Canonical rescaling of a scalar cosmological perturbation ($v=a\delta \chi$)

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 $v/a$.

In the de Sitter approximation, a canonically rescaled scalar has superhorizon powers $(-\tau {)}^{1/2\pm \nu }$. The physical field has powers $(-\tau {)}^{3/2\pm \nu }$. For $H^2<m^2<2H^2$, both physical branches decay, while the slower rescaled branch grows. The generic late-time dimensionless power spectrum is proportional to $(k/(aH){)}^{3-2\nu }$.