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Long-wavelength metric backreactions in slow-roll inflation

B. Losic, W. G. Unruh

TL;DR

The paper analyzes second-order corrections to linear cosmological perturbation theory in a slow-roll inflationary background, focusing on superhorizon (IR) backreactions. By computing the cumulative backreaction on the homogeneous sector via the eigenvalues of the total stress-energy, it shows that IR contributions behave like a negative cosmological constant in the chosen gauge and can dominate over linear terms for many slow-roll scenarios. The authors demonstrate that this backreaction grows with the number of e-foldings and may signal a breakdown of the linearized approximation in sufficiently slow-roll spacetimes. They also discuss gauge dependence, the role of UV–IR coupling, and the implications for interpreting local measurements of the cosmological constant in an inflationary context.

Abstract

We examine the importance of second order corrections to linearized cosmological perturbation theory in an inflationary background, taken to be a spatially flat FRW spacetime. The full second order problem is solved in the sense that we evaluate the effect of the superhorizon second order corrections on the inhomogeneous and homogeneous modes of the linearized flucuations. These second order corrections enter in the form of a {\it cumulative} contribution from {\it all} of their Fourier modes. In order to quantify their physical significance we study their effective equation of state by looking at the perturbed energy density and isotropic pressure to second order. We define the energy density (isotropic pressure) in terms of the (averaged) eigenvalues associated with timelike (spacelike) eigenvectors of a total stress energy for the metric and matter fluctuations. Our work suggests that that for many parameters of slow-roll inflation, the second order contributions to these energy density and pressures may dominate over the first order effects for the case of super-Hubble evolution. These results hold in our choice of first and second order coordinate conditions however we also argue that other `reasonable` coordinate conditions do not alter the relative importance of the second order terms. We find that these second order contributions approximately take the form of a cosmological constant in this coordinate gauge, as found by others using effective methods.

Long-wavelength metric backreactions in slow-roll inflation

TL;DR

The paper analyzes second-order corrections to linear cosmological perturbation theory in a slow-roll inflationary background, focusing on superhorizon (IR) backreactions. By computing the cumulative backreaction on the homogeneous sector via the eigenvalues of the total stress-energy, it shows that IR contributions behave like a negative cosmological constant in the chosen gauge and can dominate over linear terms for many slow-roll scenarios. The authors demonstrate that this backreaction grows with the number of e-foldings and may signal a breakdown of the linearized approximation in sufficiently slow-roll spacetimes. They also discuss gauge dependence, the role of UV–IR coupling, and the implications for interpreting local measurements of the cosmological constant in an inflationary context.

Abstract

We examine the importance of second order corrections to linearized cosmological perturbation theory in an inflationary background, taken to be a spatially flat FRW spacetime. The full second order problem is solved in the sense that we evaluate the effect of the superhorizon second order corrections on the inhomogeneous and homogeneous modes of the linearized flucuations. These second order corrections enter in the form of a {\it cumulative} contribution from {\it all} of their Fourier modes. In order to quantify their physical significance we study their effective equation of state by looking at the perturbed energy density and isotropic pressure to second order. We define the energy density (isotropic pressure) in terms of the (averaged) eigenvalues associated with timelike (spacelike) eigenvectors of a total stress energy for the metric and matter fluctuations. Our work suggests that that for many parameters of slow-roll inflation, the second order contributions to these energy density and pressures may dominate over the first order effects for the case of super-Hubble evolution. These results hold in our choice of first and second order coordinate conditions however we also argue that other `reasonable` coordinate conditions do not alter the relative importance of the second order terms. We find that these second order contributions approximately take the form of a cosmological constant in this coordinate gauge, as found by others using effective methods.

Paper Structure

This paper contains 15 sections, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Long-wavelength second order perturbation theory
  3. Linear and second order coordinate transformations; gauge fixing
  4. Total energy density and pressure at second order
  5. Eigenvalues of the total stress energy
  6. Linear contributions to the energy density and pressure
  7. Second order energy density and pressure perturbations at fixed $k$
  8. IR (super-Hubble) contributions from the backreactions
  9. Comment on backreaction on inhomogeneous second order modes
  10. Summary and Conclusion
  11. Acknowledgements
  12. Four-point correlation functions
  13. Second order equations of motion and correlation functions at second order
  14. Derivation of Inequalities (60) and (73)
  15. Scalar metric fluctuations in de-Sitter

Figures (1)

  • Figure 1: All linearized modes in the shaded region, spanning from $k=aH$ to $k=k_{min} << aH$ in spatial scale, are taken to seed second order modes at a particular value of $k$. The thick black boundary indicates the Hubble scale, where the amplitudes of the linearized fluctuations freeze out during slow-roll. We compare the amplitudes of the linearized fluctuations at the thick black boundary to that of the second order fluctuations at $k= \tilde{k}$ and $k=0$ (and ignore the influence of suitably renormalized subhorizon ($k> aH$) modes at second order) during slow-roll.