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Physics, Astronomy & Cosmology / Re: Does the cosmological principle dictate isotropy?
« on: 29/06/2021 11:15:24 »
Hi.
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.
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.
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 ].
- - - - - - - - -
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.
- - - -
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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