[Eternal Student]

Hi everyone.

Yor_on said:

I got somewhat stuck on this one ES   "  The existence of structure in the universe essentially violates the exact statement of the Cosmological principle and forces us to consider it as a statistical description only.  "

   Sorry it's taken me a while to reply.   It also looks like you've gone back and edited your post so that there isn't much need to say anything anymore.
    I probably shouldn't have said "an exact statement of the cosmological principle" but instead called it something like the strongest or most idealised version of the cosmological principle.   The moment we have identified a structure (a cluster of matter in some arrangement) that exists in some parts of the universe but not others, then the homogeneity of the universe has been lost and we can easily identify points in space where isotropy will be lost (pick a point close to the boundary of such a structure).
    A perfectly isotropic and homogeneous universe could be totally empty but it could have a constant non-zero density everywhere.   For example, an FRW universe models the contents of space as a homogeneous ideal fluid.  This seems to be what you (Yor_on) have said in your latest edit.
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I've got to go and do some more stuff.  Bye for now and best wishes.

[yor_on]

Interesting Halc, and yep ES. First a ideal empty space for isotropy,  and then a ideal fluid for homogeneity :)
Hopefully being what you intended.

[Eternal Student]

Hi again.
   It looks like there are plenty of good replies and answers here already.   I've read through most of this discussion again and it still seems that a few things might still have been left unanswered or unclear.   I'm going to break this post into smaller chunks because otherwise it will become a monologue and may not be useful to you (ChiralSPO) anyway.


Your (ChiralSPO) first post said the following:   

As I understand it, the main idea underlying the cosmological principle is that no part of the universe is "special"

That seems a reasonable axiom, but where I have difficulty is that this appears to be widely interpreted as implying that the universe must be isotropic on some grand scale.

  ..and I'm not sure that we (previous contributors) have really explained why isotropy is generally implied. 
Are you (ChiralSPO) now able to see why people tended to think that isotropy (on a large scale in the universe) follows from the idea that no point in space is special?

I think what my question down to is this: does the definition of "local anisotropy" scale with the scale that is being considered?

    This is very difficult to answer.   If we go to small scales, then we are in the territory of quantum mechanics and most reasonable ideas go out of the window.   If there is a fundamental Planck length then we cannot meaningfully divide space into smaller regions and calculate average densities or consider structures below this scale.
   We may hit a similar upper size limit for scale:  If the universe has positive curvature then it has finite extent.  Once we are averaging over the whole of space and considering structures of that size, there is no larger scale we can go to.
    Ignoring these limitations and trying to answer in the spirit in which the question was asked -  Well possibly.  If the universe is infinite and the distribution of matter in it was random then somewhere within it there is a structure (a region of above average density) that has a size exceeding any fixed number you want to set.  We then only need to consider a point in space that is close to the boundary of that structure and take averages over lengths comparable to the size of that structure.

[chiralSPO (OP)]

Thanks all for your patient and enlightening discussion. I think I understand now:

The axiom that holds that fundamental laws of physics are the same at every location seems reasonable and is well-supported by observation and experimentation.

From there we can say that the universe is homogenous by translational symmetry (no special spot). And isotropy from rotational symmetry (no special direction).

As far as the distribution/organization of matter: we can say that there is a probability distribution function of finding matter somewhere that is essentially the same everywhere. This does not mean there is no variation, but just that the variation is equal everywhere (to first approximation—then we can start to account for deviations from this random noise distribution by saying that although the rules are the same everywhere, the history is not.)

Please let me know if this is incorrect.

[Eternal Student]

Hi Chiral.

Well I quite like your summary.  Well done.

This part could be softened a bit, if required:

As far as the distribution/organization of matter: we can say that there is a probability distribution function of finding matter somewhere that is essentially the same everywhere. This does not mean there is no variation, but just that the variation is equal everywhere (to first approximation—then we can start to account for deviations from this random noise distribution by saying that although the rules are the same everywhere, the history is not.)

    We only require that the distribution was random (and had the same probability distribution function everywhere) at one time.
    Typically, we assume that just after the Big Bang the distribution of matter was like this*.  Then we need only assume that the laws of physics apply homogeneously and isotropically - so that the universe will tend to have evolved in much the same way everywhere and in every direction.     I think it was Alancalverd? who first mentioned that matter will tend to clump together and more generally structures tend to form which can start to break the homogeneity and isotropy.  However, these changes take time and there has only been a finite amount of time (14 billion years) since the Big Bang.  This is not enough to have lost all the homogeneity across a universe that is of Astronomical size.  We may very well need to to be taking averages over larger distances as time evolves.   (This is returning to your (Chiral) statement that the history of different regions can be different but just noting that very distant regions must have been causally disconnected and would have evolved independently thus automatically preserving the cosmological principle over large enough distances).

* Actually we can go back a little earlier than when there was matter in the universe.  The early universe was thought to be filled with radiation and that radiation was (almost) uniform across space.  The (almost) uniformity of that radiation and the assumption of physics applying everywhere and in all directions equally is enough to indicate that matter would be (almost) uniform across the universe.
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It seems a shame not to add anything new to the discussion, so let's do that now...
    I find it interesting that Inflation may be both the villain (causing anisotropy) and also the saviour (preserving isotropy). 
It could be the CAUSE of local anisotropy since it is thought that quantum fluctuations may have been magnified and thus caused the anisotropy in the early universe to begin with.
It is the SAVIOUR of what statistical isotropy remains because it resulted in such rapid expansion of the universe that distant regions have evolved independently.

OK.  Bye and best wishes.