Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 3

With an approximately constant Hubble parameter $H$, the scale factor is $a(t)=e^{Ht}$ after a choice of normalization. Integrate $d\tau =dt/a(t)$ and choose the integration constant so that the future endpoint is $\tau =0$:Substituting $dt=ad\tau$ in the FLRW metric gives $d{s}^2=a^2(\tau )(-d{\tau }^2+d{\mathbf{x}}^2)$. This is Conformal time during de Sitter expansion.

For the FLRW metric in conformal time, $\sqrt{-g}=a^4$, $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}{\delta }^{ij}$. The equation for a minimally coupled massless scalar field is consequentlyFor a Fourier mode of comoving wavevector $\mathbf{k}$, the Laplacian contributes $-k^2$, where $k=|\mathbf{k}|$. ThusSince $a^{\prime }/a=-1/\tau$, this yieldsThe absence of a curvature-coupling term is material: a conformally coupled scalar field obeys a different mode equation.

Denote the comoving box volume by $\mathcal{V}$, keeping it distinct from the inflaton potential $V(\varphi )$. For the Canonically rescaled de Sitter scalar mode $u_k=a\delta {\varphi }_k$, the equation in part (b) becomesDirect differentiation verifies the positive-frequency solutionFor $|k\tau |\gg 1$, $u_k$ tends to $e^{-ik\tau }/\sqrt{2k\mathcal{V}}$, the positive-frequency flat-space mode. It therefore selects the incoming Bunch-Davies vacuum. The Bunch-Davies mode normalization in a finite comoving volume is fixed by the Wronskian conditionAfter cosmological horizon exit, $|k\tau |\ll 1$, the mode tends to the time-independent value $iH/\sqrt{2k^3\mathcal{V}}$. The complex mode and its Hermitian conjugate together make the real quantum field.

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.