# How Flat is Our Universe Really?

1. Jul 12, 2012

### marcus

"How Flat is Our Universe Really?"

http://arxiv.org/pdf/1207.3000v1.pdf
Interesting paper, trying to get a model-independent grip on largescale spatial curvature using the available datasets.

I've acquired respect for Bruce Bassett from his other papers over the years, so am inclined to at least look this over and get the gist.

Abstract summary:
http://arxiv.org/abs/1207.3000
==quote==
How Flat is Our Universe Really?
Distance measurement provide no constraints on curvature independent of assumptions about the dark energy, raising the question, how flat is our Universe if we make no such assumptions? Allowing for general evolution of the dark energy equation of state with 20 free parameters that are allowed to cross the phantom divide, w(z) = 1, we show that while it is indeed possible to match the first peak in the Cosmic Microwave Background with non-flat models and arbitrary Hubble constant, H0, the full WMAP7 and supernova data alone imply −0.12 < Ωk < 0.01(2σ). If we add the HST H0 prior, this tightens significantly to Ωk = 0.002 ± 0.009. These constitute the most conservative and model-independent constraints on curvature available today, and illustrate that the curvature- dynamics degeneracy is broken by current data, with a key role played by the Integrated Sachs Wolfe effect rather than the distance to the surface of last scattering. If one imposes a quintessence prior on the dark energy (−1 ≤ w(z) ≤ 1) then just the WMAP7 and supernova data alone force the Universe to near flatness: Ωk = 0.013 ± 0.012. Finally, allowing for curvature, we find that all datasets are consistent with a Harrison-Zel’dovich spectral index, ns = 1, at 2σ.
4 pages, 3 figures.
==endquote==

Last edited: Jul 12, 2012
2. Jul 12, 2012

### Mark M

A quick question:

If dark energy really is a cosmological constant, shouldn't we expect the universe to have a slight positive curvature, undetectable in our observable universe? Considering the de Sitter solution to the EFE is positively curved, shouldn't we expect $\Lambda$ to contribute positively to the total curvature?

3. Jul 12, 2012

### George Jones

Staff Emeritus
Re: "How Flat is Our Universe Really?"

de Sitter is a maximally symmetric spacetime that has constant, positive spacetime curvature. For cosmology "flat", refers to curvature of spatial slices.The spatial curvature of spatial slices for de Sitter can be positive, zero, or negative, depending on how de Sitter is foliated.

Now I am going to think out loud, so what I write might be complete rubbish.

I think because of this symmetry, there exist different congruences of integral curves (comoving observers) for which the universe is spatially homogeneous and isotropic. These different congruences give rise to the different types of foliations.

Adding normal matter breaks some of the symmetry, and only one type of foliation remains (which one depends on amounts), and, with the right combination of cosmological constant and matter density, it is possible to have flat spatial slices.

4. Jul 12, 2012

### Mark M

If it requires fine tuning of the matter density to get a foliation that allows you to define a family of spacelike surfaces in the which we get a spatially flat slice, why would we expect the universe to be exactly flat? Since the expansion provided by inflation would have made a curved universe very, very close to flat, wouldn't it be more realistic to say that the global topology is very close to flat because of of inflation, but not exactly flat?

5. Jul 12, 2012

### George Jones

Staff Emeritus
Re: "How Flat is Our Universe Really?"

Our universe is only a FLRW universe on average at large scales, so, with or without inflation, I don't think that our universe can be exactly spatially flat.

6. Jul 12, 2012

### Mark M

Thanks, that's what I was thinking. The reason I mentioned inflation was that it would make any global curvature undetectable. The reason I asked the question is that I have often heard people generalize that flatness of the observable universe to the universe as a whole.