Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors
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| Format: | Preprint |
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2025
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| _version_ | 1866910159756853248 |
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| author | Prokopec, Tomislav Rigopoulos, Gerasimos |
| author_facet | Prokopec, Tomislav Rigopoulos, Gerasimos |
| contents | We examine the behaviour of the gauge invariant scalar field perturbations in an analytic inflationary model that transitions from slow-roll to an ultra-slow-roll (USR) phase. We find that the numerical solution of the Mukhanov-Sasaki equation is well described by Hamilton-Jacobi (HJ) theory, as long as the appropriate branches of the Hamilton-Jacobi solutions are invoked: Modes that exit the horizon during the slow-roll phase evolve into the USR as described by the first HJ branch, up to a subdominant $\mathcal{O}(k^2/H^2)$ correction to the Hamilton-Jacobi prediction for their final amplitude that we compute, indicating the influence of neglected gradient terms. Modes that exit during the USR phase are described by a separate HJ branch once they become sufficiently superhorizon, obtained by the shift $\left(ε_1,ε_2\right) \simeq \left(0,-6+Δ\right) \rightarrow \left(\tildeε_1,\tildeε_2\right)\simeq (0,-Δ)$ and corresponding to a slow-roll solution (very close to de Sitter) supported by the same potential. This transition is similar to the conveyor belt concept put forward in our previous work [1] and suggests that the limit $ε_2\rightarrow -6$ is unphysical as an asymptotic value for the background/long wavelength solution. We further discuss implications for the validity of the stochastic equations arising from the Hamilton-Jacobi formulation. Our work suggests that if Hamilton-Jacobi attractors are appropriately used, they can successfully describe the dynamics of long wavelength inflationary inhomogeneities for potentials with USR regions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors Prokopec, Tomislav Rigopoulos, Gerasimos General Relativity and Quantum Cosmology Cosmology and Nongalactic Astrophysics We examine the behaviour of the gauge invariant scalar field perturbations in an analytic inflationary model that transitions from slow-roll to an ultra-slow-roll (USR) phase. We find that the numerical solution of the Mukhanov-Sasaki equation is well described by Hamilton-Jacobi (HJ) theory, as long as the appropriate branches of the Hamilton-Jacobi solutions are invoked: Modes that exit the horizon during the slow-roll phase evolve into the USR as described by the first HJ branch, up to a subdominant $\mathcal{O}(k^2/H^2)$ correction to the Hamilton-Jacobi prediction for their final amplitude that we compute, indicating the influence of neglected gradient terms. Modes that exit during the USR phase are described by a separate HJ branch once they become sufficiently superhorizon, obtained by the shift $\left(ε_1,ε_2\right) \simeq \left(0,-6+Δ\right) \rightarrow \left(\tildeε_1,\tildeε_2\right)\simeq (0,-Δ)$ and corresponding to a slow-roll solution (very close to de Sitter) supported by the same potential. This transition is similar to the conveyor belt concept put forward in our previous work [1] and suggests that the limit $ε_2\rightarrow -6$ is unphysical as an asymptotic value for the background/long wavelength solution. We further discuss implications for the validity of the stochastic equations arising from the Hamilton-Jacobi formulation. Our work suggests that if Hamilton-Jacobi attractors are appropriately used, they can successfully describe the dynamics of long wavelength inflationary inhomogeneities for potentials with USR regions. |
| title | Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors |
| topic | General Relativity and Quantum Cosmology Cosmology and Nongalactic Astrophysics |
| url | https://arxiv.org/abs/2507.04114 |