Conformal and Minimally Coupled Scalars in FLRW

Conformal coupling supplies a decisive negative control for cosmological particle-production calculations. In four-dimensional conformally flat FLRW, a free massless scalar at the site value $\xi=-1/6$ has exactly flat rescaled dynamics. Minimal coupling instead exposes the field to the potential $-a''/a$. Neither statement, by itself, selects a state.

Required background. FLRW mode quantization fixes the Wronskian, and curvature-coupled scalars fixes the site sign. Helpful background. Conformal transformations explain the field rescaling, while adiabatic states explain approximate positive-frequency data.

Curvature coupling as an effective potential

For $\chi=a\phi$,

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

The mass term and the curvature-coupling term are physically distinct. A mass produces the time-dependent contribution $a^2m^2$ even at conformal coupling. The term containing $a''/a$ vanishes at $\xi=-1/6$ even in a rapidly changing geometry.

For a massless conformally coupled field,

$$\chi_k''+k^2\chi_k=0, \qquad \chi_k=\frac{A_ke^{-ik\eta}+B_ke^{ik\eta}}{\sqrt{2k}}, \qquad |A_k|^2-|B_k|^2=1.$$

The geometry generates no mixing between the two frequency signs: $A_k$ and $B_k$ are constants. The conformal vacuum is the additional choice $A_k=1$, $B_k=0$. An already excited normalized state with $B_k\ne0$ remains excited; the Wronskian does not turn it into the conformal vacuum.

Four exact controls

The radiation and de Sitter limits isolate what follows from the equation and what follows from a state choice.

Coupling and geometry	Rescaled mode equation	Controlled conclusion
Conformal, massless; radiation	$\chi_k''+k^2\chi_k=0$	The conformal positive-frequency mode stays a plane wave.
Conformal, massless; de Sitter	$\chi_k''+k^2\chi_k=0$	The same negative control holds despite nonzero curvature.
Minimal, massless; radiation with $a''=0$	$\chi_k''+k^2\chi_k=0$	A selected radiation positive-frequency mode stays a plane wave within that era.
Minimal, massless; exact de Sitter	$\chi_k''+(k^2-2/\eta^2)\chi_k=0$	Long modes are amplified, but there is no intrinsic de Sitter out-particle number.

In exact de Sitter, $a=-1/(H\eta)$ for $\eta<0$. The mode that approaches positive frequency as $-k\eta\to\infty$ is

$$\chi_k(\eta)= \frac{e^{-ik\eta}}{\sqrt{2k}} \left(1-\frac{i}{k\eta}\right).$$

It has Wronskian $i$. For $-k\eta\ll1$,

$$|\phi_k|=\frac{|\chi_k|}{a} \longrightarrow \frac{H}{\sqrt{2k^3}}, \qquad \mathcal P_\phi(k)=\frac{k^3}{2\pi^2}|\phi_k|^2 \longrightarrow\left(\frac{H}{2\pi}\right)^2.$$

This is a late-time correlation statement for the selected short-distance state. Exact de Sitter has no stationary future region in which the same modes become ordinary out oscillators, so the frozen spectrum must not be relabeled as a unique late-time particle number.

The contrast is the heart of Parker’s result: expansion-induced mixing depends on the field equation and on declared in/out data, not on the word “expansion” alone Parker 1969, §§III–IV.

A conformal negative-control calculation

Let $a(\eta)$ be any smooth positive scale factor that is constant or adiabatic in the far past and future. Initialize the massless conformal mode by

$$\chi_k^{\rm in}=\frac{e^{-ik\eta}}{\sqrt{2k}}.$$

Because its exact equation is flat for all $\eta$, the evolved solution is the same plane wave, and projection onto the identical future basis gives

$$\alpha_k^{\rm out}=1, \qquad \beta_k^{\rm out}=0.$$

This result is independent of how quickly $a$ changes. It is therefore an unusually strong code test: a residual $\beta_k$ diagnoses an incorrect curvature sign, a discontinuous or inconsistent matching rule, a solver error, or a comparison of different state conventions.

Parker’s conformal transformation analysis gives this noncreation result directly for the conformally transported state Parker 1973.

Now repeat the calculation from general normalized data $(A_k,B_k)$. The output has the same coefficients. Thus the invariant negative-control statement is

$$B_k^{\rm out}-B_k^{\rm in}=0,$$

not “every normalized conformal mode is the vacuum.” This distinction repairs a common confusion between absence of dynamically generated mixing and absence of pre-existing excitations.

For a minimally coupled massless scalar, the same plane-wave control holds only on an interval with $a''=0$. If radiation eras are joined through a region where $a''/a\ne0$, that transition can mix the modes. Increasing the differentiability of the joining profile should soften the large-$k$ tail; a hard tail can reveal unresolved derivatives rather than physical ultraviolet production.

Particles, squeezing, and local stress

Three observables that are often grouped under “production” answer different questions.

• A Bogoliubov coefficient compares two declared complex mode bases, usually in controlled in/out regimes.
• Squeezing describes phase-sensitive covariance in a chosen mode pair and remains meaningful without an out region.
• $\langle T_{\mu\nu}\rangle_{\rm ren}$ is a local composite, defined by a state and a renormalization prescription rather than by an instantaneous number operator.

The conformal scalar makes the separation unavoidable. In the conformally transported Minkowski state, no state-dependent frequency mixing is generated. Nevertheless, renormalization can leave local curvature terms and the conformal anomaly in the stress tensor Fulling, Parker, and Hu 1974, Brown and Cassidy 1977. An anomalous stress is not a hidden population of out particles.

For minimal coupling, the $-a''/a$ potential amplifies long wavelengths in de Sitter. The spatial zero mode and infinite-volume infrared limit need separate state and regulator choices; in particular, the massless minimally coupled field has no de Sitter-invariant Fock vacuum of the usual type Allen 1985, §§III–IV. The $k\to0$ formula should not be silently promoted to a globally defined de Sitter vacuum.

Common pitfalls

Using “same Wronskian” to mean “same state.” The Wronskian normalizes every pair $(A_k,B_k)$ satisfying $|A_k|^2-|B_k|^2=1$. Positive frequency must be selected separately.

Calling every nonzero stress contribution a particle energy. Local curvature polarization and the conformal anomaly can remain when the conformal Bogoliubov coefficient is exactly zero.

Exercises

1. Separate initial excitation from created excitation

For a massless conformal scalar, begin with $\chi_k=(A_ke^{-ik\eta}+B_ke^{ik\eta})/\sqrt{2k}$. Show that an arbitrary scale factor changes neither coefficient. What is the created occupation relative to the same future basis?

Solution

At $m=0$ and $\xi=-1/6$, the exact equation is $\chi_k''+k^2\chi_k=0$ for every $a(\eta)$. Hence $A_k$ and $B_k$ are constants. The state may have initial occupation $|B_k|^2$, but the geometry adds no new mixing: the out coefficient equals the in coefficient. Relative to the same basis, the created excess is therefore

$$|B_k^{\rm out}|^2-|B_k^{\rm in}|^2=0.$$

2. Check the de Sitter spectrum

Verify the Wronskian of the displayed minimal de Sitter mode and derive its superhorizon dimensionless spectrum.

Solution

Direct differentiation gives $\chi_k\chi_k^{*'}-\chi_k'\chi_k^*=i$. When $-k\eta\ll1$,

$$\chi_k\simeq-\frac{i}{\sqrt{2k^3}\,\eta}.$$

Since $a=-1/(H\eta)$, its physical mode has magnitude $|\phi_k|\to H/\sqrt{2k^3}$. Therefore

$$\mathcal P_\phi(k)=\frac{k^3}{2\pi^2}|\phi_k|^2 \to\frac{H^2}{4\pi^2}.$$

This calculation concerns $k>0$ modes in the selected short-distance state; it does not settle the global zero-mode or infrared problem.

References

• Allen, Bruce. “Vacuum States in de Sitter Space.” Physical Review D 32 (1985): 3136–3149. DOI.
• Brown, Lowell S., and J. P. Cassidy. “Stress Tensors and Their Trace Anomalies in Conformally Flat Space-Times.” Physical Review D 16 (1977): 1712–1716. DOI.
• Fulling, Stephen A., Leonard Parker, and B. L. Hu. “Conformal Energy-Momentum Tensor in Curved Spacetime: Adiabatic Regularization and Renormalization.” Physical Review D 10 (1974): 3905–3924; erratum 11 (1975): 1714. DOI.
• Parker, Leonard. “Quantized Fields and Particle Creation in Expanding Universes. I.” Physical Review 183 (1969): 1057–1068. DOI.
• Parker, Leonard. “Conformal Energy-Momentum Tensor in Riemannian Space-Time.” Physical Review D 7 (1973): 976–983. DOI.
• Parker, Leonard, and David Toms. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press, 2009, ch. 2. Chapter DOI.