An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation
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| Format: | Preprint |
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2024
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| _version_ | 1866910360942936064 |
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| author | Zhao, Ji-Xiang Li, Nan |
| author_facet | Zhao, Ji-Xiang Li, Nan |
| contents | Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation ${\mathcal R}_k$ and its power spectrum ${\mathcal P}_{\mathcal R}$ can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of ${\mathcal R}_k$ in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of ${\mathcal R}_k$ and its time derivative ${\mathcal R}_{k,N}$ on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, ${\mathcal R}_k$ on five typical scales are investigated in order. Our analytical approximation predicts that ${\mathcal R}_k$ first revolves around the origin in the complex plane, but if it crosses the horizon around the start of the USR stage, there appears a subsequent linear evolution towards or away from the origin. This behavior naturally explains the shape of ${\mathcal P}_{\mathcal R}$ from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of ${\mathcal P}_{\mathcal R}$ is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation Zhao, Ji-Xiang Li, Nan Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation ${\mathcal R}_k$ and its power spectrum ${\mathcal P}_{\mathcal R}$ can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of ${\mathcal R}_k$ in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of ${\mathcal R}_k$ and its time derivative ${\mathcal R}_{k,N}$ on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, ${\mathcal R}_k$ on five typical scales are investigated in order. Our analytical approximation predicts that ${\mathcal R}_k$ first revolves around the origin in the complex plane, but if it crosses the horizon around the start of the USR stage, there appears a subsequent linear evolution towards or away from the origin. This behavior naturally explains the shape of ${\mathcal P}_{\mathcal R}$ from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of ${\mathcal P}_{\mathcal R}$ is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics. |
| title | An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation |
| topic | Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2403.01416 |