This paper claims possible observable signatures from superHubble modes. I'm finding it pretty hard to follow (probably my fault because I'm not familiar with the formalism).
I'd have thought what you want to look at is the variance of the conditional probability distribution [tex]P(H\rho)[/tex], where both H and [tex]\rho[/tex] are the fully perturbed values. Then you could tell if there were significant deviations from the Friedmann equation. This would be equivalent to doing their spatial averaging procedure if the hypersurface chosen for the average has uniform density. But it doesn't look as though they are doing that. I can see the density perturbation will be negligible on superhorizon scales in most frames for adiabatic modes, but surely not when there are isocurvature modes?
If they are computing the equaltime variance of H, which is how I understand them, I'm not sure how useful this is. It's true that given a value of our proper time the measured value of H will depend on the superHubble modes. But surely it isn't a local observable? We can't send observers into different Hubble volumes and then let them compare measurements of H after a certain time: we can't tell if our one measurement of H is due to superHubble modes, or just different background values.
To put it another way: what's wrong with the separate universes picture? (e.g. astroph/0003278) i.e. well separated Hubble volumes evolve independently consistently with the Friedmann equation; the global (unobservable superhorizon) curvature perturbation changes because the evolution in the separate universes is different; but within each universe the observable local Hubble rate does nothing unusual. The superhorizon evolution is only observable if the modes reenter the horizon.
If I misunderstood, please could someone help me pin down what the observable is they claim they are computing? (i.e. variance of H in what frame/conditional on what?)
PS. There is an obvious sign typo on [tex]\dot{H}[/tex] in Eq. 4.
[astroph/0410541] Cosmological influence of superHubble perturbations
Authors:  Edward W. Kolb, Sabino Matarrese, Alessio Notari, Antonio Riotto 
Abstract:  The existence of cosmological perturbations of wavelength larger than the Hubble radius is a generic prediction of the inflationary paradigm. We show that superHubbleradius (superHubble) perturbations have a physical influence on local observables (e.g., the local expansion rate) if the Universe is filled with more than one fluid or if isocurvature perturbations are present. 
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