Methods using SNe Ia
The χ2-minimization technique has traditionally been used for estimating cosmological
parameters and SN Ia standardization parameters from SN Ia light-curve fit data (Riess et al., 1998; Perlmutter et al., 1999; Kessler et al., 2009a; Marriner et al., 2011; Conley et al., 2011; Betoule et al., 2014). χ2-minimization relies on constructing an optimal
statistic which compares observations to theory, and using that statistic to evaluate the likelihood of the data given a proposed model. In the conventional framework, the best-fit model will be that which minimizes the value of the statistic and maximizes the value of the likelihood. However, this is not necessarily the case in the SN Ia cosmology problem, which requires a complex statistic to fully capture survey systematics.
Theχ2 statistic is frequently used in regression analyses to measure how well a data
random variableχ2is defined as χ2≡ N
∑
i=1 (xobsi −xmodi )2 σi2 , (2.8)wherexmodi is a function of the model parameters andσi2is the uncertainty of the data. If
the dataxobsi are independent with Gaussian errors, then χ2 follows a chi-squared distri-
bution χν2 with meanν, where ν indicates the number of degrees of freedom. When the
data are a good fit to the model, we expectχ2/ν ≈1. χ2/ν>>1 orχ2/ν<<1 indicate
the model is not a good fit to the data and should be rejected. The likelihood,L, is defined in terms of thisχ2, i.e.,
L ≡p(D|θ) (2.9)
L= (2π σ2)−N/2exp −χ2/2. (2.10)
In this description of the likelihood, the parameter setθ which minimizes χ2also maxi-
mizes the likelihood. The particularθmaxwhich satisfies this condition is known as amax-
imum likelihood estimator. Values ofθmaxobtained using aχ2-minimization (maximum-
likelihood) technique are single-valued and are assumed to have Gaussian uncertainties. In practice, the likelihood is evaluated by sampling p(D|θ)over the model parameter
space. In simple cases, this can be done by evaluating the likelihood across a grid of points in parameter space. However, as the dimensionality of the problem increases and/or the likelihood function becomes more complex, more sophisticated sampling mechanisms are required. It becomes more efficient to explore regions of parameter space near the peak of the likelihood distribution, rather than sampling across every possible point inθ.
One technique commonly used to explore the parameter space is Markov Chain Monte Carlo (MCMC). MCMC algorithms construct a “chain” of points in parameter space, where the position of each element in the chain is only informed by the position of its predecessor. For example, in the subclass of “random walk” MCMC algorithms, the chain “moves” by drawing new steps from a proposal distribution and comparing the likelihood of the new step to that of the previous step. Features of the proposal distribution can be
altered to adjust parameters such as the step size. A crucial property of an MCMC chain is that it ultimately evolves to a stationary or “target state” distribution independent of the starting point. If the chain has converged properly, this “target state” distribution is proportional to the probability distribution of interest. Several MCMC algorithms exist, with a variety of proposal distributions and other tunable parameters. Popular MCMC algorithms include the Metropolis-Hastings (Metropolis et al., 1953), Gibbs sampling, and ensemble sampling (Goodman and Weare, 2010).
In the context of SN Ia cosmology, theχ2statistic takes the form
χ2= (µobs−µmod)TC−1(µobs−µmod), (2.11)
whereµmodis the vector of theoretical distance moduli evaluated using the set of cosmo-
logical model parameters (Eq. 1.15).µobsis the set of observed distance moduli computed
for each SN Ia using the observed light curve and subsequent light-curve fit parameters (e.g., using Eq. 2.4).Cµ is the total covariance matrix which is often a linear combination of measurement uncertainties (Cobs), the intrinsic Hubble diagram dispersion, redshift
uncertainties, and other systematics. For example, the full covariance matrix used in the Betoule et al. (2014) analysis is given by
C =Cη+diag
5σz zlog10
2
+diag(σlens2 ) +diag(σcoh2 ). (2.12)
In this formalism, Cη includes contributions from systematics: uncertainties stemming from the error propagation of light-curve fitting, i.e.,Cobs; light-curve model uncertainties (which will depend on the regression cofficients α and β) and selection bias uncertain-
ties estimated from rigorous simulations of the SN Ia sample; uncertainties of the SN Ia host-galaxy masses; corrections for Milky Way extinction; peculiar velocities; and sam- ple contamination from core-collapse SNe. The other terms account for uncertainties in cosmological redshift, the variation in SN Ia magnitudes due to gravitational lensing, and any remaining intrinsic scatter not captured by other terms, respectively. Clearly, com- puting the full covariance matrix is nontrivial. Furthermore, this technique assumes that contributions to the covariance matrix are fixed across all of parameter space. While this
is a justified assumption for global systematics including Milky Way extinction and core- collapse contamination, it is possible that uncertainties related to selection effects vary across parameter space.
Althoughχ2-minimization has been widely adopted and tested by the SN Ia commu-
nity, a few significant issues remain:
1. Unlike Eq 2.8, the SN Ia χ2 includes parameters inθ, namelyα andβ, in the un-
certainty (C); i.e., the contribution toC from light-curve fitting is dependent onα
andβ. From Eqs. 2.11, it is apparent that certain values of (α,β) may maximize
the covariance, and thus minimize theχ2, but also maximize the difference between µmod andµobs. Becauseα andβ act as both range and location parameters, their
errors are not necessarily Gaussian. This means thatχ2/ν ≈1 is no longer a sat-
isfactory measure of goodness-of-fit. Furthermore, the value which minimizes the
χ2may no longer maximize the likelihood.
2. Although the intrinsic dispersion is treated as a model parameter, there is much va- riety in the way in which it is estimated. In many cases, the χ2 is first minimized
andσint is adjusted until χ2/ν ≈1. This is precisely what is done in Conley et al.
(2011), where a single σint2 is determined for each of the SN survey samples. Be-
toule et al. (2014) add additional degrees of freedom to their χ2 by splitting the
SN Ia sample into redshift bins and calculating a minimizedχ2per bin. They then
use these minimized χ2 values to iteratively determine the intrinsic scatter. Fit-
ting for the intrinsic dispersion in this way means that only a single number can be estimated without any uncertainty.
3. Many of the χ2-minimization analyses involve a combination of parameter infer-
ence techniques. A single analysis may includeχ2-minimization, iterative updates,
and marginalization to infer best-estimates of the parameters of interest. Therefore, it is difficult to compare uncertainties from these analyses to those that use more standard sampling techniques.