Quantum Fields in Cosmology

In an expanding universe, quantum-field modes behave as time-dependent oscillators. Their evolution connects vacuum choice and particle diagnostics to renormalized stress, relic production, and possible gravitational backreaction.

Enter this chapter

Begin with FLRW Fields and Mode Quantization to derive canonically normalized modes on an expanding background.

Helpful background. Particle Creation in Time-Dependent Backgrounds and Adiabatic States, WKB Order, and Regularity develop the definitions used by the chapter’s opening pages.

Topics

1. FLRW Fields and Mode Quantization — Derives the scalar oscillator and Wronskian.
2. Conformal and Minimally Coupled Scalars in FLRW — Isolates curvature coupling and expansion history.
3. Spinor and Gauge Fields in Expanding Universes — Preserves tetrad, gauge, and polarization constraints.
4. Adiabatic Particle Number in Cosmology — Treats a controlled but basis-dependent diagnostic.
5. Adiabatic Subtraction for Local Observables — 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-State Diagnostics — 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 Backreaction in FLRW — Solves the constrained mean system.

How the ideas fit together

Mode mixing can define particle production when an appropriate comparison basis is available Parker 1969, §§ II–IV, pp. 1059–1067. Local stress nevertheless requires ultraviolet subtraction, which answers a different question from choosing an adiabatic state or particle basis.

A production spectrum becomes a relic abundance only after the expansion history, era matching, dilution, and subsequent interactions are specified. If the produced stress significantly affects that history, the fixed-background calculation must be replaced by a coupled evolution. Finite mode occupations alone do not establish a finite renormalized source.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Task	Required data	Quantity computed	Controlled statement	Decisive check	Failure or further method needed
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

Where to go next

Continue to Gravitational Vacuum Decay and Cosmological Transitions to distinguish a decay exponent, a survival observable, and a cosmological history.

References

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