# Modelling the WMAP large-angle anomalies as an effect of a local density inhomogeneity

## Abstract

We investigate large-angle scale temperature anisotropy in the Cosmic Microwave Background (CMB) with the Wilkinson Microwave Anisotropy Probe (WMAP) data and model the large-angle anomalies as the effect of the CMB quadrupole anisotropies caused by the local density inhomogeneities. The quadrupole caused by the local density inhomogeneities is different from the special relativity kinematic quadrupole. If the observer inhabits a strong inhomogeneous region, the local quadrupole should not be neglected. We calculate such local quadrupole under the assumption that there is a huge density fluctuation field in direction , where the density fluctuation is , and its center is away from us. After removing such mock signals from WMAP data, the power in quadrupole, , increases from the range to . The quantity S, which is used to estimate the alignment between the quadrupole and the octopole, decreases from to , while the model predict that , . So our local density inhomogeneity model can, in part, explain the WMAP low- anomalies.

## 1 Introduction

Although the WMAP data are regarded as a dramatic confirmation of standard inflationary cosmology (Vale [2005]; de Oliveira-Costa & Tegmark [2006]; Gaztañaga et al. [2003]; Gordon et al. [2005]), some anomalous features have emerged (Inoue & Silk [2006]; Campanelli et al. [2006]; Dominik et al. [2004]). Firstly, the amplitude of the quadrupole is substantially less than the expectation from the best-fit standard model (Abramo et al. [2006]; de Oliveira-Costa et al. [2004]; Efstathiou [2004]), which was found by COBE a decade ago (Bennett et al. [1996]) and confirmed by WMAP (Spergel et al. [2003]). Secondly, the quadrupole and octopole indicate an unexpectedly high degree of alignment (Spergel et al. [2003]; de Oliveira-Costa et al. [2004], [2006]; Schwarz [2004]; Land et al. [2005]; Hansen et al. [2004]a; Eriksen et al. [2004]).

Recently, many efforts have been devoted to explain the origin of the anomalies. They can be systematic error, statistical flukes, improper subtraction of known foreground, or an unexpected foreground (Copi et al. [2004], [2005], [2006]). The WMAP team claims that there are no unexpected systematic errors (Bennett et al. [2003]; Finkbeiner [2004]), and Copi et al. ([2004], [2005], [2006]) noted that the anomalies are most unlikely to be due to residual foreground contamination. Several authors attempted to explain the anomalies in terms of a new foreground (Abramo et al. [2006]; Gordon et al. [2005]; Bennett et al. [2003]; Finkbeiner [2004]; Prunet et al. [2005]; Rakic et al. [2006]).

Abramo et al. ([2006]) showed circumstantial evidences that an extended foreground near the dipole axis could distort the CMB. They proposed that the possible physical mechanism, which can produce such a foreground, is the thermal Sunyaev-Zeldovich (SZ) effect. But the SZ model, as presented by them, cannot account for the anomalous quadrupole and octopole successfully. Therefore, they thought that the Ress-Sciama (RS) effect (Rakic et al. [2006]), or the combination of SZ effect and RS effect may be responsible for the foreground. Many other authors suggested that the large-angle anomalies are affected by local inhomogeneities (Tomita [2005]a, [2005]b; Vale [2005]). However, when they applied a model in which the Local Group is falling into the center of the Shapley supercluster, the discrepancy between the observed data and the model prediction became even worse. (Rakic et al. [2006]; Inoue et al. [2007]).

Inoue et al. (2007) explored the large angular scale temperature anisotropies due to homogeneous local dust-filled voids in a flat Friedmann-Robertson-Walker universe. They found that a pair of voids with radius and density contrast might help explain the observed large-angle CMB anomalies. While Wu & Fang ([1994]) explored the possibility that the CMB is affected by local density inhomogeneities basing on Tolman-Bondi model. They calculated the quadrupole amplitude of the local collapse model with the general relativity (GR). The results show that the CMB anisotropies from the local quadrupole contribution can be different from the special relativity (SR) kinematic quadrupole by a factor as large as 3, which depends on the size and density fluctuation of the region the observer inhabits. Therefore, if we live in a large density fluctuation area, the local quadrupole might be significant in the CMB observations.

The goal of this paper is to examine whether such local quadruple could account for the observed large-angle CMB anomalies in WMAP data. Our analysis is based on the 1-year, 3-year and 5-year WMAP Internal Linear Combination maps (Spergel et al. [2006]; Hinshaw et al. [2006]) (henceforth ILC1, ILC3 and ILC5). We try to remove the mock CMB foreground caused by the effect described in Wu & Fang ([1994]) for each observed CMB map under the assumption that we are in a huge density fluctuation area. The parameters of the area we adopted are based on Kocevski & Ebeling ([2006]) and Watkins et al. ([2008])’s work. We reanalyze the WMAP data by using the multipole vector framework in Section 2. In Section 3, we review the estimate of the foreground of Wu & Fang ([1994]) and present the result of our examination. We conclude in Section 4.

## 2 Large-angle anomalies of CMB

In this section, we re-investigate the anomalies reported from the WMAP maps on very large angular scale. As we already remarked, the angular power in quadrupole, , is less than expected. To measure , we expand the temperature anisotropy in terms of spherical harmonics (Campanelli et al. [2006]; Copi et al. [2004])

(1) |

And the angular power spectrum is defined as

(2) |

A simple way to quantify the peculiar alignment of the quadrupole and octopole is to use the multipole vectors. In the multipole vector representation, the multipole of the CMB, , can be written in terms of a scalar and unit vectors (Dominik et al. [2004]; Copi et al. [2005], [2006])

(3) |

For the statistical comparison, we use the area vectors

(4) |

The alignments between the quadrupole area vector and the three octopole area vectors can be evaluated by the magnitudes of the dot products between and each

(5) | |||

The widely used estimator that checks for alignments of the quadrupole and octopole planes is the average of the dot products (Dominik et al. [2004]; Abramo et al. [2006]; Katz & Weeks [2004]; Schwarz [2004])

(6) |

Given a CMB map, the harmonic components can be easily extracted
with the HEALPix^{1}

In order to compare with the standard model,
mimic CMB maps are generated with Monte Carlos (MC)
simulation based on theoretical CMB power spectrum predicted by
model, which is generated by
CAMB^{2}

Fig 1 is the histogram of the S statistics generated from Gaussian random, statistically isotropic MC mock maps. The average value of S from MC simulations is , which is much lower than the S statistics from WMAP data, that is for ILC1, for ILC3, and for ILC5. The final rank in table 1 lists the odds of finding a value among the MC maps larger than the one observed, from which one can see that the probabilities are for ILC1, for ILC3, and for ILC5. This means that the alignment between quadrupole and octopole for each WMAP map is significant.

These alignments could be explained by an unexpected foreground caused by a local collapse due to the second-order effect of the density fluctuation area (Wu & Fang [1994]). In next section, we will briefly discuss this foreground.

## 3 Hypothetical foreground induced by super large structure

The CMB temperature anisotropy produced by a locally spherical collapse can be modeled basing on a Tolman-Bondi universe solution (Wu & Fang [1994]).

Because we are interested in the effect of a local density fluctuation, in the following we only consider the case of , where , is the distance between the observer and the center of the perturbation, , is the size of the perturbed region. When the initial density perturbation is assumed to be constant in the region , the first-order solution consists mainly of two parts: a monopole term and a dipole term which we are familiar with. The second-order solution of is (Wu & Fang [1994])

(7) |

where and is the redshift at decoupling time , and is the incidence angle of the photon.

When the terms of the order of and are taken into account, the quadrupole anisotropy caused by local density fluctuation should be (Wu & Fang [1994])

(8) |

and

(9) | |||||

The first term in the left-hand side of equation (8) is the SR kinematic quadrupole anisotropy. Equation (8) tells us that if higher orders are involved, the SR kinematic quadrupole may not always be a good approximation of the quadrupole produced by a local collapse. The local quadrupole anisotropy strongly dependents on the size, matter density in the peculiar field, and the position of the observer. Fig 2 shows the quadrupole amplitude as a function of the distance between the observer and local gravitational field . The SR kinematic quadrupole is denoted by solid curve, and the local quadrupole is denoted by dotted curve. We assume to satisfy . The quadrupole showing in Fig. 2 is along the center of the perturbation. Fig 3 shows the relationship between the amplitude of local quadrupole and the radius of the local gravitational field for . changes from to . Because the distance of the observer to the center of the collapse should at least be greater than the distance to the Great Attractor, which is estimated to be . Therefore, it would be reasonable to take the lower value of which is about 2 times of the distance to the Great Attractor and the higher value of which is about the size of horizon (Wu & Fang [1994]). We find that the influence of on the amplitude of local quadrupole is about one magnitude larger than the influence of . When is fixed, the results change little with . Fig 4 shows the corrected of ILC5 as a function of when . It turns out that for all values of .

In order to explain the large-angle anomalies we propose a model that we are in a large density fluctuation area. As Kocevski & Ebeling ([2006]) suggests that of the Local Group’s (LG) peculiar velocity is induced by more distant overdensities between 130 and 180 away. Watkins et al. ([2008]) also notes that the bulk flow within a Gaussian window of radius is toward , , and roughly of the LG’s motion is due to sources at greater depths. Interestingly, we find that a region with a density fluctuation over a distance away on the direction of may be responsible for the origin of the anomalies on large angular scales. We compute the mock foreground (equation (8)) using these parameters. Fig 5 shows the map of the contribution of CMB anisotropies caused by the local density fluctuation.

After subtracting such a mock foreground from the CMB sky maps of the WMAP observation, we find that the power in quadrupole will dramatically increase and the alignment of the quadrupole and octopole plane will be weakened. In Table 2 we compare the quadrupole and S obtained from the ”foreground-corrected” WMAP data to those obtained from fiducial model. The powers in quadrupole of the three WMAP maps increase to , , , respectively, which is apparently in much better agreement with the model. Furthermore, from the S statistics, one can see that the frequencies of finding a simulation with a S value larger than that from WMAP seem to converge to for ILC1, for ILC3, for ILC5. Therefore, if such a large scale structure exist, the foreground model presented here can not be neglected.

We evaluate the probability that the primary quadrupole is cancelled
by the local quadrupole. We generate CMB maps, which have
random quadrupole orientations, with the HEALPix software, and the
input theoretical power spectra, , are generated by the
CAMB package. Then we combine the foreground with the random,
statistically isotropic CMB maps. We find that about of
the quadrupole are consistent with the observed WMAP five year
values, that is
^{3}

## 4 Conclusions

In this paper, we have re-investigated the anomalies in WMAP data. The power in quadrupole is found to be for ILC1, for ILC3 and for ILC5, while the power in quadrupole for the standard model is . It is obvious that the power in quadrupole is less than the expected. By comparing the distribution of the S statistics from WMAP data to those from MC simulation mimic CMB maps, we found that they are consistent at the level of for ILC1, for ILC3 and for ILC5. These results indicate that the quadrupole and octopole planes are aligned strongly.

We provide a possible explanation for the anomalies in WMAP data by using the foreground model caused by a large density fluctuation. The model depends on the matter distribution, and the position of the observer. So we assumed that there is a large-scale structure in direction , the center is away from us, and the density fluctuation is . After subtracting the mock foreground caused by such area from the WMAP data ILC1, ILC3 and ILC5, we found that the power in quadrupole, , increases to level, and the S decreases to level, which agrees with the prediction from the standard model. To conclude, the local gravitational collapse might be responsible for explaining the origin of the large-angle CMB anisotropy.

Recently, it has been suggested by many researchers that the local inhomogeneities can account for the large angular scales anomalies (Tomita [2005]a, [2005]b; Vale [2005]). However, none of the proposed models can successfully explain the anomalies (Inoue & Silk [2006]). Because it is well known from the GR that in a linear approximation, the behavior of a comoving object in an expansion or collapsing metric can not be equivalently described as SR Doppler motion if the higher orders are involved. The amplitude of the kinematic quadrupole is about of the cosmic quadrupole (Wu & Fang [1994]). Therefore the CMB quadrupole anisotropy calculated as an effect of a local density inhomogeneity can not be approximated by a SR effect, which is the main reason why we have derived different results from others.

However, many other specific features of the anomalies have been discovered, such as anomalously cold spots on angular scales (Vielva et al. [2004]; Cruz et al. [2005]), and asymmetry in the large-angle power between opposite hemispheres(Eriksen et al. [2004]; Hansen et al., [2004]b; Sakai & Inoue [2008]). We have not interpreted these anomalies with our model explicitly, so further research is expected.

## Acknowledgement

We would like to thank Wen Xu, Huan-Yuan Shan, Xiao-Chun Mao, Xin-Juan Yang, Nan Li and Qian Zheng for helpful comments and discussions. And we are grateful to WMAP team for providing such a superb data set. We also thank Wen Xu for careful reading on the draft manuscript.

S | P(S) | ||

ILC1 | 204.4 | 0.744 | 0.8% |

ILC3 | 260.3 | 0.700 | 2.1% |

ILC5 | 254.1 | 0.726 | 1.2% |

1071.5 | 0.412 | 50.0% |

S | P(S) | ||

ILC1-corr | 1064.2 | 0.317 | 75.1% |

ILC3-corr | 1034.2 | 0.371 | 61.5% |

ILC5-corr | 1022.3 | 0.368 | 62.2% |

1071.5 | 0.412 | 50.0% |

### Footnotes

- http://healpix.jpl.nasa.gov/
- http://camb.info/
- http://lambda.gsfc.nasa.gov/product/map/

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