= 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> $\chi(z)=c\int_0^z dz'/H(z')$. Photon energies and arrival rates each bring one factor of $1+z$, yielding the <luminosity distance> $d_L=(1+z)\chi$. Thus
$$
\boxed{\frac{F(z_1)}{F(z_2)}=
\left[\frac{(1+z_2)\int_0^{z_2}dz'/H(z')}
{(1+z_1)\int_0^{z_1}dz'/H(z')}\right]^2.}
$$
For the one-component constant-$w$ universe, $H(z)=H_0(1+z)^{3(1+w)/2}$. Its <luminosity distance> is
$$
d_L(z)=\frac{2c(1+z)}{H_0(1+3w)}
\left[1-(1+z)^{-(1+3w)/2}\right]\quad(w\ne-1/3),
$$
with 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. \b[Flux ratios constrain the expansion history through an integral of $1/H(z)$, rather than measuring $w$ directly without model assumptions.]
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