Comparison of piecewise-constant methods for dark energy

We compare four different methods that can be used to analyze the type Ia supernovae (SnIa) data, ie to use piecewise-constant functions in terms of: the dark energy equation of state $w(z)$, the deceleration parameter $q(z)$, the Hubble parameter $H(z)$ and finally the luminosity distance $d_L$. These four quantities cover all aspects of the accelerating Universe, ie the phenomenological properties of dark energy, the expansion rate (first and second derivatives) of the Universe and the observations themselves. For the first two cases we also perform principal component analysis (PCA) so as to decorrelate the parameters, while for the last two cases we use novel analytic expressions to find the best-fit parameters. In order to test the methods we create mock SnIa data (2000 points, uniform in redshift $z\in[0,1.5]$) for three fiducial cosmologies: the cosmological constant model ($\Lambda$CDM), a linear expansion of the dark energy equation of state parameter $w(a)=w_0+w_a(1-a)$ and the Hu-Sawicki $f(R)$ model. We find that if we focus on the two mainstream approaches for the PCA, i.e. $w(z)$ and $q(z)$, then the best piecewise-constant scheme is always $w(z)$. Finally, to our knowledge the piecewise-constant method for $H(z)$ is new in the literature, while for the rest three methods we present several new analytic expressions.