Probing the early universe by stochastic gravitational wave background

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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.