Dipole leakage and low CMB multipoles

A number of studies of WMAP-7 have highlighted that the power at the low multipoles in CMB power spectrum are lower than their theoretically predicted values. Angular correlation between the orientations of these low multipoles have also been discovered. While these observations may have cosmological ramification, it is important to investigate possible observational artifacts that can mimic them. The CMB dipole which is almost 550 times higher than the quadrupole can get leaked to the higher multipoles due to the non-circular beam of the CMB experiment. In this paper an analytical method has been developed and simulations are carried out to study the effect of the non-circular beam on power leakage from the dipole. It has been shown that the small, but non-negligible power from the dipole can get transferred to the quadrupole and the higher multipoles due to the non-circular beam. Simulations have also been carried out for Planck scan strategy and comparative results between WMAP and Planck have been presented in the paper.


Introduction
The standard model of cosmology emerging from recent observations is a remarkable success of theoretical physics. It can explain the cosmological observations up to an extremely high precision using a handful set of parameters. However, there are some effects that seem to be anomalous in the standard cosmological model. One of such observational fact is the power anomaly at the low multipoles of the CMB power spectrum. It has been seen that the power at the low multipoles are lower then their theoretical predictions and there are possibly correlation between the orientations of the low multipoles. Many researchers have tried to explain the phenomenon such as in [1,2], but the phenomenon is not satisfactorily explained. In this paper we have analyzed the effect of the non-circular beam as origin of CMB anomalies at low multipoles.
In most of the CMB experiments such as WMAP, the beam shape of the detectors are noncircular about the pointing direction. However in the data analysis techniques the beam is assumed to be circularly symmetric. The CMB dipole is almost 550 times stronger than the quadrupole. Therefore due to the non-circularity of the beam some power from dipole may leak to the quadrupole and immediate higher multipoles. However, assuming a circular beam in the data analysis technique, this leakage of the power from dipole will not be accounted for. Therefore, the contribution of this effect will contaminate the resultant map, generated by this inadequate data analysis technique. Different effects of non-circular beam have been discussed by different authors in [4,5]. However, this particular effect of leakage from the dominant dipole has not been analyzed yet.
In this paper, analytical methods have been developed to calculate the amount of power leakage from dipole and simulations with the actual scan pattern of WMAP have been carried out showing the order of power leakage for different WMAP beams. Our results show that the amount may not be sufficient to explain the anomalies, but the power transfer does have a measurable effect on the quadrupole. In anticipation, a similar simulation has also been carried out with Planck scan pattern. As the data for Planck beams are not available publicly we provide only upper limits to the beam non-circularity parameters beyond which the dipole leakage would cause detectable effect in the power spectrum.

Analytical description of the beam convolution
This section describes the formalism employed for measuring the power leakage from dipole to the quadrupole and higher multipoles during scanning the sky with a non-circular beam. The measured sky temperature in a CMB experiment is a convolution of the true sky temperature with the beam function. If the measured temperature along γ i is expressed byT (γ i ), where as the sky temperature along γ is T (γ) theñ Here T N is the noise in the scan procedure. Since we deal with low multipoles, the noise is ignored in our analysis. Here the beam function B(γ i , γ) represents the sensitivity of the telescope around the pointing direction γ i . This is a two point function and can be expanded in terms of spherical harmonics as Since we intend to measure the power leakage from dipole to the quadrupole and it is convenient to consider only the sky dipole map and check the amount of power leakage from the dipole to higher multipole due to the non-circular beam. A dipole map T (γ) can be written as a sum of all the spherical harmonics with l = 1, i.e. T (γ) = 1 m=−1 a 1m Y m 1 (γ). However, its always possible to choose a coordinate system such that a 1,1 and a 1,−1 modes vanish and the sky temperature can be expressed only as In such a case the measured sky temperature along the γ i direction can be expressed as The above expression can not be directly used for measuring the sky temperature because it contains the beam harmonic coefficient b 10 (γ i ) , which is a function of γ i . It is convenient to orient the beam along some fixed direction of the sky, say along theẑ direction, and consider the multipole b lm (ẑ) to characterise the beam [6]. The spherical harmonic coefficients of the beam b lm (γ i ) at any particular direction γ i can obtained by using Wigner-D functions as Substituting explicit expressions in terms of trigonometric functions for d 1 0,0 (θ i ) , d 1 0,−1 (θ i ) and d 1 0,1 (θ i ), in the above and using eq(3), the expression for the scanned temperature can be written asT Here ρ i is the orientation of the semi-major axis of the beam at the i th scan point. The function b i (z) and b r (z) can be defined as follows. In eq(4) b 1,1 and b 1,−1 are complex quantities. Since the beam is real, the beam spherical harmonic coefficients should satisfy the relation b * 1,1 = −b 1,−1 . Here, b r and b i has been defined as b 1,1 = b r + ib i , i.e. the real and the imaginary part of b 1,1 . From eq(5) we can calculate the power that gets leaked to the quadrupole or the higher multipoles. It can be seen from eq(5) that the term with b 10 (z) will not contribute to any power leakage from dipole. Therefore the dipole to quadrupole power transfer is caused only by the terms multiplied with b r (z) or b i (z). Hence, if the experimental beam is designed in such a way that the b r (z) or b i (z) components of the beam are completely negligible than it is possible to completely eliminate the dipole to higher multipole power transfers.

Dipole leakage from WMAP scan pattern
The WMAP satellite follows a unique scan pattern, in which pixels near the two poles are scanned for large number of times, whereas those which are near the equator are scanned for least number of times. The satellite scans the sky temperature in five different frequency bands, named as K, Ka, Q, V and W in a differential measurements of a pair of horns. Amongst them Q and V band have two detectors each and W band has four detectors. Each of these detectors has a pair of horns, both of which are are about 70. The simulation has been carried with a dipole map, similar to the known CMB dipole. We assumed that the shape of the two beams for a pair of horns of a detector are almost same, i.e. the parameters b r , b i and b 10 are same for both the beams of a detector.
All the simulations have been carried out on Healpix map with N side = 256 resolution. The simulation gives us three independent maps from the three independent components of eq(5). The figures from the three maps are shown in the Fig.1. All these maps are shown in ecliptic coordinate system. These maps can be multiplied with b 10 , b r and b i and then summed up to get the final scanned map from a detector and hence the amount of power transfer can be calculated. The values of the beam spherical harmonic coefficients i.e. b r , b i and b 10 along with the amount of power leakage are listed in the table 1.
The analysis shows that the temperature leaked into the quadrupole from the dipole is less than 2µK compared to the CMB quadrupole measured by WMAP of ∼ 6µK. Although the dipole power leakage is small, the amount of power leakage is sufficient to suggest a deeper analysis of the WMAP data for this effect.

Dipole leakage from Planck scan pattern
The Planck satellite has only one beam for each detector and thus instead of the differential measurement it measures the actual temperature of the sky. The Planck satellite beam is approximately 85 o off symmetric axis and the precession angle for the satellite is around 7.5 o . The precession rate is taken as one revolution per six month and the spin rate as 180 o /min.
An analysis similar to that of WMAP has been carried out for Planck satellite also. Three independent maps are calculated from the Planck scan strategy are shown in Fig.2. As the Table 1. The dipole coefficients, b r and b i (real and imagionary part of b 1,1 ) of the beam spherical harmonics for different WMAP beams estimated from the publicly available beam maps [7]. The quadrupole and octapole temperature are calculated from the simulation considering the dipole temperature as 3.358 mK. data of the Planck beam map is not publicly available therefore the amount of power leakage expected from the dipole can not be computed for the Planck scan pattern. But, the analysis shows that if the b i and b r are of the order of 5 × 10 −4 or less then it will cause a temperature leakage less than < 1µK. Therefore, that amount of leakage may be safely ignored for beams satisfying the above limits.

Conclusion
An analytical formalism has been developed to use in simulations of scan strategy to estimate the leakage of power in the dipole anisotropy to the quadrupole and higher multipoles. It has also been shown that the power leakage only depends on the two spherical harmonic coefficients of the satellite beam (b i and b r ) and therefore if the beam has been designed in such a way that these two parameters of the beam are small enough then power leakage will be negligible. For WMAP, the amount of the power leakage is found to be small but not insignificant compared to the low value of quadrupole measured. The simulations also show that the amount of power leakage depends on the scan pattern. For identical beam shapes, the amount of power leakage is more in WMAP scan strategy compared to Planck scan strategy.