Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: chiralSPO on 25/06/2021 15:44:13
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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.
It seems more reasonable to me to extend the cosmological principle to say that no location in the universe is "special" and no scale in the universe is "special."
We observe structure in the universe at every scale. The structure isn't identical, or even necessarily analogous (don't worry, I'm not going to compare atoms to solar systems :P ), and the forces that dictate the organization are different at different scales (?? < strong force < electroweak < gravity < ??). But there is always some sort of structure, some sort of organization, and at any given location, there is localized anisotropy.
I am reminded of turbulence, which produces eddies and swirls anywhere from the micron-scale to the galactic scale.
So can we be shocked every time there is evidence that the observable universe has anisotropy?
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It is observably anisotropic but apparently random. The randomness confers "nonspeciality" to the extent that in any sufficiently large sample is likely to contain pretty much the same amount of stuff as any other but not necessarily in the same state of order.
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Hi.
Does the cosmological principle dictate isotropy?
This depends on your definition of the Cosmological principle. The "Copernican principle" seems to be well defined and the Cosmological principle is loosely based on it.
I can find some references indicating that the term "Cosmological Principle" was first used and developed by Newton but I can also find some definitions from Google which just directly include the requirement that space is isotoropic in their definition. Obviously if we use a definition where isotropy is built-in to the definition then it's non-negotiable that the principle implies isotropy.
As regards developing some Mathematics, it is useful to assume space is isotropic. This is one reason why many people will just accept that the Cosmological principle includes or provides that condition.
Unless you want to provide a definition of the Cosmological Principle that you are using, I'm going to assume that we are talking only in the most general terms and not using any one text book definition of the Cosmological principle.
It seems more reasonable to me to extend the cosmological principle to say that no location in the universe is "special" and no scale in the universe is "special."
The Cosmological Principle does not imply anything like this. If anything it implies precisely the opposite: There is a preferred scale. The universe is the same everywhere and all directions when viewed on the largest of scales. If we wanted to try and make this more precise we might say that it only becomes true in some limiting process as we start taking some average property (e.g. density of matter) over distances or volumes of space that tend to infinity. It should be possible to prove such a statement from a more general statement about the randomness of the distribution of matter in the early universe and a general idea that the laws of physics applied in much the same ways across the universe regardless of location or direction.
You (CiralSPO) are proposing a very different and novel concept, if I've understood you correctly.
So can we be shocked every time there is evidence that the observable universe has anisotropy?
My personal opinion - no. Our models were always based on the isotropy of space and we don't actually have isotropy at any real place in space that we know about. Locally there is some anisotropy as you mentioned and often it's glaringly obvious.
I think it's more surprising when the scale over which the anisotropy is detected is large, especially if the anisotropy remains undiminished when we re-evaluate over a larger scale. Sometime of course, we just can't re-evaluate over a larger scale. The very tiny irregularities that we observe in the CMB is one such example. We can't push the CMB further away and observe it again BUT we can try to average it over a larger volume of space by another method. The Planck satellite aims to produce an image of the CMB with a resolution of about 5-10 arc minutes and with this resolution anisotropy in the CMB should be seen easily. The older COBE data only resolved the sky enough to see structures in the CMB on the scale of about 7 degrees and hardly any anisototropy is observed then.
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This question is a fine example of the temptation to confuse a model with reality.
The statement that "no part of the universe is special" is speculative since we haven't explored the entire universe. It simply reflects the convenient axiom that the laws of physics are universal and timeless.
That axiom is comforting, and like Newtonian physics, has served us very well to date - we haven't actually been anywhere or seen anything inconsistent with what we observe anywhere else. It may even be true, but it is inconsistent with the notion of a big bang unless we admit that, like Newton's mechanics, we haven't completely discovered the way that things work.
Unfortunately we seem to have evidence of the echoes of a big bang, with a fairly well established date and some increasingly credible theories about how we got from there to here. Problem is we don't understand "there", so the axiom remains a model, not a fact.
Gravitation has a lot to answer for. There was a lot of interest in the 1960s in Voronoi polyhedra and short-range order in liquid water: the bent polar molecule can form all sorts of temporary polymers. Now consider random chunks of solid or gaseous matter floating about in a void and subject to simple (Newtonian!) laws of gravitation and conservation of momentum. If the bits aren't flying apart too quickly, some will congregate into spinning discs, so the older the universe becomes, the more local "order" will prevail but there is no reason why any local collection should look or behave exactly like any other. Macroscopically isotropic matter becomes mesoscopically anisotropic.
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Does the cosmological principle dictate isotropy?
Like Alan's "Model" and Eternal Student's "Largest of Scales", I view the Cosmological Principle as an assumption to make the mathematics and calculation of General Relativity and Cosmology more feasible.
Effectively, it says "Space is the same everywhere, after you take out all the local influences...".
It also implies that you are talking about a particular epoch in the universe (which is impossible for us to measure, since we can't see outside our own light cone). The universe does change over time, eg becoming less dense, and the effective temperature of the SMBR declining.
- It is assumed to be the same everywhere, even though we can't observe it.
- And computer simulations of the universe suggest that it is true (computer simulations can "see" the whole universe at once, without visibility limitations due to light cones)
The Cosmological Principle has been a useful calculation tool, even before:
- Hubble discovered other galaxies (1924), or the expansion of the universe (1929),
- or Vera Rubin discovered early hints of Dark Matter (1960s),
- or discovery of cosmic voids (1978)
- or discovery of accelerating expansion (1990s)
...And it has also been a useful philosophical tool for people who still think that they are the center of the universe...
See: https://en.wikipedia.org/wiki/Timeline_of_knowledge_about_galaxies,_clusters_of_galaxies,_and_large-scale_structure
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Talking of local influences, there are suggestions that teh filaments of teh cosmic web are spinning...
Does the cosmological principle dictate isotropy?
Like Alan's "Model" and Eternal Student's "Largest of Scales", I view the Cosmological Principle as an assumption to make the mathematics and calculation of General Relativity and Cosmology more feasible.
Effectively, it says "Space is the same everywhere, after you take out all the local influences...".
It also implies that you are talking about a particular epoch in the universe (which is impossible for us to measure, since we can't see outside our own light cone). The universe does change over time, eg becoming less dense, and the effective temperature of the SMBR declining.
- It is assumed to be the same everywhere, even though we can't observe it.
- And computer simulations of the universe suggest that it is true (computer simulations can "see" the whole universe at once, without visibility limitations due to light cones)
The Cosmological Principle has been a useful calculation tool, even before:
- Hubble discovered other galaxies (1924), or the expansion of the universe (1929),
- or Vera Rubin discovered early hints of Dark Matter (1960s),
- or discovery of cosmic voids (1978)
- or discovery of accelerating expansion (1990s)
- And it has also been a useful philosophical tool for people who still think that they are the center of the universe...
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Talking of "local" effects that are very "large"... some researchers have suggested that the filaments of the cosmic web are spinning.
https://www.space.com/largest-spinning-structures-universe-discovered
Could this be evidence for cosmic strings?
https://en.wikipedia.org/wiki/Cosmic_string#Gravitation
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Hi and thank you for providing some references to follow.
Talking of "local" effects that are very "large"... some researchers have suggested that the filaments of the cosmic web are spinning.
https://www.space.com/largest-spinning-structures-universe-discovered
Could this be evidence for cosmic strings?
https://en.wikipedia.org/wiki/Cosmic_string#Gravitation
?? I.d.k. ?? I don't see the connection between spinning filaments of the cosmic web and cosmic strings.
Filaments are just bits of matter distributed so that they look like tendrils connecting galaxies, clusters of galaxies or super clusers together. That some of them are spinning requires some explanation but there are ways this can be done.
Cosmic strings are topological defects in space. Other than the words "string" and "filament" being similar I don't see how one needs to relate to the other. However, I'm sure there are ways of explaining the spinning filaments using cosmic strings or connecting the two ideas in some way.
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To me the isotropy of a universe connects intuitively to a lot of other assumptions science make as conservation laws existing 'everywhere', or 'constants', and the way we can explain f.ex a 'photon recoil' from it. If it wasn't I would start to wonder about those expectations too.
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I don't see the connection between spinning filaments of the cosmic web and cosmic strings.
Cosmic Strings (if they exist) would be very thin, stretching over vast distances through space. The are also potentially very massive, perhaps amounting to as much as 10% the mass of matter in the universe (as an upper bound).
Cosmic strings could be the gravitational backbone around which the galaxy clusters are orbiting, forming filaments of galaxies. But there is no solid evidence that they exist (to date).
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Thank you all for engaging in such a rich discussion!
I am fully on board with the "simple model" viewpoint provided by alancalverd. I can certainly see that is it convenient and reasonable to have a simple model assume isotropy as the default and perturb from there as much as is necessary to get the accuracy and precision required for the question asked.
Eternal Student's "preferred scale" is precisely what I am having difficulty seeing as an accurate depiction (but I too am an eternal student and happy to have my worldview altered by new ideas, so prove me wrong, and I'll learn from it!)
What is this preferred scale? Is there an unambiguous cutoff (that can be calculated like an even horizon?) Or is it just the idea that in the limit of larger and larger volumes, randomness balances out? Something else entirely? As evan_au points out, there are many *ahem* lines of evidence of anisotropy on a cosmic scale. I would predict base on everything I know about the subatomic to the intergalactic scales, that any scale we picture the universe at is going to have a similar range of degrees of structure.
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When I read this I can't help connect it to the homogeneity and isotropy expected of our universe.
http://www.astro.ucla.edu/~wright/cosmo_02.htm
It's somewhat of a principle.
" I am still confused the homogeneity of space and isotropy of space. What is the difference? What are the examples of each of them?
Simply put, ‘homogeneity’ of space implies you can choose any point in space as the origin of your reference frame. In other words,translating your origin to a new point in space won't change the physics you observe. Homogeneity,therefore, is related to translational invariance.
‘Istropy’ on the other hand,suggests you can orient your reference frame in any manner i.e. choose any direction as [math]x,y[/math] and [math]z[/math] and it won't change the physics. Isotropy is thus related to rotational invariance.
As it turns out, these invariances are related to the two of the conservation laws in physics: translational invariance implies conservation of linear momentum while rotational invariance implies conservation of angular momentum.
An example of inhomogeneity is a fluid of varying density(i.e. density varying as function of distance) such as our atmosphere.
An exmaple of anisotropy would be a crystal in which physical properties such as thermal and electrical conductivity depend on the direction of propagation of heat and current. "
https://www.quora.com/What-is-homogeneity-and-isotropy-of-space-time-with-example
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Myself I think it's a sane approach to the universe. Assuming that what you find here should be applicable on 'there' too. That is what we do with constants, and with conservation laws too. Then you have relative motion that 'moves us' through the universe, well presumably, and everything seems to work the same way as always when it comes to experiments and physics. Assuming the opposite moves us to something that we don't know, where whatever physics we find here being a local expression, on a 'global scale'.
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As for the scale the homogeneity in the link I gave first, relating to relativity, was defined this way
" To say the Universe is homogeneous means that any measurable property of the Universe is the same everywhere. This is only approximately true, but it appears to be an excellent approximation when one averages over large regions. Since the age of the Universe is one of the measurable quantities, the homogeneity of the Universe must be defined on a surface of constant proper time since the Big Bang. Time dilation causes the proper time measured by an observer to depend on the velocity of the observer, so we specify that the time variable t in the Hubble law is the proper time since the Big Bang for comoving observers"
And in this https://assets.cambridge.org/97805218/47049/excerpt/9780521847049_excerpt.pdf
" It turns out that the scale at which the galaxy distribution becomes homogeneous is difficult to determine. From the analysis of the Sloan Digital Sky Survey (SDSS) it has been concluded that the irregularities in the galaxy density are still on the level of a few per cent on scales of 100 h−1 Mpc (Hogg et al., 2005). Fortunately, we know that the geometry of the Universe shows only small deviations from the homogeneous and isotropic background, already on scales of a few Mpc..
The geometry of the Universe can be tested with the peculiar motion of galaxies, with lensing, and in particular with the cosmic microwave background (CMB). The small deviations from homogeneity and isotropy in the CMB are of uttermost
importance since, most probably, they represent the ‘seeds’, which, via gravitational instability, have led to the formation of large-scale structure, galaxies and eventually solar systems with planets that support life in the Universe. ( If ‘analyticity’ is not assumed, the matter distribution could also be fractal and still statistically isotropic around each point. For a detailed elaboration of this idea and its comparison with observations see Sylos Labini et al. ) "
Actually both analyze it through the lens of relativity. And myself I think those two goes together as one principle, homogeneity and isotropy, when describing this universe. If one accept that, the way isotropy may differ becomes a connection to the Big Bang.
Maybe something of a simplification but if you find something to be homogeneous it should be isotropic too, and it doesn't state anything about a dimensionality, you standing on a sphere will find isotropy in all directions without a 'edge'.
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Turned the words around above, so I needed to correct it. You don't get homogeneous from isotropic but you can get isotropic if you find it to be homogeneous. Or maybe both can be correct? Depending on definitions?
https://theeternaluniverse.blogspot.com/2009/08/what-is-homogeneity-and-isotropy.html
" Is there a space that is isotropic and not homogeneous? The answer is no, isotropy implies homogeneity so I can't give a cool example of this unfortunately. " Well, that's how I thought first.
Then you take this.
" If a region is homogenous then it's automatically isotropic around every point. If every point is the same then you can stand anywhere you like and look any direction you like and you're going to see points that are just like where you're standing. Therefore every direction is going to look the same.
The reverse does not apply. If what you see is isotropic then all you can say is it looks the same. Often, if your view is isotropic, then your surroundings are circularly or spherically symmetric, but there's no guarantee."
https://www.quora.com/What-is-homogeneity-and-isotropy-of-space-time-with-example
Soo :) Pick a choice.
( well, I connect it to a lot other things in physics, and they all fit together to me. So I will call them two sides of the same coin. Which should mean that the sentence becomes ' Maybe something of a simplification but if you find something to be Isotropic /or/ homogeneous it should be homogeneous /or/ isotropic too' )
Physics :)
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Hi.
Eternal Student's "preferred scale" is precisely what I am having difficulty seeing as an accurate depiction.....
I think you (we) need to establish more firmly what definition or notion of the cosmological principle you are using before I (we) can discuss many specific details.
Definition from Wikipedia:
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale
Definition from "The Origins of the Universe for Dummies". [ p. 102, Pub: John Wiley & Sons Ltd, Date: 2007, Auth: Pincock & Frary ].
This fundamental rule says that the universe, at least on large scales, looks the same in all directions (hence it's isotoropic) and has the same propetires in every place (it's homogenous).
More academic defintions may include the following:
(Strong Cosmological Principle). All physical quantities measured by a comoving observer are spatially homogeneous and isotropic.
(Statistical Cosmological Principle). The distribution of light and matter in the Universe is statistically isotropic around any point, apart from anisotropies of local origin.
(A Minimal or Weak Cosmological Principle).There exists a class of observers that see a statistically isotropic Universe, apart from anisotropies of local origin.
[Definitions based on "Thoughts on the Cosmological Principle", Dominik Schwawz, 2007. ArXiv version available at this URL https://arxiv.org/pdf/0905.0384.pdf ].
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So, depending on which notion of the Cosmological Principle you are using, the isotropy of space isn't optional. It doesn't "follow" from the Cosmological Principle, for many deifnitions it just is part of the definition.
What is this preferred scale? Is there an unambiguous cutoff (that can be calculated like an even horizon?) Or is it just the idea that in the limit of larger and larger volumes, randomness balances out?
It would be best to consider the notion on a statistical basis and therefore there is no clearly defined "cut-off" but instead the mean distribution of matter tends to become more symmetic as the volume over which averaging is done increases. However, there have been some attempts to put numbers on the sort of scales that would be required. Yor_on (earlier) has suggested some numerical values. I've found some estimates that are slightly larger than those (however, in Astronomical terms, getting agreement within a couple of orders of magnitude is acceptable):
A volume independent mean density seems to exist on scales larger than 100 Mpc, but this issue remains controversial. [Reference as above, Dominik Schwarz].
The existance of structure in the universe essentially violates the exact statement of the Cosmological principle and forces us to conisder it as a statistical description only. As ever larger structures are identified, Astronomers are forced to push the scale higher and exceed that of the largest known structures when the Cosmological Principle is stated. This is a problem very much like trying to specify the Range of some statistical data as a measure of spread - as more data is gathered the Range can only increase. It is preferable to use some other measure of spread such as the standard deviation from the mean.
A weak Cosmological principle can be stated in a similar way: We do not demand that the Cosmological principle applies at all points in space, only that most points in space will satisfy the principle in some specified way (e.g. that measurement of properties like the density of matter would conform to an average within some tolerance limit). For some models, this sor of statistical formulation of the Cosmological principle is sufficient.
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Returning to the OP
I'm going to stop writing. I think we have drifted off the original issue that Chiral wanted to discuss. Sorry.
It is equally interesting to consider that large scale structure and microscopic structure are related: Cosmology involving Inflation suggests that the lack of isotropy in the CMB could be the result of what were quantum fluctations but have been blown up or inflated to Astronomical propotions. Tiny differences in temperature and matter density throughout the universe are precisely what caused planets and stars to form. If the universe had been perfectly homgeneous and isotropic there probably wouldn't be any such structures.
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Hi :-)
I think you (we) need to establish more firmly what definition or notion of the cosmological principle you are using before I (we) can discuss many specific details.
Indeed, this may well be part of my confusion. Thank you for providing these options to compare.
As is often the case, I take issue with some of the aspects/implications of the less technical definition as defined by the wikipedia quote provided, but find the more technical definitions less problematic. Both the statistical and weak "flavors" of the cosmological principle strike me as reasonable. (I'm not sure about the strong definition, I don't think I know quite what they mean by "all physical quantities measured")
If I understand correctly, these more technical versions say that any region of the universe all have the same laws of physics (I'll buy that!) and that the distribution of matter is not necessarily equal, but the same probability density function applies universally. (this seems a reasonable approximation, but I still have some questions...see below)
When considering matter distribution we are still left with the distinction between "local" anisotropies and "large enough regions" (thank you to Eternal Student and Yor_on for providing some approximate numbers on this latter point). As Eternal Student points out, we continue to identify larger and larger structures—which is not out of line with having some distribution function that applies everywhere (if the universe is flat and unbounded and has the same statistical distribution everywhere, the sizes of "structures" is also unbounded--infinite monkeys on typewriters and all...).
I think what my question down to is this: does the definition of "local anisotropy" scale with the scale that is being considered? ie if we are considering orbital mechanics in a solar system, we care about stellar- and planetary-sized anisotropies; and if we care about galactic motions, then individual galaxies or even superclusters could be considered "local anisotropies." Is this a correct interpretation?
And if so, would it be reasonable to say that "local anisotropies" can be arbitrarily large? And if so, is that useful?
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I got somewhat stuck on this one ES " The existance of structure in the universe essentially violates the exact statement of the Cosmological principle and forces us to conisder it as a statistical description only. "
How did you mean there? As in, only in a totally empty space can there be a true isotropy?
Or was it something else you though about?
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Thinking about it again I think I get you. 'That perfect uniform gas universe' is what you refer to as a example of a ideal cosmological principle, right?
syntax
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I guess that's up to the definitions and scale you use Chiral. But it is still more homogeneous and isotropic than the opposite, well, as far as I know. If it wasn't one could get preferred directions in space, and relativity too should be able to be questioned, I think?
https://physics.stackexchange.com/questions/24881/what-is-meant-when-it-is-said-that-the-universe-is-homogeneous-and-isotropic
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It gets confusing very quickly if you look at mathematical models. They can allow all sorts of things if I remember right, as a 'rotating universe', or as the link refers to, as something not being isotropic still obeying GR. http://www.scholarpedia.org/article/Bianchi_universes
Or https://physics.stackexchange.com/questions/1048/what-if-the-universe-is-rotating-as-a-whole
But if it would rotate, wouldn't it have a preferred direction too?
" It would be a weird place. Being non-isotropic would mean there is a preferred direction in space, for example, if the entire universe were rotating. Then the axis of rotation is a special direction. You would then see Coriolis- and Centrifugal-like effects, and the motions of planets might not even obey Newton’s laws.
Strictly speaking, it IS non-homogeneous already so that is less of an issue. Homogeneity is only an approximation that applies on a very large scale. It would be weirder if the universe actually were perfectly homogeneous as it would then be a uniform gas — no planets, no stars, no galaxies, no nothing. " by Paul Camp, Ph. D. in theoretical physics and 30 years experience
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I think what my question down to is this: does the definition of "local anisotropy" scale with the scale that is being considered? ie if we are considering orbital mechanics in a solar system, we care about stellar- and planetary-sized anisotropies; and if we care about galactic motions, then individual galaxies or even superclusters could be considered "local anisotropies." Is this a correct interpretation?
Any structure at all (my car, solar system, galaxy, supercluster) is by definition a local anisotropy, and the cosmological principle has nothing to say about the dynamics of that structure other than to say that it should behave similar to an equivalent structure elsewhere.
And if so, would it be reasonable to say that "local anisotropies" can be arbitrarily large?
I would say no, since I cannot think of any structure larger than about 500 Mpc. OK, a number of 100 Mpc was quoted above, but the Shapley supercluster is centered about 200 Mpc away and we're gravitationally bound to it, making us effectively part of one bound structure. This is not true of more distant collections of mass. So it becomes unreasonable beyond a certain scale to speak of a thing being a single anistropy instead of multiple smaller ones.
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I cannot think of any structure larger than about 500 Mpc
Perhaps we could also say that we can't observe any larger structures (with our current technology).
- We see distant quasars that happen to be beaming radiation in our direction, but that is hardly a representative sample...
- The James Webb telescope should give us a much bigger light-bucket, collecting red-shifted light from the distant universe (if/when it is ever successfully commissioned!)
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I cannot think of any structure larger than about 500 Mpc
I take that back. We're part of Laniakea at best, and that structure is only about 160 Mpc, and even that doesn't count as a 'local influence', but rather a collection of smaller local influences since Laniakea at large isn't a bound object and will break up in the long run. If the clusters comprising Laniakea are not bound to each other, we're certainly not bound to something larger like Shapley.
Perhaps we could also say that we can't observe any larger structures (with our current technology).
- We see distant quasars that happen to be beaming radiation in our direction, but that is hardly a representative sample...
Quasars are relatively small objects, and do not currently exist. My definition of local influence' seems reasonably to be an object that is bound in some way.
I don't think bigger or more detailed telescopes are going to find larger things. Just more smaller ones, or more distant ones. The most distant objects known are about 3, 6, 13, or 35 BLY away (same object, four different ways of expressing distance to it) and a new telescope might break that record (at least one of the two larger numbers, not the smaller ones), it isn't going to find a bigger one.
All that said, you look at a picture of the structure of say a region of 400 Mpc (below) and it is difficult to express the concept of homogeneity to it. I also find the image misleading since it shows objects (lines) being ejected by the dipole repeller and making their way all the way to the Shapley concentrate eventually, which they never will since regions that large cannot be gravitationally bound.
(https://q-mag.org/_media/img/medium/dipole-repeller.png)
The yellow arrow is apparently us and our peculiar velocity, and we'll never make it there either.
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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.
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Interesting Halc, and yep ES. First a ideal empty space for isotropy, and then a ideal fluid for homogeneity :)
Hopefully being what you intended.
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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.
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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.
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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.