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Primordial Black Holes (PBHs) may be produced by gravitational collapse in regions with a large amplitude of density contrasts in the early Universe. They may provide the seeds for galaxy formation, account for a population of the LIGO-Virgo events, and the candidates of cold dark matter. Measurements or constraints on their abundance can be regarded as a probe of the inflationary models. In this seminar, I will show this by studying an inflation model that glues two linear potentials of different slopes, which is a simple toy model to enhance the inflationary scalar perturbations at small scales naturally. The enhanced perturbation can produce PBHs, together with gravitational waves induced through higher-order gravitational interactions between enhanced scalar and tensor fluctuations. Moreover, a very important issue about PBH is how to accurately estimate its abundance, as it is the observable quantity that can be detected directly. Apart from the Press Schechter method and the Compaction function method, people are developing a more precise method with using of the peak's theory. In the latter half of this seminar, I will talk about how to use the peak's theory to estimate the PBH abundance, especially for perturbations with a finite-width.
Recently, there has been increasing interest in the physical degrees of freedom (DoFs) of generalized theories in the teleparallel framework and the strong coupling issues with the vanishing coefficient of kinetic terms at a linear level in homogeneous or static backgrounds. We take f(T) gravity as an example. On the one hand, such phenomena are frequently observed in backgrounds with high symmetry, which plays a vital role in reducing the number of propagating modes at lower orders. On the other hand, such strong coupling in f(T) gravity may only exhibit strong nonlinear effects at extremely high energy scales for subhorizon modes. Furthermore, we expand the scope by introducing an extra scalar field non-minimally coupled to $f(T)$ gravity, aiming to address or alleviate the aforementioned strong coupling behavior.
Torsion and non-metricity can be used to describe gravity instead of curvature, providing new perspectives on nature of gravity. A general theory of gravity based on torsion and nonmetricity such as f(T) and f(Q) gravity can have multiple branches. The covariance of the f(T) theory can only be ensured by considering spin connections. While different ways of determining the spin connection give rise to different branches of covariant f(T) theory. In f(Q) gravity many studies have been conducted within the coincident gauge. Notably, in the case of flat FLRW metric, there exist three branches of connections that satisfy cosmological symmetry, with the coincident gauge being merely one among them. It is therefore worthwhile and interesting to study the nature and the relationship between the different branches in torsion and non-metricity gravity.