Precision cosmology made more precise

So far, the standard attitude to solve the Friedmann equations in the simultaneous presence of radiation $R$, matter $M$ and cosmological constant ${\Lambda}$ is to find solutions $R_R (t)$, $R_M (t)$ and $R_{\Lambda} (t)$ separately for each individual component alone, and next to join them together, thereby obtaining a piecewise solution $R_{\rm pw} (t)$. We instead find the exact and analytic solution $R (t)$ of the same equations in flat space. Moreover, we quantify the error made when $R_{\rm pw} (t)$ is used in place of $R (t)$.