We consider the Lemaˆıtre metric, which is the inhomogeneous, spherically symmetric metric, containing a non-static, comoving, perfect fluid with non-zero pressure. We use it to generalise the metric of the cosmos algorithm, first derived for the zero-pressure Lemaˆıtre-Tolman (LT) metric, to the case of non-zero pressure and non-zero cosmological constant. We present a method of integration with respect to the null coordinate w, instead of comoving t, and reduce the Einstein’s Field Equation (EFEs) to a system of differential equations (DEs). We show that the non-zero pressure introduces new functions, and makes several functions depend on time that did not in the case of LT. We present clearly, step by step an algorithmic solution for determining the metric of the cosmos from cosmological data for the Lemaˆıtre model, on which a numerical implementation can be based. In our numerical execution of the algorithm we have shown that there are some regions which need special treatment : the origin and the maximum in the diameter distance. We have coded a set of MATLAB programs for the numerical implementation of this algorithm, for the case of pressure with a barotropic equation of state and non-zero Λ. Initially, the computer code has been successfully tested using artificial and ideal cosmological data on the observer’s past null cone, for homogeneous and non-homogeneous spacetimes. Then the program has also been generalized to handle realistic data, which has statistical fluctuations. A key step is the data smoothing process, which fits a smooth curve to discrete data with statistical fluctuations, so that the integration of the DEs can proceed. Since the algorithm is very sensitive to the second derivative of one of the data functions, this has required some experimentation with methods. Finally, we have successfully extracted the metric functions for the Lemaˆıtre model, and their evolution from the initial data on the past null cone.
Reference:
Alfedeel, A. 2013. The impact of inhomogeneity on the analysis of cosmological data. University of Cape Town.
Alfedeel, A. H. A. (2013). The impact of inhomogeneity on the analysis of cosmological data. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/4952
Alfedeel, Alnadhief Hamed Ahmed. "The impact of inhomogeneity on the analysis of cosmological data." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2013. http://hdl.handle.net/11427/4952
Alfedeel AHA. The impact of inhomogeneity on the analysis of cosmological data. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2013 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/4952