[Shape-univ] Re: loose thoughts on topology and nongaussianity

[Boud Roukema]

Hi Bartek, shape-univ,

On Mon, 22 May 2006, blew wrote:

> Hi Boud,
>
> I've  just had such a though.
>
> If we assumed some multiply-connected space-time within some model (eg.
> dodecahedral) then my question is whether is leads to nongaussian (NG)
> features in CMB. I think it definitely should.

In principle, yes. In practice, maybe.

> Such spacetime, gives a modified power spectrum of fluctuations, limited to
> fundamental domain size and there should be repercusions of this size
> in harmonic modes that are natural multiple of that size. Then given a feature
> at one length scale should automatically give features at other scales -
> hence vialation of gaussianity. This could be investigated by bi- or
> tri-spectrum optimalized for these particular perturbation modes that come
> from the model. Given NG simulations one could estimate the sensityvity of
> this approach.
> This is also a test of the multiply connected space hypothesis.
>
> Is that right ?

In principle, yes.

Hajian & Souradeep (2006) measure the bispectrum from several different
version of WMAP-1st-year and claim no significant deviations from gaussianity:
http://arxiv.org/abs/astro-ph/0501001

But they don't model the *expected* signature of the PDS (poincare dodecahedral
space) in 0501001.

In Hajian & Souradeep (2003), they model the expected signature from various
specific versions of the T^3 model:
http://arxiv.org/abs/astro-ph/0301590

i've only looked at this quickly, but it seems that they do not necessarily
expect a strong (or any non-zero) signal.

In any case, back in 2003, there was little discussion of the PDS - the paper
is dated 11 Aug 2003, and the Luminet et al. paper came out around Oct 2003.
Hmmm... i guess at least, there was no pressure for them to study PDS. :P


So working out what bispectrum is expected from the PDS has (maybe) not been
done yet.

Hmm: better check the Aurich, Lustig, Steiner and Gundermann papers - they
might have tried this.


> What I would need is to have a full fourier space of perturbations with
> topology encoded in it.

The PDS is a positive curvature model: you can't do fourier analysis in curved
space. You need the full set of eigenmodes of the PDS itself.

Tarou-san has done lots of cosmic topology eigenmode modelling - if
you want to do something like this, you might want to visit him:

Kaiki Taro Inoue
kinoue phys kindai ac jp


Or maybe your idea is more like making simulations, then making
measurements of the bispectrum parameters, and then comparing them to
the analytical calculations in Hajian & Souradeep (2003)?  Hmmm... well,
this would only function as a check that HS2003 have not made any errors.

But then you could presumably think of something new and interesting
as a followup step - you can add stuff to simulations which can be (in
some sense) difficult to add to analytical calculations.


pozdr
boud