# Define and measure inhomogenities and anisotropies

1. Jul 23, 2013

### tom.stoer

The cosmological principle says that on large scales the universe is homogeneous and isotropic. Therefore there should be a way to define and measure inhomogenities and anisotropies.

Regarding the definition I see the following problem:

Usually we would like to define

$M=\int_V dV\,\mu$
$P^i=\int_V dV\,\pi^i$
$L^i=\int_V dV\,\lambda^i$

and

$\bar{\mu} = \frac{M}{V}$
$\tilde{\mu}=\mu-\bar{\mu}$
...

But we know that these integrals cannot be defined mathematically for arbitrary spacetimes (not asymptotically flat, ...) For the mass we have some definitions like Komar mass, for momentum and angular momentum it becomes even more complicated.

So how do we define inhomogenities and anisotropies mathematically?

A related problem is the definition of "large scales ...". What does that mean exactly? How would one classify and distinguish self-similar/ fractal-like / scale-free structures (with voids of every size)?

Having a mathematical definition at hand the problem is to measure it. Obviously we do not have data on a space-like section, but light rays (from galaxies and CMB). Therefore an appropriate definition should be based on light-like data samples.

The next question is whether we have data analysis for inhomogenities (fluctuations in mass or energy density, ...) and for anisotropies (fluctuations in momentum and angular momentum density, polarization, ...)

A remark regarding CMB and Planck data: of course they tell us something about inhomogenities - but only for a very special data set, namely the visible celestial sphere centered at the earth. As indicated by the integrals mentioned above I would like to have a more general definition using e.g. arbitrary volumes. I think this is not possible based on CMB.

Last edited: Jul 23, 2013
2. Jul 23, 2013

### Mordred

this is an interesting topic, for a long time I considered 100 Mpc to be a value for a homogenous and isotropic scale. However recent papers with regards to unusual large scale structures have indicated that a larger scale is needed. Last one I read indicated 120 Mpc. Essentially the scale needed is simply on where homogenous and isotropy is achieved with the averages I mentioned above. However some regions can be thus defined with smaller or larger scales, depending on distribution.

As far as a mathematical definition, it would depend on the average distribution, desired scale and what you are modelling I would think.

to expand on that, take your lawn and the blade of grass distribution if your lawn is evenly distributed ie a good green lawn with no dead spots the scale would be much smaller than a lawn that is poorly cared for. Essentially its when you can obtain a decent, statistical average distribution.

Last edited: Jul 23, 2013
3. Jul 24, 2013

### Chronos

A real problem is defining the range space. The easy approach is to model it as an indefinite integral.

4. Jul 24, 2013

### tom.stoer

Afaik there is no natural expression for a scalar mass density in GR in general. Therefore there is no scalar mass defined via a volume integral. All expressions I know are restricted to stationary or asymptotically flat spacetimes

http://en.wikipedia.org/wiki/Mass_in_general_relativity

So there is no definition of mass in realistic cosmological models, and therefore there is no definition for its inhomogenities, either.

Looking at 4-momentum (especially energy) or angular momentum is problematic as well. The integral

$\int_V dV\,T^{00}$

has no well-defined transformation properties. Using a (timelike) Killing vector field one could define

$Q^\mu[\xi] = \int_V dV\,T^{\mu\nu}\,\xi_\nu$

but for realistic models there is no such Killing vector field.

So there is no definition of energy, momentum (and angular momentum) in realistic cosmological models, and therefore there is no definition for its inhomogenities, either. All expressions I know are again valid only for spacetimes with special symmetries (the Kiling vector fields) or are quasi-local expressions using surface integrals (like Hawking energy) which are not applicable in this context.

Now even if there would be such expressions based on local densities, the problem is to relate them to observations. All these expressions use some kind of space-like slicing, but our observations are based on null-lines. So one would have to introduce something like "retarded" density

$\mu(r) \;\to\; \mu_\text{red}(r) = \mu(r-ct(r))$

where t is the (unknown!) light propagation time from r to 0 in an (unknown!) background geometry.

Last edited: Jul 24, 2013
5. Jul 24, 2013

### Mordred

As far as modelling anisotropic models there have been countless examples if one looks for them.

here is some examples from a quick search.

http://arxiv.org/abs/gr-qc/0412078
http://iopscience.iop.org/0067-0049/148/1/175
http://arxiv.org/abs/gr-qc/0308059
http://iopscience.iop.org/0004-637X/643/2/616
http://www.worldscientific.com/doi/abs/10.1142/S0218271803003761
http://iopscience.iop.org/1538-4357/545/1/L5
http://www.jetp.ac.ru/cgi-bin/dn/e_037_05_0739.pdf
http://prl.aps.org/abstract/PRL/v102/i11/e111301
http://arxiv.org/abs/gr-qc/0304078
http://arxiv.org/abs/1212.3569
http://arxiv.org/abs/0712.2291

keep in mind some of the articles I posted are controversial but they do show examples of modelling anisotropy

edit just saw your post as I was posting mine I'll have to give the above some thought I have seen examples of anistropic GR models but will have to dig them up assuming I can find them and assuming that is what your looking for

Last edited: Jul 24, 2013
6. Jul 24, 2013

### tom.stoer

Thanks.

I know how to model or construct anisotropic universes.

My question is how to extract information regarding anisotropies from a general spacetime which is not explicitly known. The idea is to use quantities like mass or energy-momentum (densities) and to calculate a measure for anisotropies.

But due to the above mentioned obstacles I do not see how this could work in principle.

7. Jul 24, 2013

### Mordred

8. Jul 24, 2013

### Chronos

Tom, I agree. There is no known realistic way to model volumes in GR - nor energy content.

9. Jul 24, 2013

### tom.stoer

So does that mean we cannot define a measure for anisotropies? Not even in principle? Or are there other expressions I am not aware of?

10. Jul 24, 2013

### Chalnoth

I don't think this is possible to do. The very idea of "anisotropies" involves imagining you have some base space-time, and the anisotropies are deviations from that base. If you don't nail down what you mean by the base space-time, there is no way to determine which parts you're seeing are the anisotropies.

11. Jul 24, 2013

### tom.stoer

I tend to agree, but this is horrible.

- we believe that (beyond a certain scale) anisotropies will be neglectable
- we definitly observe anisotropies (galaxies, clusters, voids, CMB, ...)
but the discussion shows that we are not able to define anisotropies mathematically, and that we are therefore not able to check quantitatively (!) whether the observed anisotropies violate the cosmological principle and to which extent

I would say this is not science

12. Jul 24, 2013

### WannabeNewton

13. Jul 24, 2013

### tom.stoer

Thanks a lot.

This was not precisely what I was looking for, but it seems that at least with my questions I am on the right track. It seems to make to use a background metric and calculate deviations from this background in perturbation theory. The anisotropies are not measured by directly calculating averages, but by the convergence of the approximations.

Next question would be how relate Wald's formalism to averaging and to observations.

14. Jul 24, 2013

### Mordred

15. Jul 24, 2013

### Chalnoth

Nah, it's not a big deal.

First, it is perfectly valid to take a parameterization for the base, and then carefully work through the results of what happens when you have perturbations from that base. It is very true that the choice of base space-time is arbitrary, but the results won't match up if that choice is a bad one.

It's worth pointing out that many people have suspected that subtle errors in how we handle anisotropies are the reason why we've measured an accelerated expansion. This was very much a valid line of inquiry, but it has come up empty. The only way for anisotropies to explain our observations is if we live almost identically in the center of a massive void. But this possibility is ruled out by detailed observations anyway. See here:
http://arxiv.org/abs/1007.3725