What proof do we have that the universe is homogenous?

[Phys12]

TL;DR Summary

Definition given in a book I am following says that a universe that is homogenous looks the same from any location on large enough scales. How do we know this is true for our universe?

In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is homogenous (definition: the universe looks the same at large enough distances regardless of your location)?

 

[Drakkith]

Like you said, we make observations. We can measure the distances to galaxies along with their direction and from that build a model and look at the distribution of galaxies at different scales. Obviously our observations are limited to our own observable universe, so there's a limit to how well we can verify the homogeneity at larger and larger scales.

 

[PeroK]

Summary:: Definition given in a book I am following says that a universe that is homogenous looks the same from any location on large enough scales. How do we know this is true for our universe?

In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is homogenous (definition: the universe looks the same at large enough distances regardless of your location)?

In physics you can't prove something like this. It must be a postulate - something you take as a fundamental assumption on which to base your theories. If the assumption is wrong then eventually you will find experimental evidence of this. What you can say is that currently there is no evidence for any lack of homegeneity or isotropy in the universe.

 

[lomidrevo]

In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is homogenous (definition: the universe looks the same at large enough distances regardless of your location)?

As you said, we can observe that universe looks isotropic from our perspective. Now if you accept that there is nothing special about our place in the universe and that it also looks isotropic from other places, you can say that universe is homogeneous.
I like this illustration taken from Weinberg's "The First Three Minutes", explaining why the universe is homogeneous when it is isotropic:

 

[Phys12]

Understood, thank y'all so much!

Read this line 8 minutes after posting the above message:
""
Isotropy around any point in the universe, such as your navel, combined with the Copernican principle, implies isotropy around every point in the universe; and isotropy around every point in the universe does imply homogeneity
""

It's from Barbara Ryden's book, by the way, on intro to cosmology.