## Wednesday, January 23, 2013

### homogeneous and anisotropic cosmology

can a cosmological model homogeneous but anisotropic? what would that construction imply? are there observable consequences, and which ones would that be?

(question courtesy of R. Schmidt, Centre for Astronomy, Heidelberg)

bonus question: what's the angle between an edge and the space diagonal in an $n$-dimensional unit hypercube? what's the limit of this angle in the case $n\rightarrow\infty$?

for the solution CQW shows here the angle in question (in units of $\pi$) as a function of dimension $n$:

1. Handwaving on the main question:

Yes it can. Why not? I could e.g. imagine a tiny (i.e. dynamically insignificant) homogeneous magnetic field. Don't ask me how to produce that in the big bang, but I don't see why it should violate GR.

It definitely cannot be the other way round, i.e. inhomogeneous and (strictly) isotropic, because the inhomogeneity will create local anisotropies, viz. the local gradients of the inhomogeneity. Of course it still can be inhomogeneous on small scales but (essentially) isotropic on large scales.

On the bonus question, I have a pictorial guess (no strict proof; it would be nice if somebody could confirm or refute my guess):

atan(sqrt(n-1)). Thus the limit would be 90 degrees.

I *love* spirou's mathematical games!
Cheers,
Uli Bastian

2. bonus question: thanks for the compliment! in fact, you can define the unit vectors $\vec{e}_i=(0,\ldots,1,\ldots,0)$ with a $1$ at the position $i$ that point along an edge of the hypercube, and construct the diagonal $\sum_i\vec{e}_i$. then, one can use $$\cos\alpha = \frac{\vec{e}_i\cdot\sum_i\vec{e}_i}{\left|\sum_i\vec{e}_i\right|} = \frac{1}{\sqrt{n}}$$ such that really $\cos\alpha\rightarrow 0$ as $n\rightarrow\infty$, and $\alpha\rightarrow\pi/2$.

3. The standard answer is to think of the bark of a tree, which is the same everywhere, but has vertical structure. Now, that never convinced me very much because it's not my idea of homogeneous.

Gödel found a much better answer: A homogeneous universe that rotates with a fixed rotation direction around every point. There are of course many other examples and a whole slew of literature, but for me this is the archetype of the anisotropic homogeneous universe.

Observable consequences? Yes. For example, if your model has background radiation (which only expanding Gödel types can have, not his original one), the radiation (e.g. CMB) may show spiral patterns. This has been searched for, but never found, leading to relatively tight constraints on rotating big bang models that affect the background radiation.

Quite interesting though, light bundles may also be affected by this, leading to a detectable weak lensing signal. Perhaps this can still be discovered, depending on the rotation.