v = 0 \, .\end{aligned}$$ For the scale factor $a\sim \tau^q$, the solution of the Mukhanov-Sasaki equation is given by $$\begin{aligned}
v (\tau) = c_1(k) \sqrt{\tau} J_{\alpha} (k\tau) +  c_2(k) \sqrt{\tau}
Y_{\alpha} (k\tau),\quad \alpha\equiv\ q-\frac{1}{2}  ,\end{aligned}$$ where $J(x)$ and $Y(x)$ are Bessel functions of the first and second kind, respectively. On sub-Hubble scales, the solutions are oscillatory, on super-Hubble scales they can be approximated by a power law. To see this, we note that the expansion of the Bessel function solutions for small argument, $x\ll 1$, is $$\begin{aligned}
J_\alpha (x)  &= \sum_{m = 0}^\infty
\frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{x}{2}
\right)^{2m+\alpha}, \\ Y_\alpha (x)  &=
\frac{\cos(\alpha\pi)}{\sin(\alpha\pi)} J_\alpha(x) -
\frac{1}{\sin(\alpha\pi)}J_{-\alpha}(x)  \nonumber \\ &=
\frac{\cos(\alpha \pi)}{\sin(\alpha \pi)} \sum_{m = 0}^\infty
\frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{x}{2}
\right)^{2m+\alpha} - \frac{1}{\sin(\alpha\pi)}\sum_{m = 0}^\infty
\frac{(-1)^m}{m!\Gamma(m-\alpha+1)} \left( \frac{x}{2}
\right)^{2m-\alpha},\end{aligned}$$ we can express the mode function in terms of a series expansion $$\begin{aligned}
v(\tau) = \sum_{m=0}^{\infty} d_{1m} (k) \tau^{q+2m} +
\sum_{m=0}^{\infty} d_{2m} (k) \tau^{1-q+2m}  ,
\label{vtaum}\end{aligned}$$ where $d_{1m}(k)$ and $d_{2m}(k)$ are given, respectively, by $$\begin{aligned}
d_{1m}(k) & = \left[ c_1(k) + \frac{\cos(\alpha\pi)}{\sin(\alpha\pi)}
  c_2(k) \right]  \frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left(
\frac{k}{2} \right)^{2m+\alpha}, \\ d_{2m}(k) & =
-\frac{c_2(k)}{\sin(\alpha\pi)} \frac{(-1)^m}{m!\Gamma(m-\alpha+1)}
\left( \frac{k}{2} \right)^{2m-\alpha}  .\end{aligned}$$ Since we are interested in those modes that went classical (crossed the Hubble radius), such 