Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

Despite some of its successful features, the standard big bang cosmology su{uFB00}ers from many problems. The Flatness and Horizon problems are two of its main un- solved problems. In{uFB02}ationary cosmology is an alternative framework to standard cosmology within which the in{uFB02}ation solves some of these problems e{uFB03}ciently. Many scenarios of in{uFB02}ation are invented. The original version of in{uFB02}ation was based on adding a scalar {uFB01}eld to the energy-momentum at the early Universe whose dynamics causes the in{uFB02}ation. Other alternatives depend on modifying the geometrical part of the general theory of relativity. The f(R) modi{uFB01}ed theory of gravity is one of the most straightforward generalizations of such examples. However f(R) serves again as the general relativity minimally coupled with a geometry-inspired scalar degree of freedom. In{uFB02}ation in f(R) theories is one of the active topics of research these days. In{uFB02}ation in the framework of f(R) modi{uFB01}ed gravity is revisited. We study the conditions that f(R) should satisfy in order to lead to a viable in{uFB02}ationary model in the original form and in the Einstein frame. Based on these criteria we propose a new logarithmic model as a potential candidate for f(R) theories aiming to de- scribe in{uFB02}ation consistent with observations from planck satellite (2015). The model predicts scalar spectral index 0.9615 <ns <0.9693 in agreement with observation and tensor-to-scalar ratio r of order 10-3. Furthermore, we show that for a class of models, a natural coupling between in{uFB02}ation and a scalar boson is generated through the minimal coupling between gravity and matter {uFB01}elds and a reheating temperature less that 109 GeV is obtained

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