Quantum Fields in Cosmology

Quantum fields in an expanding universe are time-dependent oscillators constrained by covariance, canonical normalization, state admissibility, and local renormalization. A mode basis can define a useful particle diagnostic without defining a unique vacuum; a finite power spectrum does not replace a renormalized stress tensor; and a relic abundance is predictive only after the expansion history, matching, dilution, and backreaction are controlled.

Helpful background. Particle creation in time-dependent backgrounds supplies Bogoliubov evolution; adiabatic states supplies ultraviolet control; stress-tensor subtraction schemes supplies local renormalization; and cosmological backreaction benchmarks supplies self-consistency tests.

The FLRW calculation contract

On spatially flat FLRW,

$$ds^2=a^2(\eta)(d\eta^2-d\mathbf x^2), \qquad \mathcal H=\frac{a'}a, \qquad R_{\rm site}=-6\frac{a''}{a^3}.$$

The last sign follows the site’s Riemann convention. For $P=\Box+m^2+\xi R$, set $\chi=a\phi$ in four dimensions. Its normalized modes satisfy

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

Thus the site conformal coupling is $\xi=-1/6$; for $m=0$ the curvature potential vanishes. These formulas fix the mode dynamics and symplectic normalization. They do not select positive frequency at every time.

The separation between canonical mode evolution and asymptotic particle interpretation is developed explicitly by Parker 1969, §§II–IV, pp. 1059–1067. The observable-specific adiabatic construction of a conserved homogeneous stress tensor is given by Parker and Fulling 1974, §§II–IV, pp. 344–352.

Every calculation in the chapter should name:

• the scale factor and differentiability class;
• field spin, mass, curvature or background coupling, and canonical variable;
• initial state or density matrix and its ultraviolet order;
• observable—particle diagnostic, correlator, local composite, abundance, or mean stress;
• subtraction and finite local gravitational couplings;
• matching history, physical cutoff hierarchy, and numerical error;
• the first hierarchy that is no longer controlled.

The structure map should be read from modes and state through an explicitly chosen observable, then through subtraction or late-time transfer, and only afterward into relic or backreaction conclusions.

Cosmological QFT separates mode evolution, state choice, basis-dependent particles, local renormalization, relic transfer, and mean backreaction. Schematic; not to scale.

Routes through the chapter

1. FLRW fields and mode quantization derives the scalar oscillator and Wronskian.
2. Conformal and minimally coupled scalars isolates curvature coupling and expansion history.
3. Spinor and gauge fields preserves tetrad, gauge, and polarization constraints.
4. Adiabatic particle number treats a controlled but basis-dependent diagnostic.
5. Adiabatic subtraction renormalizes local observables to the required order.
6. Vacuum choice and initial-state effects separates ultraviolet admissibility from infrared preparation.
7. Initial density matrices and boundary EFT encodes mixed and excited states with boundary counterterms.
8. Bunch–Davies, Euclidean, and alpha states tests de Sitter analyticity and singularities.
9. Trans-Planckian initial-state sensitivity states the EFT ceiling.
10. Gravitational production of massive relics converts late-time modes into abundance.
11. Mode matching across cosmological eras preserves canonical data through transitions.
12. Renormalized stress and FLRW backreaction solves the constrained mean system.

Domain and failure conditions

This is the canonical comparison table for the chapter.

Task	Required data	Quantity computed	Controlled statement	Decisive check	Failure or handoff
Scalar modes	$a(\eta),m,\xi$, canonical variable	Normalized $v_k$	Field algebra and two-point function	Wronskian	A time-dependent basis is not a preferred particle notion
Conformal comparison	Spin/coupling and conformal state	Rescaled mode equation	No production for massless conformally invariant free fields	Flat oscillator after rescaling	Anomaly or interactions can leave local stress
Spinor/gauge modes	Tetrad, spin structure, gauge, constraints	Physical mode functions	Fermionic or transverse field algebra	Anti/commutator and constraint propagation	Gauge/tetrad artifacts do not define production
Adiabatic particles	WKB order and asymptotic regime	$n_k^{(r)}=\lvert\beta_k^{(r)}\rvert^2$	Late-time occupation with a remainder estimate	Stability across orders	Intermediate-time number is basis dependent
Local subtraction	State, modes, observable, finite couplings	$\langle\phi^2\rangle_{\rm ren}$ or $\langle T_{\mu\nu}\rangle_{\rm ren}$	Local covariant observable	Second order for $\phi^2$, fourth for $T_{\mu\nu}$ in 4D; conservation	Subtraction is not state preparation
Gaussian state	Initial mode covariance and UV asymptotics	Power/correlation/stress differences	Admissible family of states	Hadamard or sufficient adiabatic order	UV regularity does not fix infrared data
Boundary EFT	Initial slice, density kernel, cutoff, operators	Modified propagator/correlator	Expansion in physical scales over $\Lambda$	Positivity and boundary RG	Unbounded density matrix or unsuppressed operators
de Sitter state	Patch, analytic continuation, mass/coupling	Two-point function	BD/Euclidean equivalence in its domain	Hadamard wavefront and loop counterterms	Formal alpha states add singularities
Trans-Planckian effect	UV completion encoded by boundary coefficients	EFT correction with error	Model-dependent suppressed sensitivity	Slice independence after running; stress bound	No generic prediction above the cutoff
Relic production	Smooth history, late adiabatic basis, entropy history	$n$, $\rho$, or yield $Y$	Model abundance with matching error	Smooth-transition and WKB convergence	A discontinuity-dominated yield is unphysical
Era matching	Canonical data and junction regularity	Symplectic transfer matrix	Preserved Wronskian and phase	Smooth-family convergence	Artificial junction particles
Backreaction	Renormalized $\rho,p$, state–geometry data, couplings	Coupled $a(\eta)$ and modes	Mean semiclassical solution	Friedmann constraint and conservation	Scheme, state, truncation, or numerical error dominates

The failure map emphasizes four category errors: treating particle number as local stress, changing the state while claiming a subtraction change, promoting cutoff sensitivity to a UV prediction, and inserting a nonconserved stress into the Friedmann equation.

Each cosmological conclusion has a distinct failure test; passing mode normalization alone does not license a particle, relic, or backreaction claim. 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.
• Parker, L., and S. A. Fulling, “Adiabatic Regularization of the Energy-Momentum Tensor of a Quantized Field in Homogeneous Spaces,” Physical Review D 9, 341–354 (1974), doi:10.1103/PhysRevD.9.341.