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The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant $H$. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

In the incoming Bunch-Davies vacuum, only $⟨a_{\mathbf{k}}{a}_{\mathbf{q}}^{†}⟩={\delta }_{\mathbf{kq}}$ contributes to the coincident vacuum expectation value. ThusReplacing the box sum by $\mathcal{V}\int d^{3}k/(2\pi {)}^3$ givesThe subhorizon vacuum contribution is the flat-space zero-point term; at fixed comoving $k$ it redshifts as $a^{-2}$. The other term gives frozen scalar power on superhorizon scales. Specifically, the dimensionless cosmological power spectrum per logarithmic interval isIts late-time value is independent of $k$, producing a scale-invariant inflationary power spectrum. Each fixed Fourier mode freezes separately as $\tau \to 0$.

The coincident integral is formal: the vacuum part diverges at large $k$, and an unlimited scale-invariant part diverges at small $k$. The conclusion concerns the power in each finite logarithmic band. A finite inflationary history supplies a largest generated wavelength, as described by the infrared qualification of a scale-invariant covariance; taking the late-time limit inside an unregulated integral would not be justified.

For a statistically homogeneous isotropic dimensionless field, write its two-point autocorrelation function of a random field as $C(r)$ andThe formal real-space scale invariance condition is $C(\lambda \mathbf{r})=C(\mathbf{r})$ for all $\lambda >0$. Changing variable $\mathbf{q}=\lambda \mathbf{k}$ in the first expression and equating the nonzero-mode Fourier transforms gives ${\lambda }^{-3}P(q/\lambda )=P(q)$, equivalently $P(\lambda k)={\lambda }^{-3}P(k)$. Taking $\lambda =k/k_0$ yieldsThus a scale-invariant inflationary power spectrum has equal variance per logarithmic wavenumber interval, rather than equal power per Fourier volume.

There is an important mathematical qualification to the literal covariance condition. A nonzero $k^{-3}$ power spectrum over all scales has $C(0)=A\int dk/k/(2{\pi }^2)$, divergent at both endpoints; it is not the covariance of a finite-variance ordinary field. Moreover, exact dilation invariance of a continuous isotropic $C(r)$ makes it constant for $r>0$ and, by continuity, at zero: its spectrum can then consist only of a zero-mode delta measure. The nontrivial cosmological statement consequently concerns nonzero modes, with cutoffs or subtraction of an unobservable constant. For example the finite subtracted covarianceis invariant under simultaneous rescaling of $r,r_0$. This is the precise infrared qualification of a scale-invariant covariance; a regulated unsubtracted $C(r)$ generally shifts by an additive constant under dilation. The formal derivation above gives the intended nonzero-mode scaling, with this qualification rather than an impossible finite-variance premise.