 Canonical rescaling of a scalar cosmological perturbation

Neglecting metric perturbations, a free massive scalar field fluctuation obeys $\delta {\chi }^{\prime \prime }+2\mathcal{H}\delta {\chi }^{\prime }-{\nabla }^2\delta \chi +a^2{m}^2\delta \chi =0$. Rescaling it by the scale factor removes the conformal-time friction and gives $v^{\prime \prime }-{\nabla }^{2}v+(a^2{m}^2-a^{\prime \prime }/a)v=0$. Growth of $v$ need not mean growth of the physical fluctuation $v/a$.