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One-loop correction to primordial tensor modes during radiation era

Markus B. Fröb, Dražen Glavan, Paolo Meda, Ignacy Sawicki

TL;DR

This work challenges the standard assumption of superhorizon tensor perturbation conservation during the radiation era by modeling the thermal plasma as a bath of free photons and computing the one-loop corrections to the primordial tensor power spectrum. Using a stochastic gravity framework and the Yang-Feldman formalism, the authors identify a non-suppressed, secular enhancement at one loop arising from radiation-exchange between tensor modes and the photon bath, while thermally induced GW production is subdominant on observable scales. A resummation via an effective local equation with λ = 2/5 reveals a power-law growth of the tensor power spectrum with a positive exponent, signaling a breakdown of perturbation theory and the necessity of nonperturbative treatment. These results imply that interpreting CMB tensor constraints and inflaton properties may require more sophisticated modeling of the radiation-era background and its backreaction on long-wavelength tensor modes, including possible plasma interactions beyond the noninteracting photon model.

Abstract

The ability to infer properties of primordial inflation relies on the conservation of the superhorizon perturbations between their exit during inflation, and their re-entry during radiation era. Any considerable departure from this property would require reinterpreting the data. This is why it is important to understand how superhorizon perturbations interact with the thermal plasma driving the radiation dominated Universe. We model the plasma by free photons in a thermal state and compute the one-loop correction to the power spectrum of primordial tensor perturbations. This correction grows in time and is not suppressed by any small parameter. While one-loop result is not reliable because it invalidates perturbation theory, it signals potentially interesting effects that should be investigated further.

One-loop correction to primordial tensor modes during radiation era

TL;DR

This work challenges the standard assumption of superhorizon tensor perturbation conservation during the radiation era by modeling the thermal plasma as a bath of free photons and computing the one-loop corrections to the primordial tensor power spectrum. Using a stochastic gravity framework and the Yang-Feldman formalism, the authors identify a non-suppressed, secular enhancement at one loop arising from radiation-exchange between tensor modes and the photon bath, while thermally induced GW production is subdominant on observable scales. A resummation via an effective local equation with λ = 2/5 reveals a power-law growth of the tensor power spectrum with a positive exponent, signaling a breakdown of perturbation theory and the necessity of nonperturbative treatment. These results imply that interpreting CMB tensor constraints and inflaton properties may require more sophisticated modeling of the radiation-era background and its backreaction on long-wavelength tensor modes, including possible plasma interactions beyond the noninteracting photon model.

Abstract

The ability to infer properties of primordial inflation relies on the conservation of the superhorizon perturbations between their exit during inflation, and their re-entry during radiation era. Any considerable departure from this property would require reinterpreting the data. This is why it is important to understand how superhorizon perturbations interact with the thermal plasma driving the radiation dominated Universe. We model the plasma by free photons in a thermal state and compute the one-loop correction to the power spectrum of primordial tensor perturbations. This correction grows in time and is not suppressed by any small parameter. While one-loop result is not reliable because it invalidates perturbation theory, it signals potentially interesting effects that should be investigated further.

Paper Structure

This paper contains 13 sections, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Standard results for tensors in the early universe
  3. Setting up one-loop equation for tensor perturbations
  4. Perturbing equations
  5. Vector field strength correlator
  6. Transverse traceless projection
  7. Tensor perturbation correlator
  8. Computing diagrams
  9. Thermal GW production
  10. Radiation exchange
  11. Resummed computation
  12. Discussion
  13. Loop integrals

Figures (2)

  • Figure 1: 1PI 4-vertex diagram contributing to the graviton equation of motion. The double wavy line stands for the statistical two-point function of the photon, while curly lines denote amputated gravitons. This diagram is conventionally defined as the expectation value of the second variation of the Maxwell action with respect to the metric, which does not correspond directly to the contribution in (\ref{['HartreeTerm']}), that is defined by perturbing the Einstein equation with the square root of the metric stripped off. This detail is immaterial in practice as the same additional term not included (\ref{['HartreeTerm']}) is also omitted in the left-hand side when perturbing the Einstein tensor.
  • Figure 2: One-loop corrections to the graviton two-point function. Curly represent gravitons, while wavy lines represent photons. Double lines stand for statistical two-point functions, while single lines stand for retarded propagators. Diagram $A$ depicts the so-called thermally induced gravitational wave production. Diagrams $B$ and $C$ depict contributions dubbed radiation exchange in Ota:2023iyh.