Seeking understanding regarding Ω_b, Ω_c, and Ω_m

[Buzz Bloom]

TL;DR Summary

The source is https://en.wikipedia.org/wiki/Lambda-CDM_model#Parameters. The three parameters are presented, and I fully expected that Ω_m = Ω_b + Ω_c. However this does not happen, which I find to be quite confusing.

Ω_b = Baryon density parameter = 0.0486±0.0010.
Ω_c = Dark matter density parameter = 0.2589±0.0057.
Ω_m = Matter density parameter = 0.3089±0.0062.
But
Ω_b + Ω_c = 0.3078±0.0067.

I would much appreciate any explanation about why this unexpected relationship with the three parameters is presented, especially without any explanation at all (that I could find) in the Wikipedia article.

 

[Vanadium 50]

Ω_m - (Ω_b + Ω_c) = 0.001 ±0.0022 (or more), So what's the problem?

 

[Buzz Bloom]

Hi @Vanadium 50:

Thank you for your comment. The "problem" is my confusion about how the independent measurements of the three variables can result in a disagreement in the addition of two variables (Ω_b and Ω_c) whose sum does not equal the third variable (Ω_m) which is by definition: the sum of Ω_b and Ω_c equals Ω_m. Thus, there are two plausible values for Ω_c (ignoring standard deviations). Once value is approximately 0.309, and the other is 0.308. One way (which I do not particularly like) that comes to mind for resolution is that the combination of the three results is that
Ω_m = 0.3085
with its error range unequal between + and -.
If I want to calculate the age of the universe to the best possible precision available, one plauisible result would be the integration
T = (1/H_0) {INTEGRATE](0,1) da/( a SQRT[ Ω_m/a^3 + (1-Ω_m) ] ).
For this example, the key problem is: What value of Ω_m gives the most accurate result for the age of the universe?

Regards,
Buzz

 

[PeterDonis]

The "problem" is my confusion about how the independent measurements of the three variables can result in a disagreement in the addition of two variables (Ω_b and Ω_c) whose sum does not equal the third variable (Ω_m) which is by definition: the sum of Ω_b and Ω_c equals Ω_m.

You are contradicting yourself. First you say that all three variables are independently measured. Then you say that by definition the third variable is the sum of the first two. Those two statements are inconsistent with each other. They can't both be true.

Since the first two variables, as measured, do not exactly equal the third variable, it is obvious that the first of the two statements, that all three variables are independently measured, is the correct one, and that the second statement, that the third is by definition the sum of the first two, is false. Instead, adding the first two together and comparing the result to the third is a sanity check on the measurements; if the sum of the first two is within the error bars of the third (which is in fact the case, as @Vanadium 50 pointed out), the measurements pass the sanity check.

 

[Orodruin]

What is confusing you is that the measurements are independent but that the parameters are theoretically related. Compare to measuring the sides of a right angle triangle: We can measure each side independently but we expect to obtain $c^2 = a^2 + b^2$. Checking whether we do get this or not (within error bars) is a good check if we want to verify whether or not the triangle is indeed a right angle triangle.

Same here, there are different ways of measuring these parameters but we still want to check that we get something consistent (and we do, within error bars). If this was not the case we would have to consider if there could be additional contributions to $\Omega_m$ or if we are double counting somewhere depending on the sign of the discrepancy.

Edit: Minor typo corrected.

 

[Buzz Bloom]

I thank all of you for the clarifying the relationships regarding the three parameters. What remains unclear to me is if I want to use a value for Ω_m to calculate the age of the universe, based on the three Omega values what is the value most likely to give me the most accurate age?

By the way, if I correctly understand the math regarding the sum of two variables together with their respective standard deviations, then adding together Ω_b and Ω_c is:

Ω_m_bc = Ω_b + Ω_c = 0.0486 + 0.2589 +/- SQRT(0.0010^2 + 0.0057^2) = 0.3078 +/- 0.0058.​

Then, to combine the two values

Ω_m_adj = (1/2) x (Ω_m + Ω_m_bc)​

= 0.30835 +/- SQRT(0.0062^2 + 0.0058^2)​

= 0.30835 +/- 0.0085.​

If any of you recognize an error, please let me know how to correct it.

 

[Orodruin]

+/- SQRT(0.0010^2 + 0.0057^2)

This is only true if the errors are statistically independent and do not include correlations. You should only be combining measurements if you understand whether this is the case or not.

 

[PeterDonis]

What remains unclear to me is if I want to use a value for Ω_m to calculate the age of the universe, based on the three Omega values what is the value most likely to give me the most accurate age?

The actual measured value. Then you're only dealing with uncertainty in one measurement, instead of two.

 

[Orodruin]

The actual measured value. Then you're only dealing with uncertainty in one measurement, instead of two.

Well, this depends. Theoretically the uncertainty in the other two measurements could be significantly smaller leading to a smaller overall uncertainty in $\Omega_m$ than the actual direct measurement - under the assumption that those contributions indeed are the only ones. This happens, for example, in the measurement of the W mass. The error in the direct measurement is larger than the error in the value implied by assuming that the standard model is correct and using other measurable quantities and their errors.

Whether it is favourable to use two uncertainties instead of one depends on the values of those uncertainties.

 

[Buzz Bloom]

Thank you very much @PeterDonis and @Orodruin. I conclude now that I do not have sufficient detailed understanding of the processes leading to the Means and Standard Deviations. Therefore, the best I can do is just use the Ω_m values. Intuitively I am guessing that there is some way to combine the Ω_m values with the values of the other two variables, but without the proper advanced knowledge, I will just use the Ω_m values.

 

[kimbyd]

Bear in mind that these are not uncorrelated Gaussian random variables. When these measurements are actually done, they measure the entire probability distribution using a type of MCMC method. Those parameter values are a simplification of the full quality of the data, which you can see represented in Fig. 5 on page 15 of this paper:
https://arxiv.org/abs/1807.06209

Note that there are three related parameters pictured:
$\Omega_b h^2$: the baryon matter density
$\Omega_c h^2$: the dark matter density
$\Omega_m$: the total matter density fraction

The reason for the $h^2$ here is that what is measured better by the data is the actual matter densities of these two components, and this changes the parameter from a density fraction to a density.

Based upon what they show, $\Omega_m$ is not actually measured independently. It is a function of the other two parameters and $h$. My guess as to why these parameter values can't be used to compute one another is that it's probably down to the distributions not being strictly Gaussian, so that the mean value of A plus the mean value of B is simply not equal to the mean value of A+B.

 

[Orodruin]

My guess as to why these parameter values can't be used to compute one another is that it's probably down to the distributions not being strictly Gaussian, so that the mean value of A plus the mean value of B is simply not equal to the mean value of A+B.

E(A+B) = E(A) + E(B) holds regardless of correlations between A and B and whether or not they are Gaussian.

They could be the max likelihood values reported rather than means. That could give such an effect.

 

[kimbyd]

E(A+B) = E(A) + E(B) holds regardless of correlations between A and B and whether or not they are Gaussian.

They could be the max likelihood values reported rather than means. That could give such an effect.

Yeah, you're right. Now I'm a bit curious as to what's going on here. They report both the best-fit and marginalized mean values for these parameters in Table 1 on page 15, but the marginalized means don't strictly add either (ignoring the errors for now):

$\Omega_b h^2 = 0.02237$
$\Omega_c h^2 = 0.1200$
$\Omega_m h^2 = 0.1430$
$(\Omega_b + \Omega_c) h^2 = 0.14237$

They definitely state that the first two parameters are free parameters, while the third is derived. Perhaps it has to do with the numerical method to derive the marginalized mean not being strictly linear even though the marginalized mean is, in the ideal case, a linear function of the underlying parameters. Or they mistakenly wrote down those results from two different runs of the MCMC chain into the same table.

The discrepancy is much smaller than the error in the parameter, so I'm not concerned at the result's validity. But I am curious as to why.

 

[ohwilleke]

Is it possible that Ω_m is defined to include Ω_b, Ω_c, and Ω_v (i.e. the fraction of the mass-energy density of the universe that is made up of neutrinos)?

The magnitude of the difference (which is very small relative to Ω_b and Ω_c, and statistically indistinguishable from zero), is consistent with that possibility, and some cosmology models are fitted with this component, while other cosmology models are fitted without neutrino mass.

It is also worth noting when Ω_tot is not fixed to be exactly 1 as a theoretical assumption, that the value for it is 0.9993±0.0019, which is on the same order of magnitude as the sum discrepancy.

 

[kimbyd]

It coming from neutrino density could make sense. I don't think it can be a rounding error, given how the numbers are presented.

Edit: Ah! In fact, that's exactly what it is! They write:

"Note that Ωm includes the contribution from one neutrino with a mass of 0.06 eV."

Mystery solved!