Probing the early universe by stochastic gravitational wave background

2022年9月与学生完成论文：《Starobinsky’s Linear Potential Model中的Primordial Black Hole Formation》，皮诗、王佳宁，e-Print: 2209.14183，被JCAP录用。我们重新研究了分段线性势的Starobinsky模型，并找到了曲率扰动功率谱最大值的精确位置和高度，这对确定原始黑洞（PBH）的形成是有用的。在2022年11月，我与Misao Sasaki教授完成了一篇论文：“曲率摄动的对数对偶性”，Shi Pi and Misao Sasaki, e-Print: 2211.13932。我们研究了具有分段二次势的单场膨胀。利用（经典）δΝ形式，我们发现曲率扰动一般可以用膨胀场扰动δφ的对数函数和来表示，而由于运动方程的二阶性质，每个对数表达式都有一个对偶表达式。我们阐明了单一对数占主导地位的条件，它产生了著名的R的PDF的“指数尾”或类似甘贝尔分布的尾。我们的公式统一了以往关于曲率摄动的全非线性pdf的所有工作，包括USR膨胀、慢滚膨胀、常滚膨胀、曲率情景等，并揭示了新的解析可解模型。它对PBH形成和引力波谱的精确计算有一定的指导意义。

In September 2022 I finished a paper with my student: "Primordial Black Hole Formation in Starobinsky's Linear Potential Model", Shi Pi and Jianing Wang, e-Print: 2209.14183, accepted by JCAP. We revisit Starobinsky model with a piecewise linear potential, and find the exact position and height of the maximum of the power spectrum of the curvature perturbation, which is useful for determining the primordial black hole (PBH) formation.In November 2022 I finished a paper with Prof. Misao Sasaki: "Logarithmic Duality of the Curvature Perturbation", Shi Pi and Misao Sasaki, e-Print: 2211.13932. We studied single-field inflation with a piecewise quadratic potential. By using (classical) δΝ formalism, we found that in general the curvature perturbation can be expressed by a sum of logarithmic functions of the inflaton field perturbation δφ, while each logarithmic expression has a dual expression due to the second-order nature of the equation of motion. We clarified the condition under which a single logarithm dominates, which yields either the renowned "exponential tail" of the PDF of R or a Gumbel-distribution-like tail. Our formula unifies all previous works on the fully nonlinear PDFs of the curvature perturbation, including USR inflation, slow-roll inflation, constant-roll inflation, curvaton scenario, etc., and reveals new analytically solvable models. It sheds light on accurate calculation of the PBH formation and the induced gravitational wave spectrum.