Quote by Chalnoth As I keep saying, the population of galaxies at z=1 is, for instance, very different from the population of galaxies at z=0.01. The reason for this is that there are more older galaxies in the nearby universe, and more younger galaxies in the early universe. Clusters are more numerous and larger in the nearby universe. Active galactic nuclei are more common a bit further away (that is, AGN's are typically characteristic of younger galaxies).
I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1.
Besides the examples you are giving about galactic age shouldn't affect the universe density at large scale. The clusters part I would have to check it. I'm not sure that has been statistically shown to happen.

Recognitions:
 Quote by AWA I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1.
You said it. I didn't agree to it.

What you have to bear in mind is that there is a fundamental difference between the typical length scale of homogeneity and actually statistically demonstrating it (given an appropriate threshold). To statistically demonstrate it, you need a region much larger than the scale of homogeneity (because you have to show that all such regions of said size are statistically identical, to within some pre-defined threshold).

It's not such a surprise to me that there's argument about this, because it's mathematically a difficult thing to demonstrate, and the threshold of homogeneity is arbitrary anyway.

 Quote by AWA Besides the examples you are giving about galactic age shouldn't affect the universe density at large scale. The clusters part I would have to check it. I'm not sure that has been statistically shown to happen.
Well, density is one of the most significant things to evolve with redshift. The average density of the matter in our universe at z=1 is eight times the density today. Baryon Acoustic Oscillation observations, which measure the typical separation between galaxies at different redshifts, are a good measurement of how this density changes with redshift.

 Quote by AWA Sure, the distance-redshift relation is not linear, so?.

So it depends on input parameters.

 Quote by AWA Borrowing a little from cosmology textbook stuff, this is from Hobson's General Relativity: "In general relativity the concept of a ‘moment of time’ is ambiguous and is replaced by the notion of a three-dimensional spacelike hypersurface. To define a ‘time’ parameter that is valid globally, we ‘slice up’ spacetime by introducing a series of non-intersecting spacelike hypersurfaces that are labelled by some parameter t. This parameter then defines a universal time in that ‘a particular time’ means a given spacelike hypersurface. We may construct the hypersurfaces t = constant in any number of ways. In a general spacetime there is no preferred ‘slicing’ and hence no preferred ‘time’coordinate t. According to Weyl’s postulate, there is a unique worldline passing through each (non-singular) spacetime point. The set of worldlines is sometimes described as providing threading for the spacetime." In our case one of the three spatial dimensions of the spacelike hypersurfaces represents visually (thanks to light's nature) the timelike worldline passing thru each spacetime point defined by a specific redshift.

Thanks. Very nice quotation.

 Quote by AWA I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1..
There is the source of your confusion. Please look here . You are comparing homogeneity of two different time slices, which, obviously, when compared are not the same, and talking about 'not yet' homogeneous universe. Universe is just as much homogeneous now as it ever was, just on different scale.

 Quote by Calimero There is the source of your confusion. Please look here . You are comparing homogeneity of two different time slices, which, obviously, when compared are not the same, and talking about 'not yet' homogeneous universe. Universe is just as much homogeneous now as it ever was, just on different scale.
Thanks, cool site.
A couple of remarks: first, that is a simulation school exercise and they stress it that as a simulation it is not expected to match the real universe.
Second, according to GR (and as pointed out in the quote from my last post) you can slice up spacetime arbitrarily (general covariance, remember?) : "We may construct the hypersurfaces t = constant in any number of ways. In a general spacetime there is no preferred ‘slicing’ and hence no preferred ‘time’coordinate t."
Given a statistically significant number of slices you should find homogeneity across the sufficiently long time-like worldline formed by the statistically large stack of spacelike slices if each of the different time spacelike slices is itself homogenous. If you don't agree with this, I should remind you that GR is to this day the best theory to understand the universe that we have.

 Quote by Chalnoth The average density of the matter in our universe at z=1 is eight times the density today.
Please, back up that figure with some reliable reference. Specifying how do you exactly measure the universe density time spacelike slice at precisely z=1.

Recognitions:
 Quote by AWA Please, back up that figure with some reliable reference. Specifying how do you exactly measure the universe density time spacelike slice at precisely z=1.
Matter density scales as $a^{-3}$. But $1+z = 1/a$, so matter density scales as $(1+z)^3$.

Basically, proposing that the matter density wasn't eight times its current value at z=1 requires proposing a universe that has radial-dependent density for an equal-time slicing. This is, in principle, a rather difficult thing to accurately determine, but suffice it to say our theories that use a homogeneous matter distribution work, while alternative theories proposed to explain certain unpleasant aspects of the homogeneous theories don't.

 Quote by Chalnoth Matter density scales as $a^{-3}$. But $1+z = 1/a$, so matter density scales as $(1+z)^3$.
Right, that is a model-dependent calculation, not an observation. That is what is apparently contradicted (and thus where I see the paradox) by the hypothetical future SDSS 3D galaxy map up to a high redshift, that we expect to be homogenous. But perhaps, I'm misunderstanding something and cosmologists don't expect to find that map statistically homogenous. If so, please explain.
 Quote by Chalnoth Basically, proposing that the matter density wasn't eight times its current value at z=1 requires proposing a universe that has radial-dependent density for an equal-time slicing.
I see what you mean, yes, that seems to follow logically.
The universe is a strange place, and is full of apparent contradictions, wish we knew it better, but let's be humble (and honest), we are barely starting to grasp it.
 Quote by Chalnoth This is, in principle, a rather difficult thing to accurately determine, but suffice it to say our theories that use a homogeneous matter distribution work, while alternative theories proposed to explain certain unpleasant aspects of the homogeneous theories don't.
Ultimately, it seems to come down to a practical matter.
BTW, I don't know what are those theories proposed to explain unpleasant aspects of homogenous theories, and what these unpleasnt aspects are. Would you elaborate a little on this?

Recognitions:
 Quote by AWA Right, that is a model-dependent calculation, not an observation. That is what is apparently contradicted (and thus where I see the paradox) by the hypothetical future SDSS 3D galaxy map up to a high redshift, that we expect to be homogenous. But perhaps, I'm misunderstanding something and cosmologists don't expect to find that map statistically homogenous. If so, please explain.
You have to look into it in a bit more detail. For example, if you combine nearby measurements of the Hubble constant and supernovae with WMAP data, you end up with a nearly-flat universe.

If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$.

There are all sorts of different ways you can do this sort of experiment, but the basic idea here is that if you make a series of assumptions, and multiple independent experiments measure the same set of parameters based upon those assumptions, you gain confidence that those assumptions are, in fact, true, at least in an approximate sense. One of those foundational assumptions is homogeneity.

 Quote by AWA Ultimately, it seems to come down to a practical matter,it seems. BTW, I don't know what are those theories proposed to explain unpleasant aspects of homogenous theories, and what these unpleasnt aspects are. Would you elaborate a little on this?
In this case, some have attempted to explain away the accelerated expansion by proposing a universe that has radial-dependent density. It turns out that such proposals are ruled out by observation.

 Quote by Chalnoth You have to look into it in a bit more detail. For example, if you combine nearby measurements of the Hubble constant and supernovae with WMAP data, you end up with a nearly-flat universe. If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$. There are all sorts of different ways you can do this sort of experiment, but the basic idea here is that if you make a series of assumptions, and multiple independent experiments measure the same set of parameters based upon those assumptions, you gain confidence that those assumptions are, in fact, true, at least in an approximate sense. One of those foundational assumptions is homogeneity.
This is all understood and fine, I'm just taking that assumption to its last logical consequences if we take relativity seriously, and if we agree that if you observe long distances spaces you are also observing the past, one cannot be homogenous if the other isn't too, and viceversa. As they say, you can't have one without the other.

This leads to some contradiction with standard cosmology, so when in doubt, of course we choose standard cosmology, right?

 Quote by Chalnoth You have to look into it in a bit more detail. If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$.
Searching with the word BAO in arxiv, the first random paper I read casts shadows over BAO signal measures.: http://arxiv.org/PS_cache/arxiv/pdf/...009.1232v1.pdf

Recognitions:
 Quote by AWA This is all understood and fine, I'm just taking that assumption to its last logical consequences if we take relativity seriously, and if we agree that if you observe long distances spaces you are also observing the past, one cannot be homogenous if the other isn't too, and viceversa. As they say, you can't have one without the other. This leads to some contradiction with standard cosmology, so when in doubt, of course we choose standard cosmology, right?
This is getting tiring. We do not observe homogeneity in the radial direction. We don't expect to, because the radial direction is also looking backwards in time.

What we observe instead is a universe that looks like the nearby universe is a later version of the far away universe. In other words, it's as if looking outward in space is looking through a succession of homogeneous equal-time slices. This is the standard cosmology. This is what we observe. None of our observations contradict this, and it is fully self-consistent.

As for the BAO paper, if you look at their data, the apparent $3\sigma$ deviation is represented in figure 6, where you can clearly see that the discrepancy comes down to the signal being rather noisier than their simulations estimate, which would be indicative of not properly taking something into account in the simulations.

A perhaps better paper for this particular issue is this one:
http://arxiv.org/abs/0705.3323

...because this shows the combination of WMAP, supernova, and BAO data. The relevant plot is fig. 13, where you see that the contours all converge on the same point in parameter space when these data are taken together.

 Quote by Chalnoth This is getting tiring. We do not observe homogeneity in the radial direction.
Oh, but you don't have to answer if you get tired. There are more people in this forum.

 Quote by Chalnoth We do not observe homogeneity in the radial direction.We don't expect to
Are you sure? we expect to find spatial homogeneity. why on earth would you want to leave out one spatial dimension just beats me. I mean that's pretty bizarre, how do you keep one spatial direction inhomogenous and the others homogenous, and still keep isotropy?

 Quote by Chalnoth What we observe instead is a universe that looks like the nearby universe is a later version of the far away universe. In other words, it's as if looking outward in space is looking through a succession of homogeneous equal-time slices.
See posts 49 and 55.

Recognitions:
 Quote by AWA Are you sure? we expect to find spatial homogeneity. why on earth would you want to leave out one spatial dimension just beats me. I mean that's pretty bizarre, how do you keep one spatial direction inhomogenous and the others homogenous, and still keep isotropy?
1. Many cosmologists, early on, expected homogeneity in both time and space. This was disproven when Hubble measured the expansion of the universe.
2. Since when we look far away, we are looking back in time, we do not expect to see homogeneity in that direction, because an expanding universe changes with time.

 Quote by AWA See posts 49 and 55.
So, you're still confused about the simultaneity thing? The expansion of the universe itself creates a notion of "universal time". If you use coordinates that move with the expansion, then observers that are stationary with respect to the coordinate system each see the universe as being isotropic from their point of view.

This is, ultimately, what we mean by spatial homogeneity: if I go anywhere else in the visible universe, and adjust my velocity to move along with the local matter there, the universe will look isotropic to me. A homogeneous universe is defined as one in which you can do this: you can move anywhere within the universe, set your velocity to some value, and see an isotropic universe. You can then define the time coordinate globally in such a way that at the same time, separated observers see the same properties of the universe (such as the CMB temperature). In these coordinates, the properties of the universe are the same everywhere in space, but change with time.

You can change to a different set of coordinates, of course, and things won't necessarily look constant in space any longer. You'll still get the right answers for any observable you calculate, but you won't see the homogeneity.

 Quote by Chalnoth Since when we look far away, we are looking back in time, we do not expect to see homogeneity in that direction, because an expanding universe changes with time.
This indeed is getting repetitive,once again this makes no sense in GR. You have some source where this is explicitly stated? that homogeneity is forbidden in one spatial direction?

 Quote by Chalnoth This is, ultimately, what we mean by spatial homogeneity: if I go anywhere else in the visible universe, and adjust my velocity to move along with the local matter there, the universe will look isotropic to me. A homogeneous universe is defined as one in which you can do this: you can move anywhere within the universe, set your velocity to some value, and see an isotropic universe. You can then define the time coordinate globally in such a way that at the same time, separated observers see the same properties of the universe (such as the CMB temperature). In these coordinates, the properties of the universe are the same everywhere in space, but change with time. You can change to a different set of coordinates, of course, and things won't necessarily look constant in space any longer. You'll still get the right answers for any observable you calculate, but you won't see the homogeneity.
You won't? spatial homogeneity is not a physical observable? it is just a convenient perspective only watchable with some privileged coordinates?

Recognitions:
 Quote by AWA You won't? spatial homogeneity is not a physical observable? it is just a convenient perspective only watchable with some privileged coordinates?
Yes, spatial homogeneity is only something that is watchable in some privileged coordinates. The only sort of homogeneity that would be visible in any coordinates is space-time homogeneity. We don't get to do that for homogeneity that is only in space, unfortunately.

This doesn't mean that spatial homogeneity is meaningless, however. Yes, it only appears in some special choice of coordinates. However, it isn't something that you can do in any sort of universe you might conceive. Remember the definition I laid down previously: if, at any point in space, one can construct a hypothetical observer that will see an isotropic universe, then we can call that universe homogeneous in space.

I could easily construct a universe that doesn't have this property. For instance, if we imagine a universe that is very dense in the direction of both poles of the Earth, but has very little matter in the directions outward from the Earth's equator, that would be a very anisotropic universe. The north/south direction would be picked out as a special direction. But what's more, there is no choice of observer located on Earth that could see that distribution as being isotropic.

In the end, this model of a homogeneous universe isn't a direct observable (because we can't move far enough away to check isotropy from different spatial locations), but it does have observable consequences. Namely, it states that the expansion of the universe should follow the Friedmann equations. When we measure the expansion of our universe using many different sorts of observations, and continually come up with the same answer every time, we gain confidence that the Friedmann equations are valid, at least approximately, which means we gain confidence that our universe is genuinely homogeneous in space (for a specific choice of coordinates).

 Quote by Chalnoth Yes, spatial homogeneity is only something that is watchable in some privileged coordinates. The only sort of homogeneity that would be visible in any coordinates is space-time homogeneity. We don't get to do that for homogeneity that is only in space, unfortunately. This doesn't mean that spatial homogeneity is meaningless, however. Yes, it only appears in some special choice of coordinates. However, it isn't something that you can do in any sort of universe you might conceive. Remember the definition I laid down previously: if, at any point in space, one can construct a hypothetical observer that will see an isotropic universe, then we can call that universe homogeneous in space. I could easily construct a universe that doesn't have this property. For instance, if we imagine a universe that is very dense in the direction of both poles of the Earth, but has very little matter in the directions outward from the Earth's equator, that would be a very anisotropic universe. The north/south direction would be picked out as a special direction. But what's more, there is no choice of observer located on Earth that could see that distribution as being isotropic. In the end, this model of a homogeneous universe isn't a direct observable (because we can't move far enough away to check isotropy from different spatial locations), but it does have observable consequences. Namely, it states that the expansion of the universe should follow the Friedmann equations. When we measure the expansion of our universe using many different sorts of observations, and continually come up with the same answer every time, we gain confidence that the Friedmann equations are valid, at least approximately, which means we gain confidence that our universe is genuinely homogeneous in space (for a specific choice of coordinates).
Ok, I see now clearly the source of our disagreement and of my "false paradox". Actually there is no paradox at all.

I guess the moral of the story is that one must not take GR to seriously because that is considered naive at best and against standard cosmology at worse.
But all books on GR stress general invariance (covariance), all of them say we can choose coordinates arbitrarily, wich also means of course we can privilege some coordinates for the sake of convenience, but that convenience in no way means the results be get with that coordinates are physically real unless they can be reproduced with other choices of coordinates and metrics. I guess I also took too seriously the interchangeability of spacetime dimensions that relativity teaches us.

Now you tell me that spatial homogeneity, even though it is a property as physical as it can be, only appears with a determinate choice of coordinates that produce a certain privileged slicing of spacelike hypersurfaces, and that this homogeneity disapears if we try to make it coordinate invariant when we change the coordinates, appearing instead a sort of statistical homogeneity wrt both space and time (spacetime) and inhomogeneity or radial density dependance in the purely spatial hypersurface, and both of this things are forbidden by standard cosmology and astronomical observations and I have to take your word on this, no matter what GR says because you know more than me and standard cosmology says so and I'm a responsible citizen.

I declare the paradox solved unless someone else finds this a bit odd too or has some new input. Thanks a lot.

 Recognitions: Science Advisor General Relativity itself respects general covariance. But the specific distribution in our universe does not. In fact, it's pretty easy to prove that normal matter/radiation cannot respect general covariance, because the only covariant stress-energy tensor is one that behaves like vacuum energy. Therefore, the very existence of matter ensures that the universe will look different in different coordinate systems. The general covariance of General Relativity ensures that you get the same results for the behavior of said matter no matter what coordinate system you use. And the math will be made easier if we use coordinates that follow any symmetries that exist in the matter distribution. For instance, if you are doing physics on the surface of the Earth and not moving very far, it is convenient to approximate the Earth as a perfectly-flat surface. This set of coordinates will start to be wrong if we move too far along the Earth's surface or too far above it, but it is a convenient choice as long as we don't do these things. If you are instead, for example, attempting to put a satellite into low-Earth orbit, it becomes convenient to use spherical coordinates centered at the center of the Earth, and ignore the effect of bodies further away. This set of coordinates will start to be wrong if you get too close to the Moon, or far enough from the Earth that the Sun's gravity becomes more important. If you are instead interested in describing the motions of the planets, it becomes convenient to use spherical coordinates centered on the Sun. And so on and so forth. You can use General Relativity in each case. In each case, exploiting the symmetries of the physical matter distribution makes the math easier. This is what we are doing when we use FRW coordinates: we are exploiting a particular symmetry of the average matter distribution of our universe, namely spatial homogeneity. Allowing our coordinate choice to follow this symmetry makes the math easier.