The dark matter density of the Universe

Clusters provide a very nice way to estimate the total amount of dark matter in the Universe. The argument goes like this.

Suppose the Universe starts-out smooth, with a (nearly) constant ratio $\omega$ of dark matter to gas:

$$\omega = \rho_{\hbox{\rm dark matter}}/\rho_{\hbox{\rm gas}}\,.$$ (9.9)

As a cluster starts to form, the density of dark matter and of gas will increase. But the potential well of the cluster is so deep, that it is probably a good approximation to assume that none of the dark matter, nor the gas, can ever escape the gravitational pull of the cluster. This means that the ratio of the total dark matter to total gas mass in the cluster

$${M_{\hbox {\rm cluster-in-dark-matter}}\over M_{\hbox{\rm cluster-in-gas}}} \approx \omega\,.$$ (9.10)

We could do slightly better by also including the mass in stars - but that is a small correction anyway.

So we can find $\omega$ by determining $M_{\hbox{\rm cluster-in-dark-matter}}$ from dynamics, and $M_{\hbox{\rm cluster-in-gas}}$ from the X-ray emissivity. The result is $\omega\approx 6$.

Next we need an estimate of the mean baryonic density, $\rho_{\hbox{\rm gas}}$. An elegant way is by determining the Deuterium abundance of gas^9.4. Deuterium is produced in the Big Bang and destroyed in stellar burning. So if we find Deuterium, we know it is left over from the Big Bang. Analysis of the spectra of quasars (we will discuss quasars soon) allows us to measure the Deuterium abundance quite accurately. How does this help us? The missing link is that the gas density $\rho_{\hbox{\rm gas}}$ determines how much Deuterium is produced in the Big Bang. Figure 9.2 shows how the abundance (with respect to hydrogen) of elements produced in Big Bang nucleo-synthesis, as function of the baryon density. The result is that the mean baryon density corresponds to of order $\rho_{\rm gas}/\hbox{$m_{\sc\rm p}$}\approx 2.2\times 10^{-7}$ hydrogen atoms per cm$^{-3}$ (i.e. $3.75\times 10^{-31}$ g cm$^{-3}$. So the density of paper/density of the screen you are reading this on, is about $10^{30}$ times higher than the mean!

Figure 9.2: The production of various elements during Big Bang nucleo-synthesis as a function of the baryon density, taken from http://astron.berkeley.edu/~mwhite/darkmatter/bbn.html.

And so we're done: the Deuterium abundance determines the gas density through the known nuclear reactions that occurred during Big Bang nucleo-synthesis. And the dark matter density is $\omega\,\rho_{\hbox{\rm gas}}$ with $\omega\sim 6$ determined from clusters.