Inflationary Fluctuations as Curved-Spacetime Fields

Inflationary fluctuations are quantum fields on a time-dependent background, but the observable scalar mode is not generally a test-field fluctuation. In single-field inflation, the lapse and shift constraints combine the inflaton perturbation with scalar metric perturbations into the gauge-invariant Mukhanov–Sasaki variable. Canonical vacuum normalization at $k/a\gg H$ then fixes the amplitude whose growing solution becomes the late-time curvature spectrum.

Required background. Compactly, the mode equation is a hyperbolic initial-value problem and its Wronskian fixes the quantum commutator; the two-point function is a distribution whose coincidence limit is not used here. Partial Differential Equations and Green Functions supplies that Cauchy and distribution framework. Helpful background. Flat-Spacetime Quantum Field Theory provides oscillator quantization; Causality and Global Structure establishes the inflationary patch and Cauchy surfaces; Quantum Fields in Curved Spacetime provides state selection and ultraviolet admissibility.

Test fields and constrained perturbations

Use

$$ds^2=a^2(\eta)(-d\eta^2+d\mathbf x^2), \qquad \mathcal H=\frac{a'}a=aH,$$

with comoving Fourier wave number $k$ and physical wave number $k/a$. A test scalar perturbation on this prescribed metric has canonical variable $q=a\,\delta\phi$ and an equation of the form

$$q_k''+\left[k^2+a^2m^2+(6\xi-1)\frac{a''}{a}\right]q_k=0.$$

That is the correct problem for a spectator whose stress–energy does not perturb the metric at the retained order. It is not the complete scalar sector when the background is driven by the same inflaton being perturbed.

Write the scalar metric perturbations as

$$\begin{aligned} ds^2=a^2\bigl[& -(1+2A)d\eta^2+2\partial_iB\,d\eta\,dx^i\\ &+\{(1-2\psi)\delta_{ij} +2\partial_i\partial_jE\}dx^i dx^j\bigr]. \end{aligned}$$

Under the time shift $\eta\mapsto\eta+T$, the perturbations transform as

$$\delta\phi\mapsto\delta\phi-\phi_0'T, \qquad \psi\mapsto\psi+\mathcal H T.$$

Therefore

$$\mathcal R=\psi+\frac{\mathcal H}{\phi_0'}\delta\phi, \qquad Q=\delta\phi+\frac{\phi_0'}{\mathcal H}\psi$$

are gauge invariant, with $Q=(\phi_0'/\mathcal H)\mathcal R$. This construction and the scalar constraint structure are developed in Mukhanov, Feldman, and Brandenberger 1992, Part I, §3, p. 213 onward, and §6, pp. 234–246.

This use of comoving slicing assumes $\phi_0'\ne0$. On that slicing $\delta\phi=0$ and $\mathcal R=\psi$; on spatially flat slicing $\psi=0$ and $Q=\delta\phi$. With the spatial metric written as $a^2(1-2\psi)\delta_{ij}$, the variable often defined by $g_{ij}=a^2e^{2\zeta}\delta_{ij}$ obeys $\zeta=-\mathcal R$ on comoving slices at linear order. Their power spectra agree, but their signs in nonlinear relations must be translated.

The Mukhanov–Sasaki action

For Einstein gravity with one canonical background scalar, solve the linearized lapse and shift constraints and define

$$v=aQ=z\mathcal R, \qquad z=\frac{a\phi_0'}{\mathcal H} =\frac{a\dot\phi_0}{H}.$$

Using the reduced Planck mass $M_{\mathrm{Pl}}^{-2}=8\pi G$, the quadratic scalar action reduces to

$$S^{(2)} =\frac12\int d\eta\,d^3x \left[(v')^2-(\nabla v)^2+\frac{z''}{z}v^2\right].$$

Varying $v$ and Fourier transforming gives

$$v_k''+\left(k^2-\frac{z''}{z}\right)v_k=0, \qquad v_kv_k^{*\prime}-v_k^*v_k'=i.$$

The derivation explains both the canonical normalization and the effective time-dependent potential. Replacing $v$ by $a\,\delta\phi$ without specifying a gauge discards the gravitational constraints and can change the claimed spectrum.

For tensors write $g_{ij}=a^2(\delta_{ij}+h_{ij})$ with $\partial_i h_{ij}=h_{ii}=0$ and $h_{ij}=\sum_{\lambda=1}^2e_{ij}^{\lambda}h_\lambda$. Choose $e_{ij}^{\lambda}e_{ij}^{\lambda' *}=\delta_{\lambda\lambda'}$. The quadratic tensor action is $M_{\mathrm{Pl}}^2\int a^2[(h_{ij}')^2-(\partial_\ell h_{ij})^2]/8$, so the exact canonical variable for each polarization is $u_\lambda=aM_{\mathrm{Pl}}h_\lambda/2$. It satisfies

$$u_{\lambda,k}''+\left(k^2-\frac{a''}{a}\right)u_{\lambda,k}=0.$$

Scalar and tensor perturbations thus share the curved-spacetime oscillator structure, while their pump fields $z$ and $a$ differ.

Short-wavelength state normalization

When

$$\frac{k}{aH}=\frac{k}{\mathcal H}\gg1,$$

and the background varies slowly compared with the physical frequency, the $k^2$ term dominates. The adiabatic positive-frequency condition is

$$v_k(\eta)\longrightarrow\frac{e^{-ik\eta}}{\sqrt{2k}}.$$

This boundary condition fixes the Wronskian and the amplitude of the growing solution. It is often called the Bunch–Davies condition in an approximately de Sitter past, but the name should not conceal its hypotheses: the patch must extend far enough into an adiabatic subhorizon regime, and an alternative initial state changes the mode by normalized Bogoliubov coefficients.

For nearly constant slow-variation parameters

$$\epsilon=-\frac{\dot H}{H^2}, \qquad \delta=\frac{\ddot\phi_0}{H\dot\phi_0},$$

$z''/z$ is approximately proportional to $\eta^{-2}$, so the normalized solution is a Hankel function. The connection from quantum normalization to the growing superhorizon mode is derived in Mukhanov, Feldman, and Brandenberger 1992, Part II, §9, p. 257, §10, pp. 260–271, and §11, pp. 272–276.

From modes to power spectra

Use $\mathcal R(\mathbf x)=\int d^3k\,\mathcal R_{\mathbf k}e^{i\mathbf k\cdot\mathbf x}/(2\pi)^3$ and define the dimensionless curvature spectrum by

$$\langle\mathcal R_{\mathbf k}\mathcal R_{\mathbf k'}\rangle =(2\pi)^3\delta^3(\mathbf k+\mathbf k') \frac{2\pi^2}{k^3}\mathcal P_{\mathcal R}(k).$$

Since $\mathcal R_k=v_k/z$,

$$\mathcal P_{\mathcal R}(k) =\frac{k^3}{2\pi^2}\left|\frac{v_k}{z}\right|^2.$$

To obtain the amplitude, use $z^2=2a^2\epsilon M_{\mathrm{Pl}}^2$ from the background Einstein equation and, at leading slow-variation order,

$$v_k\simeq\frac{e^{-ik\eta}}{\sqrt{2k}} \left(1-\frac{i}{k\eta}\right), \qquad a\simeq-\frac{1}{H\eta}.$$

In the late-time limit $|k\eta|\ll1$, $|v_k/z|^2\simeq H^2/(4\epsilon M_{\mathrm{Pl}}^2k^3)$. Thus the frozen spectrum, with the slowly varying background parameters evaluated near $k=aH$, is

$$\mathcal P_{\mathcal R}(k) \simeq\frac{H_\ast^2} {8\pi^2\epsilon_\ast M_{\mathrm{Pl}}^2}.$$

With the polarization convention above, define $\mathcal P_T=\sum_\lambda k^3|h_{\lambda,k}|^2/(2\pi^2)$. The same oscillator solution and $h_\lambda=2u_\lambda/(aM_{\mathrm{Pl}})$ give

$$\mathcal P_T(k) \simeq\frac{2H_\ast^2}{\pi^2M_{\mathrm{Pl}}^2}.$$

The star denotes where background parameters are evaluated; the mode amplitude in these formulas is its frozen value. Substituting $|k\eta|=1$ directly into the leading mode above would retain a transient term and give twice that amplitude squared. Horizon crossing is a matching convention, not particle creation at a particular instant. The scalar and tensor spectra and their gauge-invariant interpretation are presented in Mukhanov, Feldman, and Brandenberger 1992, §§12–13, pp. 277–291.

Superhorizon conservation and its limits

From $v=z\mathcal R$, the exact equation can be written

$$(z^2\mathcal R_k')'+k^2z^2\mathcal R_k=0.$$

For $k/\mathcal H\to0$,

$$\mathcal R_k=C_1+C_2\int^\eta\frac{d\eta'}{z^2(\eta')}.$$

In an attractor single-field background the second mode decays, so $\mathcal R$ is conserved. This check explains why a late-time spectrum can be read from the growing mode. Conservation need not hold for entropy perturbations, nonattractor evolution, singular matching, or additional fields.

For example, in an approximately de Sitter phase with a nearly flat scalar potential, $\ddot\phi_0+3H\dot\phi_0\simeq0$ gives $\dot\phi_0\propto a^{-3}$ and $z\propto a^{-2}\propto\eta^2$. Then $\int d\eta/z^2\propto\eta^{-3}\propto a^3$: the second solution grows despite the presence of only one field. Conversely, the strict $\epsilon=0$, $\dot\phi_0=0$ limit has no scalar clock, so the apparent $1/\epsilon$ divergence of the slow-roll formula is not a valid de Sitter curvature-spectrum prediction.

The treatment is linear and fixed-background apart from the constrained first-order metric perturbations. It does not include the stress–energy backreaction of produced quanta, nonlinear non-Gaussian evolution, reheating transfer functions, or observational parameter inference. Those require additional dynamics rather than another choice of Fourier normalization.

Diagnostic: identify the physical scalar

Under a time shift, suppose $\delta\phi\mapsto\delta\phi-\phi_0'T$ and $\psi\mapsto\psi+\mathcal HT$. Show that $\mathcal R=\psi+\mathcal H\delta\phi/\phi_0'$ is invariant. Why is quantizing $a\,\delta\phi$ alone generally insufficient for the inflaton perturbation?

Solution

The transformed combination is $\psi+\mathcal HT+\mathcal H(\delta\phi-\phi_0'T)/\phi_0'=\mathcal R$, so the two gauge shifts cancel exactly. For a spectator test scalar, $a\,\delta\phi$ is a legitimate canonical field on a prescribed geometry. For the inflaton, however, scalar metric perturbations are constrained by the same matter fluctuation; $\delta\phi$ alone changes under a time slicing. Solving the lapse and shift constraints yields the canonical gauge-invariant variable $v=z\mathcal R$. Quantizing a gauge-dependent test-field variable can therefore assign physical significance to a coordinate perturbation and produce the wrong pump term.

References

• Mukhanov, Viatcheslav F., H. A. Feldman, and Robert H. Brandenberger. “Theory of Cosmological Perturbations.” Physics Reports 215 (1992): 203–333. DOI.