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\section{Abstract}
We present a study of cosmic microwave background (CMB) temperature fluctuations in spherical spaces, which are the models of the universe where the space is a 3-dimensional sphere or its quotient. We give an overview of the preliminary machinery used in the study of CMB temperature fluctuations for spherical space. We examine the behavior of isospectral but not isometric spherical forms. We explore the problem of testing for anisotropies in the mean of CMB temperature fluctuations for spherical spaces.
We present a study of cosmic microwave background (CMB) temperature fluctuations in spherical spaces, which are the models of the universe where the space is a 3-dimensional sphere or its quotient. We give an overview of the preliminary machinery used in the study of CMB temperature fluctuations for spherical space. Next, we examine the behavior of isospectral but not isometric spherical forms. Finally, we explore the problem of testing for anisotropies in the mean of CMB temperature fluctuations for spherical spaces.
\section{Introduction}
The cosmic microwave background (CMB) is the electromagnetic radiation left from the Big Bang. Temperature fluctuations provide an idea of the early state of the universe, as well as possible hints on its current shape. Observations of space missions such as COBE, WMAP and PLANK have revealed an unexpectedly low variance in CMB anisotropies, which are small variations in the radiation, at really large angular scales \cite{aurich_2012}. This goes against the expected results of the infinite and flat universe from the standard $\Lambda$CDM model.