 Jeans length and gravitational time conventions

The physical Jeans length is ${\ell }_J=c_s\sqrt{\pi /(G{\rho }_0)}$. A pressureless uniform sphere of initial density ${\rho }_0$ has free-fall time of a uniform sphere $t_{\mathrm{ff}}=\sqrt{3\pi /(32G{\rho }_0)}$, giving the title relation. Thus a full-wavelength sound-crossing time and this exact free-fall convention agree in scaling, not numerical value. Using the scale $k_J^{-1}$ instead gives $1/(c_s{k}_J)=1/\sqrt{4\pi G{\rho }_0}$, the inverse long-wave static Jeans growth-rate spectrum limit. At the threshold itself pressure balances gravity and the net growth rate vanishes; the pressureless free-fall time must not be confused with that marginal mode's infinite growth time.