, we obtain unbiased best-fit values for the stretch and colour response parameters, and, respectively (see Table 4). These differ by about 15% from the (biased) values given in A06. The absolute magnitude M is consistent with A06.
The parameters describing the zero-point shifts ( to ) are all consistent with zero (Table 4). Except for they have zero mean and rms of about. The influence of the derivative (Eq. (8)) on the second term of the likelihood (Eq. (9)) is small in comparison with the first term. Nevertheless, correlations with other parameters are introduced as is discussed in the next section. The mean of the z-band zero-point shifts is negative (although not significantly so) and its variance smaller than the expected value of. It is also this parameter which shows the highest correlation with cosmological parameters.
Table 4: Mean and 68% errors for SNIa internal parameters, for a flat wCDM model. The values of are in units of 0.01 mag.
In most cases the zero-point shifts are uncorrelated with other parameters. However, some pairs of one or more nuisance parameters show correlations, most notably, the absolute SNIa magnitude M is correlated both with the B- and V-band uncertainty and, respectively, see Fig. 5. This correlation can be explained by looking at the rest-frame colour parameter,, which contains the B-V colour excess at the time of the B-band maximum. Redefining the B- or V-band zero-points causes a systematic change in c which is compensated by a corresponding change in M. The slope of the measured -correlation is about 1.5. The correlation between M and is much weaker and has a smaller slope of -0.3 to -0.4. The correlation between M and the difference is therefore on the order of the best-fit value of, which is expected from Eq. (6). The difference in correlation can be explained by the fact that the observed magnitudes of each object translate into different rest-frame bands for the light-curve fit depending on the object's redshift.
Further, the z-band zero-point offset ( ) shows a correlation with both and w (Fig. 6). We find and. The latter value can be compared to A06 (Table 5) who varied the zero-points to infer the influence on cosmological parameters; our value is of the same order of magnitude as the one from A06 but has opposite sign. For a fixed we obtain, about half of the value cited in A06 (Table 5).
Correlations between the universal absolute SNIa magnitude M and the B-band (, left panel) and V-band (, right panel), respectively. The zero-point shift parameters are given in units of 0.01 mag.
Correlations between the z-band zero-point offset ( ) and ( left panel) respectively w ( right panel).
A06 discussed further sources of systematic uncertainties which can affect cosmological results from SNIa. Some of those uncertainties could be parametrised and invoked in a joint analysis with cosmology as presented in this paper. We state a few more examples here and leave a thorough analysis for future work.
For example, the difference between measured rest-frame U-band magnitude and the one ``predicted'' from the light-curve fit using only two bands can be included in the likelihood function.
In addition, the Malmquist bias can be modelled as a function of redshift. The inclusion of this bias will be necessary when redshift-dependent dark-energy models are to be tested. This requires accurate knowledge of the sample and of selection effects.
Furthermore, weak gravitational lensing of distant SNIa can be modelled using the dark-matter power spectrum, either from (non-linear) theoretical prescriptions (Ménard & Dalal 2005) or numerical simulations (Martel & Premadi 2008).
To calculate CMB temperature and polarisation power- and cross-spectra we use the publicly available package CAMB (Lewis et al. 2000). The likelihood is evaluated using the public WMAP5 code (Dunkley et al. 2009). We include all ``standard'' components which are the same for three- and five-year. Those are the low- TT and TE/EE/BB spectra, the high- TT and TE spectra, and the point-source TT correction. The low- ( for TT, for polarisation) likelihoods are calculated using Gibbs sampling (Dunkley et al. 2009; Page et al. 2007). The high- sampling uses pseudo- according to Hinshaw et al. (2007).
In contrast to the published WMAP5 results, we do not include corrections due to SZ. The SZ amplitude is unconstrained by WMAP5 and is not degenerate with other parameters (Dunkley et al. 2009). A recent analysis found no biases for WMAP5 when ignoring the contribution of the thermal SZ effect from clusters (Taburet et al. 2009).
We assume a flat dark-energy cold dark matter cosmology (wCDM) with the parameter vector. Clustering of dark energy is not taken into account. For CMB, instead of the normalisation defined at a scale of 8 Mpc/h,, the parameter which is sampled is, the curvature perturbations amplitude at the pivot scale. As stated in Sect. 2.1 we use the Smith et al. (2003) fitting formula to model the non-linear power spectrum needed for weak lensing. We do not include a massive neutrino component to the mass-energy tensor and assume as the effective number of massless neutrinos (Mangano et al. 2002), which is the preferred value for WMAP5 (Dunkley et al. 2009). Tereno et al. (2008) obtained constraints on the neutrino mass using various probes including CFHTLS-Wide weak lensing, see also Gong et al. (2008) and Ichiki et al. (2009). Additional parameters are (a,b,c,c0) from lensing and from SN Ia. These parameters are described in the respective subsections of Sect. 2; including systematics we sample a total of 22 parameters.
We use an adaptive Metropolis-Hastings algorithm (Metropolis et al. 1953; Hastings 1970) to generate Monte Carlo Markov Chains (MCMC) as sample of the posterior. The Fisher matrix evaluated at the maximum-likelihood (ML) parameter serves as initial multi-variate Gaussian proposal. The ML point is estimated by a conjugate-gradient search (Press et al. 1992). For lensing and SNIa the typical chain length is 100 000 with an acceptance rate of about 15-20%. Every 1000 steps the chain covariance is updated and replaces the previous proposal. The proposal variance is multiplied by a factor which is optimal for a Gaussian posterior (Dunkley et al. 2005; Hanson & Cunningham 1998).
For WMAP5 on the other hand, we choose a different strategy. With acceptance rates as stated above we found the posterior not well sampled and as a consequence the parameter uncertainties largely underestimated. We refrain from updating the proposal but kept the initial guess, which is larger than the Fisher matrix, for the whole duration of the chain. This choice results in small acceptance rates of 3% to 10% but guarantees an unbiased sampling of the posterior.
The experiments are combined by multiplying the corresponding likelihoods or, equivalently, by summing up the log-likelihoods. The confidence intervals are obtained by creating histograms of the parameter vectors of the chain and estimating regions with 68%, 95% and 99.7% of the density.
In particular for WMAP5 the calculation of the converged chain is very time-consuming and takes a few days to a week on a fast multi-core machine. To overcome this and other problems related to MCMC, such as the difficulty to assess convergence and the high correlation of a Markov chain, a new method called Population Monte Carlo (PMC, Cappé et al. 2007,2004) has been developed which is based on adaptive importance sampling. In a companion paper, we test this method with simulations and apply it to cosmology posteriors (Wraith et al. 2009).
68% and 95% confidence levels for lensing only, assuming wCDM (solid red lines), CDM (dashed red), and CMB only, for wCDM (solid blue), CDM (dashed blue curves).
The main contribution from cosmic shear (without tomographic information), in combination with CMB anisotropy measurements, are constraints on the normalisation (Contaldi et al. 2003; Tereno et al. 2005). For a CDM model, the degeneracy direction between and is orthogonal between lensing and CMB. Lensing is sensitive to the amount of structure and therefore, a high normalisation has to be countered by a smaller amount of the total matter. On the other hand, an overall increase of the CMB angular power spectrum by a higher normalisation has to be compensated by more matter to dampen the peaks.
This picture changes if the dark energy equation-of-state is a free parameter. Whereas the lensing contours for and do not broaden much, the CMB constraints increase dramatically. Most importantly, they increase along the lensing-direction of degeneracy, see Fig. 7. The result is that adding lensing to CMB data will not improve the constraints in the plane as much as it would do in the framework of a cosmological constant model.
Lensing without tomography is not very sensitive to w, thus leaving it a free parameter has a minor influence on the other parameters. For CMB however, strong correlations enter through the angular diameter distance which CMB measures only at z=1100. Those degeneracies (see Fig. 9) can be broken by adding an additional distance measurement at low redshift as a lever, e.g. using BAO or SNIa, (Blake &