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  • Öge
    Dynamical system analysis of cosmological inflation models with axion-like-particles (ALP)
    (Graduate School, 2022-01-13) Çağan, Sermet ; Arapoğlu, Savaş A. ; 509181126 ; Physics Engineering
    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.
  • Öge
    Nucleosynthesis in alternative theories of gravity
    (Graduate School, 2022-06-22) Bulunur, İlayda ; Özdemir, Neşe ; 509191116 ; Physics Engineering
    Big Bang nucleosynthesis (BBN) is one of the most reliable tools for testing standard model cosmology, as well as alternative models, well-known models are Brans-Dicke's theory of gravity, quintessence models, and higher-dimensional models. Standard BBN employs general relativity and the standard model of particle physics, thus, relying solely on one adjustable parameter; the baryon number density. Predicted primordial abundances based on SBBN are calculated with the help of BBN codes that contain well-established thermonuclear reactions network involved during the early evolution of the universe and presented as a function of the baryon number density. Observations from CMB and large-scale structure distributions indicate that the baryon number density can be restricted to a small range, allowing us to derive the basic relationship between predicted primordial abundances and new parameters emerging from alternative models of cosmology. All modifications to SBBN enforce the expansion rate of the early evolution of the universe to change, resulting in new relic abundances that differ from element abundances predicted by SBBN. Hence, we can parameterize the deviations from SBBN by introducing the $S$ parameter as $S\equiv H'/H $ where $H'$ is the modified Hubble parameter, $H$ is the Hubble parameter in the first Friedmann equation derived from the Einstein equation inserting the FRW metric. $S$ is constrained with the range of $0.85 \leq S \leq 1.15$ to obtain the simple relations between relic abundances and free parameters of the alternative models. Therefore, with this range of $S$, we can bound for free parameters of non-standard cosmological models. This thesis focuses on two models; Brans-Dicke's theory of gravity and its extensions with self-coupling potentials, and five-dimensional pure gravity which has an extra curled and compact dimension. Both theories have two free parameters. For the five-dimensional pure gravity, the parameters are the scale factor of the extra dimension, $b(t)$, and the length of the extra dimension, $l_c$ whereas the Brans-Dicke theory has parameters $w$ and $\beta$ that comes from the evolution function of the scalar field as $\phi(t) = \phi_i e^{-\beta(t-t_i)}$. To constrain these parameters, we used predicted primordial element abundances, leftover in the first three minutes of the universe, as a function of the number baryon density and expansion rate factor, $S$. In our five-dimensional model, the scale factor $b$ and the length of the extra dimension, $l_c$, directly impact on the synthesis of light elements. Since the range of $S$ is kept limited, that is, the deviation from SBBN is minimal, it is anticipated that its effect decreases as time passes. Therefore, first, it is assumed that the evolution of an extra dimension is $b(t)=b_0e^{-\beta t}$. In that case, predicted $^4 {He}$ mass fraction $Y_p$, $De$ abundance, $y_D$ and $Li$ abundance as a function of $\beta$ and $l_c$ can be obtained and compared with the data inferred from observations. The Big Bang Nucleosynthesis (BBN) bounds on the parameters of the five-dimensional theory of gravity as $\beta \sim 2$x$10^{-2}$, $10^{-7} \lesssim l_c \lesssim 10^{-2}$. It can be seen that $\beta$ works only in a limited range while $l_c$ is suitable in an extensive range. Our motivation for an extra dimension comes from the string theory, which suggests that the extra dimension should be too small to be not detected in a large scale. Hence, it can be concluded that our results are compatible with our motivation. Also, we investigate another possibility that the evolution of the scale factor of an extra dimension as $b=b_0 t^{-p}$. In that case, $p$ is restricted on $p\sim 0.5$ while the broad range of $l_c$ satisfies the theory, $10^{-7} \lesssim l_c \lesssim 10^{-2}$. For Brans-Dicke theory of gravity, first, we studied the effects of the BD scalar field in the absence of potential, $V(\phi)$, on Big Bang Nucleosynthesis. Inserting the FRW metric to the Brans-Dicke field equation, we obtained the modified Hubble parameter of the theory, which depends on various parameters $(\phi,\Dot{\phi},w,\rho)$. Therefore, these parameters can directly alter the synthesis of primordial elements. Within the allowed range of $S$, it is assumed that the effects of a scalar field diminish over time as $\phi(t)=\phi_i e^{-\beta(t-t_i)}$, where $t_i$ is the initial cosmic time. These parameters can be constrained by using $^4 {He}$ mass fraction, $De$, and $Li$ abundances. It is found that $\beta$ is limited in the range of $10^{-5}-10^{-6}$, and for $w$ is $10^{-3}-10^{-2}$. Also, we have obtained the initial value of a scalar field extremely large value as $\phi_i = 1.3$x$10^7$. Next, we looked for alternative models which include scalar field potential, $V(\phi)$, to be compatible with data from BBN. The scalar field potential is taken polynomial function as $V(\phi) = V_0 \phi^n$. In all cases, from $n=-1$ to $n=3$, the same conclusion as the previous model without scalar field potential has been achieved; the theory is highly dependent on the initial condition of the scalar field and requires a considerably large value of $\phi_i$.
  • Öge
    Theoretical and observational aspects of inflationary cosmology
    (Graduate School, 2022-06-22) Akın, Kemal ; Arapoğlu, Savaş ; 509181112 ; Physics Engineering
    A testable theory of the universe has come up with Einstein's theory of general relativity. Combination of the theory with fundamental physics has provided significant understanding of the universe in the light of modern cosmological observations. On the other hand, the success of the hot Big Bang and Λ-CDM relies on the existence of dark energy and dark matter which are beyond the standard model of particle physics. Another required extension is inflationary mechanism which was suggested as a resolution to shortcomings Big Bang such as horizon and flatness problems. However, the biggest success of the inflationary paradigm is to explain the generation of initial perturbations that are responsible for the structure formation in the universe. A scalar field, called inflaton, leads to a exponential expansion in the early stage of the universe. Although inflation is a very strong theory for the early universe, direct test of the theory is not possible due to extremely-high energy scales. Instead, inflationary models are tested against observations come from imprints of the primordial density perturbations. An important pair of parameters that comes from the observations are spectral index n_s and tensor-to-scalar ratio r. Many inflationary models rely on slow--roll mechanism in which the inflaton slowly rolls through its potential minima so that equation of state parameter satisfies the acceleration condition ω < -1/3. Slow--roll parameters are used to dictate such behaviour to the inflaton field. Besides, perturbations and therefore inflationary observables can be expressed in terms of slow--roll parameters. Despite the fact that various minimally coupled single field models are consistent with current observations, quantum field theory in curved space anticipates a non-minimal coupling of scalar field to curvature scalar R. In this study, inflationary dynamics within the context of general relativity and scalar--tensor theories of gravitation is investigated. In the minimally coupled case, inflaton with a potential of the form φ^n is studied. In the non-minimally coupled case, same model with a coupling F(φ) = 1 + ξ φ^2 to curvature scalar is examined. In order to study inflation in scalar--tensor theories, usually conformal transformations are used, and for convenience the analysis is performed in Einstein Frame. In addition to standard Einstein frame analysis, we also perform the analysis in the Jordan frame. The predictions of the models are compared with PLANCK dataset using CosmoMC.
  • Öge
    D = 3 string theory review and closed string spectrum
    (Graduate School, 2023-07) Turhan, Şafak ; Özkan, Mehmet ; 509201102 ; Physics Engineering
    String theory is a framework in which all matter and force particles are mathematically represented by tiny vibrating strings. One of the most remarkable aspects of the theory is that it is a theory of quantum gravity, in string theory gravity emerges as in the scope of the closed string spectrum. Another quite intriguing property of string theory is the fact that it "dictates" the necessity of a specific space-time dimension, namely the critical dimension, in order for the theory to preserve the Lorentz invariance. It is exactly this aspect that the thesis will build up to and offer another way out other than the renowned 26-dimensional space-time. The thesis, as is customary, will start with a brief investigation of relativistic point particle. The reason laying behind this is that the string case will be treated in a very similar fashion. The relativistic point particle action and the equations of motion will be calculated. The action of the point particle can be generalized to the p-brane action, which is simply the action of a p-dimensional membrane. By making use of the generalized action, the action of the string will then be examined. In the scope of the thesis, the focus will be on the free bosonic strings. The string motion is represented by the worldsheet of the string, the 2-dimensional space-time surface which the string sweeps throughout its motion. The parametrization will be made by specifying the string coordinates by $\sigma$ and a time parameter $\tau$. The string action is derived by considering the area of the worldsheet and is called the Nambu-Goto action, from which will be moved on to obtain the conjugate worldsheet momentum in the Hamiltonian formalism. That conjugate momentum will give rise to two constraints, which then will give rise to the final form of the action that will be examined. The action contains a constant $T$, analogous to the mass $m$ in the relativistic point particle case, which is called the string tension. The equations of motion will again be derived and then the open string boundary conditions will be analysed: Dirichlet and Neumann boundary conditions. For the closed string, the periodicity condition will be introduced together with the reparameterization invariance of the string action. After all of the above-mentioned calculations, the conserved currents and charges will be calculated and after that, a switch to the light-cone coordinates will be made. It will be the spin part of the Noether charge that will be used to check the Lorentz invariance of the theory later on. Then, the string wave equation will be calculated. Moving on from the wave equation, the Fourier mode expansion will be written. Following the calculation of the conserved currents, the mass-shell condition will be derived. The canonical quantization procedure will then be conducted and it will appear that the oscillator modes of the pre-calculated mode expansion will correspond to the annihilation and creation operators when the quantization is made. All other operators will be quantized as well. Finally, it will be possible to check the Lorentz invariance of the theory by looking at the commutation relations of Lorentz charges. This will be equivalent to examining only the commutators of the spin parts of the Lorentz charges and requiring them to be equal to zero will give rise to the critical dimension of D = 26. But also, it will be shown that for the special case of D = 3 this commutation relation also vanishes i.e. preserves the Lorentz invariance. At last, the focus can be directed on the spectrum of the D = 3 theory. Even more specifically, to the D = 3 closed string. The Poincaré invariants will be calculated and then the level-matching condition will be shown. It will then be possible to obtain the states corresponding to different levels in terms of the creation and annihilation operators from before. Eventually, a set of calculations will be conducted to find the spins of different levels and then we will end up with a set of numbers that depend on the normal-ordering constant $a$. After examining the final results, it will be apparent that the spectrum gives rise to anyonic states at some levels regardless of the choice of $a$, states which has spin $s$ where $2s$ is not an integer. An effort to evaluate this result will be made and further areas for possible contributions will be discussed.