Is the CMB asymmetry due to the kinematic dipole?
aa r X i v : . [ a s t r o - ph . C O ] J a n Is the CMB asymmetry due to the kinematic dipole?
P. Naselsky, W. Zhao, J. Kim and S. Chen
Niels Bohr Institute and DISCOVERY Center, Blegdamsvej 17, 2100 Copenhagen, Ø,Denmark
ABSTRACT
Parity violation found in the Cosmic Microwave Background (CMB) radia-tion is a crucial clue for the non-standard cosmological model or the possiblecontamination of various foreground residuals and/or calibration of the CMBdata sets. In this paper, we study the directional properties of the CMB parityasymmetry by excluding the m = 0 modes in the definition of parity parameters.We find that the preferred directions of the parity parameters coincide with theCMB kinematic dipole, which implies that the CMB parity asymmetry may beconnected with the possible contamination of the residual dipole component. Wealso find that such tendency is not only localized at l = 2 ,
3, but in the extendedmultipole ranges up to l ∼ Subject headings: cosmic microwave background radiation — early universe —methods: data analysis — methods: statistical
1. Introduction
Symmetry of the physical process in our Universe and particular mechanisms of itsviolation is golden mind of the modern physics. Since pioneering Lee and Yang investiga-tions of the parity symmetry in the weak interaction, the principle of symmetry is deeplyincorporated into the modern particle physics, including the Higgs mechanism of symme-try breaking, in chemistry, in physics of condensed matter and, in general, in the theory ofthe phase transition. Passing from the microscopic physics to the properties of the spaceand time at large, we have to admit that the Cosmic Microwave Background (CMB) radi-ation anisotropy provides invaluable test for the investigation of parity at the megascopicscales above the scale of inhomogeneity ∼ o ≤ Θ ≤ o (Schwarz et al. 2004; Copi et al. 2010), and newly discoveredanomaly of the correlations at 1 o ≤ Θ ≤ o (Kim & Naselsky 2011). In combination withthe widely discussed anomalies of the CMB map and the power spectrum (Eriksen et al.2004) (see for review (Bennett et al. 2011)), investigation of the origin of these anomaliescould put a new light on the physics of the early Universe, the methods of the foregroundsreductions and calibration of the CMB data sets, improving our knowledge of the mostfundamental cosmological parameters and the theory of inflation.The local motion of an observer through the CMB frame produces the so-called Kine-matic Dipole (KD) anisotropy of the CMB, which is the most powerful component of thesignal and fitted out from the CMB data before cosmological analysis. In this paper, we aregoing to show that some of the discovered features of the Wilkinson Microwave AnisotropyProbe (WMAP) CMB TT anisotropy, including the odd-parity preference of the powerspectrum, could have common origin associated with the KD of the CMB. Previously, thepossible contamination of the CMB by KD has been assessed by multipole vector statistic(alignment of the quadrupole and octupole components) (Gordon et al. 2005; Peiris & Smith2010). Additionally, we will show that the low multipole anomalies are associated with otheranomalies such as the lack of angular correlation, the even/odd-parity asymmetry and theplanarity of the multipole l = 5. First, we will focus on the properties of the CMB TTcorrelation function C (Θ) = ∆ T (ˆ n )∆ T ( ˆ n ′ ) at angle Θ = arccos(ˆ n · ˆ n ′ ) = π and show that C (Θ = π ) is connected to the parity parameter g ( l ). We will estimate the power spectrumwithout the m = 0 mode so that the rotational invariance of the angular power spectrummay not be automatically satisfied. As well known, these m = 0 modes pick up a certaindirection (i.e. the z -axis direction in the spherical coordinate system where a lm are defined(Gordon et al. 2005)). Using these estimators, we will investigate the possible preferred di-rection in the CMB field. If our Universe is, indeed, statistically homogeneous and isotropicwith Gaussian seed perturbations, we should have no or little parity preference, given forour estimators associated with the angular power spectrum. Using the WMAP 7-yr Inter-nal Linear Combination (ILC7) map, we compute the parity parameters for coordinates ofvarious orientations. We find some level of alignment between the KD direction and the ori-entation, in which the parity asymmetry is greatest. Therefore, the CMB parity asymmetrymay be related to the systematics associated with KD, which may be also responsible for thealignment problem of quadrupole and octupole (Gordon et al. 2005; de Oliveira-Costa et al.2006).The outline of the paper is the following. In Section 2 we introduce the basic char-acteristics of the CMB parity asymmetry. In Section 3, we investigate the orientations of 3 –maximum parity asymmetry and compare them with the CMB kinematic dipole. In Section4, we summarize our investigation.
2. Odd-multipole preference of the CMB power spectrum
The temperature fluctuations of CMB anisotropy, can be conveniently decomposed asfollows: ∆ T ( θ, φ ) = ∞ X l =0 l X m = − l a lm Y lm ( θ, φ ) , (1)where a lm are the coefficients of decomposition: a lm = | a lm | exp( iφ lm ), with φ lm as the phase.Under the assumption of total Gaussian randomness, as predicted by the large class ofinflationary models, the amplitudes | a lm | are distributed according to Rayleigh’s probabilitydistribution function and the phases of a lm are supposed to be evenly distributed in therange [0 , π ] (Bardeen et al. 1986).For any signals T (ˆ n ) defined on the sphere, one can extract symmetric (∆ T + (ˆ n ) =∆ T + ( − ˆ n )) and antisymmetric (∆ T − (ˆ n ) = − ∆ T − ( − ˆ n )) components, where∆ T ± (ˆ n ) = X l l X m = − l a lm Γ ± ( l ) Y lm (ˆ n ) , (2)and Γ + ( l ) ≡ cos ( πl ), Γ − ( l ) ≡ sin ( πl ), Y lm (ˆ n ) = ( − l Y lm ( − ˆ n ). Naive expectation, wherethe concordant ΛCDM cosmological model with initial statistically isotropic and Gaussianadiabatic perturbations is assumed, is the absence of any features distinct between even andodd multipoles. However, in reality this statement needs more accurate clarification. Inparticular, for the curvature perturbations beyond the present horizon the power spectrumis given by P ( k ) ∝ k − n s , where n s ≃ .
96 is the spectral index of the primordial densityperturbations (Komatsu et al. 2011). Thus, the variance of the metric perturbations σ ∼ R k P ( k ) dk ∝ k n s − has very weak power-law ( n s <
1) or logarithmic divergence ( n s ≃ k min →
0. Since the low multipole range of the CMB temperature anisotropy is determinedby the ordinary and integrated Sachs-Wolfe effects, these peculiarity of the power spectrumof metric perturbations are crucial for the two-point correlation function: C th (Θ) ≡ < ∆ T (ˆ n )∆ T ( ˆ n ′ ) > = ∞ X l = l min l + 14 π C th ( l ) P l (cos Θ) , (3) 4 –where C th ( l ) are the theoretical power spectrum, P l (cos Θ) are the Legendre polynomials,cos Θ = n · n ′ , and < .. > denotes the average over the statistical ensemble of realizations.Using Eq. (3), we may easily show, for the largest angular distance: C th (Θ = π ) = ∞ X l = l min l + 14 π C th ( l )(Γ + ( l ) − Γ − ( l )) . (4)As clear in Eq. (3), the natural way to estimate the relative contribution of even and oddmultipoles to the correlation function is to define the statistic g ( l ) = P ll ′ = l ′ min l ′ +14 π C ( l ′ )Γ + ( l ′ ) P ll ′ = l ′ min l ′ +14 π C ( l ′ )Γ − ( l ′ ) , (5)where l ′ min = 1 or 2 (see discussion in the forthcoming sections). Note that this statisticdiffers from the g ( l ), widely used in (Kim & Naselsky 2010a,b, 2011), in the sense that l ′ ( l ′ + 1) is replaced by 2 l ′ + 1. Then, from Eqs. (4) and (5) we get the estimator of thequantity C th (Θ = π ): C (Θ = π ) = P − ( l ) [ g ( l ) − ,P ± ( l ) = l X l ′ = l ′ min l ′ + 14 π C ( l ′ )Γ ± ( l ′ ) . (6)Thus, if g ( l ) = 1, the corresponding correlation function is C (Θ = π ) = 0. In reality,the theoretical correlation function shows some parity asymmetry. For instance, the dipolecomponent naturally contributes the odd parity, while the quadrupole contributes the evenparity. When the background cosmological parameters correspond to the concordant ΛCDMmodel, the properties of theoretical correlation function C th (Θ = π ) depends on the value of l ′ min , which is clearly shown in Fig. 1. We find that the odd parity is perferred when the odd l ′ min is chosen, while the even parity is perferred when the even l ′ min is chosen. However, theobserved data show the different tendency for the parity asymmetry, i.e. the parity violationcomparing with the theoretical predictions. From Fig. 1, we find that the odd parity isalways perferred for all the cases with l ′ min ≤ C th (Θ = π ) is not included in Eq. (3), and the lowerlimit is set to l ′ min = 2. In Fig. 2 we carefully calculate the WMAP7 observational correla-tion functions by considering the KQ75 mask and a theoretical prediction with 1 σ interval,where the cosmic variance effect is taken into account. Consistent with Fig. 1, we expectthe theoretical value of C th ( θ = π ) is positive, and the zero value is included at 68% C.L.Therefore the Universe has tendency to be parity asymmetric g ( l ) > l ′ min = 2,with very low chance of g ( l ) <
1. However, the WMAP7 data show that C ( θ = π ) <
3. Directional statistic of the parity asymmetry
As shown in the previous section, for the random Gaussian statistically isotropic andhomogeneous perturbations of the CMB, the correlation function C (Θ) is fully determinedby the power spectrum C ( l ), which is rotationally invariant. Statistical invariance meansthat for any rotations of the reference system of coordinate, the power spectrum and thecorrelation function are invariant. The idea of the method, proposed in this section, is toreplace the power spectrum C ( l ) in Eq. (3) by a rotationally variant power spectrum D ( l ),defined as D ( l ) ≡ l + 1 l X m = − l | a lm | (1 − δ m ) , (7)where δ mm ′ is the Kroneker symbol.As it is seen from the definition given by Eq. (7), the relative difference between D ( l )and C ( l ) is given by ∆( l ) ≡ D ( l ) − C ( l ) C ( l ) = − a l / P m | a lm | . So we have ∆( l ) ∼ O ( l ) forrandom Gaussian CMB field. Thus, the major difference ∆( l ) comes from l = 2 and l = 3modes, while for l ≥ D ( l ) in any coordinate system. Imaginingthe Galactic coordinate system is rotated by the Euler angle ( ψ, θ, φ ), and the coefficients a lm ( ψ, θ, φ ) in this new coordinate system can be calculated by a lm ( ψ, θ, φ ) = l X m ′ = − l a lm ′ D lmm ′ ( ψ, θ, φ ) , (8)where a lm ≡ a lm (0 , ,
0) are the coefficients defined in the Galactic coordinate system, and D lmm ′ ( ψ, θ, φ ) is the Wigner rotation matrix (Edmonds 1985). Similar to Eq. (7), we can 6 –define the power spectrum D ( l ; ψ, θ, φ ). It is easy to find that D ( l ; ψ, θ, φ ) is independent ofthe angle ψ , so in this paper we only consider two Euler angle ˆ q ≡ ( θ, φ ) and set ψ = 0. Ifwe consider ˆ q as a vector, which labels the z -axis direction in the rotated coordinate system,then ( θ, φ ) is the polar coordinate of this direction in the Galactic system .Now, we can define the rotationally variable parity parameter G ( l ; ˆ q ) by replacing C ( l )in Eq. (5) with D ( l ; ˆ q ), and estimate the maxima and minima of G ( l ; ˆ q ) for different Eulerangles ˆ q . By the definition, the parity parameter G ( l ; ˆ q ) depends on the coefficients a l (ˆ q )as follows: G ( l ; ˆ q ) = P + ( l ) − X + ( l ; ˆ q ) P − ( l ) − X − ( l ; ˆ q ) , (9)where X ± ( l ; ˆ q ) ≡ π P ll ′ =2 a l ′ (ˆ q )Γ ± ( l ′ ), and a l (ˆ q ) = X mm ′ a lm a ∗ lm ′ D l m (ˆ q ) D l ∗ m ′ (ˆ q )= 4 π l + 1 X mm ′ ( − m + m ′ a lm a ∗ lm ′ Y ∗ lm (ˆ q ) Y lm ′ (ˆ q ) . So the cross-term a lm a ∗ lm ′ is responsible for the angular dependency of the parity parameter G ( l ; ˆ q ). We can also calculate the difference between G ( l ; ˆ q ) and g ( l ) by G ( l ; ˆ q ) − g ( l ) g ( l ) ≃ X − ( l ; ˆ q ) − X + ( l ; ˆ q ) /g ( l ) P − ( l ) . (10)From the relation ∆( l ) = O ( l ), we know that X ± ( l ; ˆ q ) ≪ P − ( l ), and G ( l ;ˆ q ) − g ( l ) g ( l ) ≪ l >
3. So, we conclude that G ( l ; ˆ q ) mainly stands for the amplitude of the original parityparameter g ( l ). At the same time, due to the rotational variance of G ( l ; ˆ q ), we can studythe possible preferred direction, which may reveal hints on the origin of the observed parityasymmetry in CMB field.Let us show that G ( l ; ˆ q ) map depends on the angular ˆ q . As we have mentioned, ˆ q labelsthe z -axis direction in the rotated coordinate system, and ( θ, φ ) is just the polar coordinateof this direction in Galactic coordinate. We plotted the parameter G ( l ; ˆ q ) as a function ofˆ q for 3 ≤ l ≤
22, and found that G ( l ; ˆ q ) have the similar morphology for l ≥
4, whichis clearly shown in Fig. 3. (Note that, the morphology of l = 3 map is different, whichmay relate to the unsolved low quadrupole problem as well as the alignment of quadrupole Throughout this paper, we use the polar coordinate ( θ, φ ) in the Galactic system, which relates to theGalactic coordinate ( l , b ) by l = 90 o − θ and b = φ . q , wherethe parity parameter G ( l ; ˆ q ) for each l is minimized (note that different from the problemin (Gubitosi et al. 2011), here as the widely discussed aligment problem of quadrupole andoctupole in (de Oliveira-Costa et al. 2006), the uncertainties of the preferred directions aredifficult to be defined), which are very close with each other for l ≥
4. So, we can choose thespecial direction ˆ q (note that − ˆ q is another equivalent preferred direction), where all theparameters G ( l ; ˆ q ) are minimized or maximized. We picked out these regions, and plottedthem in the Galactic coordinate system in Fig. 4. It is interesting to find that the preferreddirections ˆ q , where parity violation is largest, are coincident with the WMAP7 KD direction(Jarosik et al. 2011), while the preferred directions ˆ q , where parity asymmetry is smallest,are nearly perpendicular to the KD direction. If we assume the parity asymmetry in theCMB has the cosmological origin, it is very hard to explain these coincidences. So, thecoincidence of the preferred direction ˆ q with the WMAP7 KD direction implies that theCMB parity asymmetry may relate to the possible contamination of residual WMAP KDcomponent.Although we will not detailedly study the physical mechanism in this paper, we couldprovide some possible explanations for this coincidence problem. It is noticed that there isnot a great deal of residual dipole in the WMAP data, so a possible explanation could beconnected to the use of the dipole as a photemetric calibrator in the WMAP data set.Another possible explanation is related to the contaminations generated by the collectiveemission of Kuiper Belt Objects (KBOs) and other minor bodies in the solar system wherethe KD direction is localized. Since the emission of KBOs is nearly independent of thefrequency in the WMAP frequency range, this contamination is very hard to be removed inthe WMAP data analysis. In (Maris et al. 2011; Hansen et al. 2011b), it was discussed thatthis foreground residual could leave significant parity asymmetry in the CMB data.Besides, the explanation may also relate to the measure deviation of the WMAP kine-matic dipole, which could be caused by the measure error in dipole direction, antenna point-ing direction, sidelobe pickup contamination, and so on. In (Liu & Li 2011), it was foundthat this KD deviation could generate the artificial CMB anisotropies in the low multipoles.If this is true, these artificial components may account for the CMB parity violation.In order to cross-check this result, we consider another rotationally variant estimator,which is proposed by (de Oliveira-Costa et al. 2006)˜ D ( l ) ≡ l + 1 l X m = − l m | a lm | . (11)If our Universe is statistically isotropic, the ensemble average of this estimator is related to 8 –the power spectrum as follows: h ˜ D ( l ) i = l ( l + 1)3 h C l i . (12)As well discussed, this statistic has also chosen the preferred direction, i.e. the z -axis direc-tion. In addition, this statistic favors high m s and so it works well in searches for planarity.In a quantum mechanical system, this quantity also corresponds to the angular momentumalong the z -axis direction (Edmonds 1985). Due to the rotational variance of this quantity,we can define ˜ D ( l ; ˆ q ) and the corresponding parity parameter ˜ G ( l ; ˆ q ), where ˆ q is the Eulerrotation angle. We notice that in the definition of ˜ G ( l ; ˆ q ), due to the factor m in ˜ D ( l ), theweight of higher multipoles are much greater than the lower ones. So in this paper, we onlyconsider the parity parameter ˜ G ( l ; ˆ q ) for l ≤
10, where the parity violation is obvious. Weplot the quantity ˜ G ( l ; ˆ q ) as a function of ˆ q for low multipoles in Fig. 5, and find that thesemaps have the quite similar morphology, especially for 4 ≤ l ≤
10. The preferred direc-tions ˆ q (similar, − ˆ q is another equivalent preferred direction), where the parity parameter˜ G ( l ; ˆ q ) for each l is minimized, are also listed in Table 2. Again, we find the similar results:for 4 ≤ l ≤
10, the bluer regions (larger parity violation) are excellently coincident to theWMAP7 KD direction, while the redder regions (smaller parity asymmetry) are nearly per-pendicular to the KD direction. So this cross-check agrees with our previous finding: theCMB parity asymmetry in the low multipoles may connect with the possible contaminationof WMAP KD component.
4. Conclusion
In this paper, we have investigated the directional properties of CMB parity asymmetry.In order to break the rotational invariance of the CMB power spectrum, we defined two dif-ferent power spectrum estimators so that a special direction is picked out. By rotating theseestimators with respect to the Galactic coordinate system, we studied the correspondingparity parameters as functions of the preferred directions ˆ q , where the parity parameters areminimized or maximized. We found that these preferred directions are aligned (parallel orperpendicular) with the WMAP7 kinematic dipole direction, which implies that the CMBparity asymmetry may be produced by the systematics associated with kinematic dipole.This study also shows that the effect of the WMAP kinematic dipole may extend to thehigher multipoles l ∼ REFERENCES
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11 – C ( θ = π ) [ m K ] l ′ min Fig. 1.— The theoretical (blue curve) and observed (red curve) values of C ( θ = π ) as afunction of l ′ min , where the WMAP7 power spectra C ( l ) have been used as the observeddata. The error bars indicate the 1 σ confident levels caused by cosmic variance.. 12 –
30 60 90 120 150 180−600−400−2000200400600800100012001400 θ [ ° ] C ( θ ) [ µ K ] Λ CDM Λ CDM with ILC dipoleILCforeground reduced V band
Fig. 2.— The TT correlation function estimated from WMAP7 observational data setsand a theoretical prediction with 1 σ interval (shaded with Cyan color). We estimated thetheoretical prediction respectively with l ′ min = 2 and l ′ min = 1, where we used the residualdipole anisotropy of the ILC7 map. The foreground-contaminated region in the data sets isexcluded by the WMAP KQ75 mask. . 13 –Fig. 3.— The parameter G ( l ; ˆ q ) based on the estimators in Eq. (7) as a function of ˆ q ≡ ( θ, φ )for l = 3 , , ,
21 (left panels) and l = 4 , , ,
22 (right panels). 14 –Fig. 4.— The ILC7 dipole component in the Galactic coordinate system. The functions G ( l ; ˆ q ) (4 ≤ l ≤
22) minimize at the white regions, and maximize at the black regions. Notethat the center direction of the white regions are ˆ q = (46 . o , . o ) and − ˆ q , while thoseof the black regions are ˆ q = (50 . o , . o ) and − ˆ q .. 15 –Fig. 5.— The parameter ˜ G ( l ; ˆ q ) based on the estimators in Eq.(11) as a function of ˆ q ≡ ( θ, φ )for l = 3 , , , l = 4 , , ,
10 (right panels). 16 –Table 1: The WMAP7 kinematic dipole direction is compared with the preferred directionˆ q = ( θ, φ ), where the parity parameter G ( l ; ˆ q ) (based on the estimator in Eq. (7)) isminimized. Note that − ˆ q is another preferred direction. θ [ o ] φ [ o ] cos α a KD 41.74 263.99 —— l = 3 85.22 204.61 0.400 l = 4 46.59 280.89 0.975 l = 7 48.19 279.14 0.976 l = 8 48.99 277.03 0.979 l = 11 49.77 277.73 0.976 l = 12 49.77 277.73 0.976 l = 21 51.32 283.36 0.957 l = 22 50.50 284.06 0.957 a α is the angle between ˆ qq
10 (right panels). 16 –Table 1: The WMAP7 kinematic dipole direction is compared with the preferred directionˆ q = ( θ, φ ), where the parity parameter G ( l ; ˆ q ) (based on the estimator in Eq. (7)) isminimized. Note that − ˆ q is another preferred direction. θ [ o ] φ [ o ] cos α a KD 41.74 263.99 —— l = 3 85.22 204.61 0.400 l = 4 46.59 280.89 0.975 l = 7 48.19 279.14 0.976 l = 8 48.99 277.03 0.979 l = 11 49.77 277.73 0.976 l = 12 49.77 277.73 0.976 l = 21 51.32 283.36 0.957 l = 22 50.50 284.06 0.957 a α is the angle between ˆ qq and the KD direction.
17 –Table 2: The WMAP7 kinematic dipole direction is compared with the preferred directionˆ q = ( θ, φ ), where the parity parameter ˜ G ( l ; ˆ q ) (based on the estimator in Eq. (11)) isminimized. Note that − ˆ q is another preferred direction. θ [ o ] φ [ o ] cos α a KD 41.74 263.99 —— l = 3 86.42 206.02 0.401 l = 4 45.80 303.20 0.890 l = 5 48.19 305.86 0.867 l = 6 52.08 274.22 0.975 l = 7 57.91 279.84 0.939 l = 8 39.20 255.57 0.994 l = 9 37.20 252.90 0.989 l = 10 40.30 249.17 0.985 a α is the angle between ˆ qq