Infrared Growth and Secular Effects

Infrared enhancement and secular growth are related but distinct. Infrared sensitivity concerns small spatial wave number in a specified state and regulator; secular dependence concerns accumulation with time or e-fold number in a specified perturbative observable. At the evidence cutoff 2026-08-19, controlled calculations establish both effects in particular light-field models, but they do not support a field-independent claim that every de Sitter observable grows or that one resummation gives a universal late-time state.

Required background. Compactly, an infrared prescription fixes the distributional meaning of a two-point function at small momentum, while retarded kernels and initial data fix the time integrals appearing in loops. Partial Differential Equations and Green Functions supplies distributions, propagator support, and initial-value dependence. Helpful background. Flat-Spacetime Quantum Field Theory provides loop power counting and renormalization; Causality and Global Structure establishes the cosmological patch and its boundaries; Quantum Fields in Curved Spacetime provides state, observable, and in-in prescriptions.

Two questions hidden inside one logarithm

For a massless minimally coupled spectator scalar in a Poincaré patch, modes that have crossed the Hubble scale give schematically

$$\langle\phi_L^2(t)\rangle \sim\frac{H^2}{4\pi^2} \ln\left(\frac{a(t)}{a(t_0)}\right).$$

Here $\phi_L$ denotes the coarse-grained long-wavelength field, with the displayed contribution coming from the band $a_0H\lesssim k\lesssim aH$; it is not the ultraviolet-divergent bare coincidence limit of the complete field. The lower limit remembers an initial time or comoving infrared cutoff. At one time, the logarithm is the integral $\int dk/k$ over this band. Across times, it is secular because the band grows with the e-fold number $N=\ln(a/a_0)$. The state, field algebra, patch, and infrared prescription remain essential.

A small positive effective mass

$$M^2=m^2+12\xi H^2$$

changes $\nu=\sqrt{9/4-M^2/H^2}$ and introduces the small exponent

$$\Delta=\frac32-\nu\simeq\frac{M^2}{3H^2}.$$

For a free field with $0\lt M^2\ll H^2$ and constant $H$, the superhorizon mode decays as $a^{-\Delta}$. Summing modes that enter the long-wavelength band between $N'=0$ and $N'=N$ gives, at leading order in the light-mass approximation,

$$\begin{aligned} \langle\phi_L^2(N)\rangle_{\rm new} &\simeq\frac{H^2}{4\pi^2}\int_0^N e^{-2\Delta(N-N')}dN'\\ &=\frac{H^2}{8\pi^2\Delta}\left(1-e^{-2\Delta N}\right). \end{aligned}$$

For $\Delta N\ll1$ this recovers $H^2N/(4\pi^2)$; for $\Delta N\gg1$ it approaches $3H^4/(8\pi^2M^2)$. Initial long-wavelength variance supplies an additional decaying term. The limits $M\to0$ and $N\to\infty$ therefore do not commute. This free-field calculation explains saturation on a scale $N\sim\Delta^{-1}$; it does not establish the endpoint of an arbitrary interacting theory. The conformally coupled massless field, with $\nu=1/2$, has no analogous superhorizon enhancement in its conformal state.

Perturbative bounds and what they do not prove

Weinberg’s late-time power counting studies in-in correlation functions for interactions satisfying specified restrictions on their scale-factor dependence. Under those conditions, loop corrections cannot grow as positive powers of $a$; the worst permitted behavior is polynomial in $\ln a$. The assumptions and argument are given in Weinberg 2005, §II and §§IV–VII.

This is a useful negative result: it excludes a broad class of faster late-time divergences. It does not establish that every logarithm is nonzero, gauge invariant, observable, or perturbatively harmless. A term

$$\lambda^L[\ln(a/a_0)]^p$$

can become order one for sufficiently many e-folds even though it grows more slowly than any positive power of $a$. Nor does the bound decide whether cancellations occur after constraints, composite-operator renormalization, or a relational observable are included. The explicit loop organization and examples in Weinberg 2006, §§II–V and §VI must therefore be read with the chosen fields, gauge, state, and correlator attached.

Results and limitations

Assumptions and observable	Position or controlled result	Limitation or adverse test
Free massless minimally coupled spectator; Poincaré patch; finite initial time or comoving infrared cutoff; equal-point field variance	Superhorizon modes give growth proportional to $\ln(a/a_0)$	A small positive $M^2$, conformal coupling, another state, or a shift-invariant derivative observable changes or removes the growth
In-in correlators whose interactions satisfy Weinberg’s scale-factor bounds; fixed perturbative order	Late-time corrections are bounded by powers of $\ln a$, not positive powers of $a$	The result does not prove cancellation, gauge invariance, or validity when the logarithm compensates the small coupling
Gauge-fixed inflaton or metric correlators with constraints retained to the stated order	Some secular logarithms survive explicit loop calculations; others cancel or move under field and coordinate redefinitions	A coordinate correlator is not automatically a relational observable, and a change of gauge can expose an artifact
Weakly interacting light scalar in a declared initial state; leading-log hierarchy	Dynamical renormalization-group or stochastic methods can resum selected powers and produce effective decay, mass generation, or a stationary distribution	The closure is model- and observable-dependent; subleading logs, derivative interactions, non-Gaussian state data, and backreaction may lie outside it
Massive principal-series or massless conformal field in its distinguished state	Correlators decay or retain conformal behavior rather than showing the minimally coupled secular pattern	This adverse case rules out universal growth but does not settle light minimally coupled interactions

The matrix supports credible competing positions only after their domains are separated. One position treats surviving secular logarithms in a fixed gauge-invariant observable as genuine perturbative evolution requiring resummation. Another emphasizes that many proposed effects are state, gauge, coordinate, or regulator artifacts and demands a local or relational observable before assigning physical meaning. A third accepts model-specific nonperturbative late-time limits—stochastic, dynamical-RG, large-$N$, or Schwinger–Dyson—without identifying any one closure as universal.

Resummation is an observable-specific approximation

Suppose fixed-order terms organize as

$$F(N)=F_0\left[1-c_1\lambda N+c_2(\lambda N)^2+\cdots\right].$$

A multiplicative leading-log evolution equation may resum the sequence into $F(N)\sim F_0e^{-c_1\lambda N}$. That ansatz predicts $c_2=c_1^2/2$ and fixes every higher leading coefficient; the one-loop term by itself proves none of those relations. This can extend the useful time range only when the hierarchy is justified. It must reproduce the known fixed-order coefficients, preserve the relevant symmetries and conservation laws, and state which subleading terms were discarded.

Burgess and collaborators demonstrate such secular power counting and dynamical-RG resummation in controlled de Sitter scalar examples in Burgess et al. 2010, §§2.1–2.4, §§3.1–3.3, §§4.1–4.3, and §6. Their calculation is evidence for that method in its declared hierarchy, not a theorem that stochastic, kinetic, and dynamical-RG equations agree for every field or observable.

The decisive comparison is operational. A candidate resummation should match the first perturbative logarithms, reproduce a protected conformal or massive limit, remain stable under the declared regulator removal, and predict the same quantity that the original series computed. Resumming a gauge-fixed field correlator and then interpreting it as a local stress tensor fails the last test.

Gauge, state, and regulator discriminators

For scalar cosmological perturbations, the linear Mukhanov–Sasaki variable is gauge invariant, but nonlinear composite observables require higher-order gauge transformations and renormalization. Graviton two-point functions can also contain large-gauge or coordinate-sensitive pieces. A robust claim states whether the object is a local curvature scalar, a relationally located correlator, a detector response, a smeared field, or a coordinate-space mode amplitude.

State dependence enters through the initial density operator and time $t_0$. A boundary counterterm or nonadiabatic initial state can generate time dependence that resembles bulk secular evolution. Regulator dependence enters differently: a fixed comoving cutoff, a fixed physical cutoff, dimensional regularization, and a finite spatial volume do not hold the same modes fixed as $a$ grows. A result is credible only after renormalized quantities are compared at fixed physical conditions.

Loop order and time domain must be recorded together. Fixed-order perturbation theory is controlled when every enhanced combination—such as $\lambda N^p$—is small. Beyond that range, its breakdown establishes the need for a new approximation, not the truth of a particular resummed endpoint.

What remains unresolved

No single result in the checked corpus settles which relational gravitational observables exhibit lasting secular evolution at arbitrarily late times, how competing resummations compare beyond their shared leading terms, or when fixed-background effects become large enough to require metric backreaction. The answer also remains open across masses, nonminimal couplings, derivative interactions, non-Gaussian states, and global versus Poincaré de Sitter geometries.

A decisive adverse case must survive reformulation. If a logarithm vanishes after passing from a coordinate perturbation to a gauge-invariant observable, it cannot support a universal physical-growth claim. Conversely, if an independently renormalized relational observable retains the same coefficient across gauges and regulators, dismissing it as “just coordinates” is no longer adequate.

What could change these conclusions

The conclusions could change after any of the following: a theorem sharpens the allowed state, mass, coupling, geometry, or observable class; an independent multiloop calculation changes the leading or subleading late-time coefficient; a regulator or initial-boundary analysis invalidates a published hierarchy; two resummation schemes are proved equivalent or demonstrably inequivalent for the same observable; or a gauge-invariant adverse calculation resolves one of the open discriminators.

Diagnostic: decide whether to resum

A one-loop calculation in comoving gauge finds $\lambda\ln^2(a/a_0)$ in a coordinate-space inflaton correlator. At $\lambda\ln^2(a/a_0)\sim1$, may one conclude that a physical observable diverges and apply any leading-log resummation? State the checks needed before either conclusion.

Solution

The order-one combination shows only that the fixed one-loop truncation has lost control. First identify a gauge-invariant or relational observable and repeat the constraint and composite-operator renormalization analysis; the coordinate correlator alone is insufficient. Then declare the initial state, initial time, mass and coupling, infrared regulator, and whether wave numbers are held comoving or physical. A resummation is justified only if its hierarchy reproduces the known early-time coefficient, preserves the relevant symmetries, and survives a regulator or gauge comparison. A conformal or sufficiently massive adverse case should also remove the claimed enhancement. These checks can establish genuine secular evolution, cancellation, or merely the need for a different approximation, but not a universal divergence from one logarithm.

References

• Burgess, C. P., R. Holman, L. Leblond, and S. Shandera. “Breakdown of Semiclassical Methods in de Sitter Space.” Journal of Cosmology and Astroparticle Physics 2010, no. 03 (2010): 033. DOI. Open PDF, arXiv:0912.1608.
• Weinberg, Steven. 2005. “Quantum Contributions to Cosmological Correlations.” Physical Review D 72: 043514. DOI.
• Weinberg, Steven. 2006. “Quantum Contributions to Cosmological Correlations. II. Can These Corrections Become Large?” Physical Review D 74: 023508. DOI.