To find the maximum likelihood estimator (MLE) from the data, we must define the appropriate likelihood:

i.e. we have to first specify our model of the data. For a given SN Ia, the true data are drawn from some global distribution. These values are contaminated by various sources of noise, yielding the observed values . Assuming the SALT2 model is correct, only the true values obey equation (1). However when the experimental uncertainty is of the same order as the intrinsic variance as in the present case, the observed value is not a good estimate of the true value. Parameterising the cosmological model by θ, the likelihood function can be written as13:

which shows explicitly where the experimental uncertainties enter (first factor) and where the variances of the intrinsic distributions enter (second factor).

Having a theoretically well-motivated distribution for the light curve parameters would be helpful, however this is not available. For simplicity we adopt global, independent gaussian distributions for all parameters, M, x 1 and c (see Fig. 1), i.e. model their probability density as:

Figure 1 Distribution of the SALT2 stretch and colour correction parameters for the JLA sample11 of SN Ia, with our gaussian models superimposed. Full size image

All 6 free parameters are fitted along with the cosmological parameters and we include them in θ. Introducing the vectors Y = {M 1 , x 11 , c 1 , … M N , x 1N , c N }, the zero-points Y 0 , and the matrix , the probability density of the true parameters writes:

where |…| denotes the determinant of a matrix. What remains is to specify the model of uncertainties on the data. Introducing another set of vectors , the observed , and the estimated experimental covariance matrix Σ d (including both statistical and systematic errors), the probability density of the data given some set of true parameters is:

To combine the exponentials we introduce the vector and the block diagonal matrix

With these, we have and so . The likelihood is then

which can be integrated analytically to obtain:

This is the likelihood (equation (3)) for the simple model of equation (4), and the quantity which we maximise in order to derive confidence limits. The 10 parameters we fit are . We stress that it is necessary to consider all of these together and Ω m and Ω Λ have no special status in this regard. The advantage of our method is that we get a goodness-of-fit statistic in the likelihood which can be used to compare models or judge whether a particular model is a good fit. Note that the model is not just the cosmology, but includes modelling the distributions of x 1 and c.

With this MLE, we can construct a confidence region in the 10-dimensional parameter space by defining its boundary as one of constant . So long as we do not cross a boundary in parameter space, this volume will asymptotically have the coverage probability

where is the pdf of a chi-squared random variable with ν degrees of freedom, and is the maximum likelihood.

To eliminate the so-called ‘nuisance parameters’, we set similar bounds on the profile likelihood. Writing the interesting parameters as θ and nuisance parameters as ϕ, the profile likelihood is defined as

We substitute by in equation (10) in order to construct confidence regions in this lower dimensional space; ν is now the dimension of the remaining parameter space. Looking at the Ω m − Ω Λ plane, we have for {0.68 (“1σ”), 0.95 (“2σ”), 0.997 (“3σ”)}, the values respectively.

Comparison to other methods

It is illuminating to relate our work to previously used methods in SN Ia analyses. One method14 maximises a likelihood, which is written in the case of uncorrelated magnitudes as

so it integrates over μ SN to unity and can be used for model comparison. From Equation (3) we see that this corresponds to assuming flat distributions for x 1 and c. However the actual distributions of and are close to gaussian, as seen in Fig. 1. Moreover although this likelihood apparently integrates to unity, it accounts for only the data. Integration over the x 1 , c data demands compact support for the flat distributions so the normalisation of the likelihood becomes arbitrary, making model comparison tricky.

More commonly used1,8 is the ‘constrained χ2’