tokens: lower. If the unknown energy is from dark energy, this implies the existence of inhomogeneity of the dark energy (de Putter et al. 2010 ; Blomqvist 2008 , 2010 ). Appendix A NPB Correction for Patches with Beam Convolution Fourier transform presumes that the signal is periodic such that the two ends are continuous; thus, when one deals with real data, it is inevitable to encounter jump discontinuity, which induces extra power in the power spectrum. Although the data are mostly discrete, the NPB effect nevertheless persists and should be taken into account as systematic error. A patch taken from a full-sky map has such an intrinsic effect, which is uncorrectable because NPB is part of the morphology. The effect, however, is further aggravated when there is beam convolution. As the statistics and comparisons in this paper are made between beam-convolved Planck data and simulations with the best-fit ΛCDM model (without beam convolution), it is necessary to study the effect because the power spectrum of a beam-convolved patch has more induced power than the smoothed power spectrum of the nonperiodic patch. To estimate the correction, we simulate 50 full-sky CMB maps and take 30 deg 2 patches centered at the equator (in total 600 patches). We take the central 20 deg 2 part directly from the 30 deg 2 patches and denote their mean power spectrum as . Then we use 5′ to convolve the 30 deg 2 patches and choose the central 20 deg 2 part, the mean power spectrum of which is denoted . The correction is therefore , where b is the beam transfer function. One finds that the extra power due to 5′ beam convolution is a rather smooth curve and can be best fitted to be ( μ K 2 ) from l ≥ 300. And for the 143 GHz band map with beam transfer function 730 (Adam et al. 2016a ) the correction for NPB is ( μ K 2 ). Appendix B XPS on the <italic>Planck</italic> Data Throughout the paper we use XPS on the Planck data: four CMB maps, the 143 GHz channel map, and the foreground-subtracted maps at 100, 143, and 217 GHz from the SEVEM method, to extract the peak position. The XPS is a quadratic estimator between two maps that can provide an unbiased estimate of the underlying power spectrum of the correlated signals and, at the same time, reduce the uncorrelated ones. We take from the Planck Legacy Archive the FULL MISSION RINGHALF1 and FULL MISSION RINGHALF2 maps for SMICA, NILC, SEVEM, Commander, and the 143 GHz map. Since the patches from RINGHALF1 (denoted α ) and RINGHALF2 (denoted β ) contain the same CMB signal but uncorrelated noise, we can employ the XPS on their Fourier modes and ,where ∗ denotes complex conjugate and the angle brackets denote average over all for integer k . However, as Chiang Chen ( 2011 , 2012 ) pointed out, lack of an ensemble and nonzero chance correlation with the so-called “uncorrelated” signal results in some residual from the XPS, which has the following relation:where and are the (uncorrelated) noise power spectrum from RINGHALF1 and
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