 Solution

The flux relation is $F=L/(4\pi d_L^2)$ with luminosity distance $d_L=(1+z)\chi (z)$. In a spatially flat universe,Thus Type Ia supernova cosmology measures the distance-redshift curve. Equation (2) makes $H(z)$ depend on an integral of $w(z)$, while $d_L$ introduces a second integral. Fitting predicted distances to many supernova fluxes over a range of redshifts therefore constrains parameters or bins describing $w(z)$, although these integrations smooth fine redshift structure.

The common luminosity $L$ need not be known to constrain the shape of $w(z)$. An unknown $L$ multiplies every inferred distance by the same factor and is degenerate with the overall scale $H_0^{-1}$, or equivalently with the supernova absolute magnitude. Relative distances at different redshifts still determine the shape of the expansion history. An external calibration is needed to determine the absolute distance scale.