= Solution
<Type Ia supernovae> are <standardizable candles>. Multi-epoch photometry measures peak brightness, light-curve shape and color; spectroscopy identifies the supernova type and supplies a redshift, preferably also using the host. Correcting the empirical width–luminosity relation and attenuation reduces the spread of absolute magnitudes. A nearby sample fixes a distance-ladder intercept, while distant events test the shape of the <Hubble diagram>. The observable relation is
$$
\boxed{\mu=m-M=5\log_{10}\left(\frac{d_L}{10\,\mathrm{pc}}\right).}
$$
An unknown common $M$ remains degenerate with the Hubble-constant intercept unless externally calibrated; relative distance-versus-redshift shape can still constrain acceleration.
For a cosmological spacetime described by an <FLRW metric>, the <deceleration parameter> is $q=-a\ddot a/\dot a^2$, and the low-redshift luminosity-distance expansion is
$$
d_L(z)=\frac{cz}{H_0}\left[1+\frac{1-q_0}{2}z+O(z^2)\right].
$$
A range of redshifts therefore tests the expansion history beyond the leading linear Hubble law. Within conventional matter-plus-cosmological-constant models, standardized distant supernovae fainter than a specified decelerating prediction imply larger luminosity distances and can favor $q_0<0$. For late-time matter and a cosmological constant, $q_0=\Omega_{m,0}/2-\Omega_{\Lambda,0}$. A flat $(0.3,0.7)$ example has $q_0=-0.55$. One brightness residual alone does not establish acceleration across arbitrary expansion histories or arbitrary attenuation models.
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2013/paper-1-supernova-distance-models.png]
{title=Illustrative luminosity-distance curves: expansion histories change standardized supernova magnitudes}
The <supernova discovery of accelerating expansion> combined repeated imaging, difference-image candidate searches, spectroscopic classification and calibrated light curves. In the High-Z survey, CCD discovery images and later host images separated transient light from its host; point-spread-function fitting and artificial-star recovery tests checked the photometry. The measurements require K corrections between received and emitted passbands and account for the observed $(1+z)$ stretching of light-curve times.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[Sections 2.1–2.2, discovery imaging, spectroscopy and photometric controls]{ref}
Relevant systematic tests include dust and color laws, possible population evolution, flux-limited selection, calibration across filters and telescopes, gravitational lensing and host contamination. An accelerating interpretation is assessed by fitting a specified cosmological family and its nuisance parameters, checking those effects and comparing independent observations. The two principal discovery teams provided independent distant-supernova analyses, including the 42-event Supernova Cosmology Project result.<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[abstract and Section 2]{ref}
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