= Matter bound from a Friedmann bounce
{title2=$\Omega_m\leq2/[z_*^2(z_*+3)]$}
In a dust-plus-cosmological-constant universe with an <FRW metric> with $a_0=1$, a nondegenerate minimum at $a_*<1$ has $H(a_*)=0$ and $\ddot a(a_*)>0$. Combining these conditions with $\Omega_m+\Omega_\Lambda+\Omega_k=1$ gives the strict version of the displayed bound; equality is the limiting stationary case. A source at <cosmological redshift> $z$ requires $z_*=a_*^{-1}-1\geq z$, so a high-redshift source bounds the allowed present <matter density parameter> even without directly observing the bounce.
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