Assumptions in dark-energy density parameter measurement

[center o bass]

I've understood that the main evidence for a non-zero ##\Omega_\Lambda## comes from supernova 1a measurements where one measures the redshifts along with the luminosity distances (equivalently magnitude) of the supernovae and, plots them against each other, then compares the result with theoretically derived curves for different values of cosmological parameters and find the parameter choices corresponding to the best fit. (One example of such plots are shown here.)

From what I have read, the only two parameters that are being varied is ##\Omega_m## and ##\Omega_\Lambda## from which one gets the result approximately ##\Omega_\Lambda \approx 0.7## and ##\Omega_m \approx 0.3## supporting the claim that the universe is flat since ##\Omega_\Lambda + \Omega_m \approx 1##.

But here is the thing: since ##\Omega_k## was not varied, did we not already assume ##\Omega_k = 0## in the first place? Might it not have been the case that by also varying ##\Omega_k## this could've lead to a better fit with other parameter values than in the above result?

Or is there a good argument for why one can neglect ##\Omega_k## in the curve-fitting procedure?

 

[PeterDonis]

is there a good argument for why one can neglect ##\Omega_k## in the curve-fitting procedure?

As I understand it, it's because even if ##\Omega_k## isn't exactly zero, it's so close to zero (based on measurement that it can't significantly affect the analysis. The fact that the analysis ends up with ##\Omega_{\Lambda } + \Omega_m = 1## serves as a sanity check on the analysis; if that sum came out significantly different from 1, that would indicate a problem, since the analysis would be inconsistent with other measurements that indicate that the universe is extremely close to being spatially flat.

 

[Chalnoth]

I've understood that the main evidence for a non-zero ##\Omega_\Lambda## comes from supernova 1a measurements where one measures the redshifts along with the luminosity distances (equivalently magnitude) of the supernovae and, plots them against each other, then compares the result with theoretically derived curves for different values of cosmological parameters and find the parameter choices corresponding to the best fit. (One example of such plots are shown here.)

From what I have read, the only two parameters that are being varied is ##\Omega_m## and ##\Omega_\Lambda## from which one gets the result approximately ##\Omega_\Lambda \approx 0.7## and ##\Omega_m \approx 0.3## supporting the claim that the universe is flat since ##\Omega_\Lambda + \Omega_m \approx 1##.

But here is the thing: since ##\Omega_k## was not varied, did we not already assume ##\Omega_k = 0## in the first place? Might it not have been the case that by also varying ##\Omega_k## this could've lead to a better fit with other parameter values than in the above result?

Or is there a good argument for why one can neglect ##\Omega_k## in the curve-fitting procedure?

Supernova measurements, especially early ones, didn't actually constrain [itex]\Omega_k[/itex] very well. You could certainly let it vary, but the error bars on it are huge.

The way to resolve the discrepancy is to combine supernova data with other data, such as CMB data: the CMB constrains the curvature to be very close to flat. The basic picture here is that combining the CMB with nearby supernovae gives you a very long lever arm with which to measure curvature. To see why the long lever arm helps, consider the Earth: it's difficult to notice the curvature of the surface of the Earth while standing on the ground. But get far enough away, such as in low Earth orbit, and the curvature becomes quite apparent.

 

[Chalnoth]

By the way, this is illustrated by this plot of the various errors of some of the different measurements:
http://www.astro.virginia.edu/class/whittle/astr553/Topic01/t1_cos_combined.gif

The clusters are nearly-vertical because measurements of galaxy clusters mostly provide an estimate of the matter density, but don't give much information about the cosmological constant. The CMB's errors run very close to the "Flat" line because the CMB mostly gives information about the geometry of the universe, but very little information about the total density. The supernova data runs at nearly right angles to the CMB data because the supernova data hardly constrains curvature at all, but does give a tight constraint to the ratio between matter and dark energy.

 

[sardar]

Thank you for your question. I would like to address your concerns about assumptions in the measurement of the dark energy density parameter.

Firstly, it is important to understand that the method used to measure the dark energy density parameter is based on the assumption that the universe is homogeneous and isotropic on large scales. This is known as the cosmological principle and is supported by observations such as the Cosmic Microwave Background radiation.

In addition, the assumption of a flat universe, where ##\Omega_k = 0##, is also supported by various observations, such as measurements of the cosmic microwave background and large-scale structure of the universe. Therefore, it is a reasonable assumption to make when measuring the dark energy density parameter.

Furthermore, while varying ##\Omega_k## may lead to slightly different best-fit values for ##\Omega_m## and ##\Omega_\Lambda##, the overall result of a flat universe with a non-zero dark energy density remains consistent. This is because the effects of varying ##\Omega_k## are relatively small compared to the effects of varying ##\Omega_m## and ##\Omega_\Lambda##.

It is also worth noting that the measurement of the dark energy density parameter is not solely based on supernova 1a measurements. Other observations, such as the large-scale structure of the universe and the cosmic microwave background, also support the existence of dark energy.

In conclusion, while it is important to consider all possible assumptions and sources of uncertainty in scientific measurements, the current evidence for a non-zero dark energy density parameter is supported by multiple observations and is consistent with our understanding of the universe as a whole.