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The validity of separate-universe approach in transient ultra-slow-roll inflation

Rathul Nath Raveendran

TL;DR

This work addresses the breakdown of the separate-universe approach during transitions in transient ultra-slow-roll inflation by tracking the evolution of the comoving curvature perturbation $${\cal R}$$ and its conjugate momentum $${\Pi}$$ within a controlled two-transition USR model. It demonstrates that gradient terms $\nabla^2$ become temporarily important at both the slow-roll to USR transition and the return to slow-roll, undermining homogeneous evolution in those intervals. The authors show that the homogeneous solution for $${\Pi}$$ captures the first transition while the homogeneous solution for $${\cal R}$$ becomes valid across the second, and they interpret the $${\Pi}$$-driven curvature-like contribution to the energy density as a modification of the local Hubble parameter, valid only when the first slow-roll parameter $${\epsilon_1}$$ is small and strictly constant. These results support the use of an extended $\delta N$ framework under curvature-like corrections in specific regimes and motivate a Hamiltonian perturbation theory viewpoint that treats $${\cal R}$$ and $${\Pi}$$ on equal footing for more general inflationary histories.

Abstract

We investigate the breakdown of the separate-universe approximation during transitions in transient ultra-slow-roll inflation by analyzing the evolution of the comoving curvature perturbation ${\cal R}$ and its conjugate momentum $Π$. It is well known that spatial gradient terms lead to a failure of this approximation, particularly at the transition from slow-roll to ultra-slow-roll phase. We show that a similar breakdown also occurs during the second transition back to slow-roll when considering the evolution of $Π$. Interestingly, while the homogeneous solution for $Π$ accurately captures the dynamics across the first transition, it is the homogeneous solution for ${\cal R}$ that becomes valid across the second. Furthermore, we demonstrate that the spatial curvature term introduced in the extended $δN$ formalism of \cite{Artigas:2024ajh} can be interpreted as arising from the contribution of $Π$ to the energy density perturbation. Importantly, this modification of the local Hubble parameter is valid only when the first slow-roll parameter is both small and strictly constant.

The validity of separate-universe approach in transient ultra-slow-roll inflation

TL;DR

This work addresses the breakdown of the separate-universe approach during transitions in transient ultra-slow-roll inflation by tracking the evolution of the comoving curvature perturbation and its conjugate momentum within a controlled two-transition USR model. It demonstrates that gradient terms become temporarily important at both the slow-roll to USR transition and the return to slow-roll, undermining homogeneous evolution in those intervals. The authors show that the homogeneous solution for captures the first transition while the homogeneous solution for becomes valid across the second, and they interpret the -driven curvature-like contribution to the energy density as a modification of the local Hubble parameter, valid only when the first slow-roll parameter is small and strictly constant. These results support the use of an extended framework under curvature-like corrections in specific regimes and motivate a Hamiltonian perturbation theory viewpoint that treats and on equal footing for more general inflationary histories.

Abstract

We investigate the breakdown of the separate-universe approximation during transitions in transient ultra-slow-roll inflation by analyzing the evolution of the comoving curvature perturbation and its conjugate momentum . It is well known that spatial gradient terms lead to a failure of this approximation, particularly at the transition from slow-roll to ultra-slow-roll phase. We show that a similar breakdown also occurs during the second transition back to slow-roll when considering the evolution of . Interestingly, while the homogeneous solution for accurately captures the dynamics across the first transition, it is the homogeneous solution for that becomes valid across the second. Furthermore, we demonstrate that the spatial curvature term introduced in the extended formalism of \cite{Artigas:2024ajh} can be interpreted as arising from the contribution of to the energy density perturbation. Importantly, this modification of the local Hubble parameter is valid only when the first slow-roll parameter is both small and strictly constant.

Paper Structure

This paper contains 7 sections, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Evolution of perturbations
  3. separate-universe Approach and Spatially-Homogeneous ${\cal R}$ and $\Pi$
  4. Homogeneous solutions from linear perturbation theory
  5. Homogeneous solution of $\Pi$ and ${\cal R}$
  6. Transient Ultra-Slow-Roll inflation
  7. Conclusion

Figures (3)

  • Figure 1: Evolutions of $z"/(z \,a^2 \,H^2)$, $\theta"/(\theta \,a^2 \,H^2)$ and $k^2/(a^2 \,H^2)$ as functions of the number of e-folds $N$. The black dotted vertical lines indicate the two transition points in the background evolution. The blue dashed vertical lines mark the moments when $k^2 = z"/z$, while the orange vertical lines indicate when $k^2 = \theta"/\theta$. The figure shows that the evolution of ${\cal R}$ crosses the $z"/z$ and becomes affected by the $k^2$ term twice during the first transition. Similarly, the evolution of $\Pi$ crosses $\theta"/\theta$ and becomes influenced by the $k^2$ term twice during the second transition. These crossings signal the breakdown of the separate-universe approximation during the respective transition phases.
  • Figure 2: Evolutions of $2 \, \epsilon_1 k^{3/2}\vert{\cal R}\vert$ and $\vert\Pi\vert\, M_{\rm Pl}/(k^{3/2}a^3\, H\, )$ in Fourier space. The figure clearly demonstrates that the momentum term becomes dominant and cannot be neglected during the ultra-slow-roll phase in the expression for $\delta \rho$ given in \ref{['eq:drho-in-Pi-dphi']}.
  • Figure 3: Evolutions of $\vert{\cal R}\vert$ and $\vert\Pi\vert$ are shown as solid lines, obtained by numerically solving \ref{['eq:eom-R-Pi']}. These are plotted alongside the corresponding homogeneous solutions, initialized at different times, represented by data points. The light-blue solid line denotes the exact evolution of ${\cal R}$, while the blue data points represent the homogeneous solution ${\cal R}_{\rm h}$ when initial conditions are set before the first transition. The cyan data points show ${\cal R}_{\rm h}$ when initialized after the first transition. Similarly, the orange solid line indicates the exact evolution of $\Pi$. The pink data points correspond to the homogeneous solution $\Pi_{\rm h}$ initialized before the first transition, and the red data points correspond to $\Pi_{\rm h}$ when initialized after the second transition.