Fractional One-Loop Correction in Power Spectrum

• The paper reveals that fractional one-loop corrections introduce logarithmic scale-dependent deviations to the tree-level power spectrum in slow-roll inflation.
• The methodology employs the in-in Schwinger–Keldysh formalism with cubic and quartic interaction vertices, rigorously regularizing UV and IR divergences via renormalization.
• The corrections, while negligible in standard models, can become significant in extended inflation scenarios, serving as a diagnostic for the limits of perturbative cosmological predictions.

A fractional one-loop correction in the power spectrum refers to the next-to-leading-order quantum (or non-linear classical) correction that modifies the tree-level (leading-order) prediction for two-point correlation functions in cosmological perturbation theory. These corrections, arising from both self-interactions and gravitational couplings, quantify the deviation of the physical power spectrum from its leading behavior, providing a measure of the robustness and limitations of perturbative approaches to cosmological fluctuations.

1. Perturbative Framework and Formalism

Fractional one-loop corrections to the power spectrum are computed by systematically expanding the action for fluctuations about the classical inflationary background and evaluating higher-order contributions to two-point functions. In canonical slow-roll inflation, the inflaton field is decomposed as $$\phi(t, \mathbf{x}) = \phi_0(t) + \delta\phi(t, \mathbf{x})$$, and the metric is expressed in ADM form, typically in the spatially flat gauge. The quadratic action for $\delta\phi$ yields the tree-level, nearly scale-invariant spectrum

$P_0(k) = \frac{H_*^2}{2 k^3}$

evaluated at horizon crossing ($-k\eta_* \simeq 1$).

The leading quantum corrections—one-loop contributions—are computed using the in-in (Schwinger-Keldysh) formalism, where path integrals are evaluated along the closed time path, accommodating causal expectation values appropriate for cosmological perturbations. Interactions appear as cubic and quartic vertices in the fluctuation action, with cubic terms typically $O(\epsilon^{1/2})$ and quartic terms $O(\epsilon^0)$ for slow-roll parameter $\epsilon$ (0707.3377). Time-derivative couplings and gravitational interactions necessitate the careful treatment of ghost fields and canonical momentum shifts to maintain consistency in the diagrammatic rules and account for possible cancellations.

2. Origin and Structure of One-Loop Corrections

The correction to the two-point function at one-loop is represented schematically by diagrams in which the external legs (representing long-wavelength fluctuations) are attached to interaction vertices whose remaining fields contract in a closed loop. The relevant contractions incorporate both quantum “q-loop” contributions—arising near horizon crossing due to interference among quantum modes—and classical “c-loop” effects, where integrated long-wavelength modes alter the observable on superhorizon scales via the $\delta N$ expansion formalism (0707.3378).

A critical mathematical feature is the appearance of infrared (IR) and ultraviolet (UV) divergent contributions in the loop integrals over internal momenta:

• UV divergences manifest as power-law terms, which are systematically removed by renormalization (e.g., via subtraction or dimensionally regulated counterterms).
• IR divergences, by contrast, are captured as logarithms, e.g., terms proportional to $\ln k$ or $\ln (k\ell)$ (where $\ell$ is the IR cutoff set by the size of the comoving box).

After regularization and renormalization, the net (schematic) one-loop power spectrum correction for the minimally coupled scalar in slow-roll inflation is

$$P^{(1\text{loop})}(k) = P_0(k)\left[1 - (4/3) P_0(k)\ln k + \dots \right]$$

where the coefficient of the logarithm is robust and scheme-independent, while constant power-law terms are absorbed into the renormalized parameters (0707.3377).

3. Infrared Logarithms and Physical Interpretation

A signature of the fractional one-loop correction is the emergence of logarithmic scale dependence, sourced by long-wavelength (IR) fluctuations:

• For quantum mechanical (“q-loop”) contributions, the correction for the two-point function takes the form

$$\langle \delta\phi(\mathbf{k}_1)\delta\phi(\mathbf{k}_2)\rangle_{\ast} \approx (2\pi)^3\delta(\mathbf{k}_1 + \mathbf{k}_2)P_0^2/\pi^2 \left[-\frac{4}{3}\ln k + \beta + \dots\right]$$

where $\beta$ is a scheme-dependent constant and “$\dots$” stands for renormalized, cutoff-dependent terms (0707.3377).

• For classical (“c-loop”) corrections, convolutions of tree-level spectra in the $\delta N$ expansion encode the contributions of superhorizon modes over the phase space, typically involving integrals that are sensitive to IR regularization.

These logarithmic terms can be significantly enhanced when the range between the IR and UV cutoffs is large, such as in models with a large number of $e$-folds between the scale of CMB observations and the fundamental UV scale of inflation. In such a case, the effective one-loop and potentially higher-loop corrections may become sufficiently large to invalidate the simple perturbative expansion. This necessitates either a resummation (e.g., via renormalization group techniques) or a reinterpretation of the underlying perturbative approach (0707.3378).

4. Implications and Model Dependence

The magnitude and phenomenological impact of the fractional one-loop correction depend on model parameters and cosmological history:

• In minimal or “small-field” inflationary models with $\sim$60 $e$-folds, the logarithmic enhancement is modest, and the correction remains well below observational thresholds.
• For large-field or extended models (including those motivated by string landscapes or with many $e$-folds), large logarithms $\sim \ln(k\ell)$ can amplify the correction, especially for red-tilted spectra or spectra growing strongly at large scales (0707.3378).
• The correction is also typically suppressed by the small amplitude of the primordial power spectrum ($P_0 \lesssim 10^{-10}$ in canonical slow-roll scenarios). However, if IR logarithms become large due to cumulative effects over many $e$-folds, the correction can approach or exceed percent-level significance, influencing precision CMB parameter estimation.

To ensure that perturbation theory remains valid, it is necessary to verify that the fractional correction remains subdominant, i.e., $|(4/3)P_0 \ln k| \ll 1$. When this fails, the standard tree-level result loses its predictive power, and a revised theoretical framework or reevaluation of initial conditions (e.g., the time at which correlators are evaluated in the $\delta N$ formalism) becomes necessary.

5. Regularization, Renormalization, and Diagrammatic Consistency

The architecture of the fractional one-loop calculation requires:

• Proper identification of interaction vertices, especially those involving time derivatives, and consistent inclusion of “ghost” fields to deal with nontrivial kinetic structure. In the canonical single-field slow-roll case, these ghost contributions cancel at one-loop (0707.3377).
• UV divergences are consistently removed by standard renormalization procedures, ensuring that the physically significant, universal logarithmic (IR) terms remain as the key corrections.
• The use of de Sitter (or nearly de Sitter) mode functions with appropriately chosen in-vacuum boundary conditions (often set via contour rotation in the complex conformal time $\eta$ plane).

Only with these diagrammatic and path-integral refinements is the coefficient of the logarithmic correction unambiguously fixed, as was demonstrated in the thorough "in-in" Schwinger–Keldysh calculation and subsequent cross-checks (0707.3377).

6. Observational and Theoretical Consequences

Fractional one-loop corrections to the power spectrum are a critical diagnostic of the reliability of perturbative predictions in early-universe cosmology. The conclusions in the single-field slow-roll scenario are:

• The fractional correction is given by

$\frac{\Delta P}{P_0} \sim -\frac{4}{3} P_0 \ln k,$

so the one-loop corrected power spectrum is

$$P^{(1\text{loop})}(k) = P_0(k)\left[1 - \frac{4}{3} P_0(k)\ln k + \cdots \right].$$

• For canonical parameter values, this correction is extremely small; however, if accumulated over many $e$-folds, it could become relevant.
• The precise evaluation, in both quantum and classical sectors and with proper treatment of non-linearities, is essential for robustly comparing models to high-precision CMB data and constraining departures from slow-roll, multi-field, or non-canonical inflation.

Table: Schematic Summary of One-Loop Effects

Physical regime	Functional form of correction	Importance (typical models)
Slow-roll, 60 $e$-folds	$-(4/3)P_0\ln k$	Negligible
Large-field/$N \gg 60$	$-(4/3)P_0 N$	Can become significant
Red-tilted or growing spectra	$-P_0 (\ln k)^p$ ($p>1$)	Potentially large
Cumulative IR effects	Proportional to $\ln(k\ell)$	Possible dominance

Fractional one-loop corrections thus encapsulate a sensitive probe of the interplay between quantum effects, gauge structure, and the limits of perturbative control in cosmological power spectra. Their robust calculation, ensured by advanced techniques in the in-in formalism and careful renormalization, strengthens confidence in tree-level inflationary predictions and provides a diagnostic for scenarios where loop effects may signal new physics or the breakdown of classical intuition.