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Ranku
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Does the mapping of CMBR measure spatial flatness or spacetime flatness of the universe?
Elaborating a bit on what @Orodruin wrote: suppose that spacetime is flat, i.e., that the curvature tensor ##R_{\alpha \beta \mu \nu} = 0## at all spacetime events. Then, contractions of the spacetime curvature tensor are also all zero, i.e., ##R_{\mu \nu} = 0## and ##R=0##, and Einstein's equationRanku said:Does the mapping of CMBR measure spatial flatness or spacetime flatness of the universe?
Could you clarify the difference between intrinsic and extrinsic curvature?George Jones said:zero intrinsic curvature, but they non-zero extrinsic curvature
Your initial question has been answered already, but I thought I'd add something:Ranku said:Does the mapping of CMBR measure spatial flatness or spacetime flatness of the universe?
kimbyd said:The CMBR measurements don't actually measure space to be flat on their own. There is a degeneracy between the rate of spatial expansion and spatial curvature. Measurements from the nearby universe resolve this degeneracy. So it's the combination of CMBR and nearby measurements that show the universe as being spatially-flat, with Baryon Acoustic Oscillations being the most common and highest-precision.
My understanding is that the tension in expansion rate could be entirely resolved by having a slightly non-flat universe. But I'm not 100% sure on that, as I haven't looked at it in detail. But I seem to remember it being explicitly mentioned as a possible explanation in the papers pointing out the discrepancy.phyzguy said:I didn't realize this. So to what extent can non-flatness resolve the H0 tension? I notice in the H0LiCOW thread that assuming slightly negative curvature of 0.01 resolves about half of the disconnect. So what keeps one from going further? If Ωk = -0.02, does that resolve the whole disconnect? There must be other constraints that prevent one from moving it that direction.
Could you clarify what is meant by 'degeneracy' between the rate of spatial expansion and spatial curvature?kimbyd said:There is a degeneracy between the rate of spatial expansion and spatial curvature.
Fig. 29 (page 40) of this paper shows the degeneracy:Ranku said:Could you clarify what is meant by 'degeneracy' between the rate of spatial expansion and spatial curvature?
So degeneracy essentially means that the correlation between ##\Omega_k## and ##H_0## and ##\Omega_m## is weakened.kimbyd said:Fig. 29 (page 40) of this paper shows the degeneracy:
https://arxiv.org/abs/1807.06209
It's a little more complicated, as the matter density fraction is also a part of it. The basic way to understand this is that the CMB data very tightly constrains ##H_0^2 \Omega_m## (this is the matter density), but curvature is largely (though not entirely) degenerate with both ##H_0## and ##\Omega_m##. So if the curvature parameter is off, then you get a different median estimate for both curvature and matter density fraction.
Interesting, though, is the fact that even with this degeneracy, the CMB data really has a hard time fitting the higher nearby estimates of ##H_0## (it permits lower values of ##H_0##, but not higher). So the tension is more significant than I thought in my earlier post, after looking at the most recent data available in that paper.
Sort of. A degeneracy means that given two of these values, you can extract the third with relatively high precision. A complete degeneracy would mean that given two values, the third is exactly-specified. An approximate degeneracy, as in this case, just means that given two of them, the third is tightly-constrained. Such approximate degeneracies are very common in observational science.Ranku said:So degeneracy essentially means that the correlation between ##\Omega_k## and ##H_0## and ##\Omega_m## is weakened.
Hi kimbyd:kimbyd said:So it's the combination of CMBR and nearby measurements that show the universe as being spatially-flat, with Baryon Acoustic Oscillations being the most common and highest-precision.
Yup. That's an accurate way to state it.Buzz Bloom said:Hi kimbyd:
I am wondering if you might agree that the following is an acceptable rephrasing of the quote above.
So it's the combination of CMBR and nearby measurements that show the universe as being so close to spatially-flat that current methods of analysis are not able to distinguish it from non-flatness, with Baryon Acoustic Oscillations being the most common and highest-precision.My understanding is that the current uncertainty of the value of Ωk allows for about a 10% possibility that |Ωk| might be larger than 0.005.
Reference:
Regards,
Buzz
CMBR stands for Cosmic Microwave Background Radiation. It is a type of electromagnetic radiation that is present throughout the entire universe and is a remnant of the Big Bang.
CMBR is one of the key pieces of evidence for the flatness of the universe. The distribution of CMBR across the sky is extremely uniform, which suggests that the universe is spatially flat.
In the context of cosmology, a flat universe means that the geometry of the universe is Euclidean, or "flat." This means that parallel lines will never intersect and the angles of a triangle will add up to 180 degrees.
The flatness of the universe is measured through various techniques, including analyzing the distribution of CMBR, measuring the average density of matter in the universe, and studying the expansion rate of the universe.
A flat universe has significant implications for the future of our universe. It suggests that the expansion of the universe will continue indefinitely, and may eventually lead to a "heat death" scenario where all matter and energy is evenly distributed and the universe reaches a state of maximum entropy.