Anisotropic Universe and Friedmann Equations

[Vick]

TL;DR Summary

What will the Friedmann Equations be if we assume an anisotropic universe?

The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic.

But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how would Friedman Equations be modified.

I guess we would have to redefine the Friedmann metric tensor. But how?

The Friedmann metric tensor is:

$$g = -dt \otimes dt + (- \frac {a(t)^2} {kr^2-1}) dr \otimes dr + r^2 a(t)^2 d\theta \otimes d\theta + r^2 a(t)^2 sin(\theta)^2 d\phi \otimes d\phi$$

And the Friedmann Equations are:
The first one:

$$ H^2 \equiv (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$

and the second one:

$$ \frac{\ddot a}{a} = - \frac{4\pi G}{3} (\rho + \frac{3p}{c^2}) + \frac{\Lambda c^2}{3} $$

And the derived Hubble parameter is:

$$ H_0 = \sqrt{(\Omega_c +\Omega_b)a^{-3} + \Omega_{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{DE} a^{-3(1+w)}} $$

So how exactly would those equations need to be modified to account for anisotropy and an axial pole?

 

[PeterDonis]

what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how would Friedman Equations be modified

It's not really a matter of modifying the Friedmann Equations, it's a matter of starting over again with the Einstein Field Equation with different assumptions.

If you don't make any assumption at all about symmetry, I'm not sure how you would obtain any solution; the distribution of stress-energy could be anything.

If you are assuming that the universe is still axisymmetric, and that it is still homogeneous (so the axisymmetry is the same everywhere), that would constrain the stress-energy tensor and a solution might still be possible. There are known axisymmetric solutions to the EFE, but I don't know if any of them describe an expanding universe.

 

[kimbyd]

Summary: What will the Friedmann Equations be if we assume an anisotropic universe?

The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic.

But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how would Friedman Equations be modified.

I guess we would have to redefine the Friedmann metric tensor. But how?

The Friedmann metric tensor is:

$$g = -dt \otimes dt + (- \frac {a(t)^2} {kr^2-1}) dr \otimes dr + r^2 a(t)^2 d\theta \otimes d\theta + r^2 a(t)^2 sin(\theta)^2 d\phi \otimes d\phi$$

And the Friedmann Equations are:
The first one:

$$ H^2 \equiv (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$

and the second one:

$$ \frac{\ddot a}{a} = - \frac{4\pi G}{3} (\rho + \frac{3p}{c^2}) + \frac{\Lambda c^2}{3} $$

And the derived Hubble parameter is:

$$ H_0 = \sqrt{(\Omega_c +\Omega_b)a^{-3} + \Omega_{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{DE} a^{-3(1+w)}} $$

So how exactly would those equations need to be modified to account for anisotropy and an axial pole?

As PeterDonis said, you have to start over. Unfortunately, nobody knows how to do that well. There are exact spherically-symmetric solutions to the Einstein Field Equations that might be usable, but there's no way to do it in general except through approximations.

This is the basic concept behind perturbation theory as it's used in cosmology, where you have a background spacetime that is homogeneous and isotropic, but layered on top of that are some inhomogeneous fluctuations. It's a big, complex topic. But it underlies a lot of important cosmology regarding the formation of structure in the universe.

 

[Vick]

I was hoping for some attempts at doing it mathematically. Maybe starting with a try on the metric tensor and also as PeterDonis suggested try to imagine what the $T_{\mu\nu}$ might be.

 

[kimbyd]

I was hoping for some attempts at doing it mathematically. Maybe starting with a try on the metric tensor and also as PeterDonis suggested try to imagine what the $T_{\mu\nu}$ might be.

I wouldn't even want to attempt the extremely complicated math involved in doing this on a forum post. Once you step away from highly-symmetric space-times, the Einstein field equations get monstrously complicated.

 

[Vick]

I wouldn't even want to attempt the extremely complicated math involved in doing this on a forum post. Once you step away from highly-symmetric space-times, the Einstein field equations get monstrously complicated.

Oh! What I meant was to just give me some hints and clues (in math) as to what the metric tensor and the $T_{\mu\nu}$ should be for an anisotropic and axially aligned universe, and I'll attempt the rest of the computation on my own.

 

[kimbyd]

What do you mean by axially-aligned? And what specific type of asymmetry are you considering?

 

[Vick]

What do you mean by axially-aligned? And what specific type of asymmetry are you considering?

As stated above in the original question:

the clustering of masses are aligned to an arbitrary axis (axial pole)

 

[kimbyd]

As stated above in the original question:

That's not very explicit.

 

[Vick]

That's not very explicit.

Masses in the universe will be aligned to an arbitrary axis, hence giving the universe a sense of polar axe.

 

[PeterDonis]

Masses in the universe will be aligned to an arbitrary axis, hence giving the universe a sense of polar axe.

What does this mean mathematically? For example, what kind of symmetry transformations would leave the universe looking the same?

In the homogeneous and isotropic FRW spacetimes, the set of those symmetry transformations is: all spatial translations, and all spatial rotations about any axis. Obviously those can't all leave your hypothetical universe looking the same. But which ones would leave it looking the same?

 

[Vick]

I am not sure! When we say "looking the same" must mean our own perception as to what the universe looks like and how we have modeled it within the framework of the Friedmann Equations (Friedmann metric). However what if, our perception is faulty to a certain extent, and that we would need a new model for expansion, anisotropic and inhomogeneity for very large scales and isotropic and homogeneity for local scales. See this article about the Planck results, the first 4 paragraphs are of interest: anisotropic .

But I think a spherical symmetry can still be used for it.

 

[PeterDonis]

When we say "looking the same" must mean our own perception as to what the universe looks like

More or less; but "what the universe looks like" can be characterized by things like the density of matter, so it can be given a concrete meaning.

and how we have modeled it within the framework of the Friedmann Equations (Friedmann metric)

No, that's not necessary; you don't have to have an Friedmann metric to define things like the density of matter. You just have to think about, for the kind of model you are describing, how would the density of matter vary in space? What kinds of transformations could you do that would leave the distribution of density of matter in space unchanged?

what if, our perception is faulty to a certain extent, and that we would need a new model for expansion, anisotropic and inhomogeneity for very large scales and isotropic and homogeneity for local scales

This is something different from what you've been saying up to now. If the universe has a preferred axis, that would be true on all scales.

I think a spherical symmetry can still be used for it.

No, it can't, because spherical symmetry means there is no preferred axis: rotations about any axis leave everything looking the same. But you are saying you want to construct a model where there is a preferred axis.

 

[Vick]

No, it can't, because spherical symmetry means there is no preferred axis

Ok. But then I don't know. That is why I'm asking in this thread.

This is something different from what you've been saying up to now

No, I'm just adding or theorizing based on the Planck article: which says that for large enough scales the isotropic properties start to break down (in my own words). So I was thinking that it must be the universe looks homogeneous and isotropic in local scales, but for very large scales we see anisotropic and a possible inhomogeneity, but in any case the universe is expanding and in acceleration.

I don't have a clue as to what the metric should be in these cases.

 

[PeterDonis]

I'm just adding or theorizing based on the Planck article

The Planck article says nothing about an "axial pole" so I don't know where you got that from. If you're interested in spacetimes with an "axial pole" that would be a separate discussion from a discussion of the Planck results.

If you are interested in what kinds of spacetime models cosmologists are looking at because of what they see in the Planck data, the article mentions the Bianchi models; I think they're talking about these:

http://www.scholarpedia.org/article/Bianchi_universes
But, as the article notes, nobody understands at this point how to construct a model that looks like one of these Bianchi models on very large scales but looks like an FRW model on smaller scales.

 

[Vick]

The Planck article says nothing about an "axial pole" so I don't know where you got that from. If you're interested in spacetimes with an "axial pole" that would be a separate discussion from a discussion of the Planck results.

Can't I mixed the two? The one with the Planck reports and one of my own curiosity? Anyway, I prefer to talk about the axial pole and the anisotropic aspects of a hypothetical but expanding universe!

What kind of symmetry do you think would go for these criteria?

 

[PeterDonis]

Can't I mixed the two? The one with the Planck reports and one of my own curiosity?

Not in the same thread, that just makes the discussion unfocused.

I prefer to talk about the axial pole and the anisotropic aspects of a hypothetical but expanding universe!

Then let's keep this thread focused on that and you can start a separate thread if you want to discuss the Planck results and implications based on the link you gave earlier.

What kind of symmetry do you think would go for these criteria?

The obvious symmetry would be axial symmetry, meaning that rotation about some fixed axis in space leaves everything looking the same. Unfortunately pretty much anything you will find about axially symmetric spacetimes in the literature will be about Kerr spacetime, i.e., the spacetime of a rotating black hole, which is axially symmetric but also stationary, not expanding.

 

[Vick]

Not in the same thread, that just makes the discussion unfocused.
Then let's keep this thread focused on that and you can start a separate thread if you want to discuss the Planck results and implications based on the link you gave earlier.
The obvious symmetry would be axial symmetry, meaning that rotation about some fixed axis in space leaves everything looking the same. Unfortunately pretty much anything you will find about axially symmetric spacetimes in the literature will be about Kerr spacetime, i.e., the spacetime of a rotating black hole, which is axially symmetric but also stationary, not expanding.

But we need a spacetime that is expanding. I've gone through some of them and I found out about
Taub–NUT space. Do you think this will do?

 

[PeterDonis]

Taub–NUT space. Do you think this will do?

Taub-NUT space is a vacuum solution, so it can't model a universe with matter in it.

 

[kimbyd]

Back to the original issue, the problem, fundamentally, is that you're asking us to define something that you haven't clearly specified. And the details really do matter here.

For example, here are three types of axis-aligned universes that one could write down on paper:
1) Matter has a tendency to be aligned along parallel rods. The universe is still homogeneous, but the way in which the matter is distributed picks out a particular direction. This kind of universe expands just like ours does (following the FRW equations), because it is homogeneous on large scales. The preferred direction would indicate that there were some interesting physics at work in the very early universe that aligned those initial perturbations along a particular direction.
2) There's a central line through the universe that is its maximum-density region. Density of the universe drops gradually as you move away from this line. This is a cylindrically-symmetric universe which does not behave like FRW. The Einstein equations would have to be reworked with this symmetry set up.
3) The universe is rotating around an axis. This is another type of cylindrical symmetry that would not behave like FRW. It might start homogeneous, but the rotation may force this universe out of homogeneity on large scales.

 

[Vick]

Back to the original issue, the problem, fundamentally, is that you're asking us to define something that you haven't clearly specified. And the details really do matter here.

For example, here are three types of axis-aligned universes that one could write down on paper:
1) Matter has a tendency to be aligned along parallel rods. The universe is still homogeneous, but the way in which the matter is distributed picks out a particular direction. This kind of universe expands just like ours does (following the FRW equations), because it is homogeneous on large scales. The preferred direction would indicate that there were some interesting physics at work in the very early universe that aligned those initial perturbations along a particular direction.
2) There's a central line through the universe that is its maximum-density region. Density of the universe drops gradually as you move away from this line. This is a cylindrically-symmetric universe which does not behave like FRW. The Einstein equations would have to be reworked with this symmetry set up.
3) The universe is rotating around an axis. This is another type of cylindrical symmetry that would not behave like FRW. It might start homogeneous, but the rotation may force this universe out of homogeneity on large scales.

It looks like 2 or 3 but, I need more info about 3. Are the universes in 2 and 3 still expanding?

 

[PAllen]

Just to add a little here:

Suppose one assumes just axial symmetry about a single axis, as naturally generalized from spherical symmetry about a point. This is not enough to make looking for solutions much easier than the fully general Einstein Field equations. To reach a plausibly tractable family of solutions involving matter (as well as vacuum, if desired) , one needs to further assume non-rotation and stationary character. In this case, you can characterize the solutions in terms of 3 general functions of radial and axial coordinates, with explicit formulas for the Ricci tensor in terms of these (and thus the stress energy tensor). You can then further impose the dominant energy condition for classical plausibility. [For a typical textbook discussion of these issues, see Synge, "Relativity, the General Theory", section VIII.1; remarkably, MTW and Carroll have no discussion of general families axisymmetric solutions. Wald does have such a discussion, with more rigor and a different emphasis than Synge, in particular, allowing rotation plus matter - but with the result that solutions are hard find, despite still assuming stationary + axisymmetry].

However, all of this is useless for cosmology because of the stationary assumption. In short, any way of doing anisotropic cosmology is way beyond the level of a standard graduate GR textbook (and I have no idea of what research may have been done in this area).

 

[Vick]

Well, I need the metric for the following assumptions:
1) Universe is expanding as it currently does.
2) Universe started from a Big Bang
3) Universe is anisotropic at large scales with a central axe
4) Universe is non-rotating
5) Universe looks homogeneous from local vantage points.
6) Non-locality (QM) is valid.
7) present density parameters (dark energy, matter, neutrino) are valid

Any specific works that includes the above?

 

[PAllen]

My guess is no GR solution can meet all of those. Actually, the quantum nonlocality inherently rejects GR, or any classical field theory at all. Even removing this quantum criterion, my guess is that no GR solution is possible. You have imposed too many conditions to satisfy them all.

 

[PeterDonis]

Non-locality (QM) is valid.

GR is a classical theory; it doesn't model quantum phenomena.

 

[Vick]

GR is a classical theory; it doesn't model quantum phenomena.

Yes I know.

 

[PeterDonis]

Yes I know.

So does that mean you are removing item 6 from your list?

 

[Vick]

So does that mean you are removing item 6 from your list?

Let's say it's an underlying assumption. Basically, yes, we struck it from the explicit list.