Test of consistency between Planck and WMAP

Within the context of the concordance model of cosmology we test the consistency of the angular power spectrum data from WMAP and Planck looking for possible systematics. The best fit concordance model to each observation is used as a mean function along with a Crossing function with an orthogonal basis to fit the data from the other observation searching for any possible deviation. We report that allowing an overall amplitude shift in the observed angular power spectra of the two observations, the best fit mean function from Planck data is consistent with WMAP 9-year data but the best fit mean function generated from WMAP-9 data is not consistent with Planck data at the $3\sigma$ level. This is an expected result when there is no clear systematic/tension between two observations and one of them has a considerably higher precision. We conclude that there is no clear tension between Planck and WMAP 9-year angular power spectrum data from a statistical point of view (allowing the overall amplitude shift). Our result highlights the fact that while the angular power spectrum from cosmic microwave background observations is a function of various cosmological parameters, comparing individual parameters might be misleading in the presence of cosmographic degeneracies. Another main result of our analysis is the importance of the overall amplitudes of the observed spectra from Planck and WMAP observations. Fixing the amplitudes at their reported values results in an unresolvable tension between the two observations at more than $3\sigma$ level which can be a hint towards a serious systematic.

Figure 1

In this plot the $3\sigma$ contours of the Crossing hyperparameters are plotted when we modify the WMAP 9-year best fit spectrum with a Crossing function (here Chebyshev polynomials) and compare with the WMAP 9-year data itself. All the plots show that the fiducial model (corresponding and $C_{1,2}=0$) lies well inside the $1\sigma$ contours of $C_1-C_2$. The plot in the left panel is obtained when we marginalized over $C_0$ and in the right panel we fixed $C_0=1$. The fiducial model remains nearly at the center of the $1\sigma$ C.L. in both cases. The reduced size of contours in the right panel is due to the fact that when we fix the amplitude the degeneracies decrease. In this particular analysis we have fixed ${\ell }_{\max }=1200$, however for plots hereafter we have used ${\ell }_{\max }=2500$, the maximum multipole covered by Planck.

Figure 2

Plots obtained using Planck best fit concordance model as the mean function to fit WMAP-9 and Planck data considering a Crossing function of the second order. We have used Chebyshev polynomials as the Crossing functions. Plots show that allowing the overall amplitude to vary (left panel) the two observations are in good agreement with each other having a large overlap of the confidence contours at $1\sigma$ level. The second plot is obtained by keeping $C_0=1$ (fixed amplitude) and varying the first and second order hyperparameters. In this case we can see a clear discrepancy between the two data sets. Our results indicate that if both observations insist on their reported values of the overall amplitude of the angular power spectrum, the inconsistency between the two data sets would be unresolvable.

Figure 3

WMAP 9-year data for temperature autocorrelation is plotted with different theoretical $C_{\ell }$’s. The WMAP best fit is plotted in red while the Planck best fit is plotted in blue. Note that the mismatch in power is evident near the first acoustic peak. It is shown that we can fit the WMAP data pretty well with the Planck best model modified by the Crossing function (in green).

Figure 4

Plots obtained using WMAP 9-year best fit spectrum as the mean function to fit WMAP-9 and Planck data considering a Crossing function of the second order. We have used Chebyshev polynomials as the Crossing functions. Plots clearly dictate that the WMAP-9 best fit model is not consistent with Planck data. When we fix the overall amplitude $C_0=1$, the fiducial model becomes further away from the confidence contours (right panel). Looking at these results still we cannot conclude that the two data sets are inconsistent. In fact the proper overlap of the confidence contours in the left panel (allowing a shift in the overall amplitude) reflect the consistency of the two data sets.

Figure 5

Planck data and various theoretical $C_{\ell }$’s are plotted here. The color codes of WMAP-9 and Planck best fit are the same as in Fig. 3. The green line corresponds to the WMAP-9 best fit model modified with best fit Crossing function. It can be clearly seen that the amplitude difference is appropriately addressed with the Crossing function. In the inset, Planck data for low-$\ell$ ($\ell =2–49$) is plotted. High-$\ell$ binned spectrum is plotted in the main plot which provides an idea of the differences between Planck and WMAP mean functions and the modified spectrum.

Figure 6

Samples of Crossing and inverse Crossing functions within the $2\sigma$ allowed range of Crossing hyperparameters are plotted. The largest multipole range covered by WMAP is indicated by the vertical line.