= Solution
Type Ia supernovae are <standardizable candles>. Repeated multiband <photometry> measures peak <apparent magnitude>, light-curve width and colour; spectra identify the supernova class and determine <redshift>, often from the host <galaxy>. Nearby objects calibrate the relation between light-curve shape, colour and absolute <luminosity>. Applying those relations produces a corrected <apparent magnitude> $m_{\rm corr}$, not an assumption that every event has identical <luminosity>. The <distance modulus> then gives
$$
\mu=m_{\rm corr}-M=5\log_{10}\left(\frac{d_L}{\mathrm{Mpc}}\right)+25.
$$
A common uncertainty in $M$ is degenerate with the absolute distance scale, but does not erase the redshift-dependent shape of the <Hubble diagram>.
<luminosity distance> probes the expansion history. In a spatially flat model,
$$
d_L(z)=(1+z)c\int_0^z\frac{dz'}{H(z')},\qquad d_L(z)=\frac c{H_0}\left[z+\frac{1-q_0}{2}z^2+O(z^3)\right],\qquad q_0=-\frac{a\ddot a}{\dot a^2}\bigg|_0.
$$
For fixed low-redshift <calibration>, a negative $q_0$ increases the second-order distance relative to a decelerating model. Objects at a given <redshift> are therefore fainter. A <Hubble diagram> spanning enough <redshift> can distinguish this curvature from the common magnitude intercept. In matter-plus-cosmological-constant models, $q_0=\Omega_m/2-\Omega_\Lambda$; fitting the distances can test whether $q_0<0$. The two original supernova teams found distances supporting an accelerating expansion in that framework.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[abstract and cosmological interpretation of standardized supernova luminosity distances]{ref}<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[abstract; cosmological likelihood constraints and tests of systematic uncertainty]{ref}
The comparison requires matched rest-frame passbands and <K corrections>, correction for <redshift> time dilation in the <light curves>, a common <photometric calibration>, and explicit uncertainty propagation. Dust <extinction> can imitate dimming; multicolour and <infrared> observations test reddening and the adopted <extinction> law. Evolution in the progenitor population or host-galaxy dependence can alter standardization, so one compares matched populations and fits host-dependent terms where warranted. Flux-limited discovery introduces <Malmquist bias> and other selection effects, which must be modelled through the survey detection efficiency. <peculiar velocities> matter at low <redshift>, while <gravitational lensing> and intrinsic scatter broaden the distant sample. Classification and outlier tests reduce contamination by other transient types.
Independent data reductions, nearby anchors, spectroscopic checks and a fit over the full <redshift> range distinguish an expansion-history signal from one instrumental offset. Complementary cosmological observations help separate matter <density>, curvature and dark-energy parameters. \b[The acceleration inference comes from the relative distance-redshift relation, conditional on standardization and an expansion model.] Distant supernovae integrate the past expansion history; they do not directly measure the instantaneous second time derivative of today's <scale factor> without such modelling.
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