Too few spots in the cosmic microwave background

We investigate the abundance of large-scale hot and cold spots in the WMAP-5 temperature maps and find considerable discrepancies compared to Gaussian simulations based on the $\Lambda \mathrm{CDM}$ best-fit model. Too few spots are present in the reliably observed cosmic microwave background (CMB) region, i.e., outside the foreground-contaminated parts excluded by the KQ75 mask. Even simulated maps created from the original WMAP-5 estimated multipoles contain more spots than visible in the measured CMB maps. A strong suppression of the lowest multipoles would lead to better agreement. The lack of spots is reflected in a low mean temperature fluctuation on scales of several degrees (4°–8°), which is only shared by less than 1% (0.16%–0.62%) of Gaussian $\Lambda \mathrm{CDM}$ simulations. After removing the quadrupole, the probabilities change to 2.5%–8.0%. This shows the importance of the anomalously low quadrupole for the statistical significance of the missing spots. We also analyze a possible violation of Gaussianity or statistical isotropy (spots are distributed differently outside and inside the masked region).

Figure 1

Coefficients $W_l$ for the top hat circle window function at scales $a=1°$ (right plot), 6° (left plot). The plots show which multipoles predominantly determine $\Delta T_{\mathrm{rms}}$. For smaller angular scale $a$, higher $l$ values enter the analysis.

Figure 2

Mean temperature fluctuation for various spot sizes and the $\Lambda \mathrm{CDM}$ power spectrum. The plots for circles and squares are visually indistinguishable. The difference between the result for circles and the result for squares is shown in the second figure.

Figure 3

The foreground-reduced $V$ map (temperature contrast in mK) and the KQ75 mask cutting out the contaminated galaxy region and point sources.

Figure 4

Exemplary decomposition of the sky into $N_{\sec }$ distinct sectors ${\mathcal{S}}_j$ for measuring $\Delta T_{\mathrm{rms}}$. For searching spots, the algorithm analyzes many more sectors $\mathcal{S}$ (those in between, sharing pixels with the illustrated sectors ${\mathcal{S}}_j$). Nonetheless, $N_{\sec }$ limits the maximum number of spots since overlapping spots are not multiply counted.

Figure 5

Mean spot abundances in 100 simulated $\Lambda \mathrm{CDM}$ full-sky maps showing the results for different window functions of scale $a=\sqrt{A}=6°$.

Figure 6

Mean spot abundances for a fixed threshold ($80\mu \mathrm{K}$) against a varying step size. 100 masked $\Lambda \mathrm{CDM}$ simulated maps were scanned for $a=6°$, the error bars quantify the statistical error.

Figure 7

Spot abundances in the CMB sky (with cosmic variance) as compared to 500 $\Lambda \mathrm{CDM}$ simulations (with statistical errors) on an angular scale of $a=6°$. The fractions of Gaussian simulations with smaller values of $s$ [Eq. (17)] are $p_s^{\mathrm{hot}}=0.2\%$ and $p_s^{\mathrm{cold}}=1.8\%$.

Figure 8

Spot abundances in five randomly chosen Gaussian simulations based on the $\Lambda \mathrm{CDM}$ best-fit power spectrum and the mean curve from Fig. 7 (hot spots).

Figure 9

The mean temperature fluctuation for different angular scales $a$ in 50 Gaussian $\Lambda \mathrm{CDM}$ simulations and the $V1$ map.

Figure 10

Spot abundances in the CMB sky (with cosmic variance) as compared to 500 simulations (with statistical errors) based on the WMAP-5 $C_l$ spectrum on an angular scale of $a=6°$. The fractions of Gaussian simulations with smaller values of $s$ [Eq. (17)] are $p_s^{\mathrm{hot}}=3.4\%$ and $p_s^{\mathrm{cold}}=13.2\%$.

Figure 11

Spot abundances in the ILC map (with cosmic variance) compared to simulations (with statistical errors) based on the measured WMAP-5 $C_l$ on an angular scale of $a=6°$ for three different parts of the sky. The corresponding values of $p_s$ are $p_s^{\mathrm{full}}=58\%$, $p_s^{\mathrm{out}}=7\%$, and $p_s^{\mathrm{in}}=96\%$.

Figure 12

Spot abundances of Gaussian simulations $k$ (errors statistical) with $\Delta T_{\mathrm{rms}}^{(k)}\le \Delta T_{\mathrm{rms}}^{V\mathrm{map}}$ in comparison to the $V$ map (with cosmic variance). For these plots, we have $p_s^{\mathrm{hot}}=68\%$ and $p_s^{\mathrm{cold}}=86\%$, showing agreement. Since we only consider Gaussian simulations with $\Delta T_{\mathrm{rms}}^{(k)}$ smaller than in the $V$ map, it is no surprise that the $p_s$ values lie above 50%.

Figure 13

The mean temperature fluctuation for large angular scales $a$. We compare the $V1$ map with $\Lambda \mathrm{CDM}$ simulations (highest $\Delta T_{\mathrm{rms}}$), simulations based on the WMAP-5 $C_l$ (a bit lower), the WMAP-1 $C_l$ for $l<33$ (still lower), and on two modified spectra. The first modification is created by setting $C_l=0$ for $l\le 3$, the second by halving the $C_l$ for $l\le 5$. The modified spectra agree well with the $V1$ map.

Figure 14

Spot abundances in the CMB sky (with cosmic variance) as compared to 100 simulations based on the two modified spectra, respectively, (errors statistical) on an angular scale of $a=6°$. The first modification yields $p_s^{\mathrm{hot}}=30\%$ and $p_s^{\mathrm{cold}}=55\%$, the second modification $p_s^{\mathrm{hot}}=37\%$ and $p_s^{\mathrm{cold}}=67\%$.