Jeans support versus forced baryon response (${\delta }_B^{\prime \prime }+\mathcal{H}{\delta }_B^{\prime }+c_s^2{k}^2{\delta }_B\simeq 4\pi G{a}^2{\bar{\rho }}_C{\delta }_C$)

When baryons are a small fraction of the gravitating matter, their perturbation is forced by the cold dark matter potential. Comparing pressure frequency $c_s^2{k}^2$ with $4\pi G{a}^2{\bar{\rho }}_C$ yields the scale ${\lambda }_J=c_s\sqrt{\pi /(G{\bar{\rho }}_C)}$ for following the growing matter mode. It is not an absolute statement that every sub-Jeans baryon perturbation is zero or cannot be forced: a slowly varying response is ${\delta }_B\simeq (k_J^2/k^2){\delta }_C$. In the pressureless limit the difference from CDM is a constant plus a decaying mode during matter domination, so baryons catch up in relative amplitude.