[Shape-univ] topology vs inflation --- back to study :)

[Boud Roukema]

hi Bartek, everybody,

On Sun, 23 Jan 2005, Bartosz Lew wrote:

> the basic question is: is the compact topology consistent with the
> inflation. my argument point out that there is a chance that the answer
> could be not necessarily :)

Let's see...

> well, one of the prominent predictions of the compact topology is that
> the power spectrum of fluctuations should have a cut off at some

IMHO we should always be careful to distinguish the 3d power spectrum
from the spherical harmonics (or angular) power spectrum.

The spherical harmonics power spectrum has a cutoff at l=1, or if you
like, 360 degrees, for *any* model of the Universe whatsoever. You
don't have any more than 4pi steradians on the sphere: it is finite.

But i think you're talking about the 3d power spectrum here, which makes
more sense.

> topological size scale since the size of the fluctuations must fit into
> the the finite size of the fundamental polyhedron. eg.
> some wavenumber k and some related with it distance -  corresponding to some eigenmode of
> the gravitational potential must be smaller than the size of the
> fundamental domain.

Correct, except you forgot the word "observable" and slightly confused
topology and manifold: you're talking about "observable compact
manifold" or "observable non-trivial topology" or "observable
multiple-connectedness"; not "compact topology".

> On the other hand the inflation predicts that the eigenmodes of the
> gravitational fluctuations are independent from scale and yield gaussian
> distribution. this is inconsistent with the compact topology idea, isn't it ?

Wrong.

AFAIK, inflation only predicts that the eigenmodes of the gravitational
fluctuations are independent from scale and yield gaussian
distribution *on the assumption that the amount of inflation has
made the size of the Universe much, much bigger than the horizon*.
(This is a necessary, not sufficient, assumption.)

Inflation and observable non-trivial topology *can* be perfectly
consistent, just as inflation and a non-zero cosmological constant can
be consistent: the discovery of the cosmological constant (or dark energy)
has *not* ruled out inflation.

In both cases, fine-tuning of inflation is needed; there needs to be
just the right amount of inflation, but not too much, so that we
happen to be in the transition to acceleration right now, or that the
size of the Universe is about the same as the horizon right now.

In this case, an inflationary model consistent with observable
non-trivial topology does not necessarily predict perfectly gaussian
fluctuation statistics on the largest scales.

> what do you think about it, Boud, anybody ?

IMHO there's no point going through statements about inflation
if you want to link fluctuation statistics and cosmic topology.

Anyway, i recommend Fig.~4 and of course the rest of the paper in:

Riazuelo et al. 2003:
http://arxiv.org/abs/astro-ph/0212223

Paulina, Cezary: tego artykuł może was pokazywać ile praca jest
potrzebny dla ,,poważnej" simulacji mapy CMB (promieniowanie tła)...
Jednakże, dla naszych celów, prostiejsza simulacja *może* być
wystarcza, ale to zależy dokładnie jak jest logika testów.

pozdr
boud