Conformal and Minimally Coupled Scalars in FLRW

Mass, curvature coupling, and the scale-factor history enter a scalar mode through different terms. In four-dimensional conformally flat FLRW, a massless scalar at the site value $\xi=-1/6$ has flat rescaled modes; a minimally coupled scalar generally feels $-a''/a$.

Required background. FLRW mode quantization fixes normalization, and curvature-coupled scalars fixes the site sign. Helpful background. Review conformal transformations and adiabatic states.

Two scalar limits

For $\chi=a\phi$,

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

At $m=0,\xi=-1/6$,

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

defines the conformal vacuum and no Bogoliubov mixing is generated by $a(\eta)$. This does not imply zero renormalized stress: the trace anomaly and local curvature polarization remain.

At $m=0,\xi=0$,

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

During radiation domination $a\propto\eta$, so $a''=0$ and the rescaled equation is again flat. In exact de Sitter $a=-1/(H\eta)$,

$$\chi_k''+\left(k^2-\frac{2}{\eta^2}\right)\chi_k=0, \qquad \chi_k=\frac{e^{-ik\eta}}{\sqrt{2k}} \left(1-\frac{i}{k\eta}\right)$$

for the short-distance positive-frequency choice. The growing $|\phi_k|=|\chi_k|/a$ freezes on superhorizon scales. Exact de Sitter has no adiabatic out region, so this behavior is a correlator statement rather than a unique late-time particle number.

First application: radiation and de Sitter checks

Evolve four cases with the same Wronskian: conformal/massless and minimal/massless fields in radiation-like and de Sitter-like expansion. The conformal modes remain plane waves in both. Minimal modes are plane waves in radiation but are squeezed by $a''/a$ in de Sitter. Turning on $m$ adds $a^2m^2$ even at conformal coupling, so expansion can then mix frequencies.

These limiting results reproduce Parker’s original conclusion that expansion-induced creation depends on the field equation and asymptotic definition, not on expansion alone Parker 1969, §§III–IV, pp. 1062–1067.

Correlators, particles, and local stress

The three conclusions often attached to the word “production” must be tested separately. First, Bogoliubov mixing compares two complex mode bases and requires suitable in/out regimes. Second, squeezing is a statement about the covariance of a chosen mode pair and can be meaningful even when no out region exists. Third, $\langle T_{\mu\nu}\rangle_{\rm ren}$ is a local composite obtained after state-independent subtraction; it is not the energy of an instantaneous particle distribution.

The conformal scalar provides a clean separation. In a conformally transported Minkowski state, its rescaled two-point function is the flat kernel, so there is no state-dependent mixing. Yet transforming and renormalizing the stress introduces local geometric terms, including the anomaly. For a minimally coupled field, the $-a''/a$ potential can amplify long wavelengths, but the $k=0$ sector and the infinite-volume infrared limit require their own state and regulator. A frozen superhorizon amplitude does not by itself define a conserved number of particles.

A useful calculation check is to take a scale factor that begins and ends in radiation-like eras with a smooth intervening pulse. The conformal massless mode must return exactly to a plane wave for every pulse shape. The minimal mode may acquire nonzero late $\beta_k$, but its ultraviolet tail must fall faster as the differentiability of the pulse is increased. Failure of the first control indicates a sign, matching, or normalization error; failure of the second indicates unresolved background derivatives.

The structure map locates coupling and spin before the particle-diagnostic branch. Inspect the conformally invariant route that bypasses state-dependent production but not local renormalization.

Curvature coupling and mass determine whether FLRW expansion appears in the canonical mode frequency; particle and stress conclusions remain separate. Schematic; not to scale.

Domain and failure conditions

See the chapter’s canonical domain table. The no-production statement assumes a free massless conformally coupled scalar, a conformal state, and conformally related complete mode regions.

Adversarial test. Set $m=0$ and $\xi=-1/6$ in one fixed geometry. Any residual Bogoliubov coefficient from the rescaled equation must come from a nonconformal initial state, a discontinuous matching rule, or a basis convention. A nonzero anomaly stress is not residual particle creation; it is a local renormalized composite observable.

The failure map forces that distinction before interpreting a spectrum.

The conformal limit is a decisive negative control: mode mixing vanishes for the conformal state, but curvature-dependent renormalized stress need not. Schematic; not to scale.

References

• Parker, L., “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183, 1057–1068 (1969), doi:10.1103/PhysRev.183.1057.