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There is a preprint today (think it also appeared in Nature Astronomy on Nov 4) which argues that a $\Lambda{\rm CDM}+\Omega_k$ model with negative $\Omega_k$ fits the Planck Legacy 2018 CMB data rather well, and resolves some of the internal tensions in the data. For instance, comparing the fiducial model fit to either low or high angular scales separately reveals some discrepancies in the cosmological parameters at the $1\sigma$ level (per parameter, I guess the overall tension is worse). These tensions seem to go away if the Universe is allowed to be closed, but with some interesting consequences: for instance, the preferred value of $H_0$ drops to about $50\,{\rm km} \,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ (for CMB constraint only)!

This leaves me wondering if there are any CMB-independent measurements with decent leverage on $\Omega_k$, at the relevant level of precision (the article favours $\Omega_k\sim-0.04$). As far as I know, lensing, BAO and SNIa don't, but there are other measurements in cosmology...

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  • $\begingroup$ Don't have a definitive answer yet, so just going to discuss in comments. I don't see how there could be any. You'd need something reasonably scale-invariant that offers a wide swath of the sky for testing values. I suppose you could make some astronomical measurements to constrain contributing values (such as measuring star distances and redshift to better constrain $H_0$), but in order to work that back to a measurement for $\Omega_k$, you'd either have to check against the CMB or never confirm the values at all. That said, I don't work outside the CMB too much; be skeptical of me. $\endgroup$
    – Jim
    Nov 8, 2019 at 13:43
  • $\begingroup$ I should be clearer. When I say "reasonably scale-invariant", I meant compared to a star, lensing, the BAO, etc..... Come to think of it, it's possible something could be found in the cosmic filament structure. That would take a lot of time to measure though $\endgroup$
    – Jim
    Nov 8, 2019 at 13:47
  • $\begingroup$ Also, I agree with you that there are some things that seem to fit better with a small but negative $\Omega_k$. Even when I was in grad school, I found examples in texts where that would have made more sense but was overlooked. Problem is that inflation does such a good job of flattening things out it's hard to tell. I always thought I was just seeing things or hadn't yet come across some well-known evidence that makes us sure it isn't closed (you know, like how we are pretty confident it isn't open because almost everything would fall apart) $\endgroup$
    – Jim
    Nov 8, 2019 at 13:52
  • $\begingroup$ It's claimed that we have entered precision cosmology era. However, all these newly popping up "tensions" keep me wondering: where the heck is the concordance in the "concordance cosmology"? $\endgroup$
    – MadMax
    Nov 8, 2019 at 14:59
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    $\begingroup$ @MadMax it's not the "most right", it's the "least wrong"? $\endgroup$
    – Kyle Oman
    Nov 8, 2019 at 15:05

2 Answers 2

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Supernovae measurements provide constraints on $\Omega_k$ independent of CMB measurements. A relatively up to date example of this are the constraints from the Pantheon supernovae sample, detail in Scolnic et al (2018). In particular, we can examine Figure 18 of that paper. Cosmological constraints from Type Ia supernovae, from Scolnic et al..

Focus for now on the red and gray shaded contours, which are the actual results from this paper. The dark and lighter contours are the $1\sigma$ and $2\sigma$ joint confidence regions for $\Omega_m$ and $\Omega_\Lambda$. This data is able to constrain cosmological parameters like this because it allows us to fit the distance-redshift relationship. Type Ia supernovae are what are known as standard candles — due to the mechanism that produces them, they occur at the same absolute magnitude. If we know how bright they actually are, and how bright they appear to us, we can work out a measure known as the luminosity distance. The specific functional form that this luminosity takes depends on whether the universe is open, closed, or flat, but the key point is that all forms involve an integral that looks like

$$ \chi(z) = \int_0^z \frac{dz}{\sqrt{\Omega_m (1+a)^3 + \Omega_\Lambda + \Omega_k(1+z)^2}} $$

By measuring enough supernovae redshifts and apparent magnitudes, we can attempt to adjust parameters in the above function so that we get a good fit with the data. This is more or less what is done to generate the above plot. These constraints depend only on "local" measurements and do not include information obtained from the CMB.

We can see a line in the above plot labeled "Flat Universe", which is the line such that $\Omega_m + \Omega_\Lambda =1$. We can see that the data are consistent with a flat universe (the line crosses though our contours) which is accelerating (the countours do no cross the accelerating/decelerating) lines.

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  • $\begingroup$ Just to clarify: from the plot it's clear that the measurement can separate $\Omega_\Lambda$, $\Omega_m$, and therefore constrains flatness under the assumption that "flat" means $\Omega_\Lambda + \Omega_m = 1$. But would this measurement know the difference between some other form of energy density (say, $\Omega_r$ - never mind for a moment that this is independently constrained to be $\sim 0$ at the present day) and a non-zero $\Omega_k$? Or is it specifically sensitive to something that scales as $(1+z)^2$, i.e. curvature? $\endgroup$
    – Kyle Oman
    May 25, 2022 at 8:42
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In this paper, we attempt to constrain the large scale curvature of the Universe using distances obtained from observations of Type Ia supernovae together with inferred ages of passively evolving galaxies and Hubble parameter estimates from the large scale clustering of galaxies. Current data are consistent with zero spatial curvature, although the uncertainty of $\Omega_{\kappa}$ is of order unity. Future data sets with on the order of thousands of Type Ia supernovae distances and galaxy ages will allow us to constrain the spatial curvature $\Omega_{\kappa}$ with an uncertainty of $<0.1$ at the 95% confidence level.

https://arxiv.org/abs/1102.4485

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