 Solution

Assume the new neutrino is stable and retains the abundance established by relativistic thermal decoupling in cosmology. While it remains relativistic, its contribution to the effective number of neutrino species isThus its effect on the radiation density and expansion during Big Bang nucleosynthesis is reduced relative to a fully heated extra neutrino species.

After it becomes nonrelativistic, its energy density is approximately $m_X{n}_X$. A two-state relativistic fermion relic has $n_X/n_{\gamma }=(3/4)(T_X/T_{\gamma }{)}^3\simeq 0.0275$. The density bound on a relativistically decoupled massive relic then givesHere ${\Omega }_{X,\max }$ is the allowed present matter-density fraction and $⟨E_{\gamma }⟩={\pi }^4{T}_{\gamma ,0}/(30\zeta (3))$. Choosing the total available matter fraction yields the basic overclosure bound. Its mass cannot be raised freely at a fixed relic number density.

There is also a perturbation constraint. Neutrino free streaming suppresses clustering below the distance traveled before the species becomes nonrelativistic. The ratio $T_X/m_X$ controls this distance, and the relic matter fraction controls the strength of the suppression. A significant hot or warm relic component must therefore satisfy the observed structure-formation constraints in addition to the abundance bound. Its allowed dark-matter contribution is limited jointly by these two effects. Numerical observational bounds depend on the chosen matter budget and perturbation data; the temperature and abundance dilution derived here are the factors needed to apply them.