The significance of the largest scale CMB fluctuations in WMAP

TL;DR

The paper investigates three large-scale CMB anomalies reported by WMAP—low quadrupole, planar octopole, and quadrupole–octopole alignment—and tests whether they indicate a non-standard cosmic topology. It quantifies anomaly significance within a standard cosmological framework, simulates small-universe toroidal models, and applies the S-statistic and circles-in-the-sky tests to compare with the data. The results do not support a simple small-universe explanation: the S-statistic shows no compelling evidence for symmetry beyond infinite-Universe expectations, and the matched-circles search finds no detections, ruling out the simplest $T^1$ geometries with cell sizes $R_x \gtrsim 1$. The analysis narrows topology-based explanations and outlines a path for a comprehensive six-parameter search, while noting foreground systematics and a posteriori statistics temper the anomalies’ cosmological significance.

Abstract

We investigate anomalies reported in the Cosmic Microwave Background maps from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite on very large angular scales and discuss possible interpretations. Three independent anomalies involve the quadrupole and octopole: 1. The cosmic quadrupole on its own is anomalous at the 1-in-20 level by being low (the cut-sky quadrupole measured by the WMAP team is more strikingly low, apparently due to a coincidence in the orientation of our Galaxy of no cosmological significance); 2. The cosmic octopole on its own is anomalous at the 1-in-20 level by being very planar; 3. The alignment between the quadrupole and octopole is anomalous at the 1-in-60 level. Although the a priori chance of all three occurring is 1 in 24000, the multitude of alternative anomalies one could have looked for dilutes the significance of such a posteriori statistics. The simplest small universe model where the universe has toroidal topology with one small dimension of order half the horizon scale, in the direction towards Virgo, could explain the three items above. However, we rule this model out using two topological tests: the S-statistic and the matched circle test.

Figures (5)

• Figure 1: The CMB maps and their $S$-maps (see §\ref{['BabaSec']}) . Top: the all-sky cleaned CMB map from tegmark03 is shown on the left and its $S$-map on the right. Middle: the quadrupole map (left) and its $S$-map (right). Bottom: the octopole map (left) and its $S$-map (right). Note that all three $S$-maps show dark spots in the supposed direction of suppression of its original maps, around "two o'clock".
• Figure 2: Cumulative histograms of $S_{\circ}$. Top-left: $S$-test for the cleaned WMAP map. Top-right: $S$-test for octopole alone. Bottom-left: $S$-test for the sum of the quadrupole and octopole. Bottom-right: $S$-test for hexadecapole alone. Curves show the fraction of 500 simulated maps that have $S_{\circ}$ below the given value. A small Universe should give a small $S_{\circ}$-value, but the observed value of $S_{\circ}^{WMAP}$ (vertical line) is seen to be significantly smaller than expected in an infinite universe only for the octopole case.
• Figure 3: Expected and observed $S_{\circ}$-values for the octopole map alone (top) and for the sum of the quadrupole and octopole maps (bottom). The solid line shows the mean of the Monte Carlo simulations and the yellow (or grey in BW) band shows the $1-\sigma$ spread. 120 simulations per $R_x$-value were performed for $R_x\le 1$ and 30 per $R_x$-value for $R_x>1$. The horizontal lines represent the observed values $S_{\circ}^{WMAP}$.
• Figure 4: An example of the curve $d(\alpha;\widehat{\bf n}_i)$ that we used to search for matched circles. This case is for the reflection axis $\widehat{\bf n}_i$ corresponding to ($l,b$)= $(-110^\circ,60^\circ)$. A pair of perfectly matched circles of angular radius $\alpha$ would give $d(\alpha)=0$. The drop towards zero on the left hand side is caused by a great ($\alpha=90^\circ$) circle being its own reflection, whereas the high values around $\alpha \sim 15^\circ$ are caused by residual foreground contamination.
• Figure 5: Result of search for matched circles of radius $0^\circ<\alpha<15^\circ$. A perfectly matched circle would show up as a pixel with zero temperature at the position of the circle center, whereas the map above shows no pixels below 83$\mu$K. Note that this map is parity-symmetric, i.e., the temperature at $\widehat{\bf n}$ equals that at $-\widehat{\bf n}$, although this symmetry is obscured by the Aitoff projection used.