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Angular separation vs separation distance with increasing z?
Suppose two objects are separated on the sky by 1 arc minute, and I can
move them in or out in z as a test example.... Then, as I look out to higher z normally I would think the separation distance would be larger. This is the case for objects here on earth....same angle between them but further away means the separation distance between them is larger But in the universe, the further out in z the objects are, the smaller the universe was, so, they are actually closer together, right? In other words, if a pair of objects at z=1 and another pair at z=6 both have the same angular separation on the sky (say 1 arc minute), is the pair at z=6 closer together? I'm guessing I should use Ned Wright's co moving distance calculator (somehow) to figure this out? rt |
#2
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Angular separation vs separation distance with increasing z?
In article ,
writes: Suppose two objects are separated on the sky by 1 arc minute, and I can move them in or out in z as a test example.... OK. Then, as I look out to higher z normally I would think the separation distance would be larger. This is the case for objects here on earth....same angle between them but further away means the separation distance between them is larger But in the universe, the further out in z the objects are, the smaller the universe was, so, they are actually closer together, right? Perhaps; details depend on the cosmological model. In other words, if a pair of objects at z=3D1 and another pair at z=3D6 both have the same angular separation on the sky (say 1 arc minute), is the pair at z=3D6 closer together? Perhaps; details depend on the cosmological model. I'm guessing I should use Ned Wright's co moving distance calculator (somehow) to figure this out? What you want is the angular-size distance. By definition, the angular size distance is the physical separation divided by the angular separation. You can use the definition above to calculate the physical separation AT THE TIME THAT THE LIGHT WAS EMITTED. If the objects are bound, then, ignoring things irrelevant here, this doesn't change with time. If they are not bound, then the CURRENT proper separation is (1+z) times that what you calculated from the observed angle and the angular-size distance. So, the physical separation when the light was emitted is the angular-size distance multiplied by the angle. If the objects are not bound, then the current separation is a factor of (1+z) larger. This is sometimes known as the proper-motion distance, although at cosmological distances proper motion is rarely detectable. You might think that this distance corresponds to the luminosity distance, since this is based on the CURRENT area of a sphere surrounding the source. However, the luminosity distance is another factor of (1+z) larger. Why? In general, by definition, the luminosity distance is inversely proportional to the square root of the observed flux. At cosmological distances, the flux is reduced by a factor of (1+z) due to the cosmological redshift, and another factor of (1+z) because the arrival rate of photons is reduced (apply the same argument to individual photons as you apply to crests of a wave in the cosmological redshift). So, the flux is reduced by (1+z)^2, meaning another factor of (1+z) in the distance. Thus, the angular-size distance is equal to the the luminosity distance divided by (1+z)^2. This is a very general result and holds in many more cosmological models than those normally considered. For a discussion, see my paper: http://www.astro.multivax.de:8000/he...fo/angsiz.html (By the way, the Fortran code I wrote in conjunction with this paper was used in the main Perlmutter et al. paper for which Perlmutter was awarded the Nobel Prize in 2011.) This also mentions another effect: the distance can be different from the standard result (e.g. what Ned's calculator calculates) if the matter in the universe is clumpy. This is often referred to as the "Dyer-Roeder distance" in the literature. This is significant only at high redshifts. However, it turns out that this doesn't seem to be the case in our universe. Even if the matter is not distributed completely smoothly, at least at high redshift (and at low redshift, the inhomogeneity effect doesn't matter) then the distance one calculates is the same as that in a homogeneous universe, at least in a statistical sense. I wrote about this as well: http://www.astro.multivax.de:8000/he...o/etasnia.html There is a famous paper by Steven Weinberg (one of the few people who has worked on pure particle physics AND pure cosmology (as opposed to astroparticle physics, inflation, big-bang nucleosynthesis, early universe, etc, which are essentially particle physics applied to cosmology)) where he shows that this follows from flux conservation, provided certain assumptions are fulfilled. The data now indicate that they are. In fact, it appears that even each individual line of sight is a fair sample of the universe in that, even if the mass along the line of sight is not distributed smoothly, the distance calculated from redshift is the same as if it were. I discuss this he http://www.astro.multivax.de:8000/he.../etasnia2.html This doesn't mean that matter is smoothly distributed. We are pretty sure it is not (cosmic web, etc). However, at low redshift the small-scale distribution of matter has a negligible effect on the distance, while at high redshift, the light has travelled far enough that it has essentially traversed a fair sample of the universe, and thus the distance is the same as if all the matter were smeared out smoothly along the line of sight. One could say that voids and overdense regions (sheets, filaments, etc) "average" out. One has to put "average" in scare quotes for reasons explained in the paper above. (The actual details are quite complicated; search for a paper by Nick Kaiser and John Peacock (arXiv:1503.08506) for the gory details.) So, although many people (Zeldovich, Dashevskii, Slysh, Sachs, Kantowski, Dyer and Roeder, Weinberg, Kayser, Linder, myself, Holz, Bergstr=F6m, Goliath, Gunnarsson, Dahlen, J=F6nsson, Goobar, M=F6rtsell, Bolejko, Ferreira, Kaiser, Peacock, Lima, Busti, Santos, Clarkson, Faltenbacher, Uzan, etc) have invested much time worrying about this, in turns out that it is probably not relevant in our universe. (No-one debates the fact that in a Dyer-Roeder situation, the Dyer-Roeder distance is appropriate, but it appears that we don't live in such a universe.) This topic is distinct from the possible influence of LARGE-SCALE inhomogeneities on cosmological distances. See papers by George Ellis, Roy Maartens, Syksy R=E4s=E4nen, Bagheri, Schwarz, etc. The jury is still out. Some think that large-scale inhomogeneities can explain cosmological observations without a cosmological constant, but I would bet quite a bit that this is not the case. (Whether I'll live long enough to collect on the bet is a different question.) |
#3
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Angular separation vs separation distance with increasing z?
Thanks,
(is this an appropriate post?) ;-) rt |
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