Perturbations and photon geodesic on cosmological models

Cosmologists have been dealing with two points since the beginning of the modern cosmology area. One point is trying to describe the evolution of the universe and the other point is to find out the past, today and the future of the universe. Evolution of cosmology has been focused on to figure out detailed answers of these questions and to guess the final fate of the universe. These new aspects have been based on cumulative corpus of modern cosmology. Although there are many differences, new aspects are the product of old fashioned cosmology. After a hundred years of Einstein's relativity theory, cosmology have been focused on many ranges such as large scale structures or high energy physics. In 1981, A. Guth pointed out the relation between high energy physics and cosmology. The wide ranged work clarified that the inflation mechanism of Guth's trying to give a great aspect to solve flatness, horizon and initial perturbation problems . Furthermore, importance of the Cosmic Microwave Background Radiation which is discoverd in 1964, have been rised up after this progressive work of Guth's. Consequences of perturbations on metric have a significant role on explanation of Cosmic Microwave Background Radiation (CMBR).In the "old-fashioned cosmology" Friedmann-Robertson Walker metric is considered unperturbatively. Unperturbed universe can not produce a physical phenomena such as CMBR. On the other hand, primordial gravitational fluctuations could not be occured. Thus, metric perturbations are necessary for modern cosmology and consequences of observations which are made by astro-physicists. Moreover, the theoretical products of Sachs-Wolfe works have revealed the secret of CMBR and have associated the knowledge of photons, which are coming from the last scattering surface (LSS), with gravitational potential in 1967. This theory have been based on calculation of photons' geodesic with perturbed Friedmann-Robertson-Walker (FRW) metric. R.Sachs and A.M Wolfe predicted,which are the fluctuations over temperature over large scale structure in order of $10^{-2}$ . These effect can be observed by frequency shifts of light or Doppler effect and the observations provide some information of gravitational fields. Furthermore, the part of Sachs-Wolfe study, provides new aspects such as examination and also validation of alternative gravitational theories or the nature of dark energy. This effect called up Integrated Sachs-Wolfe (ISW) effect. Integrated Sachs Wolfe effect is based on a unique mechanism which depends on the evolution of gravitational potential with respect to time derivative. Cosmic Microwave Background Radiation photons comoving from LSS pass on through time dependent gravitational potential. It was proven that, photons gain or lost their energy by the gravitational potential wells or peaks. The integration of time derivative of the gravitational potential can be measured and observable today. In matter dominated Universe, the gravitational potential is static. Thus, Integrated Sachs-Wolfe effect has to be vanished in matter dominated universe. This result of ISW effect triggers a new result. ISW effect must be observed on radiation dominated universe or dark energy-curvature dominated universe. Thus, ISW effect provides a great probe for understanding accelerating expansion of the Universe. In GR, theory has a significant imprint on large scale structures which is scale independent density growth factor at sub horizon scale. On the other hand, alternative gravity models, which introduce arbitrary function with respect to Ricci scalar on Lagrangian, provides scale dependent due to nature of itself. These unique features of modified gravity, provide answers to the large scale structures and have been observed on some physical phenomenon such as galaxy clusters or late ISW effect. Some theoretical physicists predicted that \emph{f(R)} theories can be observed and produce by observable early time ISW effect. In our thesis we focus on metric perturbations with respect to gravitational potential. These gravitational fluctuations cause an external effect on physical phenomenon which are related with metric perturbations. Then we study on photon geodesies in perturbed space.Finally, we examine metric perturbations in \emph{f(R)}gravity as an example of alternative gravity theories, which are candidate to explanation of dark energy and also nature of dark matter. In section one, we give some basic definitons our work and describe how modern cosmology models have motivated. In section two, we describe the physical interpretation of unperturbed, expanded Universe. Also, second section has description about how "old-fashioned Cosmology" has begun and how the Einstein-Hilbert action has a physical role. The physical meaning of variation of action gives the field equations of gravity. Historically these equations are called Einstein field equations. Furthermore, in the second part of our work, we give the perfect fluid solution of the Einstein field equation, in the flat De-Sitter space which are called Friedmann equations. In the third part of our work, we tried to give an introduction the metric perturbations. We describe perturbed FRW metric and its physical interpretations. Besides, in this section we try to understand how we transform the coordinates and how these transformations have gauge-freedom. Although, we do not describe it detailed way, we choose Newtonian gauge and we focus on consequences of this gauge. Fourth part contains the physical interpretation of perturbed universe and consequences of itself. These consequences are called Integrated Sachs Wolfe effect and it derived from photons geodesic equation, which are coming from LSS. Last part of our thesis contains alternative gravitational model. We introduce how action changes in modified gravity and also change of field equations. Furthermore, we examine to understand how potential fields act on composition of Universe, which is a new type of Poisson equation. Moreover, we have to give an answer to how ISW effect related with f(R) theory and how it can validate the other theories. Also the last part of the thesis examine f(R) theory with General Relativity. These theory contains the generalizing the Einstein-Hilbert action and gives the answer about how the gravitational fields act with R. Also we show that ISW effect has a same meaning in f(R) theory except it is act with modified R, which also gives an answer to generalize the action. Generalizing of the R action has a significant meaning and has physical role. The results of the thesis also contains how the alternative gravity models can be reproduced. The relation between these effects are constraining the limits of alternative gravity models. Numerical methods shows that the limitation of the alternative gravity models, such as f(R)theory. Also, these results are pointing out the evidence of dark energy. In this thesis we review photon geodesies, metric perturbations and their observable results. Since alternative gravity theories may have imprints on the photons geodesics, such as Sachs- Wolfe and Integrated Sachs-Wolfe effect, we will study effects of alternative models on these effects in near future.