What spectators do during inflation

Abstract: The inflaton equation of motion including one loop radiative corrections from spectator fields is obtained. We consider a massless scalar conformally coupled to gravity and a massless fermion Yukawa coupled to the inflaton as models for spectators that \emph{do not feature} gravitational particle production, their production during slow roll is solely a consequence of their coupling to the inflaton. The one-loop self energy and the fully renormalized equation of motion of the inflaton are obtained and solved explicitly for an inflaton potential $m<sup>2\varphi<sup>2/2$. The solution features Sudakov-type logarithmic secular terms, which are resumed via the dynamical renormalization group and compared to the solutions with a phenomenological friction term. During $N_e$ e-folds of slow roll inflation the inflaton evolves as $$\varphi<sup>{(0)}_{Isr}(t)\,e<sup>{\frac{m<sup>2Γ}{9H<sup>3}\,N_e(t)}$$ for the phenomenological friction term $Γ$ and $\varphi<sup>{(0)}_{Isr}(t)\,e<sup>{ΥN<sup>2_e}$ with $$Υ= -\frac{λ^2}{24π² H^2} ; \frac{y<sup>2_R}{12π<sup>2}$$ for the radiative corrections from bosonic and fermionic spectators respectively where $\varphi<sup>{(0)}_{Isr}(t)$ is the slow roll solution in absence of interactions, showing that a phenomenological friction term is not reliable. A generalization of the optical theorem to a finite time domain and cosmological expansion is introduced to obtain the distribution function $f(k,t)$ and total number of spectators produced \emph{during slow roll}. $f(k,t)$ is peaked at superhorizon scales and the total number of particles grows $\propto e<sup>{3N_e}$. A non-perturbative mean field theory is introduced to describe the self-consistent evolution of the inflaton coupled to spectators, its linearized version reproduces the self-energy, the inflaton equation of motion and the results on particle production.

• The paper shows spectator fields induce Sudakov log-squared corrections to the inflaton dynamics, overturning simple friction-based models.
• It employs the Schwinger-Keldysh approach and DRG resummation to derive non-local self-energy corrections and quantify spectator-induced particle production.
• The study links radiative backreaction from spectator fields to observable effects in reheating, isocurvature perturbations, and non-equilibrium quantum dynamics.

The Role of Spectator Fields During Inflation: Self-Energy Corrections, Sudakov Logarithms, and Particle Production

Introduction and Motivation

This work provides a systematic field-theoretic investigation of the influence of spectator fields on inflaton dynamics during the slow-roll inflationary epoch. While the inflaton is typically assumed to dominate the energy-momentum tensor, numerous other fields—whether from the Standard Model (e.g., the Higgs) or possible extensions (such as dark matter candidates)—are present. The prevalent approach in the literature replaces the impact of these "spectators" by phenomenological friction terms in the background inflaton equation of motion, an assumption inherited from flat-space S-matrix intuition. This paper critically scrutinizes that assumption by explicitly computing the one-loop self-energy corrections from two classes of spectator fields: a conformally-coupled massless scalar and a massless fermion Yukawa-coupled to the inflaton, both of which lack gravitational production in de Sitter but can be generated through direct inflaton coupling.

The analysis leverages both the Schwinger-Keldysh ("in-in") formalism for real-time nonequilibrium QFT and (as cross-checks) linear response theory and a non-perturbative mean field approach. It extends to explicit quantification of particle production using a finite-time optical theorem tailored for cosmological expansion. This allows a precise connection between quantum radiative corrections to background equations of motion, the breakdown of ergodic assumptions, and non-equilibrium production of out-of-equilibrium particle distributions during inflation.

Formulation and Methods

Model Setup

The system under study is a spatially-flat de Sitter background, with canonically normalized inflaton $\phi_I$ and two classes of spectator fields: a real scalar $\phi_s$ (conformally coupled, massless) and a Dirac (or Majorana) fermion field $\Psi$ Yukawa coupled to $\phi_I$. The action employs the Friedmann-Robertson-Walker metric, with fields rescaled to conformal time for technical clarity. The couplings are:

• Scalar: $-\lambda \phi_I \phi_s^2$
• Fermion: $-y\phi_I \bar{\Psi}\Psi$

Derivation of Backreaction

All computations are performed to one-loop in the spectator-inflaton coupling, leveraging the Schwinger-Keldysh contour to correctly capture causal correlation functions and secular (infrared-enhanced) structure absent in equilibrium/Euclidean approaches. The result is an explicitly non-local, real-time corrected equation of motion for the background inflaton condensate, where the self-energy kernels depend on the time-dependent background (notably, the scale factor) as well as the mode structure of the spectators.

Renormalizations (both mass and field) are carefully performed; for the scalar case, massless conformal coupling ensures the absence of canonical gravitational particle production, isolating the inflaton-mediated effects. In the fermionic case, field renormalization is also explicit. The DRG (dynamical renormalization group) resummation procedure is employed to account for secularly-growing terms in the inflaton evolution and to extract valid late-time behavior.

Comparison to Phenomenological Friction

For conceptual contrast, the standard phenomenological approach—that the radiative effect of decay into spectators can be encoded as a local friction term ($\Gamma\dot{\varphi}_I$) with $\Gamma$ taken from the inflaton decay rate in Minkowski space—is analysed explicitly. This allows unambiguous comparison between the correct non-local self-energy and local friction descriptions.

Finite-Time Optical Theorem and Particle Production

A central technical contribution is the generalization of the optical theorem to finite time in de Sitter, enabling explicit calculation of the momentum distribution and total number of spectator particles produced by coupling to the inflaton condensate. The mean field approach is shown to yield the same results in the Born (weak-coupling) approximation, establishing the internal consistency of perturbative and non-perturbative perspectives.

Main Technical Results

Inflaton Equation of Motion and Sudakov Logarithms

The inflaton background obeys a non-locally corrected equation of motion: $$\varphi''_I(\eta) + \text{(potential)} + \int_{\eta_0}^{\eta} d\eta' \, \widetilde{\Sigma}(\eta, \eta')\varphi_I(\eta') = 0,$$ where $\widetilde{\Sigma}$ is the appropriately renormalized self-energy due to spectator loops. Explicit forms for both scalar and Yukawa interactions are provided; the kernels are non-local in time.

Perturbative solution of this equation produces secular terms: for the phenomenological friction term, these are single-logarithmic in the number of e-folds; for the actual cQFT calculation, the leading secular corrections are of Sudakov type—double logarithms in the conformal time (i.e., $\phi_s$0, or equivalently, $\phi_s$1, where $\phi_s$2 is the number of efolds since the start of inflation).

The DRG resummation yields the large-number-of-efolds asymptotic: $\phi_s$3 Hence, the temporal structure, sign, and magnitude of radiative corrections due to spectator fields differ parametrically and functionally from friction term intuition; the friction approach fails even at qualitative level.

Canonical Field-Theoretic Comparison: Minkowski vs de Sitter

Repeating the calculation in Minkowski space (where energy is conserved) recovers the standard result: radiative corrections to the inflaton produce secular single logarithms (linear in time), which exponentiate to a simple decaying amplitude with rate $\phi_s$4 (the S-matrix rate). In de Sitter, the absence of energy conservation and global Killing vector produces non-trivial logarithmic structure and, crucially, opposite signs for bosonic and fermionic spectator corrections.

Particle Production: Optical Theorem and Distribution Functions

The generalization of the optical theorem in a finite-time, expanding universe allows explicit computation of the momentum distribution $\phi_s$5 and number of spectator particles produced: $\phi_s$6 For the slow-roll inflaton, $\phi_s$7 is sharply peaked at superhorizon $\phi_s$8 (i.e., long wavelengths), and for the total number: $\phi_s$9 with $\Psi$0 the physical volume at time $\Psi$1. The particle number grows as $\Psi$2 in de Sitter, reflecting Hubble dilution in phase space and amplification by the growing volume. In contrast, in Minkowski space, the distribution is sharply peaked around $\Psi$3 due to energy conservation.

Equivalence of Mean Field and Optical/Born Approximations

Implementing a mean field approach (solving the spectator modes in the condensate background self-consistently) and performing a Bogoliubov analysis, the Born approximation yields production rates and distribution functions identical to those obtained from the field-theoretic optical theorem. Thus, the out-of-equilibrium, nonperturbative framework recovers the perturbative result in weak-coupling.

Theoretical and Practical Implications

• Inadequacy of Local Friction Terms: The study definitively demonstrates that modeling backreaction from inflaton decays into light spectator fields as local friction is, in general, incorrect in an expanding background. Radiative memory, non-locality, and the dynamical breaking of (quasi-)energy conservation must be considered for correct inflaton dynamics and predictions for cosmological observables.
• Sudakov Logarithms and Resummation: The dynamical RG exhibits that inflationary radiative corrections generically yield Sudakov-type log-squared growth, with significant consequences for the evolution of the background, particularly if the couplings are not exceedingly small or inflation endures for many e-folds.
• Amplified Spectator Particle Production: Direct coupling to the inflaton during slow roll leads to exponential-in-volume particle production, with distributions peaked at long wavelengths. This has immediate relevance for isocurvature constraints and for reheating transitions. The distinction between gravitational and inflaton-mediated production is explicitly characterized.
• Methodological Advances: The finite-time optical theorem in a cosmological context and explicit identification of DRG-resummed asymptotics are both broadly applicable to studies of nonequilibrium quantum fields in curved backgrounds.

Future Directions

The work points toward several open research avenues:

• Covariant Energy-Momentum Conservation: The growth of the spectator field's energy-momentum during inflation and its eventual backreaction require deeper study, including in the context of renormalization and consistency with cosmological perturbation theory.
• Isocurvature and Non-Gaussianity: Profuse production of long-wavelength spectators could source non-trivial isocurvature or non-Gaussian features, depending on the precise IR structure and subsequent thermalization dynamics.
• Beyond One Loop and Strong Coupling: Extending the non-perturbative mean field approach beyond linearized (Born) order could capture late-time saturation, collective phenomena, or even phase transitions triggered by spectator production.

Conclusion

This work delivers an explicit, rigorous field-theoretic treatment of the impact of spectator fields on inflaton dynamics, dramatically establishing the deficiencies of frictional approximations and elucidating the correct quantum structure of radiative backreaction and particle production in de Sitter inflation. The analysis has broad implications not just for inflationary model building and reheating scenarios, but also for the general understanding of non-equilibrium QFT in curved spacetime.

Reference: “What spectators do during inflation” (2604.03866)