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Homogeneity is the idea that the universe looks roughly the same no matter where the observer is. This paper on whether the universe is homogeneous includes this line:

A common misconception is that ‘homogeneity is obvious from the cosmic microwave background (CMB) and the galaxy distribution’. In fact, we cannot directly observe or test homogeneity—since we observe down the past light cone, and not on spatial surfaces that intersect that light cone (figure 1).

It comes with this figure:

enter image description here

Which has the caption: Figure 1. We observe down the past light cone and therefore cannot directly confirm homogeneity. (Online version in colour.)

I don't understand this argument. My intuitive understanding for why we can't directly observe homogeneity is simply that we're on Earth and can't just fly a telescope to e.g. Andromeda to see what the universe looks like from another galaxy. The argument in the paper seems to be saying that we cannot directly observe homogeneity because everything we see is in the past; however it still seems to me that if the universe was homogeneous in the past then we would expect it to be homogeneous today. Alternatively, we could in principle measure the positions & velocities of all the other galaxies in the past, evolve them forward using GR, and therefore tell if our universe is homogeneous today.

Can someone explain how the paper's argument works?

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This is a very good question. I hope I can explain it clearly to you here. I'll try to be pedagogical.

Consider a tabletop spread of seeds. You can say that the distribution of seeds is homogogeneous at some length scale $L$ if the number of seeds per area $L^2$ (or unit volume $L^3$) is constant throughout the spread. You are allowed to make this statement because you can see the entire system from a global perspective at any instance of time. Furthermore the dynamics of this small arrangement is for this purpose instantaneous.

In cosmology, this "Newtonian" notion of instantaneity is abolished. (ignoring quantum effects) This means that no two events where one influences the other are ever spacelike seperated and therefore never on the same spacelike hypersurface. This is simple the causal nature of spacetime. (Note that the effect of time evolution in GR is basically acting a time evolution operator on every spacelike hypersurface, leading to a foliation of spacetime.)

Given this knowledge: What this means is that if you are observing any other system in the universe, it necessarily exists on a different spatial hypersurface and in the past light cone.

Coming back to our analogy of seeds on a table: Now if you try to define homogeneity here, it is like trying to compare the distribution of seeds with some population density $P_A$ on table A with the distribution of seeds with some other population density $P_B$ on some other table B. It doesn't make sense anymore.

What does alleviate this problem is isotropy. Isotropy of the universe basically states that if you look at any direction in the sky i.e. at any direction in the past light cone, changing the direction doesn't really change the observation. That means that all the points $d_1$ light years away look similar, all the points $d_2$ light years away look similar and so on and so forth. This allows us to extrapolate and make the claim that if we were capable of viewing the universe as a whole, by this logic, it should be homogeneous.

Bottom line is that you need isotropy for homogeneity. We can't observe homogeneity, we can only infer it through isotropy.

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  • $\begingroup$ Can you explain why it doesn't make sense to compare $P_A$ with $P_B$? If they're measured in the same units (sure seems like it) then it seems like they should be comparable. $\endgroup$
    – Allure
    Commented May 15, 2020 at 5:50
  • $\begingroup$ If I gave you a bottle of water, and another bottle of oil, each homogeneous on its own but I asked you to make a comment on the homogeneity of the entire system, that is ill defined. It doesn't make sense to define this question. Having the same units doesn't mean you can compare them. In the theme of your question, you can only define homogeneity on the same spacelike hypersurface. But since the observation of further away parts of the universe are further back in time, they are never on the same spacelike hypersurface. So yo cannot make a statement on homogeneity on observation alone. $\endgroup$ Commented May 15, 2020 at 6:12
  • $\begingroup$ Or better still: Think of an onion. Let's say that we sit at the center. Each successive layer is a hypersurface, and an outer layer corresponds to further away galaxies and is in the backward lightcone of an inner layer. when you look at each layer of an onion, sure it looks homogeneous but the notion of homogeneity becomes problematic if you look at its cross section. You can only assume that your inner layer is homogeneous because all other outer layers look the same in every direction i.e. isotropic. $\endgroup$ Commented May 15, 2020 at 6:25
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    $\begingroup$ I think my analogies are perhaps not the most ideal in this situation. Let me state a few facts. 1. To define homogeneity of the universe right NOW, you need to be able to observe the state of THE ENTIRE UNIVERSE as it is NOW. But the problem with this is that you observe almost all of the universe at it was in the past with respect to you (at least on length scales where the universe is homogeneous). This means that you cannot really observe homogeneity. That is basically the argument in the paper. (contd) $\endgroup$ Commented May 15, 2020 at 8:24
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    $\begingroup$ ...What you can do instead is observe all the points at a fixed distance away from you. This will be a 2-sphere. This 2 sphere is at a constant distance from you. This 2-sphere looks the same in every direction and is therefore isotropic with respect to us (i.e. homogeneous wrt itself). By repeating this for all possible 2 spheres in the history of the observable universe, the observation suggests that they are all isotropic with respect to us. So you say, "Ah! if each of these 2-spheres are isotropic wrt us, it is likely that the universe is also homogeneous right now." $\endgroup$ Commented May 15, 2020 at 8:31

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