= Solution
A <power spectrum> describes the variance of Fourier amplitudes as a function of wavenumber. In a statistically homogeneous continuum convention,
$$
\langle\Delta_M(\mathbf k)\Delta_M^*(\mathbf k')\rangle=(2\pi)^3\delta_D(\mathbf k-\mathbf k')P_M(k).
$$
The paper's $\langle|\Delta_M(k)|^2\rangle$ is the corresponding finite-volume or normalization-suppressed statement. For statistical isotropy only $k=|\mathbf k|$ remains. The dimensionless variance per logarithmic interval is $k^3P_M(k)/(2\pi^2)$, a different quantity.
The response changes around $2k^2\sim9\mathcal H^2$, or $k\sim2.1\mathcal H$: the velocity contribution to the relativistic constraint and the effective growth equation change relative importance across the horizon-scale transition. Large-scale $\Delta_M$ is approximately constant for its regular mode, while short-scale density perturbations grow gravitationally. A smooth initial potential spectrum can therefore acquire a different $k$ dependence in the density spectrum. For illustration, a constant matter-era potential gives $\Delta_M=-(5+k^2\eta^2/6)\Psi$, so sufficiently small-scale $P_M$ has four additional powers of $k$ relative to $P_\Psi$, while the very large-scale limits have the same power dependence.
This identifies a dynamical transition, not a universal prediction of the primordial spectrum or an exact spectral break at $2\mathcal H$ for arbitrary initial data. The familiar late-time matter-spectrum turnover also retains the history of horizon entry around radiation–matter equality; it is not generically tied to the current Hubble scale. Initial conditions and the intervening expansion history are needed for an actual spectrum.
Back to article page
