Cosmological Quantum Fields and Particle Creation

Quantum fields on an expanding universe combine an exactly solvable kinematic step with genuinely delicate choices. Fourier decomposition converts a free field into time-dependent oscillators, but the geometry does not by itself select their positive-frequency modes. Ultraviolet admissibility, asymptotic particle interpretation, de Sitter symmetry, inflationary gauge invariance, and infrared resummation answer different questions. This chapter develops those questions on prescribed FLRW and de Sitter backgrounds, stopping before quantum expectation values are allowed to change the metric.

Helpful background. Flat-Spacetime Quantum Field Theory provides canonical oscillator quantization and loop counting; Partial Differential Equations and Green Functions supplies Cauchy evolution, propagators, and distributions; Causality and Global Structure establishes the relevant patches, Cauchy surfaces, and cosmological horizons; Quantum Fields in Curved Spacetime provides quasifree states, Hadamard admissibility, and renormalized observables.

One geometry, three physical scales

The common baseline is a four-dimensional spatially flat FLRW metric,

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

The scale factor is normalized by $a(t_\ast)=1$ at a declared reference time. Cosmic and conformal expansion rates are

$$H=\frac{\dot a}{a}, \qquad \mathcal H=\frac{a'}a=aH.$$

The Fourier label $k$ is comoving; a comoving observer measures physical wave number $p=k/a$. Thus

$$\frac{k}{\mathcal H}=\frac{p}{H}.$$

This identity prevents two recurrent errors: calling $k$ a measured momentum and comparing $k$ directly with the cosmic-time rate $H$. The field mass $m$, expansion scale $H$, and physical momentum $p$ form the consequential hierarchy. State, coupling, and approximation statements are meaningful only after the hierarchy has been declared.

The six-stage route

The exposition follows the same normalized modes from their classical equation to the frontier of late-time perturbation theory.

1. Mode Equations on FLRW Spacetimes derives the scalar and spinor reductions, conserved Wronskians, and the distinction between comoving $k$ and physical $k/a$.
2. Adiabatic Vacua and Ultraviolet Regularity constructs finite-order WKB state data, identifies observable-dependent subtraction orders, and separates finite Sobolev control from the full Hadamard condition.
3. Cosmological Particle Creation evolves an in-basis through nonadiabatic expansion, extracts its out-basis Bogoliubov coefficient, and distinguishes the asymptotic particle number from basis-dependent instantaneous occupation.
4. Quantum Fields in de Sitter Space combines invariant two-point functions, Hadamard short-distance behavior, static-patch KMS thermality, and the massless minimally coupled obstruction.
5. Inflationary Fluctuations as Curved-Spacetime Fields replaces a gauge-dependent inflaton fluctuation by the canonical Mukhanov–Sasaki variable and derives scalar and tensor spectra.
6. Infrared Growth and Secular Effects asks which late-time logarithms survive changes of state, gauge, regulator, observable, mass, coupling, loop order, and resummation.

The order encodes dependencies. A particle coefficient needs normalized modes; a subtraction scheme needs their ultraviolet expansion; a de Sitter vacuum claim needs both symmetry and short-distance admissibility; an inflationary spectrum needs the constrained gauge-invariant variable; a resummation needs a precisely named observable whose fixed-order series it reproduces.

State selection and ultraviolet admissibility

For a scalar with

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

the canonical rescaling $\chi_k=a\phi_k$ gives

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

The Wronskian fixes normalization but leaves a normalized Bogoliubov family of pure states; mixed quasifree states also require occupation data. A finite-order WKB construction constrains their large-$k$ behavior. In four dimensions, second-order subtraction removes the ultraviolet divergences of $\langle\phi^2\rangle$, while the stress tensor requires terms through fourth order. Existence of the remaining integral also requires adequate state regularity and infrared control. This subtraction does not make a fourth-order state fully Hadamard. Junker and Schrohe give the precise Sobolev hierarchy in Junker and Schrohe 2002, Definition 3.2, Lemma 3.3, and Theorem 6.3, PDF.

The logical distinction is important. Hadamard form is a local ultraviolet admissibility condition, not a unique vacuum-selection rule. Symmetry, asymptotic positive frequency, Euclidean regularity, low-energy minimization, or an initial density operator can further select a state, but each adds hypotheses. Infrared modifications can leave the ultraviolet singularity unchanged while substantially altering long-distance correlations.

Particles, fields, and conformal checks

If the geometry approaches stationary in and out regimes, their positive-frequency bases are related by

$$u_{k,\mathrm{in}} =\alpha_k u_{k,\mathrm{out}} +\beta_k u_{k,\mathrm{out}}^*, \qquad |\alpha_k|^2-|\beta_k|^2=1.$$

The in-vacuum contains $|\beta_k|^2$ out-quanta in that mode. Between the asymptotic eras, an instantaneous adiabatic particle number changes with WKB order and should be labeled accordingly. Parker’s foundational expanding-universe calculation establishes the asymptotic mixing in Parker 1969, complete article, pp. 1057–1068.

The strongest normalization and sign check is the massless conformal limit. For a scalar with $m=0$, $\xi=1/6$, the rescaled frequency is exactly $k$. In the conformally transported Minkowski state, there is no expansion-induced mixing. This conclusion requires the conformally invariant equation, the corresponding state, and compatible global conditions. It does not extend to the massless minimally coupled scalar, whose frequency contains $-a''/a$. Nor does zero particle creation imply zero renormalized stress: curvature-dependent vacuum polarization and the trace anomaly can remain.

Particle number is not interchangeable with a local detector response or renormalized stress tensor. The latter can be meaningful when no out region exists, but it requires its own state, switching or subtraction prescription, and physical observer.

De Sitter selection and its counterexample

For four-dimensional de Sitter with $R=12H^2$, the scalar depends on

$$M^2=m^2+12\xi H^2, \qquad \nu^2=\frac94-\frac{M^2}{H^2}.$$

For admissible positive $M^2$, Euclidean regularity, de Sitter invariance, positivity, and Hadamard short-distance form select the Bunch–Davies quasifree state. Its restriction to a static patch is KMS at $H/(2\pi)$ for the static Killing flow normalized at the central geodesic. Generic alpha-states can remain formally invariant but acquire extra antipodal singularities and fail the Hadamard condition.

Dropping the positive-effective-mass hypothesis gives a discriminating counterexample. The massless minimally coupled scalar has a global zero-mode and an infrared logarithm, so it has no de Sitter-invariant Fock vacuum of the asserted ordinary kind. Allen establishes this obstruction in Allen 1985, complete article, pp. 3136–3149. One can use a noninvariant state, a shift-invariant observable algebra, or separate zero-mode quantization, but these are different statements rather than repairs to the original theorem.

Inflationary variables and spectra

For a spectator, $a\,\delta\phi$ is a legitimate test-field variable. For the inflaton that drives the background, $\delta\phi$ changes under a time-slicing transformation and couples to constrained scalar metric perturbations. Solving the lapse and shift constraints yields

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

with quadratic action

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

The short-wavelength condition $v_k\sim e^{-ik\eta}/\sqrt{2k}$ fixes the quantum normalization, and the growing superhorizon solution fixes the two-point amplitude. The gauge-invariant construction, variational reduction, and scalar and tensor spectra are developed in Mukhanov, Feldman, and Brandenberger 1992, Part I, §3, p. 213 onward, §6, pp. 234–246, and Part II, §§9–13, pp. 257–291.

This result assumes a prescribed inflationary background and linearized perturbations. Backreaction, nonlinear mode coupling, reheating transfer, and observational inference require additional dynamics. Even superhorizon conservation of $\mathcal R$ assumes an adiabatic single-clock attractor; entropy modes and nonattractor phases provide counterexamples.

Late-time claims and decisive checks

Infrared enhancement must specify the field, state, patch, mass, coupling, and regulator. Secular growth must additionally specify the observable, gauge, loop order, initial time, and time range. Under a broad class of interactions, late-time loop effects are bounded by powers of $\ln a$ rather than positive powers of $a$ Weinberg 2005, §II and §§IV–VII. This bound does not prove that every logarithm survives, remains perturbatively small, or represents a relational observable.

A proposed resummation must reproduce the early fixed-order coefficients, preserve the relevant constraints, and pass adverse limits. In the free light-field de Sitter benchmark, a small positive effective mass saturates the growing long-wavelength variance; conformal coupling removes that enhancement; a gauge change can remove a coordinate artifact; a different initial state can expose boundary evolution. These are discriminating checks within declared regimes, not a theorem that a mass regulates every interacting infrared problem.

References

• Allen, Bruce. “Vacuum States in de Sitter Space.” Physical Review D 32 (1985): 3136–3149. DOI.
• Junker, Wolfgang, and Elmar Schrohe. “Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties.” Annales Henri Poincaré 3 (2002): 1113–1181. DOI. Open PDF, arXiv:math-ph/0109010.
• Mukhanov, Viatcheslav F., H. A. Feldman, and Robert H. Brandenberger. “Theory of Cosmological Perturbations.” Physics Reports 215 (1992): 203–333. DOI.
• Parker, Leonard. “Quantized Fields and Particle Creation in Expanding Universes. I.” Physical Review 183 (1969): 1057–1068. DOI.
• Weinberg, Steven. “Quantum Contributions to Cosmological Correlations.” Physical Review D 72 (2005): 043514. DOI.

Integrative diagnostic

A student analyzes a massless scalar during inflation and makes five claims: $k$ is its physical momentum; fourth-order adiabatic data prove the state is Hadamard; masslessness guarantees no creation; $a\,\delta\phi$ is the scalar metric spectrum in any gauge; and a one-loop $\ln^2 a$ term proves a physical divergence. Correct the chain of reasoning and name the checks that would support defensible conclusions.

Solution

The physical momentum is $k/a$, while $k$ is comoving. Fourth adiabatic order can support stress-tensor subtraction but is finite Sobolev control, not by itself the full Hadamard condition. No creation follows only for a conformally invariant field in the conformally transported state; the massless minimally coupled scalar is the counterexample. For an inflaton, the canonical scalar is $v=z\mathcal R$ after solving the metric constraints, not gauge-dependent $a\,\delta\phi$. Finally, an order-one secular logarithm establishes breakdown of that fixed loop truncation. One must identify a renormalized gauge-invariant or relational observable, declare state, regulator, mass, coupling, and time range, reproduce the coefficient in an independent gauge or formulation, and validate any resummation against early-time perturbation theory and conformal or massive adverse limits.