Dynamical system analysis of cosmological inflation models with axion-like-particles (ALP)

Abstract

Inflation theory, developed in 1980 by Alan Guth, solves the two biggest problems of the standard Big Bang cosmology called flatness and horizon problem. The flatness problem essentially is a fine-tuning of the initial value of the energy density problem. The name itself comes from the relation between energy density to critical energy density ratio and curvature parameter. From current observations, we know that the deviation of the ratio of the energy density of the content of the universe to the critical energy density from unity is of the order $O\left(10^{-3}\right)$. Extrapolating this deviation back in time reveals that, in order to satisfy current observations, the value of the energy density has to be in agreement with the critical energy density of the order $O\left(10^{-62}\right)$. Therefore this extreme sensitivity to initial conditions arises the flatness problem. The horizon problem is the problem regarding the inexplicability of isotropy and homogeneity in the observed cosmic microwave background radiation (CMBR/CMB). CMB is almost uniformly in agreement on temperature distribution with $T \approx 2.7\ \text{K}$. One important fact of the CMB is that it contains regions that are separated by a distance larger than the particle horizon. Particle horizon is the definition of distance that light can reach from the start of the universe until now. Thus, regions or simply points in space-time that are separated more than the particle horizon are called causally disconnected regions. Causally disconnected points can never contact each other or ever be in contact previously. Therefore, CMB having causally disconnected patches that are almost in thermal equilibrium arises the question of how are the causally disconnected patches can reach a thermal equilibrium without the possibility of information exchange. Inflation theory solves those two major problems by introducing an exponential accelerated expansion in the very early universe before the start of the Big Bang theory. This accelerated expansion eventually reveals that there is no need for extreme fine-tuning of initial conditions on the energy density. Furthermore, the theory explains the horizon problem as rapid early accelerated expansion separates regions that were actually in causal contact but now seems to be causally disconnected, by the process called shrinking Hubble radius. There is no shortage of cosmological inflation theory models in the literature, starting from the original inflation theory model called chaotic inflation with squared potential to string theory motivated axion monodromy inflation. Axions are hypothetical pseudo-Nambu-Goldstone bosons that are emerged from solution to the CP problem, introduced by R.D. Peccei and H. Quinn in 1977. Axions in cosmology are regarded as the scalar field that enjoys the shift symmetry, i.e. $\phi \rightarrow \phi + \text{const}$ which solves the UV sensitivity of slow-roll inflation models. Cosmological inflation models can be examined by employing a mathematical method called dynamical system analysis. In this thesis, we tried to work out dynamical system analysis of two main axion-like inflation theory models in the linear stability analysis framework. In linear stability analysis, one defines meaningful model variables so that the evolution of said dynamical variables can be written in terms of the defined variables, i.e. there is no explicit dependence on the independent variables of the dynamical variables. This differential equation system building is called an autonomous equation system. Solution of the autonomous equation system yields several or no critical points of the system that the behaviour of mentioned critical points in the phase space can be understood by examining the eigenvalues of the evaluated Jacobian matrix at critical points of the autonomous system. There is more advanced method to determine the behaviour of critical points that fails to be determined in linear stability analysis but the scope of this thesis does not include them and further discussion on the reason for not including them is clarified in the thesis. We started with the linear stability analysis of a single scalar field having a natural inflation potential with several couplings to the gravity sector of the model. The analysis showed that having a cosine potential form is problematic in the definition of linear stability analysis therefore, we approximated to chaotic-like one. Results showed that in most of the configurations the critical points of the phase space behaves as an unstable point and in other cases linear stability theory fails to determine its behaviour. Moreover, we continued the analysis on the non-Abelian gauge field inflation model with extra scalar introduced to the model as an axion-like particle field with several different potential settings. We omitted the couplings to the gravity sector in this model for simplicity since most of the complexity comes from those said couplings and further difficulty comes from the fact that the model now has a multi-field form by definition. In a scenario where the extra scalar field is free, i.e. zero potential, with $F^{2}$ term has the coupling with the axionic field does not provide an inflationary period and by changing the potential to different forms, i.e. exponential, chaotic and general monomial we have found that in exponential case all critical points of the autonomous equation system becomes unstable and in chaotic-like and general monomial setting, none of the points' behaviour can be determined by linear stability analysis. The final attempt of linear stability analysis to axion-like field models was made to save the zero potential case by instead of coupling axion-like field to $F^{2}$ term we coupled it to a $F^{4}$ term which automatically solves the problem of not having an inflationary period since now the extra contribution coming from the $F^{4}$ has the equation of state parameter value of minus one. Although inflationary period is saved, linear stability method suffers from the non-minimal couplings since in order to observe the effect of newly introduced term one needs to use the same dynamical variables defined in the $F^{2}$ model, and while most of the equations can be written in required form, some explicit dependence to the coupling functions makes the model non-closed therefore none examinable with the same variables. Therefore, a direct comparison between those two models can not be made without defining a new variable set. As a result, we learned that the examination of axion-like cosmological model is not viable utilizing the dynamical system analysis with linear stability analysis constraint.