 Solution

For $k\gg aH$, the dominant term isThis is the positive-frequency harmonic-oscillator normalization with the physical box volume $a^3{L}^3$. Over a period much shorter than $H^{-1}$, the scale factor and physical frequency are effectively constant. Equivalently, the rescaled field $a{L}^{3/2}{w}_k$ approaches $e^{-ik\eta }/\sqrt{2k}$ exactly in the short-wavelength limit.

For $k\ll aH$, $w_k\to i{L}^{-3/2}H/\sqrt{2k^3}$ and ${\dot{w}}_k=O(x^2)$, so the growing field mode freezes. A single Fourier coefficient has variance proportional to $k^{-3}$, which by itself is not scale-independent. The number of modes in a logarithmic shell is proportional to $L^3{k}^3$. Their product givesThus equal logarithmic intervals have equal variance in exact de Sitter. Slow evolution of $H$ or nonnegligible effective mass produces a tilt. Integrating an exactly flat spectrum over arbitrarily many logarithmic intervals still requires infrared and ultraviolet cutoffs; the statement concerns power per interval, not a finite all-scale total variance.