Richard Feynman once said to a journalist “Listen, buddy, if I could explain it in two minutes, it wouldn’t be worth a Nobel Prize.” Similarly, a blog-comment box is probably not the right place for subtle discussions. If there were some obvious goof, yes, but not for a detailed analysis. For what it’s worth, there is *a lot* of discussion, including some by yours truly and one of the authors of the paper, at Sabine Hossenfelder’s blog.

One must also distinguish the hype surrounding such papers from the actual conclusions of the paper. Some pundit compared it to a smoky toaster leading to the headline “city on fire”, though on the other hand that doesn’t mean that we shouldn’t worry about smoky toasters.

If *anyone* claims that the supernova data don’t support current acceleration of the universe, or even (a stronger claim) a positive cosmological constant, then the question is why the concordance model has the parameters it does, even ignoring the supernova data (see above). Something else is whether the data are *consistent* with no acceleration but *also consistent* with the concordance model. In the former case, one needs to explain the conspiracy of other tests which consistently led to the *wrong* result.

*“strong assumptions like flatness”*

Saying that flatness is a strong assumption can be confusing depending on how it is interpreted. It is definitely not the same as “assuming the Einstein-de Sitter universe”, as some did 30 years ago. The latter was a result of theoretical prejudice and the desire for closed solutions. A flat universe (which incidentally does allow closed solutions for some quantities, but that hardly plays a role today), however, is an observation, not an assumption. The best such observation is that of the CMB. As mentioned above, just the CMB tells us that the sum of Omega and lambda (or Omega_m and Omega_lambda for those who like subscripts) is about 1.05 (with a 1-sigma uncertainty of about 0.023).

Why are CMB results now often presented assuming absolute flatness? Several reasons. First, there are enough parameters as it is, so why allow for an extra one even if we know that it will deviate from the strict assumption only slightly, even based on just the CMB data, and, when other data are used, the universe is compatible with perfect flatness? (Although not at all in the same category, one usually assumes that the gravitational constant doesn’t vary with time, even though *some* variation would be allowed by observations.) I think that that assumption is mostly harmless, unless one is investigating constraints on flatness as one’s primary goal. Another reason is that many people believe in inflation (for which there is some evidence, including the tilt in the power spectrum of CMB fluctuations), which generically implies a very flat universe, so flat that it is not worth the trouble to consider a deviation when analysing current observations, as the uncertainties in the observations are much larger than any possible deviation.

All the same, I think that an actual detection of curvature, especially positive curvature, would be very interesting.

]]>Obviously, the more tests involved, the tighter the constraints, and, indeed, in the old days, only a combination of constraints led to a small region in the allowed parameter space, more effectively than one might have thought, because of the (approximate )orthogonality you mention. But individual constraints have improved. Yes, combining them still results in an improvement, but individual constraints now allow small regions of parameters space, ruling out a non-positive cosmological constant.

The CMB results are usually presented differently, since lambda and Omega are “derived” rather than “basic” parameters, so one has to read between the lines. For example, *Planck* (https://arxiv.org/abs/1807.06209) finds, using only CMB data without including any other constraints, that the sum of lambda and Omega is about 1.05. Formally, a flat universe is ruled out at 1 sigma but, depending on which *Planck* data one uses, only marginally at 2 sigma and not at 3 sigma. (Combining with other tests, however, “pulls parameters back into consistency with a spatially flat universe to well within 2 sigma”.) See section 7.3 of the paper linked to above. Table 1 in that paper (section 3) quotes Omega_m as 0.3158 with an uncertainty of about 0.007, assuming flatness, which implies that lambda is 0.6842. But, as noted above, the sum of the two is about 1.05. While it would be interesting to see constraints in the lambda–Omega plane from just the *Planck* data, it is clear from these numbers that, even using just the *Planck* data and pushing the uncertainties in the “preferred” direction, a non-positive cosmological constant is ruled out at several sigma.

Is it true that dark energy is firmly established if we completely threw out the supernovae data? (And also without adopting strong assumptions like flatness)

I thought the various cosmological probes carved out orthogonal slices through parameter space so that we are left with a highly constrained set of parameters. If we erase the supernova contours will we still have a posterior on Omega_Lambda that is concentrated above zero? Can you point us to any papers/plots?

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