A question about scaling

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One wants the function $\Delta ^2$ to be such that, $\Delta^2(k,\tau) = \Delta^2(\frac{k}{\lambda ^{\frac{4}{n+3}}}, \lambda \tau )$. Now from this how does this follow that, the following holds,

$\Delta^2(k,\tau) = \Delta^2(\frac{k}{k_{NL}})$ where $k_{NL}^{n+3} \propto \tau ^{-4}$

?

Is there some uniqueness about it?