Curvature Constant

The curvature constant comes from the concept credited to Nikolai Ivanovich Lobachevski who came up with the experiment that one can tell the curvature of the universe by drawing a large enough triangle.

Consider the three angle of the triangle \(\alpha,\; \beta,\; \gamma\). Recall that in Euclidean space (or common sense) that the three angle of the triangle should add up to \(180^\circ = \pi\). In an non-Euclidean space there exist a generalized equation for the sum of these angles for an isosceles triangle is:

\[ \begin{equation} \alpha + \beta + \gamma = \pi + \frac{\kappa A}{R_0^2} \end{equation} \]

• \(A\) : Area of the triangle

• \(R_0\) : Radius of curvature of any side (notice the radius of curvature for a straight line is \(\infty\))

• \(\kappa\) : The curvature constant which is either \(0\) or \( \pm 1\)

Often times, we do not need to deal with this number much doing cosmology so we assign a different form of the curvature constant,

\[ \begin{equation} k \equiv \frac{\kappa}{R_0^2} \end{equation} \]