Neutrinos as dark matter

Adapted from IOAA 2009 (Tehran), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Assume that neutrinos have a mass of $m_{\nu }={10}^{-5}{m}_{\text{e}}$, where $m_{\text{e}}=9.109\times {10}^{-31}$ kg. How many neutrinos would each cubic metre need to hold to make up the dark matter of the Universe? Assume that the Universe is flat, that 25% of its mass is dark matter, and that $H_0=75$ km/s/Mpc. A flat universe has zero total energy in the sense of Section 13.3.

AnswerAbout $2.90\times {10}^8$ neutrinos in each cubic metre. The neutrinos left from the Big Bang number about $3.4\times {10}^8$ in each cubic metre, enough in number, but their masses are known to be far below ${10}^{-5}{m}_{\text{e}}$, so they cannot be most of the dark matter.