Spatial infinity and matter-energy density

[TrickyDicky]

Two of the posible spatial hypersurfaces in FRW cosmology(the flat one and the negative curvature one) are infinite in extensión. I'm not entirely sure how is a finite energy density for the universe obtained as energy/volume if the volume is infinite.
I know it is easily computed with the Friedmann equations from the scale factor and curvature, and the state equation, but I'm stuck at how we can get conceptually a density different from zero when the volume considered is infinite.

 

[Mordred]

The detail your missing is that the universe is homogeneous and isotropic. The FLRW metric can calculate the observable energy density. Based on the assumption that the unobservable portions will be the same as the observable portion. (we have no reason to assume it would be different). Then the energy density of the total universe will be the same as our finite portion of the observable universe.

 

[TrickyDicky]

The detail your missing is that the universe is homogeneous and isotropic. The FLRW metric can calculate the observable energy density. Based on the assumption that the unobservable portions will be the same as the observable portion. (we have no reason to assume it would be different). Then the energy density of the total universe will be the same as our finite portion of the observable universe.

But I'm not missing that detail. An infinite universe with vanishing global density can have a finite density locally and still be homogeneous in the large scale.
I'm more inclined to think that it is not valid to mix the three-dimensional concept of density- as mass(energy) per unit volumen- with the four-dimensional parameters from the FRW geometry and the Friedmann equations where the energy density is the time-time component of the SET.

 

[Mordred]

Fair enough. Even though there is no evidence of a vanishing density. Let's look at your original problem.

How does one determine a finite energy density with infinite volume. Obviously we see energy-mass locally. So the energy density is definitely not zero. So regardless of what metric is used. The resulting energy-density will need to reflect that.

The FRW assumes uniformity throughout the universe with that assumption we sample a finite volume (thus avoiding the infinity problem). Unless we have a rate of density drop off, we cannot calculate that (as per your example)
As we know the density cannot be zero, if the universe is infinite then the total energy is also likely to be infinite. neither can the density be infinite, as there is space all around us.

If you think about it. The method I described is the same as determing the density of a gas of an open system of undeterminable volume.

 

[Chalnoth]

Two of the posible spatial hypersurfaces in FRW cosmology(the flat one and the negative curvature one) are infinite in extensión. I'm not entirely sure how is a finite energy density for the universe obtained as energy/volume if the volume is infinite.
I know it is easily computed with the Friedmann equations from the scale factor and curvature, and the state equation, but I'm stuck at how we can get conceptually a density different from zero when the volume considered is infinite.

I don't think it makes sense to think of FRW as something that can realistically be an accurate description of the universe very far beyond the observable universe. It is an approximation that works extremely well within the observable universe, but it is unlikely to extend all that far beyond that.

 

[TrickyDicky]

I don't think it makes sense to think of FRW as something that can realistically be an accurate description of the universe very far beyond the observable universe. It is an approximation that works extremely well within the observable universe, but it is unlikely to extend all that far beyond that.

Well, I agree the "almost FRW" model is the best we have but it is not a model only for the observable universo, it is meant to work for the whole universe, it doesn't make sense as a model for the observable part only.
Anyway it is in the observable part where it obviously finds its problems with observations.

The first one is by construction of the model, its equations and principal assumption demand that the density parameter is independent (at large distribution scale) from spatial location, the dependence must only be temporal. This collides frontally for the purpose of direct observations and validation of the above-mentioned location density independence with the inability to separate the temporal and spatial parts in all non-static spacetimes by their own mathematical nature. In other words in such non-static spacetimes there is no well defined three-dimensional spacelike hypersurface.

This leads to the well known particularity of the FRW universe that only in preferred coordinates can we define with this model a universal time, that which sees a different density at each time t. But observationally this imposes a radial dependence of density since we observe by the finiteness of c different times with radial distance.

So we find that since we can't possibly observe the spatial features of our universo at an instant of time, we have no way to validate directly the independence of density from location(homogeneity) except indirectly and statistically. We basically infer it from the observed average isotropy. If the static models of the universe were mathematically or empirically feasible which are not (see cosmological redshift), what we observe wrt homogeneity would be compatible with the perfect cosmological principle for static cosmologies.

In this sense I find it odd that many find problems with the homogeneity assumption of the model every time we find cosmic structures(see http://en.wikipedia.org/wiki/Hercules-Corona_Borealis_Great_Wall) that are apparently too big to be compatible with large scale homogeneity. By the problem mentioned there is no scale at which we can observe homogeneity, to do it light speed should be infinite, and we know empirically and theoretically it isn't.

What those big structures if confirmed clash with is the time evolution of the universe or finiteness of time from Bing-Bang. It seems there is no time for such a huge structure at such distance to have evolved from the Big-Bang.

 

[timmdeeg]

It seems there is no time for such a huge structure at such distance to have evolved from the Big-Bang.

But in the far future while those structures and voids are growing people might find themselves living in a spherical or hyperbolic universe, depending on their local observations. And their forecast will be different and perhaps not include the fate of the universe as a whole.

 

[TrickyDicky]

But in the far future while those structures and voids are growing people might find themselves living in a spherical or hyperbolic universe, depending on their local observations. And their forecast will be different and perhaps not include the fate of the universe as a whole.

Sorry but I don't know what you are talking about nor how it relates to what I wrote.

 

[Mordred]

In this sense I find it odd that many find problems with the homogeneity assumption of the model every time we find cosmic structures(see http://en.wikipedia.org/wiki/Hercules-Corona_Borealis_Great_Wall) that are apparently too big to be compatible with large scale homogeneity. By the problem mentioned there is no scale at which we can observe homogeneity, to do it light speed should be infinite, and we know empirically and theoretically it isn't.

What those big structures if confirmed clash with is the time evolution of the universe or finiteness of time from Bing-Bang. It seems there is no time for such a huge structure at such distance to have evolved from the Big-Bang.

yes there are regions such as the one you mentioned that challenge homogeneity. However the term homogeneous has no set size scale. In practice its usual to state that the universe is homogeneous at sufficient scales where that becomes the case. At one time that scale was 100 Mpc. Obviously any smaller amount could not account for large scale structure formations and the voids between them. The great wall you mentioned although challenges that 100 Mpc statement. A larger value say 2 to 300 Mpc could compensate for that and still allow validity for the FRW metric.

The great wall however had plenty of time to form, we already know black holes and galaxies could form slightly before the last scattering or be in the early stages of formation so its had plenty of time to collect nearby structures.

Edit.tried looking for an arxiv paper I read a few months ago with recommendations on size scales for homogeneous conditions however haven't had any luck. Too many articles in my archives lol. If I recall they recommended approx 160 Mpc but not positive. If I find it I'll post it. As the reasons relate to the Great wall