Mode Equations on FLRW Spacetimes

Expansion turns each Fourier mode of a free field into an oscillator whose frequency and damping vary with time. The conversion is exact, not an approximation: it follows by writing the Klein–Gordon or Dirac equation on a prescribed FLRW geometry and choosing a canonical field variable. What remains nonunique is the state—equivalently, the normalized positive-frequency solution selected as initial data.

Required background. Compactly, a second-order mode equation needs two initial data and its conserved Wronskian fixes their symplectic normalization; distributions assemble the modes into the field two-point function. Partial Differential Equations and Green Functions supplies that initial-value, distribution, and Green-function framework. Helpful background. Flat-Spacetime Quantum Field Theory provides oscillator quantization and Fourier modes; Causality and Global Structure establishes the Cauchy surfaces on which initial data are posed; Quantum Fields in Curved Spacetime provides the state and fixed-background setting.

FLRW geometry in two time coordinates

Take a four-dimensional, spatially flat FLRW spacetime and normalize the scale factor by $a(t_\ast)=1$ at a declared reference time:

$$ds^2=-dt^2+a^2(t)d\mathbf x^2 =a^2(\eta)\left(-d\eta^2+d\mathbf x^2\right), \qquad d\eta=\frac{dt}{a(t)}.$$

Cosmic-time and conformal-time derivatives are denoted by overdots and primes. Their expansion rates are

$$H=\frac{\dot a}{a}, \qquad \mathcal H=\frac{a'}a=aH, \qquad R=6(\dot H+2H^2)=\frac{6a''}{a^3}.$$

The spatial coordinates $\mathbf x$ and their Fourier labels $\mathbf k$ are comoving. A comoving observer measures physical momentum $\mathbf p=\mathbf k/a$ and physical wavelength $2\pi a/k$. Consequently, “large $k$” is not by itself a short-distance statement at all times; the dimensionless comparison is $k/(aH)$ or $k/(am)$.

For nonzero spatial curvature, plane waves are replaced by eigenfunctions of the spatial Laplacian and $k^2$ by the corresponding eigenvalue, with curvature-dependent spectral measure. We keep the flat case so that no discrete-spectrum or topology qualification is hidden.

Scalar reduction to a canonical oscillator

For the action and signature used throughout this volume, the real scalar obeys

$$(\Box-m^2-\xi R)\phi=0.$$

In cosmic time this becomes

$$\ddot\phi+3H\dot\phi-\frac{1}{a^2}\nabla^2\phi +(m^2+\xi R)\phi=0.$$

Expand the field in comoving Fourier modes,

$$\phi(\eta,\mathbf x)=\frac1a\int\frac{d^3k}{(2\pi)^{3/2}} \left[a_{\mathbf k}\chi_k(\eta)e^{i\mathbf k\cdot\mathbf x} +a_{\mathbf k}^\dagger\chi_k^*(\eta)e^{-i\mathbf k\cdot\mathbf x}\right].$$

Before rescaling, the mode function $\phi_k=\chi_k/a$ satisfies

$$\phi_k''+2\mathcal H\phi_k' +\left(k^2+a^2m^2+a^2\xi R\right)\phi_k=0.$$

Substituting $\chi_k=a\phi_k$ cancels the first derivative and gives the canonical oscillator

$$\chi_k''+\omega_k^2(\eta)\chi_k=0, \qquad \omega_k^2=k^2+a^2m^2+(6\xi-1)\frac{a''}{a}.$$

This is how expansion enters: the physical mass becomes $am$ in conformal time, while nonconformal curvature coupling contributes $(6\xi-1)a''/a$. The derivation is consistent with the canonical treatment of perturbations in Mukhanov, Feldman, and Brandenberger 1992, Part I, §3, p. 213 onward.

Wronskian normalization and state data

The equal-time commutator $[\phi(\eta,\mathbf x),a^2\phi'(\eta,\mathbf y)]=i\delta^3(\mathbf x-\mathbf y)$ is equivalent to

$$\chi_k\chi_k^{*\prime}-\chi_k^*\chi_k'=i.$$

The mode equation conserves this Wronskian because $\omega_k^2$ is real, even when it is negative and the solutions cease to oscillate. A measurable family of such data, depending only on $k=|\mathbf k|$ and with sufficient ultraviolet and infrared control to define the two-point distribution, selects a pure homogeneous, isotropic quasifree state. General mixed quasifree states need occupation data as well. The differential equation alone does not select “the vacuum.” A different normalized solution

$$\widetilde\chi_k=\alpha_k\chi_k+\beta_k\chi_k^*, \qquad |\alpha_k|^2-|\beta_k|^2=1,$$

defines a different mode basis and generally a different state. Ultraviolet admissibility constrains the large-$k$ behavior of $\beta_k$, but it does not remove all state freedom.

Spinors and conformal rescaling

Choose a conformal tetrad and the Clifford convention $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbf1$ used on Dirac and Maxwell Fields. The Dirac equation is then $(\gamma^\mu\nabla_\mu-m)\psi=0$: there is no extra $i$ with these mostly-plus Clifford matrices. The spin connection produces a term proportional to $3\mathcal H/2$. Rescaling

$$\Psi=a^{3/2}\psi$$

removes it and yields a flat-form equation with a time-dependent mass,

$$\left(\gamma^{\hat0}\partial_\eta+\gamma^{\hat i}\partial_i-am\right)\Psi=0.$$

Hatted matrices are constant tetrad matrices. To compare with the common particle-physics equation $i\Gamma^a\partial_a-am$, use $\Gamma^a=-i\gamma^{\hat a}$, so $\{\Gamma^a,\Gamma^b\}=-2\eta^{ab}\mathbf1$. With spatial modes $e^{i\mathbf k\cdot\mathbf x}$ and a fixed helicity, absorb the helicity sign in the lower component. The two amplitudes obey

$$i\frac{d}{d\eta} \begin{pmatrix}h_k^I\\ h_k^{II}\end{pmatrix} =\begin{pmatrix}am&k\\ k&-am\end{pmatrix} \begin{pmatrix}h_k^I\\ h_k^{II}\end{pmatrix}, \qquad |h_k^I|^2+|h_k^{II}|^2=1.$$

The Hermitian matrix conserves this positive norm. Eliminating the opposite component gives $h_k^{I\prime\prime}+[k^2+a^2m^2+i(am)']h_k^I=0$ and the corresponding equation with $-i(am)'$ for $h_k^{II}$. The first-order equations are still needed: the separate second-order equations admit pairs that violate the original coupling or normalization. This is the conformal-time translation of Landete, Navarro-Salas, and Torrenti 2014, §III, Eqs. (19)–(26), PDF. For $m=0$, the rescaled spinor is exactly a flat-space field on the conformally flat patch.

Limiting and dimensional checks

If $a$ is constant, then $a''=0$ and the scalar frequency reduces to $k^2+a^2m^2$, the conformal-time form of $p^2+m^2$. If $m=0$ and $\xi=1/6$, then $\omega_k=k$ for every scale factor. A state obtained by conformally transporting the Minkowski vacuum has plane-wave modes and no expansion-induced Bogoliubov mixing. This no-creation statement requires both a conformally invariant field equation and the corresponding conformal state; an excited state, nonconformal boundary condition, nontrivial topology, mass, or different coupling can invalidate it.

Under the harmless coordinate relabeling $a\mapsto Ca$, $\mathbf x\mapsto\mathbf x/C$, conformal time changes as $\eta\mapsto\eta/C$ and the comoving label changes as $\mathbf k\mapsto C\mathbf k$. The physical momentum $\mathbf k/a$ and the ratio $k/\mathcal H$ are invariant. Every term in $\omega_k^2$ scales as an inverse conformal time squared. These checks catch the common error of calling $k$ a physical momentum or mixing $H$ with $\mathcal H$.

Diagnostic: translate a horizon-crossing condition

A mode is said to satisfy $k=\mathcal H$ at conformal time $\eta_\times$. Express the same condition in cosmic-time and physical variables. Then decide whether the massless minimally coupled scalar has the conformal no-creation property.

Solution

Because $\mathcal H=aH$ and $p=k/a$, the equality $k=\mathcal H$ is exactly $p=H$ at the same event; it compares a physical wave number with the cosmic expansion rate. The label $k$ itself remains comoving. For a massless minimally coupled scalar, $m=0$ and $\xi=0$, so $\omega_k^2=k^2-a''/a$ rather than $k^2$. Expansion therefore produces nontrivial mode evolution. Only the massless conformally coupled scalar, $m=0$ and $\xi=1/6$, has the exact flat oscillator after the $\chi=a\phi$ rescaling, and even then the no-creation claim assumes the conformally transported state and compatible global conditions.

References

• Landete, Aitor, José Navarro-Salas, and Francisco Torrenti. “Adiabatic Regularization and Particle Creation for Spin One-Half Fields.” Physical Review D 89 (2014): 044030. DOI; Open PDF.
• Mukhanov, Viatcheslav F., H. A. Feldman, and Robert H. Brandenberger. “Theory of Cosmological Perturbations.” Physics Reports 215 (1992): 203–333. DOI.
• Parker, Leonard, and Stephen A. Fulling. “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces.” Physical Review D 9 (1974): 341–354. DOI.
• Parker, Leonard. “Quantized Fields and Particle Creation in Expanding Universes. I.” Physical Review 183 (1969): 1057–1068. DOI.