SDSS-II/SNLS3 Joint Light-curve Analysis

First arxiv version.

Same as v1 with additionnal information (R.A., Dec. and bias correction) in the file of light-curve parameters.

Same as v2 with the addition of a C++ likelihood code in an independant archive (jla_likelihood_v3.tgz).

Second arxiv version:

• Add the compressed version of the JLA likelihood (see Betoule et al. 2014, Appendix E)
• Make the C++ likelihood code (jla_likelihood_v4.tgz) the main product.

Correction of the C_stat.fits matrix (from archive file covmat_v4.tgz).

Previous to this release, the matrix contained in C_stat.fits was not the covariance matrix of light-curve parameters but a modified matrix, which, in particular was not positive. Off-diagonal entries of the \(C_{stat}\) matrix had been symetrized, the way the cosmomc plugin expect covariances between 2 different parameters to be given (as in the data/jla_*_covmatrix.dat files). As an exemple, the entry corresponding to \(cov(x_{1i}, c_j)\) with \(i \ne j\) contained instead \(0.5 \left[cov(x_{1i}, c_j) + cov(x_{1j}, c_i) \right]\).

It is important to note that this has no impact on the computation of the covariance of distance modulii and thus no impact in cosmological fits because both terms indeed play symetric roles there.

Change the presentation of the covmat tarball (see Error decomposition below) to match the presentation of uncertainties given in the paper.

Since March 2014, the JLA likelihood plugin is included in the official release of cosmomc. For older cosmomc versions, the original plugin is still available: jla_cosmo_v2.tgz.

Untar the archive and copy the following files to the corresponding cosmomc directory:

• source/supernovae_JLA.f90
• data/jla.dataset
• data/JLA.paramnames
• data/jla_*

Add the JLA likelihood in source/supernovae.f90:

use JLA
...
call JLALikelihood_Add(LikeList, Ini)

Modify the "SUPERNOVAE =" line of the cosmomc source/Makefile line to compile the JLA likelihood:

SUPERNOVAE = supernovae_Union2.o supernovae_SNLS.o supernovae_JLA.o

Depending on your cosmomc installation, you may obtain the same result by extracting directly the archive in your cosmomc directory.

We deliver separate covariances matrices following the decomposition of the error on light-curve parameters proposed in Betoule et al. 2014, Eq. 11. The ordering of the \(3 \times N_{\rm SN} = 2220\) vector of light-curve parameters is: \[ \vec \eta = (m^\star_1, {X_1}_1, C_1, \cdots, m^\star_{N_{\rm SN}}, {X_1}_{N_{\rm SN}}, C_{N_{\rm SN}}). \] The covariance matrix of this vector can be assembled as: \[ C_{\vec \eta} = C_{\rm stat} + C_{\rm cal} + C_{\rm model} + C_{\rm bias} + C_{\rm host} + C_{\rm dust} + C_{\rm pecvel} + C_{\rm nonIa} \] with:

• \(C_{\rm stat} \): statistical uncertainty (including the statistical uncertainty of the SALT2 Model, but without \(\sigma_{\rm coh}\) and \(\sigma_{lens}\) of Eq. 13)
• \(C_{\rm cal} \): calibration uncertainty
• \(C_{\rm model} \): systematic uncertainty on the model (See Mosher et al. 2014)
• \(C_{\rm bias} \): uncertainty on the bias correction
• \(C_{\rm host} \): uncertainty on the fonctionnal form of the luminosity-host mass relation.
• \(C_{\rm dust} \): uncertainty on the Milky Way dust column density
• \(C_{\rm pecvel}\): uncertainty on the peculiar velocity correction (Systematic only: does not include the \(\sigma_z\) term of Eq. 13.)
• \(C_{\rm nonIa} \): potential contamination by non-Ia.

Given the values of \(\alpha\) and \(\beta\) parameters, the covariance matrix of distance modulii can be assembled following Eq. 13 of Betoule et al. (2014):

\[ C = A C_{\vec \eta} A^T + {\rm diag}\left(\frac{5\sigma_z}{z\log 10}\right)^2 + {\rm diag}(\sigma_\text{lens}^2 ) + {\rm diag}(\sigma_\text{coh}^2) \,. \]

The \(3N_{\rm SN}\times 3N_{\rm SN} = 2220\times2220\) matrices and the \(\sigma_{coh}\) and \(\sigma_{lens}\) vectors are provided in covmat_v6.tgz. A sample python script is provided to illustrate computation of \(C\) from the given products.

Photometric calibration uncertainties in the training data sample affect the recovered SALT2 model. The resulting error in turn affects coherently all the light-curve fitted with this model. We emphasize that *assuming a perfect calibration of the model when fitting for supernovae can result in a serious underestimation of the actual uncertainty*.

In order to enable the propagation of uncertainties associated with the use of the SALT2 model, we release a set of 200 random realizations of the model, drawn according to the estimate of calibration uncertainties affecting the JLA training sample:

To obtain an estimate of the covariance of fit parameters: A possibility is to fit a set of SNe light-curves with each random realization and estimate the covariance of