# How are the percentage contents of the universe calculated?

## Main Question or Discussion Point

Specifically I'd like to know about dark energy. How do we know it's 74% of the universe? I'm really looking for a specific explanation/derivation. I've searched a few places and all I've found so far has been people who say "due to Friedmann's equations" or "the WMAP measurements show that" and then there's just some hand waving and they tell you the numbers. I'm having trouble finding the complete explanation/derivation. It anyone has a link that would point me in the right direction that would be great, or just someone explaining directly right here. Thanks in advance.

## Answers and Replies

Chalnoth
Science Advisor
Specifically I'd like to know about dark energy. How do we know it's 74% of the universe? I'm really looking for a specific explanation/derivation. I've searched a few places and all I've found so far has been people who say "due to Friedmann's equations" or "the WMAP measurements show that" and then there's just some hand waving and they tell you the numbers. I'm having trouble finding the complete explanation/derivation. It anyone has a link that would point me in the right direction that would be great, or just someone explaining directly right here. Thanks in advance.
Well, the short answer is that there are a large variety of different ways. With the WMAP experiment, for instance, there are two different things to look at:

1. How do the physics up to the point the CMB is emitted affect its final properties?
2. How does the travel of the light from the emission of the CMB to now affect how it looks?

These two factors provide some rather different effects that can, in part, be measured independently. The CMB as measured by WMAP behaves as if there is no unknown energy density acting before the CMB was emitted. However, from looking at how the properties of the CMB have been affected by the transit, we get a universe that needs to have most of its energy density now in something other than normal matter.

Now, I know this is kind of hand-wavy and not terribly exact, but the fact of the matter is that the gory details of these calculations are, well, really gory. A perhaps easier way to understand the measurements of the dark energy density is to take a look at measures of the expansion rate of the universe as a function of time.

For a flat universe, this is governed by the Friedmann equation:
$$H^2 = H_0^2 \rho$$
Here, $$H$$ is the Hubble "constant" (nearly constant in space, but varies with time). $$H_0$$ is the value of $$H$$ at the current time. $$\rho$$ is the energy density of the universe (with everything included).

Now, the way this works is that the various different forms of energy density dilute differently as the universe expands depending upon their properties. Normal matter is the easiest to understand: its energy is almost entirely caused by rest mass energy, and so as the universe expands, its energy density just drops off right along with the volume increase:
$$\rho_m = \frac{\rho_{m0}}{a^3}$$

By contrast, if we take photons, those don't only lose energy by becoming more dilute, but their wavelengths are also stretched by the expansion. This causes the energy density of radiation to drop off as the fourth power of the expansion factor:
$$\rho_r = \frac{\rho_{r0}}{a^4}$$

Finally, if we have a cosmological constant, then this is just an intrinsic energy density of the vacuum which does not change at all:
$$\rho_\Lambda = \rho_{\Lambda 0}$$

There are other possibilities as well, these are just the most common. What all this means is that if we have sensitive measurements of the Hubble parameter as a function of the expansion, then we can place limits upon just how much of each type of matter there is. There are a variety of ways of measuring this. One would be to make use of supernovae, which provide us with a relationship between brightness (which tells us how far the light has traveled since the supernovae went off), and their redshift (which tells us how much the universe has expanded since then). There are other ways as well, and it turns out that the best measurements of the contents of the universe combine a wide variety of measures. For example, if we have a very accurate measure of the total density in normal/dark matter, such as from galaxy cluster counting, then we can combine that with other measurements to figure out how much density is left over that has to be in something else (like dark energy).

Thanks for the reply. You explained everything very well, however, I still would like to find the gory details even if they are not very nice to someone who wants to view them. Again thank you.

Chalnoth
Science Advisor