy. As you mentioned, this is a fundamental concept in cosmology and it is important to have a clear understanding of it. I look forward to continuing this discussion and seeing where it leads.