La Pequeña Virtud y el Talento Moral

In any astronomical survey, the targets included in the sample are subject to factors which determine which objects are observed, detected, or selected for study. The studied sample may not be entirely representative of the whole population, especially in magnitude-limited surveys where the primary criterion for detection is exceeding a brightness threshold. These selection effects can lead to potential biases in results if using the sample to directly determine intrinsic properties of objects, or to indirectly determine cosmological parameters. The most notable selection effect to consider is Malmquist bias, or the preferential selection of intrinsically brighter objects.

At higher redshifts, any disparity in detection efficiency can become more significant, and can be complicated by or entangled with factors such as population evolution with redshift. In supernova surveys, where the objects studied are used to measure the scale of the Universe at different redshifts, any brightness- or redshift-dependent biases can have a significant impact on derived parameters. It is thus imperative to estimate and correct for such biases in supernova samples. Where limited by selection effects, studying only detected supernovae (without corrections) will lead to biases in observables, which imply SNe Ia are intrinsically brighter, bluer, and slower-declining. Over the past several years various approaches have been employed to account for selection effects.

Selection efficiency

Selection functions of high redshift surveys have two components: the detection itself due to being magnitude-limited (a strict cutoff can be well-simulated with Monte Carlo if the underlying distributions are well understood), and the spectroscopic follow-up, which is influenced by many more human and logistical factors. JLA used the selection functions modelled in Dilday et al. (2008) for SDSS, and

Perrett et al. (2010) for SNLS – these are functions of both peak magnitude, and colour in the case of SDSS (which favours bluer events). These are combined in data/MC simulations, described in Mosher et al. (2014, section 6.2), Betoule et al. (2013, section 5.3).

Methods to compute and correct for biases in µ as a function of redshift, e.g. those in Mosher et al. (2014), have limitations. For instance, the values ofαand β are determined without bias corrections to begin with, and underlying colour and stretch distributions assumed in these simulations are approxi- mate. Scolnic & Kessler (2016) improve on these estimations by rigorously and iteratively determining these underlying colour and stretch distributions. They assume asymmetric underlying populations, and measure the migration of input parameters to observed distributions in surveys, given intrinsic scatter (various models), measurement noise, and detailed survey simulations. They also demonstrate that correcting for magnitude, stretch, and colour together as one parameter (distance modulus) is insufficient, and these individual biases should be treated independently. Improved understanding of underlying populations can be used for modelling more accurate corrections for biases from selection effects, through simulation.

Low-redshift selection functions

At low redshift, estimates of Malmquist bias are more uncertain. Unlike higher-redshift searches like SNLS and DES, these historically were not coordinated rolling searches detecting all transients to a given magnitude. Instead they were dependent on following up transients discovered by other sources (including amateurs). Moreover, selection is skewed, for example, by search strategies in LOSS, which targeted galaxies. It is more ambiguous than the higher-redshift surveys how appropriate data/MC simulations are. The approach in JLA was to consider both the magnitude-limited scenario, which can be modelled by data/MC simulations, and the volume-limited scenario, where selection biases have no impact, and use the difference between them as the uncertainty (Section 3.4.2) in the bias correction. The targeted discovery of supernovae in CfA3 and LOSS means they should not be magnitude limited; however, as observed in the JLA low-zsample, the colour distribution grows more blue with redshift, suggesting that some selection effect is at play. Approximating the bias correction for our low-zsupernovae in Chapter 4 using the JLA approximation is justified as our supernova sample is similarly distributed to the low-z

sample in JLA. In Pantheon, the magnitude-limited case is used as a baseline for low-redshift samples, averaging G10and C11 scatter models (explained in Section 3.4.7); we move towards using this bias correction for Chapter 6.

Modelling bias

Approaches to estimating Malmquist bias broadly involve Monte Carlo simulations of SN Ia lightcurves, using simulation packages such as SNCosmo (Barbary et al., 2016) and SNANA (Kessler et al., 2009a). Bias in observable parameters (magnitude, stretch, colour) due to selection effects can be estimated from these simulations, and corrected for prior to cosmology fitting. For these corrections to be accurate, the parent distributions of SNe Ia need to reflect what is observed in nature. These parent distributions, characterised by intrinsic scatter, colour distribution, and stretch distribution at any redshift bin, are used as inputs for simulating lightcurves; survey variables are then applied to predict what will be detected. Mosher et al. (2014) performs this process, and includes SALT2 fitting for ‘detected’ lightcurves, to infer biases in distance modulus due to selection effects, i.e. for each redshift bin, average difference between output distance moduli inferred from detected SNe Ia only and input values.

JLA (Betoule et al., 2014, section 5.3) relies on simulations in M14 to estimate bias corrections as part of end-to-end simulations of supernova surveys, which include the training of SALT2, lightcurve fitting, and bias corrections. These also incorporate the uncertainty associated with the lightcurve model (discussed in Section 3.4.7). The underlying colour and stretch distributions are approxi- mated by averaging over realisations of the simulations. These distributions were combined with realistic survey simulations and adjusted to match output (post-selection) distributions of parameters (redshift, stretch, and colour) with observables. Input and output distance moduli for these simula- tions were compared, to retrieve a bias in distance modulus binned by redshift. Several inputs for

§3.1 Observational considerations 41

intrinsic scatter models going into simulations (G10,C11, coherent); these are explained in Section 3.4.7. Simulation-based approaches to supernova cosmology (e.g. Bayesian hierarchical methods, Ap- proximate Bayesian Computation, and BEAMS (Kunz et al., 2007; Kessler & Scolnic, 2017)) are taking the approach of using statistical methods. Instead of trying to model the bias, these run a large number of Monte Carlo simulations to forward model the bias correction. These methods have a place in future dark energy studies including DES; and are further described in Section 3.2.3; some (e.g. BEAMS with bias corrections, or BBC, which will be mentioned in Chapter 6) have been validated on DES-like simulations.