La Moral de Tartufo

To best scientifically study luminous objects, their brightness must be placed on a quantitative scale. This practice, and the resultant numbers, are referred to as photometry. The factors that affect a measured physical flux are numerous and complex, including many instrumental and atmospheric factors; the latter will necessarily change over the duration of observations. It is hence necessary to calibrate observations and the systems used to measure them, through intricate methods of photometric calibration. In particular, the importance of precise and accurate photometric calibration for supernova cosmology in particular will be stressed and explained later in this section.

Historically, astronomical magnitude was defined so that stars visible with the naked eye ranged from zeroth magnitude (brightest) to fifth magnitude; this has been formalised as negative logarithmic function of brightness, related to spectral flux density fν,1defined at a single frequencyν:

m=−2.5 log₁₀fν+ZP (3.1)

where ZP is a magnitude zero point whose definition depends on the magnitude system used. A mag- nitude system is a set of filter transmission curves (passbands) and zero points. Two broad classes of magnitude systems are the AB system (defined in Oke, 1965), on an absolute flux scale, and Vega-based magnitude systems, normalised to Vega (αLyrae). In the AB system, the magnitude scale is normalised such that a theoretical object with flux of 3631 Janskys has magnitude zero. In practice, magnitudes are measured in broad-band systems, where a set of filters, each letting a range of frequencies through with some transmission efficiency2_{T(ν) is used to define the magnitude of an object in multiple colours. The} magnitude is integrated as mAB,b=−2.5 log10 R bT(ν)fν dν R bT(ν)dν + 8.90, (3.2)

over a passband b, where fν is in Janskys (with 1 Jy= 10−26erg−1s−2Hz−1). Common standard

magnitude systems are the Johnson-Cousins ‘UBVRI’ system, and the filters used in the Sloan Digital Sky Survey ugriz (Abazajian et al., 2003, 2004) – these systems and more are described in great detail in the Bessell (2005) review. The Johnson-Cousins system was first established in Johnson (1966) as a

1_{The spectral irradiancef}

ν is defined as the energy radiated per unit surface area, per unit frequency. Spherical

symmetry of a source is assumed, so this decreases with distance from the object squared, much like intensity.

2_{In Equation 3.2 it is assumed thatT(ν) is defined per unit energy rather than per photon. Alternatively the transmission} can be defined as photon-counting, requiring the transformationT(ν)7→T(ν)

ν .

series of broadband filtersUBVRIJHKLMNfrom the ultraviolet to infrared ends of the visible spectrum, extending (alphabetically) into the infrared. Fluxes and colours in this system are normalised to Vega, and have been defined in terms of catalogues of stars distributed over the sky (e.g. Cousins, 1976; Landolt, 1983; Bessell, 1990; Landolt, 1992). Later, synthetic photometry has been used as an alternate and comparison method for calibration, by integrating model (‘synthetic’) spectra with transmission functions over instrumental passbands to determine theoretical magnitudes. In Vega-based magnitude systems, broadband magnitudes are close to their AB equivalents in Equation 3.2, differing by quantities known asAB offsets.

In recent years, the standard approach has trended toward publishing photometry in telescopes’ natural systems (rather than the standard magnitude systems such as UBVRI as defined in Landolt (1992)), alongside properties of the magnitude system that would have been used for calibration to a standard system. These include transmission curves and zero points for each passband, and sometimes the standard star photometry used to determine the zero points. Doing so makes the calibration process more transparent, and reduces potential cross-calibration errors, particularly when studying composite data sets from multiple telescopes.

Photometry can be considered in absolute or relative terms. Absolute calibration is tying a set of observations, on a given night in a given magnitude system, to an absolute scale (AB or Vega). The magnitude and flux are related to absolute physical units, requiring some sources with models for physical energy or flux emitted by them. The HST CALSPEC system is based on three DA white dwarfs, which have atmospheres that could be modelled to determine their luminosities theoretically. Alternatives have been ‘direct illumination’ calibration experiments (DICE), using LED illuminators as known sources for calibration (Barrelet & Juramy, 2008; Regnault et al., 2016; Barrelet, 2016).

Relative calibration is the process of ensuring internal consistency within a set of observations taken by some instrument in some magnitude system,before tying this magnitude system to an absolute scale. Here, the absolute flux scale is not considered, because only ratios of fluxes (differences or offsets between magnitudes) are considered.

Calibration methods of wide field surveys (where observations were necessarily taken over many epochs, and over areas of the sky which do not overlap in a single exposure) have been based on the HST CALSPEC network through a calibration chain of non-variable stars. In this method, a network of ‘secondary’ standard stars are calibrated (i.e. have magnitudes in a given system determined) by the three DA white dwarfs, the ‘primary standards’. Then for each science field, a set of ‘tertiary’ standard stars have their calibration tied to the secondary standards. The absolute magnitudes of these tertiary standards are fitted to determine zero points and colour transformations of the passbands in the natural magnitude system. These tertiary standards are observed on each night of scientific observations, to transfer the calibration of the CALSPEC network to the survey observations, and to determine the zero point offsets for that night. Examples of this framework are in SDSS, SNLS, Pan-STARRS (Padmanabhan et al., 2008; Betoule et al., 2013; Scolnic et al., 2015). SkyMapper currently utilises a similar calibration chain. The calibration of DES is via the Forward Global Calibration Method (FGCM), described in Section 6.4.1. The FGCM method defines the natural standard system of DECam; AB offsets are then used to transfer the CALSPEC zero points to the DECam system.

Calibration in supernova surveys

For supernovae, photometric calibration is especially relevant and important for several reasons: the magnitude is directly used to infer distance (i.e. they are standard candles), and it evolves rapidly over a timescale of days, on nights with different atmospheric conditions. Errors in relative calibration between nights will distort the measured lightcurve, and consequently the derived distances.

§3.1 Observational considerations 39

originating from its host galaxy. Methods involve host template subtraction, difference imaging, or in DES, Scene Modelling Photometry (Section 6.4.2); these are outlined in Section 2.4.2.

Moreover, magnitude systems are often calibrated using non-variable ‘standard’ stars, with sub- stantially different spectral energy distributions (SEDs) to supernovae; extrapolating calibrations to SNe inherently involves a degree of uncertainty (this is addressed by model-based methods of calibration such as the Forward Global Calibration Method (FGCM) in DES, which is described in detail in Section 6.4.1 of Chapter 6). As found in SNLS and JLA, the challenge of photometric calibration is that it is both highly complex and subtle, and at present especially important because of its large contribution to the error breakdown in cosmological surveys; these findings have greatly influenced subsequent developments in cosmology: the cross-calibration in JLA between SNLS and SDSS, the careful calibration (Supercal) in Pan-STARRs and later, and FGCM as part of DES calibration efforts. FGCM and its associated uncertainties are discussed further in Section 6.5.2.

Ensuring accuracy and consistency in photometric calibration of a supernova survey is important for numerous reasons. When measuring dark energy, we are concerned with the shape of the Hubble diagram (the relative rate of change of expansion). While best efforts are made to perform accurate photometric calibration, offsets in the absolute scale will impact the peak SN Ia magnitude MB (a

nuisance parameter for cosmology) but not the crucial cosmological parameterswand Ωmwhich we seek

to measure. It is critical then the calibration is consistent between all supernovae within the sample, particularly between high-redshift and low-redshift supernovae. On the other hand, when measuring the Hubble constant, we require the present expansion rate in actual physical units. Thus, supernovae are anchored to an absolute scale via a distance ladder (Section 4.3), whereby SN Ia absolute magnitudes are calibrated by Cepheid variables, which are then calibrated by geometric methods. The work in Chapter 4 relies on this principle. In this context it is critical that the calibration between different rungs of the distance ladder is consistent and accurate. As Section 4.3.3 will demonstrate, the absolute SN Ia peak magnitude MB is degenerate with the Hubble constant H0, so the absolute calibration is crucial. For finding H0 the ‘location’ of the Hubble diagram on the vertical axis is most important, whereas for dark energy, the shape is most important.