Testing Lambda and the Limits of Cosmography with the Union2.1 Supernova Compilation

We present a cosmographic study designed to test the simplest type of accelerating cosmology: a flat universe with matter and a cosmological constant ($\Lambda$). Hubble series expansions are fit to the SCP Union2.1 supernova data set to estimate the Hubble Constant ($H_{0}$), the deceleration parameter ($q_{0}$), and the jerk parameter ($j_{0}$). Flat $\Lambda$CDM models always require $j_{0} = 1$, providing a single-parameter test of the entire paradigm. Because of convergence issues for $z \gtrsim 1$, we focus on expansions using the newer redshift variable $y$, and to estimate the effects of "model-building uncertainties" -- the dependence of the output results upon the fitting function and parameters used -- we perform fits using five different distance indicator functions, and four different polynomial orders. We find that one cannot yet use the supernova data to reliably obtain more than four cosmological parameters, and that cosmographic estimates of $j_{0}$ remain dominated by model-building uncertainties, in conjunction with statistical and other error sources. While $j_{0} = 1$ remains consistent with Union2.1, the most restrictive bound that we can place is $j_{0} \sim [-7.6,8.5]$. To test the future prospects of cosmography with new standard candle data, ensembles of mock supernova data sets are created, and it is found that the best way to reduce model-building uncertainties on lower-order Hubble parameters (such as $\{H_{0}, q_{0}, j_{0} \}$) is by limiting the redshift range of the data. Thus more and better $z \lesssim 1$ data, not higher-redshift data, is needed to sharpen cosmographic tests of flat $\Lambda$CDM.