Dipole leakage and low CMB multipoles
aa r X i v : . [ a s t r o - ph . C O ] S e p Dipole leakage and low CMB multipoles
Santanu Das and Tarun Souradeep
IUCAA, Post Bag 4, Ganeshkhind, Pune,411 007, IndiaE-mail: [email protected], [email protected]
Abstract.
A number of studies of WMAP-7 have highlighted that the power at the lowmultipoles in CMB power spectrum are lower than their theoretically predicted values. Angularcorrelation between the orientations of these low multipoles have also been discovered. Whilethese observations may have cosmological ramification, it is important to investigate possibleobservational artifacts that can mimic them. The CMB dipole which is almost 550 times higherthan the quadrupole can get leaked to the higher multipoles due to the non-circular beam ofthe CMB experiment. In this paper an analytical method has been developed and simulationsare carried out to study the effect of the non-circular beam on power leakage from the dipole. Ithas been shown that the small, but non-negligible power from the dipole can get transferred tothe quadrupole and the higher multipoles due to the non-circular beam. Simulations have alsobeen carried out for Planck scan strategy and comparative results between WMAP and Planckhave been presented in the paper.
1. Introduction
The standard model of cosmology emerging from recent observations is a remarkable successof theoretical physics. It can explain the cosmological observations up to an extremely highprecision using a handful set of parameters. However, there are some effects that seem to beanomalous in the standard cosmological model. One of such observational fact is the poweranomaly at the low multipoles of the CMB power spectrum. It has been seen that the power atthe low multipoles are lower then their theoretical predictions and there are possibly correlationbetween the orientations of the low multipoles. Many researchers have tried to explain thephenomenon such as in [1, 2], but the phenomenon is not satisfactorily explained. In thispaper we have analyzed the effect of the non-circular beam as origin of CMB anomalies at lowmultipoles.In most of the CMB experiments such as WMAP, the beam shape of the detectors are non-circular about the pointing direction. However in the data analysis techniques the beam isassumed to be circularly symmetric. The CMB dipole is almost 550 times stronger than thequadrupole. Therefore due to the non-circularity of the beam some power from dipole mayleak to the quadrupole and immediate higher multipoles. However, assuming a circular beamin 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 thisinadequate data analysis technique. Different effects of non-circular beam have been discussedby different authors in [4, 5]. However, this particular effect of leakage from the dominant dipolehas not been analyzed yet.In this paper, analytical methods have been developed to calculate the amount of powerleakage from dipole and simulations with the actual scan pattern of WMAP have been carriedut showing the order of power leakage for different WMAP beams. Our results show thatthe amount may not be sufficient to explain the anomalies, but the power transfer does have ameasurable effect on the quadrupole. In anticipation, a similar simulation has also been carriedout with Planck scan pattern. As the data for Planck beams are not available publicly we provideonly upper limits to the beam non-circularity parameters beyond which the dipole leakage wouldcause detectable effect in the power spectrum.
2. Analytical description of the beam convolution
This section describes the formalism employed for measuring the power leakage from dipole tothe quadrupole and higher multipoles during scanning the sky with a non-circular beam. Themeasured sky temperature in a CMB experiment is a convolution of the true sky temperaturewith the beam function. If the measured temperature along γ i is expressed by ˜ T ( γ i ), where asthe sky temperature along γ is T ( γ ) then˜ T ( γ i ) = Z B ( γ i , γ ) T ( γ ) d Ω γ + T N ( γ i ) . (1)Here T N is the noise in the scan procedure. Since we deal with low multipoles, the noiseis ignored in our analysis. Here the beam function B ( γ i , γ ) represents the sensitivity of thetelescope around the pointing direction γ i . This is a two point function and can be expanded interms of spherical harmonics as B ( γ i , γ ) = ∞ X l =0 l X m = − l b lm ( γ i ) Y ml ( γ ) . (2)Since we intend to measure the power leakage from dipole to the quadrupole and it isconvenient to consider only the sky dipole map and check the amount of power leakage from thedipole to higher multipole due to the non-circular beam. A dipole map T ( γ ) can be written asa sum of all the spherical harmonics with l = 1, i.e. T ( γ ) = P m = − a m Y m ( γ ). However, itsalways possible to choose a coordinate system such that a , and a , − modes vanish and thesky temperature can be expressed only as T ( γ ) = T Y ( γ ), where T = a , is a constant. Insuch a case the measured sky temperature along the γ i direction can be expressed as˜ T ( γ i ) = Z B ( γ i , γ ) T ( γ ) d Ω γ = Z ∞ X l =0 l X m = − l b lm ( γ i ) Y ml ( γ ) T ( γ ) d Ω γ = T ∞ X l =0 l X m = − l b lm ( γ i ) Z Y ml ( γ ) Y ( γ ) d Ω γ = T b ( γ i ) . (3)The above expression can not be directly used for measuring the sky temperature because itcontains the beam harmonic coefficient b ( γ i ) , which is a function of γ i . It is convenient toorient the beam along some fixed direction of the sky, say along the ˆ z direction, and consider themultipole b lm (ˆ z ) 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 b ( γ i ) = l X m ′ = − l b m ′ ( z ) D m ′ = b , − ( z ) D , − ( ϕ i , θ i , ρ i ) + b , ( z ) D , ( ϕ i , θ i , ρ i ) + b , ( z ) D , ( ϕ i , θ i , ρ i )= b , − ( z ) d , − ( θ i ) e iρ i + b , ( z ) d , ( θ i ) + b , ( z ) d , ( θ i ) e − iρ i . (4)ubstituting explicit expressions in terms of trigonometric functions for d , ( θ i ) , d , − ( θ i )and d , ( θ i ), in the above and using eq(3), the expression for the scanned temperature can bewritten as ˜ T ( γ i ) = T b ( z ) cos( θ i ) + √ T sin( θ i ) [ b r ( z ) cos( ρ i ) + b i ( z ) sin( ρ i )] . (5)Here ρ i is the orientation of the semi-major axis of the beam at the i th scan point. Thefunction b i ( z ) and b r ( z ) can be defined as follows. In eq(4) b , and b , − are complex quantities.Since the beam is real, the beam spherical harmonic coefficients should satisfy the relation b ∗ , = − b , − . Here, b r and b i has been defined as b , = b r + ib i , i.e. the real and the imaginarypart of b , .From eq(5) we can calculate the power that gets leaked to the quadrupole or the highermultipoles. It can be seen from eq(5) that the term with b ( z ) will not contribute to any powerleakage from dipole. Therefore the dipole to quadrupole power transfer is caused only by theterms multiplied with b r ( z ) or b i ( z ). Hence, if the experimental beam is designed in such a waythat the b r ( z ) or b i ( z ) components of the beam are completely negligible than it is possible tocompletely eliminate the dipole to higher multipole power transfers.
3. Dipole leakage from WMAP scan pattern
The WMAP satellite follows a unique scan pattern, in which pixels near the two poles arescanned for large number of times, whereas those which are near the equator are scanned forleast 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 detectorshas a pair of horns, both of which are are about 70 . o off the symmetry axis. It has a fast spinabout the symmetry axis with the spin period of around 2 . . o about the Sun-WMAP line. This precession periodis about 1 hour. The earth-sun vector rotates 360 o in a year.The simulation has been carried with a dipole map, similar to the known CMB dipole. Weassumed 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 are same for both the beams of a detector.All the simulations have been carried out on Healpix map with N side = 256 resolution. Thesimulation 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 eclipticcoordinate system. These maps can be multiplied with b , b r and b i and then summed upto get the final scanned map from a detector and hence the amount of power transfer can becalculated. The values of the beam spherical harmonic coefficients i.e. b r , b i and b along withthe amount of power leakage are listed in the table 1.The analysis shows that the temperature leaked into the quadrupole from the dipole is lessthan 2 µK compared to the CMB quadrupole measured by WMAP of ∼ µK . Although thedipole power leakage is small, the amount of power leakage is sufficient to suggest a deeperanalysis of the WMAP data for this effect.
4. Dipole leakage from Planck scan pattern
The Planck satellite has only one beam for each detector and thus instead of the differentialmeasurement it measures the actual temperature of the sky. The Planck satellite beam isapproximately 85 o off symmetric axis and the precession angle for the satellite is around 7 . 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. Threeindependent maps are calculated from the Planck scan strategy are shown in Fig.2. As the able 1. The dipole coefficients, b r and b i (real and imagionary part of b , ) of the beamspherical harmonics for different WMAP beams estimated from the publicly available beam maps[7]. The quadrupole and octapole temperature are calculated from the simulation consideringthe dipole temperature as 3.358 mK. b r b i T d /T q T q ( µK ) T d /T oc T oc ( µK ) K − . × − . × − . .
82 11250 . Ka − . × − − . × − . .
31 16493 . Q . × − − . × − . .
22 14226 . V . × − . × − . .
03 19674 . W . × − . × − .
57 0 .
10 133621 . b i and b r are of the order of 5 × − or less then it will cause a temperatureleakage less than < µK . Therefore, that amount of leakage may be safely ignored for beamssatisfying the above limits. Figure 1.
Simulated maps of cos θ , sin θ cos ρ and sin θ sin ρ component from WMAP scanpattern in the Ecliptic coordinate system. Figure 2.
Simulated maps of cos θ , sin θ cos ρ and sin θ sin ρ component from Planck scanpattern in Ecliptic coordinate system.
5. Conclusion
An analytical formalism has been developed to use in simulations of scan strategy to estimatethe leakage of power in the dipole anisotropy to the quadrupole and higher multipoles. It hasalso been shown that the power leakage only depends on the two spherical harmonic coefficientsof the satellite beam ( b i and b r ) and therefore if the beam has been designed in such a way thatthese two parameters of the beam are small enough then power leakage will be negligible. ForWMAP, the amount of the power leakage is found to be small but not insignificant comparedto the low value of quadrupole measured. The simulations also show that the amount of powerleakage depends on the scan pattern. For identical beam shapes, the amount of power leakageis more in WMAP scan strategy compared to Planck scan strategy. References [1] Moss A, Scott D and Sigurdson K 2011
J. Cosmol. Astropart. Phys.
JCAP01(2011)001[2] Feng B and Zhang X 2003
Phys. Lett. B Physique Phys Rev. D. Phys. Rev. D. Astrophys. J.28[7] The beam maps can be found on