A complete analysis of inflation with piecewise quadratic potential
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arXiv
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| Format: | Preprint |
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2024
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| author | Wang, Xinpeng Ma, Xiao-Han Sasaki, Misao |
| author_facet | Wang, Xinpeng Ma, Xiao-Han Sasaki, Misao |
| contents | We conduct a thorough study of the comoving curvature perturbation $\mathcal{R}$ in single-field inflation with two stages, represented by a piecewise quadratic potential, where both the first and second derivatives are allowed to be discontinuous at the transition point. We calculate the evolution of $\mathcal{R}$ by combining the perturbative and non-perturbative methods consistently, and obtain the power spectrum and the non-Gaussian features in the probability distribution function. We find that both the spectrum and the statistics of $\mathcal{R}$ depend significantly on the second derivatives of the potential at both the first and second stages. Furthermore, we find a new parameter constructed from the potential parameters, which we call $α$, plays a decisive role in determining various features in the spectrum such as the amplitude, the slope, and the existence of a dip. In particular, we recover the typical $k^4$ growth of the spectrum in most cases, but the maximum growth rate of $k^5(\log k)^2$ can be obtained by fine-tuning the parameters. Then, using the $δN$ formalism valid on superhorizon scales, we give fully nonlinear formulas for $\cal{R}$ in terms of the scalar field perturbation $δϕ$ and its time derivative. In passing, we point out the importance of the nonlinear evolution of $δϕ$ on superhorizon scales. Finally, using the Press-Schechter formalism for simplicity, we discuss the effect of the non-Gaussian tails of the probability distribution function on the primordial black hole formation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A complete analysis of inflation with piecewise quadratic potential Wang, Xinpeng Ma, Xiao-Han Sasaki, Misao Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology We conduct a thorough study of the comoving curvature perturbation $\mathcal{R}$ in single-field inflation with two stages, represented by a piecewise quadratic potential, where both the first and second derivatives are allowed to be discontinuous at the transition point. We calculate the evolution of $\mathcal{R}$ by combining the perturbative and non-perturbative methods consistently, and obtain the power spectrum and the non-Gaussian features in the probability distribution function. We find that both the spectrum and the statistics of $\mathcal{R}$ depend significantly on the second derivatives of the potential at both the first and second stages. Furthermore, we find a new parameter constructed from the potential parameters, which we call $α$, plays a decisive role in determining various features in the spectrum such as the amplitude, the slope, and the existence of a dip. In particular, we recover the typical $k^4$ growth of the spectrum in most cases, but the maximum growth rate of $k^5(\log k)^2$ can be obtained by fine-tuning the parameters. Then, using the $δN$ formalism valid on superhorizon scales, we give fully nonlinear formulas for $\cal{R}$ in terms of the scalar field perturbation $δϕ$ and its time derivative. In passing, we point out the importance of the nonlinear evolution of $δϕ$ on superhorizon scales. Finally, using the Press-Schechter formalism for simplicity, we discuss the effect of the non-Gaussian tails of the probability distribution function on the primordial black hole formation. |
| title | A complete analysis of inflation with piecewise quadratic potential |
| topic | Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2412.16463 |