Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 6

Holding the background metric fixed, the linearized homogeneous-field equation isConsequently each comoving Fourier coefficient obeysThe unscaled Laplacian in the starting equation must be interpreted as the physical spatial Laplacian; for the explicitly comoving wavevector it gives $k^2/a^2$, not $k^2$. Metric fluctuations have been discarded as permitted. The mode written later is a massless, minimally coupled approximation, requiring $|m_{\mathrm{eff}}^2|\ll H^2$; it is not a solution for an arbitrary potential curvature.

Take a periodic comoving box of volume $L^3$, with $\mathbf{k}=2\pi \mathbf{n}/L$. A real field has the expansionwhere $[{\hat{a}}_{\mathbf{k}},{\hat{a}}_{\mathbf{q}}^{†}]={\delta }_{\mathbf{kq}}$ and the other ladder-operator commutators vanish. The coefficient of $e^{i\mathbf{k}\cdot \mathbf{x}}$ is therefore $w_k{\hat{a}}_{\mathbf{k}}+w_k^{\ast }{\hat{a}}_{-\mathbf{k}}^{†}$. This opposite momentum is required by reality; the single-oscillator shorthand with the same label on both operators must not be read literally as every traveling-wave Fourier coefficient.

The quadratic action has canonical momentum $\hat{\pi }=a^3\delta \dot{\hat{\varphi }}$. Its equal-time field commutator requiresChoose the vacuum annihilated by all $a_{\mathbf{k}}$ and the subhorizon positive-frequency condition. For constant $H$ and negligible effective mass, put $x=k/(aH)$, $C_k=L^{-3/2}H/\sqrt{2k^3}$. The candidate is $w_k=C_k(i+x)e^{ix}$. Since $\dot{x}=-Hx$,Direct substitution makes ${\ddot{w}}_k+3H{\dot{w}}_k+(k^2/a^2)w_k$ identically zero. Its Wronskian is $2iH{C}_k^2{x}^3=i/(L^3{a}^3)$, which has the required normalization. ThusThis is the existing Bunch-Davies mode normalization in a finite comoving volume. It is exact for a massless minimally coupled test field in de Sitter and the leading approximation for a light slowly evolving field. The homogeneous zero mode needs separate treatment.

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.