Inflaton Perturbations from Slow-Roll Inflation to Ultra-Slow-Roll Inflation
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Recently, in the search for the nature of dark matter, interest in alternatives to the WIMP-paradigm has been renewed. One such an alternative is the idea that dark matter consists of primordial black holes (PBHs), which were created by the gravitational collapse of overdense regions in the early universe. It was recently realized that a previously unrecognised regime of inflation exists, in which the slow-roll (SR) approximation breaks down, giving rise to the so-called ultra-slow-roll (USR) regime. During this regime, the primordial power spectrum can experience rapid growth, and therefore give rise to more PBHs. In this thesis, we match a period of USR inflation to a period of SR inflation, and analyse the growth of the curvature perturbations. We do this with taking corrections of order epsilon, which are often neglected, into account. We also make sure the background (the scale factor and Hubble parameter) is smoothly matched. The background changes only with order epsilon. There is also a rapid growth in the curvature spectrum, which can account for a higher concentration of PBHs.