# I CMBR and flatness

#### Ranku

Does the mapping of CMBR measure spatial flatness or spacetime flatness of the universe?

#### Orodruin

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Spatial flatness.

#### George Jones

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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 equation
$$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = 8 \pi T_{\mu \nu}$$
gives that the universe is empty, i.e., $T_{\mu \nu} =0$. (Note that the converse is not true.)

In a spatially flat non-empty FLRW univesre, 3-dimensional spatial hypersurfaces of constant cosmic time have zero intrinsic curvature, but they non-zero extrinsic curvature.

#### Ranku

zero intrinsic curvature, but they non-zero extrinsic curvature
Could you clarify the difference between intrinsic and extrinsic curvature?

#### kimbyd

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Does the mapping of CMBR measure spatial flatness or spacetime flatness of the universe?

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.

#### phyzguy

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.
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.

#### kimbyd

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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.
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.

That said, I believe the BAO observations and the CMB observations combined, which is the most common method to constrain spatial curvature, measure spatial curvature in a way that should be independent of the rate of expansion. So the possibility of spatial curvature doesn't entirely remove all tension; it just changes it from tension in expansion rate to tension in curvature estimates.

#### Ranku

There is a degeneracy between the rate of spatial expansion and spatial curvature.
Could you clarify what is meant by 'degeneracy' between the rate of spatial expansion and spatial curvature?

#### kimbyd

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Could you clarify what is meant by 'degeneracy' between the rate of spatial expansion and spatial curvature?
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.

#### Ranku

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.
So degeneracy essentially means that the correlation between $\Omega_k$ and $H_0$ and $\Omega_m$ is weakened.

#### kimbyd

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So degeneracy essentially means that the correlation between $\Omega_k$ and $H_0$ and $\Omega_m$ is weakened.
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.

However, in this case the degeneracy is even tighter: all three values are closely-related. If you imagine the parameter space as a three-dimensional space, with one dimension for each of the three parameters, the CMB observations limit the results to a line that stretches through the space. So given any one of the three parameters, you can determine the other two with high precision.

This means that you can either use a measurement of $\Omega_k$ to get an accurate estimate of $H_0$ and $\Omega_m$ (such as Baryon Acoustic Oscillations), or you can use a nearby measurement of $H_0$ to get the other two (such as the nearby cepheids), or you can use a measurement of matter density (such as from large scale cluster counts). Any one of the three will result in pretty tight error bars given the CMB data.

Mathematically, it's almost as if there are three unknowns, and the CMB data supplies two equations for those three unknowns. The third equation is supplied by some other experiment.

All that said, bear in mind that this degeneracy is approximate. The CMB data can't fit every value of $H_0$, and the values measured by Riess et. al. recently seem to be outside the range the CMB data permits given current models. So something wonky is definitely going on.

#### Buzz Bloom

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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.
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

#### kimbyd

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2018 Award
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
Yup. That's an accurate way to state it.

"CMBR and flatness"

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