Observational limit from the super-horizon mode spectrum

[GeorgeDishman]

In making cosmological measurements, we are limited to the region within the particle horizon, the 'observable universe'. However, it is reasonable to assume that even if the universe is finite, it is much larger than that volume. If, for example, we measure the curvature $\Omega_K$, the value we could get even with hypothetically perfect instruments would still be only a sample of the curvature of the whole. In particular, even if the curvature of the whole universe is positive, our observable portion might be a local negative fluctuation, or vice versa.

If many such non-overlapping samples could be taken (obviously impossible by definition) and plotted as a histogram (e.g. something comparable to a Gaussian spread around some mean though maybe a bit one sided), could we calculate the expected standard deviation based on the apparent spectrum, i.e. Harrison-Zel'dovich but with a spectral index of ~0.96 as measured by Planck and predicted from inflation for super-horizon modes, or would the fact that there is no upper bound on the length mean we cannot get a useful width around whatever is the mean?

 

[Chalnoth]

In making cosmological measurements, we are limited to the region within the particle horizon, the 'observable universe'. However, it is reasonable to assume that even if the universe is finite, it is much larger than that volume. If, for example, we measure the curvature $\Omega_K$, the value we could get even with hypothetically perfect instruments would still be only a sample of the curvature of the whole. In particular, even if the curvature of the whole universe is positive, our observable portion might be a local negative fluctuation, or vice versa.

If many such non-overlapping samples could be taken (obviously impossible by definition) and plotted as a histogram (e.g. something comparable to a Gaussian spread around some mean though maybe a bit one sided), could we calculate the expected standard deviation based on the apparent spectrum, i.e. Harrison-Zel'dovich but with a spectral index of ~0.96 as measured by Planck and predicted from inflation for super-horizon modes, or would the fact that there is no upper bound on the length mean we cannot get a useful width around whatever is the mean?

From what I recall, the answer to this is that the subhorizon modes are the only things that are measurable, and the superhorizon modes get mixed in with them in a way that makes it impossible to tease out what the amplitudes of those superhorizon modes might be. This makes sense if you consider that the speed of light limitation prevents information from traveling faster than light, and any measurement of a superhorizon mode would provide information about the universe that is further away than is allowed by the speed of light.

 

[GeorgeDishman]

Yes I understand that. The way I think of it is that if we were at the peak of some Fourier component with amplitude A and a wavelength several orders of magnitude larger than the horizon on top of mean M, we couldn't distinguish that from a mean of M+A. I am not suggesting that we could ever measure super-horizon modes. (I do however wonder if the slope of such a mode might be related to the suposed spatial variation of the fine structure constant seen be Webb et al, but that's another topic entirely.)

However, what I am asking here is whether we can calculate a predicted variance of the curvature that would apply to multiple independent samples, each of the size of the observable universe, from the H-Z spectrum modified by the spectral index. I'm assuming that "(nearly) equal power on all scales" applies to wave number rather than wavelength hence the sum to zero wave number would be finite but I don't know how to convert that to a predicted Gaussian distribution. The small value ("nearly flat") observed by Planck and predicted by inflation implies the spread is going to be small (if the Copernican Principle holds) so should be reasonably approximated by a Gaussian, though of course there is a lot of interest on non-Gaussianity in the observable universe, but that's another level of detail.

I'm continuing to dig and find articles but at the moment this for example is beyond my abilities: http://www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/NG.pdf.