 Solution

For equal intrinsic bolometric luminosities $L$, the observed fluxes are $F(z)=L/[4\pi d_L(z{)}^2]$. In the flat model, a null geodesic gives the comoving distance $\chi (z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$. Photon energies and arrival rates each bring one factor of $1+z$, yielding the luminosity distance $d_L=(1+z)\chi$. ThusFor the one-component constant-$w$ universe, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Its luminosity distance iswith the continuous limit $d_L=c(1+z)\log (1+z)/H_0$ for $w=-1/3$. Measurements spanning several redshifts fit the shape of this relation and hence $w$; the common luminosity and $H_0$ cancel in the flux ratio. Equivalently, the relative distance modulus is $5{\log }_{10}[d_L(z_2)/d_L(z_1)]$.

Actual Type Ia supernovae are standardized through their light-curve shape and color rather than being identical standard candles. Scatter, dust extinction, changes in progenitor populations, photometric calibration, spectral corrections between observed bands, flux-limited selection, lensing magnification and low-redshift peculiar velocities all affect inferred distances. The assumed cosmological model is another uncertainty: with a separate matter component, $H^2/H_0^2={\Omega }_m(1+z{)}^3+(1-{\Omega }_m)(1+z{)}^{3(1+w)}$ in the flat case, so $w$ is partly degenerate with ${\Omega }_m$; allowing spatial curvature or varying $w$ adds degeneracies. Flux ratios constrain the expansion history through an integral of $1/H(z)$, rather than measuring $w$ directly without model assumptions.