= Solution
<Type Ia supernovae> have similar peak luminosities that can be standardized using their light-curve width and color, rather than assumed to be identical without correction. Their redshifts and corrected <apparent magnitudes> give a <distance modulus>,
$$
\mu=m-M=5\log_{10}\left(\frac{d_L}{10\,\mathrm{pc}}\right).
$$
Compare this <Hubble diagram> with the luminosity-distance prediction of a background cosmology. Restoring $c$, put $E(z)=H(z)/H_0$, $I(z)=\int_0^z dz'/E(z')$ and define $\mathcal S_{\Omega_k}(I)$ as $\sinh(\sqrt{\Omega_k}I)/\sqrt{\Omega_k}$ for $\Omega_k>0$, $I$ for $\Omega_k=0$, and $\sin(\sqrt{-\Omega_k}I)/\sqrt{-\Omega_k}$ for $\Omega_k<0$. Then
$$
d_L(z)=\frac{c(1+z)}{H_0}\mathcal S_{\Omega_k}(I(z)).
$$
Neglecting radiation at these redshifts, <Friedmann equations> and a separately conserved <dark energy> component with $w(z)=p/(\rho c^2)$ give
$$
E^2(z)=\Omega_{m0}(1+z)^3+\Omega_{k0}(1+z)^2
+\Omega_{{\rm de},0}\exp\left[3\int_0^z\frac{1+w(z')}{1+z'}\,dz'\right].
$$
For constant $w$, the last factor is $(1+z)^{3(1+w)}$; a <cosmological constant> has $w=-1$. Thus the redshift dependence of the measured distance constrains matter density, curvature and the dark-energy <equation of state>. At small redshift, $d_L=(cz/H_0)[1+(1-q_0)z/2+\cdots]$, showing how departures from the linear Hubble law probe the <deceleration parameter>.
The early distant-supernova analyses found larger distances than matter-only decelerating models predicted, providing evidence for accelerated expansion; see https://arxiv.org/abs/astro-ph/9805201[the 1998 supernova analysis] and https://arxiv.org/abs/astro-ph/9812133[the 1999 independent supernova analysis]. This is a statement about the distance-redshift curve under the standardization and cosmological assumptions, not a direct measurement of a fluid pressure.
The <absolute magnitude–Hubble constant degeneracy> means uncalibrated supernovae constrain the shape of $d_L(z)$ without separately determining $H_0$ and the common absolute luminosity. The integral over $H(z)$ also makes different matter, curvature and $w(z)$ histories partly degenerate. Wide redshift coverage and independent <cosmic microwave background> and <large-scale structure of the universe> information help break these degeneracies. Dust extinction, source evolution, photometric calibration, sample selection and <K corrections> must be modeled, and constraints on a time-dependent equation of state depend on how that function is parameterized. \b[Standardized supernovae test the expansion history; they do not by themselves uniquely determine every cosmological parameter.]
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