An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation: an extended study

Abstract

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 Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} and its power spectrum PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} and its time derivative Rk,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_{k,N}$$\\end{document} on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} on five typical scales are investigated in order. Our analytical approximation predicts that Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} 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 PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics.