= Solution
In the specified dust model $H_0=2/(3t_0)$, so $\rho_{\rm crit}=m_{\rm pl}^2/(6\pi t_0^2)$ and $n_B=\Omega_{B0}\rho_{\rm crit}/m_B$. <Photons> and the six <neutrino>/<antineutrino> helicity states give
$$
g_{*s,0}=2+\frac78\,6\left(\frac{T_\nu}{T_\gamma}\right)^3=\frac{43}{11},\qquad s_0=\frac{2\pi^2}{45}\frac{43}{11}T_\gamma^3.
$$
Using the supplied GeV estimates literally gives
$$
\boxed{Y_B\equiv\frac{n_B}{s}\simeq4.5\times10^{-12}}.
$$
The rounded time/<temperature> conversions are unusually crude: converting $10$ Gyr to $4.8\times10^{41}\ {\rm GeV}^{-1}$ and $3$ K to $2.6\times10^{-13}$ GeV instead gives $Y_B\sim10^{-10}$. Either set gives the same order-of-magnitude <mass> conclusion below, but the numerical abundance should identify which supplied units are used.
For a stable relic, the relevant comparison is <energy> <density>, not merely particle number. The matter <density> left over after the <baryons> is at most nine times the <baryon> <density> here. Consequently
$$
\boxed{m_YY_Y\lesssim9m_BY_B},\qquad Y_Y=n_Y/s.
$$
Put $x=(m_{Y\rm nr}/m_Y)^{1/3}$ and use $Y_Y\simeq C_Yx^{3/2}e^{-x}$ with $C_Y\simeq10^{-3}$. The condition becomes
$$
x+\frac32\log x\gtrsim\log\left(\frac{C_Ym_{Y\rm nr}}{9m_BY_B}\right).
$$
For the literal supplied numbers the right side is about $30.5$, giving $x\gtrsim25.6$ and $m_Y\lesssim40$–$50$ GeV. With the physically consistent conversions the estimate is about $60$ GeV. The intended rough conclusion is therefore \b[a sufficiently suppressed branch with $m_Y$ no larger than order $10^2$ GeV], within the toy <cosmological particle freeze-out> law. Comparing the two number yields alone gives a similar rough scale but misses the <mass> weighting. This is a <thermal relic density bound for a tunable fermion>.
For $m_Y\gtrsim m_{Y\rm nr}$, relativistic decoupling would instead leave $Y_Y\simeq135\zeta(3)g_Y/(8\pi^4g_{*s})\simeq4\times10^{-3}$, incompatible with the matter-density bound for such heavy stable particles. These are conditional bounds on a populated, stable <thermal relic>. A short lifetime or an initially suppressed nonthermal population can change the conclusion; the supplied information alone does not constrain every possible production and decay history.
Back to article page
