Inflation calculation shows how apparent changes in large-scale cosmic ripples cancel

Why the persistence of primordial ripples matters

Inflationary cosmology describes how quantum fluctuations in the early universe could seed the patterns later seen in the cosmic microwave background and the distribution of galaxies. A central quantity is the curvature perturbation, denoted by ζ, which describes spatial variations in expansion. Its power spectrum measures the strength of those variations across different wavelengths.

In suitable single-field inflation models, curvature perturbations become approximately constant after their wavelengths stretch beyond the inflationary horizon. That conservation makes it possible to connect an early-universe calculation with much later observations. But does it survive when nonlinear quantum effects are included?

Keisuke Inomata of Johns Hopkins University addresses that question in a theoretical study published in Physical Review D. Within the paper’s assumptions, the apparent late-time evolution of the superhorizon curvature power spectrum cancels at one-loop order: the first loop correction beyond the leading, or tree-level, calculation.

Extending the calculation across scales

Earlier work by Inomata assumed that the fluctuations circulating inside loop calculations had much shorter wavelengths than the curvature perturbation being studied. In the paper’s notation, their wavenumbers obeyed k much greater than q; larger wavenumbers mean shorter wavelengths.

The new analysis removes that hierarchy. It explicitly includes loop wavelengths comparable to, or longer than, the wavelength of interest. Those contributions cannot generally be dismissed individually, making their cancellation a substantive part of the result.

The calculation uses spatially-flat gauge, a coordinate choice that places the relevant fluctuations in the inflaton—the field driving inflation—while keeping the spatial metric flat. It also adopts the decoupling limit, in which the slow-roll parameter epsilon approaches zero and metric fluctuations can be neglected relative to the retained inflaton interactions. The limiting procedure preserves the curvature-fluctuation amplitude rather than simply eliminating all fluctuations.

Why the conversion to curvature matters

The method has two connected components. First, Inomata uses the in-in formalism to calculate quantum expectation values, including corrections to the inflaton’s two-point correlation and the average background evolution. The treatment includes counterterms, which are needed to handle divergent contributions consistently.

Second, he converts inflaton fluctuations into curvature perturbations using the δN formalism. This describes differences in local expansion between regions as differences in their number of expansion e-folds. Crucially, that conversion is nonlinear: keeping only a proportional relationship between the two quantities would miss terms needed at one-loop order.

The author organizes the remaining contributions into two diagrammatic groups, called cut-in-the-middle and cut-in-the-side terms, plus boundary terms. Within each group, time-dependent pieces cancel when the relevant quantum, background and nonlinear-conversion contributions are combined.

This does not mean every loop contribution vanishes. Constant corrections associated with earlier evolution can remain. The demonstrated result concerns the absence of further time dependence during the specified late period, not the absence of all quantum corrections.

A boundary that cannot be ignored

An important subtlety concerns smoothing. The δN approach averages away shorter-wavelength fluctuations so that sufficiently large regions can be treated as locally homogeneous universes.

A spatially uniform cutoff fixed in comoving coordinates leaves an apparently time-dependent boundary contribution. Inomata argues that this smoothing prescription fails to respect the local coordinate rescaling associated with a very long-wavelength perturbation.

Allowing the cutoff to respond to the local long-wavelength curvature produces an additional term that cancels the remainder. An appendix obtains the corresponding cancellation with a smoothing scale fixed in physical units, where fluctuations crossing the smoothing boundary contribute stochastic noise. Another appendix reproduces the relevant in-in decomposition using dimensional regularization rather than an explicit ultraviolet cutoff.

What the result does—and does not—establish

The proof applies to canonical single-field inflation in the decoupling limit and at one-loop order. It assumes an initial period, a sufficiently long slow-roll interval and a late period. The long-wavelength tree-level curvature perturbations must already have become approximately constant by the end of the initial period and remain so afterward. During the intervening slow-roll interval, the parameter eta, which describes the evolution of epsilon, must stay sufficiently above minus three.

Transient departures from slow roll are allowed only when those conditions remain satisfied. The paper therefore does not establish conservation for arbitrary inflationary histories, additional fields, noncanonical models or higher-loop corrections.

Models with enhanced small-scale fluctuations, sometimes studied in connection with primordial black holes and induced gravitational waves, motivate the debate. This work does not demonstrate either phenomenon observationally; it clarifies the theoretical accounting needed to assess large-scale conservation.

The original arXiv submission was posted on June 26, 2026. Version 2, posted on November 6, is identified in the arXiv record as the published version, with the journal reference Physical Review D 114, 103514 (2026).