Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space

The standard models of inflation predict statistically homogeneous and isotropic primordial fluctuations, which should be tested by observations. In this paper we illustrate a method to test the statistical isotropy of the mean of the cosmic microwave background temperature fluctuations in the spherical harmonic space and apply the method to the Wilkinson Microwave Anisotropy Probe seven-year observation data. A classical method to test a mean, like the simple Student’s t test, is not appropriate for this purpose because the Wilkinson Microwave Anisotropy Probe data contain anisotropic instrumental noise and suffer from the effect of the mask for the foreground emissions which breaks the statistical independence. Here we perform a band-power analysis with Monte Carlo simulations in which we take into account the anisotropic noise and the mask. We find evidence of a nonzero mean at 99.93% confidence level in a particular range of multipoles. The evidence against the zero-mean assumption as a whole is still significant at the 99% confidence level even if the fact is taken into account that we have tested multiple ranges.

Figure 1

The mean spectrum, $M_{\ell }$, of the overall combined map ($\mathrm{Q}+\mathrm{V}+\mathrm{W}$), defined in Eq. (12).

Figure 2

The covariance matrix of the mean spectrum, $M_{\ell }$, calculated by Monte Carlo simulations. The central red strip is diagonal element array and the surrounding blue strips indicate negative correlation. Only a part ($100\le \ell \le 200$) of the covariance matrix whose dimension is 300 is shown here.

Figure 3

The decorrelated band mean spectrum obtained by the seven-year WMAP data. The boxes and solid line are the result of the overall combined map ($\mathrm{Q}+\mathrm{V}+\mathrm{W}$) and color data points are the results of the individual frequency maps (Q, V, W). The horizontal width of boxes indicates the bin size $\Delta \ell =6$ (top panel), 10 (middle panel) and 20 (lower panel) for calculating the binned mean spectrum, ${\mathcal{M}}_i$.

Figure 4

The $p$ values of the data points in the lower panel in Fig. 3, for the bin size $\Delta \ell =20$.

Figure 5

The decorrelated band mean spectra for two different masks. The black solid line (thick boxes) is the result for the conventional mask (KQ75y7) and the red dashed line (thin boxes) is for the more expansive mask ($\mathrm{KQ}75\mathrm{y}7+\mathrm{\text{cut}}|b|\le 30°$).

Figure 6

The $p$ values of data points in Fig. 5.

Figure 7

The result from the CMB-free combination (dashed lines), over plotted on the lower panel in Fig. 3. It seems that the contribution from the noise is smaller than that of the signal in the multipole range we are discussing and the behavior of the noise at $\ell =221–240$ does not show any anomaly.