= 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> is
$$
\Delta N_{\mathrm{eff}}=(T_X/T_\nu)^4
=\left(\frac{10.75}{106.75}\right)^{4/3}\simeq0.047.
$$
Thus 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 $m_Xn_X$. A two-state relativistic fermion relic has $n_X/n_\gamma=(3/4)(T_X/T_\gamma)^3\simeq0.0275$. The <density bound on a relativistically decoupled massive relic> then gives
$$
\frac{\Omega_X}{\Omega_\gamma}
=\frac{m_X}{\langle E_\gamma\rangle}\frac{n_X}{n_\gamma},\qquad
\boxed{m_X\le\frac{\Omega_{X,\max}}{\Omega_\gamma}
\frac{\pi^4T_{\gamma,0}}{30\zeta(3)}\frac1{0.0275}.}
$$
Here $\Omega_{X,\max}$ is the allowed present matter-density fraction and $\langle E_\gamma\rangle=\pi^4T_{\gamma,0}/(30\zeta(3))$. 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.
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