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A genuine nondegenerate bounce has ${\dot{a}}_{\ast }=0$ and ${\ddot{a}}_{\ast }>0$; using ${\ddot{a}}_{\ast }\ge 0$ also includes the limiting stationary case. Put $s=a_{\ast }^{-1}=1+z_{\ast }$, and define the present matter density parameter ${\Omega }_m=8\pi G{\rho }_0/(3H_0^2)$, ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)$ and ${\Omega }_k=-k/H_0^2$. The dust-plus-cosmological-constant Friedmann equation and its acceleration equation becomeAt the bounce these giveFor a past bounce with $s>1$, eliminate ${\Omega }_k$ from the first equation:Subtracting ${\Omega }_m{s}^3/2$ and multiplying by the positive denominator givesThus the matter bound from a Friedmann bounce isStrict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.

An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_{\ast }\ge 6$. The denominator of the bound increases with positive $z_{\ast }$, soThe observational lower bound is much larger. Cluster mass-to-light ratios, inferred independently from motions, gas temperatures and gravitational lensing, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the cosmic baryon fraction from representative clusters, combined with the baryon cosmological density parameter inferred from Big Bang nucleosynthesis, provide another estimate: ${\Omega }_m\simeq {\Omega }_b/f_b$. Large-scale clustering and peculiar velocities provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for ${\Omega }_m\gtrsim 0.2$.