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I understand that, in general, homogeneity is the physical attribute of being uniform in composition (" of the same form at every point"), but I'm slightly confused when it is used in cosmology as there seems to be an overlap with the notion of isotropy, i.e. direction independence. I've read that in cosmological terms, homogeneous means that the universe "looks the same" at every point (at a given instant in time), implying that it has uniform density (on large scales) and is such that the same laws of physics apply universally. However, isotropy means that at a given point (at a given instant in time) the universe "looks the same" in every direction. This seems to overlap somewhat with the concept of homogeneity?!

Is the idea that homogeneity means that, if an observer observes the universe to look a particular way from their vantage point (i.e. that it has certain physical properties), then another observer at any other point in the universe will make the the same observations and conclude that it has the same physical properties. The point being that both of them could observe anisotropies and thus the universe could potentially be homogeneous, but not isotropic. The reverse does not hold however, as if both observe the universe to be isotropic, the it is necessarily homogeneous?!

Sorry for the long windedness of this question, it's just been bugging me for a while.

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Homogeneity in cosmology means uniformity from point to point, not only in composition or content, but in geometry as well. An empty space with a singularity is still non-homogeneous. Isotropy at every point does imply homogeneity, but we are not in a position to observe the universe from every point. Mathematically, isotropy at any two distinct points already implies homogeneity, but in practice we need them to be sufficiently far apart, and we are not in a position to observe the universe even from two points that are sufficiently far apart (recall how telescopes resolved stellar parallax only in 1838, and its absence was used as an argument against moving Earth).

On the other hand, it is easily possible for the universe to be isotropic at some point, in particular at the point where we and the Earth are, without being homogeneous. Any non-constant spherically symmetric distribution of matter will do. To get homogeneity from isotropy-here one needs to invoke the so-called Copernican principle, which states that neither the Sun nor the Earth occupy a special position in the universe. Of course, the Copernican principle is effectively the contrapositive of homogeneity, all places are the same rephrased as no place is special. See discussion on Physics Forums.

So when it is said that the universe is "homogeneous and isotropic" there is no "overlap" because only isotropy-here is meant as an assumption. On the other hand, isotropy-here (the only thing we actually observe) and homogeneity imply isotropy everywhere as a conclusion. By rejecting the Copernican principle one can construct a curious giant void cosmology for example, which explains accelerated expansion of the universe without the dark energy, but "most scientists believe that it is not reasonable to adopt a cosmological model in which the universe is simply a joke played for the benefit of mankind".

It is also possible for a space to be homogeneous but anisotropic purely geometrically, without any matter content. A simple example to visualize is a two dimensional cylinder: every point looks the same, but vertical and horizontal directions look different (globally). A three dimensional example is Poincare dodecahedral space obtained by identifying certain points on the $3$-sphere. If our universe had this shape there would be observable patterns in the CMB radiation indicating it.

There is also a difference between local and global homogeneity, vicinities of all points, may look the same, but some points may still be special from the global point of view. Take open flat disk for example, metric tensor is everywhere constant, so small neighborhoods of all points 'look the same', but the center is special, there is no global isometry that maps it to any other point. There are less visual such examples that do not have a 'boundary'.

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Here is a short summary inspired by Barbara Ryden: Homogeneity: No preferred location Isotropic: No preferred direction

And here are some examples to clarify things:

Example of homogeneous but not isotropic: A forest, it looks the same no matter where you are, but trees make the vertical direction distinct.

Example of Isotropic but not homogeneous: When you stand on top of a hill, no matter which direction you look, things look the same (that is the only hill around). But it certainly is a special location.

Example of Isotropic and homogeneous: Being lost in an ocean. No matter which way you go, no matter where you are, you can tell the difference, i.e. you are lost.

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What is the exact meaning of homogeneity in cosmology?

Conifold and Milad have adequately explained the distinction between homogeneous and isotropic, so I'll answer on a different tack:

See the Einstein digital papers where he said 'empty' space in its physical relation is neither homogeneous nor isotropic. A gravitational field is a place where space is not homogeneous. In the room you're in, the space near the floor is a little different to the space up near the ceiling. If it wasn't, there would be no gradient in gravitation potential, and your pencil wouldn't fall down. And because it falls down, space isn't isotropic.

Now take a look at the FLRW metric: "The FLRW metric starts with the assumption of homogeneity and isotropy of space". This assumption is in essence saying there's no overall gravitational field in the universe. IMHO that's fine, because let's face it, the universe didn't contract when it was small and dense. However this assumption also says galactic gravitational binding and conservation of energy has no effect on the homogeneity of space. IMHO that isn't fine, because as Einstein said the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. Papers such as Inhomogeneous and interacting vacuum energy by De-Santiago, Wands, and Wang touch on this issue. Space is dark, it has its vacuum energy, and when it's inhomogeneous, this has a mass-equivalence and a gravitational effect.

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