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,

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