Standard candles and supernova Hubble diagrams

Let m B be a fitted rest-frame peak apparent magnitude, M B a standardized absolute magnitude, x one a light-curve shape parameter, and c color. Adopt a correction plus alpha times shape minus beta times color, with Delta denoting any additional net additive correction in magnitudes. This is our declared sign convention. Published pipelines may distribute host, selection, and calibration terms differently; their tabulated corrections cannot be inserted without checking those definitions.

The distance modulus is five times the base-ten logarithm of luminosity distance divided by ten parsecs. Expressing distance in megaparsecs adds twenty-five to that logarithm. A larger modulus means a greater inferred distance. Magnitudes are logarithmic, so averaging fluxes and averaging magnitudes are different operations. Measurement covariance among peak magnitude, shape, and color must be propagated through the correction rather than treating those three fitted quantities as independent observations.

For a fixed dimensionless expansion history, luminosity distance scales inversely with the present Hubble rate H zero. Supernova apparent magnitudes therefore determine a combination of the absolute magnitude and that rate. Relative distances across redshift can constrain the expansion history even when the overall luminosity scale is unknown. Determining H zero requires an external absolute calibration, such as geometric anchors connected through another distance indicator to supernova hosts.

The ladder must propagate shared calibration rather than count each rung as independent. A photometric zero-point error can shift many objects together, leaving a narrow residual distribution while biasing its mean. Redshift also enters through observed-to-rest-frame wavelengths and time dilation: an observed light-curve duration is longer by one plus redshift. Low-redshift peculiar velocities perturb the relation between measured redshift and background distance, so simply applying c z divided by H zero to every object is insufficient.

Near a survey’s detection threshold, intrinsically brighter events or upward noise fluctuations are more likely to enter the catalog. Selection also depends on color, cadence, host background, and requirements for classification and light-curve coverage. The resulting redshift-dependent bias does not disappear merely by fitting an extra scatter parameter. Forward simulations must reproduce the discovery and analysis selection, and their uncertain population and dust models must be varied when assigning bias corrections.

A useful Hubble-diagram analysis combines statistical covariance with correlated systematic contributions from calibration, source modeling, redshift corrections, and selection. It should check residuals by survey, host properties, color, and redshift using a predefined analysis rather than tune cuts until a preferred cosmology appears. Pantheon+ provides a concrete 2022 example of explicit standardization, bias corrections, and covariance. We use that dated methodology as a reference, while the following numbers remain entirely synthetic.