Abstract
We revisit one-loop corrections to the power spectrum of curvature perturbations ζ in an inflationary scenario containing a transient ultra-slow-roll (USR) period. In ref. [1], it was argued that one-loop corrections to the power spectrum of ζ can be larger than the tree-level one within the parameter region generating the seeds of primordial black holes during the USR epoch, which implies the breakdown of perturbation theory. We prove that this is not the case by using a master formula for one-loop corrections to the power spectrum obtained in ref. [2]. We derive the same formula within the path-integral formalism, which is simpler than the original derivation in [2]. To show the smallness of one-loop corrections, the consistency relations and the effective constancy of tree-level mode functions of ζ for super-Hubble modes play essential roles, with which the master formula gives a simple expression for one-loop corrections. For concreteness, we provide a reduced set of interactions including the leading-order one, while establishing the consistency relations in a self-consistent manner. We also show how the consistency relations of various operators hold explicitly, which plays a key role in proving the absence of large one-loop corrections.
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Acknowledgments
We thank Jacopo Fumagalli, Mohammad Ali Gorji, Keisuke Inomata, Jason Kristiano, Yusuke Mikura, Yuichiro Tada, and Shuichiro Yokoyama for useful discussions. RK is supported by JSPS KAKENHI Grants No. 24KJ2108. ST is supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No. 22K03642 and Waseda University Special Research Project No. 2024C-474.
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Kawaguchi, R., Tsujikawa, S. & Yamada, Y. Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach. J. High Energ. Phys. 2024, 95 (2024). https://doi.org/10.1007/JHEP12(2024)095
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DOI: https://doi.org/10.1007/JHEP12(2024)095