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I know that it is possible to obtain the light element abundances ($^{4}\mathrm{He}, \mathrm D, ^{7}\mathrm{Li}$ etc.) and to determine the baryon-to-photon ratio $\eta$ from the total baryon abundance $\Omega_\mathrm{b}$. I know the procedure to get it and I understand it. Doing so we obtain $\eta = (6 \pm 0.5)\times 10^{-10}$ and $0.018 < \Omega_\mathrm{b} \lt 0.023$. From this outcome we know the abundance of light elements are very small compared to the total abundance $\Omega_\text{total}$, that is, $\Omega_\mathrm {b} \lt \Omega_\text{total}$, once $\Omega_\text{total} = 1$. My questions: how can I know that only $\sim 23\ \%$ of $\Omega_\text{total}$ is dark matter instead of any other amount once the dark energy were not determined here (just inside BBN)? Is really necessary to use results that come from another part of $\Lambda$CMD?

I hope I've been clear.

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The primordial abundances tell us about the density of baryons only. Before the discovery of what is now called dark energy, it was widely believed that dark matter was the gap between the baryonic density and the total energy density of the universe. In turn, the total energy density was thought to be close to (though measured to be less than) the critical density, because if it weren't it would be very unlikely to be close to critical more than 10 billion years after the big bang.

There were strong indications that non-baryonic dark matter dominated over baryonic matter from the rotation curves of spiral galaxies and the motions of galaxies in clusters.

The realisation that dark energy dominates even over dark matter (at our epoch) comes from later observations of distant supernovae and fluctuations in the cosmic microwave background, which reveal that the expansion is accelerating and the universe is geometrically flat respectively.

It is probably worth noting, as Kyle points out, that the contribution of dark energy to the total energy density is big now, but was essentially negligible at the epoch of primordial nuclei synthesis. Thus it would not be expected to have had much influence on primordial abundances.

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    $\begingroup$ Would it not be useful to mention that the DE density relative to critical at the time of BBN is basically 0? (Of course the absolute density never changes, if it's a cosmological constant.) $\endgroup$ – Kyle Oman Nov 9 '19 at 14:58
  • $\begingroup$ Thank you for answer my question. The point is: BBN do not provides any direct information about DM abundance once it is about the light elements abundance, that is, baryonic matter. Right? $\endgroup$ – Jacinto Paulo Nov 10 '19 at 19:24
  • $\begingroup$ @JacintoPaulo I am no expert on this, but I believe that one class of explanations for the slightly discrepant primordial Li abundance inferred from the oldest stars involves various exotic scenarios involving dark matter at the epoch of BBN. $\endgroup$ – Rob Jeffries Nov 10 '19 at 21:58
  • $\begingroup$ Rob Jeffries, yeah! That's true. We can really use dark matter to explain 7Li problem but I think we use only a particular sort of dark matter particle. In this way, maybe we cannot obtain the total dark matter abundance directly. $\endgroup$ – Jacinto Paulo Nov 11 '19 at 22:51

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