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## 2.1.1 FLRW Universe 

During the 1920s and 1930s, four scientists worked independently on prob- lems concerning the geometry and evolution of a homogeneous and isotropic Universe. These were Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker. The most general form of the line element in polar coordinates that satisfies homogeneity and isotropy, as well as allowing for uniform expansion is 

<img src='content_image/322154.jpg'>

where a ( t ) is the scale factor and K is the spatial intrinsic curvature. The value K = 0 corresponds to a spatially flat (Euclidean) Universe. A value of K < 0 and K > 0 correspond to a spatially open hyperbolic and spatially closed elliptical Universe respectively. This is known as the FLRW metric line element , after the authors mentioned above. Analysis of the CMB shows that our Universe is spatially flat to a very high degree of precision. For a K =0 spatially flat Universe, the line element in Cartesian coordinates is 

where the metric tensor is 

<img src='content_image/322153.jpg'>

<img src='content_image/322152.jpg'>

