Harrison–Zeldovich spectrum

Remarks

Equivalently, the dimensionless power spectrum of the primordial curvature perturbation, $\Delta^2_{\mathcal R}(k)$, does not depend on $k$; this is why the spectrum is called scale invariant. For the density, the dimensionless spectrum $\Delta^2(k) = k^3 P(k)/(2\pi^2)$ grows like $k^4$, but at the time a scale enters the horizon its density contrast is the same for every $k$.

Example: $P(k) = 2.5 \, k$ in some units has the form $A k^{n_s}$ with $A = 2.5$ and $n_s = 1$, so it is a Harrison–Zeldovich spectrum.

Non-example: a power law with $n_s = 0.965$ is called a red-tilted spectrum. It has slightly more power on large scales than the Harrison–Zeldovich spectrum, so it fails the condition $n_s = 1$.