Effects of a Cut, Lorentz-Boosted sky on the Angular Power Spectrum
Thiago S. Pereira, Amanda Yoho, Maik Stuke, Glenn D. Starkman
EEffects of a Cut, Lorentz-Boosted sky on the Angular Power Spectrum
Thiago S. Pereira,
1, 2
Amanda Yoho, Maik Stuke, and Glenn D. Starkman Instituto de F´ısica Te´orica, Universidade Estadual Paulista,Rua Dr. Bento T. Ferraz, 271, 01140-070, S˜ao Paulo, SP, Brazil Departamento de F´ısica, Universidade Estadual de LondrinaCampus Universit´ario, 86051-990, Londrina, Paran´a, Brazil CERCA/ISO, Department of Physics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, OH 44106-7079, USA. Falkult¨at f¨ur Physik, Universit¨at Bielefeld,Postfach 100131, 33501 Bielefeld, Germany
The largest fluctuation in the observed CMB temperature field is the dipole, its originbeing usually attributed to the Doppler Effect – the Earth’s velocity with respect to theCMB rest frame. The lowest order boost correction to temperature multipolar coefficientsappears only as a second order correction in the temperature power spectrum, C (cid:96) . Since v/c ∼ − , this effect can be safely ignored when estimating cosmological parameters [4–7].However, by cutting our galaxy from the CMB sky we induce large-angle anisotropies in thedata. In this case, the corrections to the cut-sky C (cid:96) s show up already at first order in theboost parameter. In this paper we investigate this issue and argue that this effect mightturn out to be important when reconstructing the power spectrum from the cut-sky data. I. INTRODUCTION
In the last decade, cosmology has become a precision science, driven largely by measurementsof the fluctuations in the Cosmic Microwave Background (CMB) temperature. These fluctuations– nearly 10 times smaller than the average temperature of the universe – are the primary windowonto most cosmological parameters [1]. Because the fluctuations are so small, one must know thecontributions to the measured temperature field from foregrounds and other contaminants. A greatdeal of effort has been put in to characterizing foreground signals from dust, synchrotron, and free-free emission [2]. Even so, when inferring cosmological parameters significant fractions of the skyare usually omitted (“cut”) from full-sky maps in order to minimize effects from non-primordialsources [3].Surprisingly, one particular known systematic effect – the distortion of the CMB radiation dueto our motion relative to the preferred cosmological frame – has been given comparatively littleattention [4–7]. It has become accepted practice to simply remove the dipole from a sky map a r X i v : . [ a s t r o - ph . C O ] S e p before calculating the power spectrum [8]. This is based on the belief that the effect on C (cid:96) , the (cid:96) th element of the angular power spectrum of the temperature anisotropy, (the central quantityin cosmological parameter estimation in the context of the canonical Lambda Cold Dark Mattermodel, aka ΛCDM) is proportional to β (cid:96) , where β ∼ − is our speed relative to the cosmologicalframe (as determined by the magnitude of that CMB dipole). In this context, the dipole is indeedthe only multipole for which the Doppler shift due to our motion is significant. Nevertheless, ithas been shown [9] that the second order ( O ( β )) Doppler effect noticeably alters the directions ofthe quadrupole multipole vectors, which characterize the shape of the quadrupole, despite the factthat it contributes negligibly to C , the strength of the quadrupole.Meanwhile, several groups have independently examined (to order β ) the effect of Lorentzboosting the CMB, and determined that there are actually significant contributions to highermultipoles [4–7]. It has also separately been shown that simply masking a map induces correlationsamong nearby multipoles [10]. These two phenomena highlight a need for characterizing the effectof masking a Lorentz-boosted temperature field, since the process will undoubtedly mix highermultipoles of the true temperature field that have boost corrections.This paper is organized as follows: in Section II we discuss the effect of boosting the multipolemoments of the CMB, in Section III we derive an expression for the boosted pseudo power spectrum,in Section IV we show preliminary numerical estimates for corrections to the power spectrum anddiscuss their significance, as well as portions of the project that will be explored in future papers. II. THE EFFECTS OF BOOSTS ON THE MULTIPOLE MOMENTS OF THEANGULAR POWER SPECTRUM OF THE CMB
If an observer in the rest frame of the CMB (denoted S ) measures a photon of frequency ν arriving along a line of sight ˆ n , then an observer in another frame S (cid:48) that is moving with respectto the CMB at velocity v ˆ v will measure the incoming photon to be arriving along a differentline-of-sight ˆ n (cid:48) with a different frequency ν (cid:48) . (Note that we will not concern ourselves here withany ambiguities in determining S associated with the existence of inhomogeneities, in particular acosmological dipole, on the assumption that that dipole is O (10 − ) smaller than the effects we willuncover.) The motion of the observer in S (cid:48) thus induces two effects: a Doppler shift in the photonfrequency and an aberration – a shift in the direction from which the photon arrives. These twoeffects can be seen explicitly in the relation between ˆ n and ˆ n (cid:48) ˆ n (cid:48) = (cid:18) cos θ + β β cos θ (cid:19) ˆ v + ˆ n − cos θ ˆ v γ (1 + β cos θ ) , (1)where β ≡ v/c and cos θ ≡ ˆ n · ˆ v . The change in observed frequency in S (cid:48) is given by a simpleLorentz transformation ν (cid:48) = γν (1 + β cos θ ) , (2)where γ = (1 − β ) − / is the standard Lorentz factor. This angle θ is related to the angle θ (cid:48) ,measured in the frame S (cid:48) , via cos θ (cid:48) = cos θ + β β cos θ . (3)Because of these two effects, we would like to boost the measured intensity, I ( ν, ˆ n ) (the incidentCMB power per unit area per unit frequency, per solid angle), and then relate the sphericalharmonic coefficients of the intensity to the traditional temperature fluctuation coefficients. First,we will need the relation between the observed intensity in each frame [11], I (cid:48) ( ν, ˆ n (cid:48) ) = γ (1 + β cos θ ) I ( ν, ˆ n ) . (4)Expanding both sides in terms of spin-weighted spherical harmonics ( s Y (cid:96)m ) and using the factthat d ˆ n (cid:48) = γ − (1 + β cos θ ) − d ˆ n [12], we get the following expression for the boosted multipolemoments, a (cid:48) (cid:96)m ( ν (cid:48) ), in terms of the rest-frame multipole moments a (cid:96)m ( ν ): a (cid:48) (cid:96) m ( ν (cid:48) ) = (cid:88) (cid:96) ,m (cid:90) d ˆ n γ (1 + β cos θ ) a (cid:96) m ( ν ) s Y (cid:96) m (ˆ n ) s Y ∗ (cid:96) m (ˆ n (cid:48) ) . (5)We use the spin-weighted spherical harmonics to keep the expression completely general, so that theeffect of the boost on the measured CMB spectrum can be explored for temperature fluctuationsas well as polarization. Note that we have also implicitly chosen a frame where β ˆv is in the ˆz direction.Expression (5) can be expanded as a series in β by means of Eqs. (1) and (2). The expressionto order β can be found in the appendix. To order β we find (consistent with [4]): a (cid:48) (cid:96)m ( ν (cid:48) ) = (cid:20) − βsm(cid:96) ( (cid:96) + 1) (cid:18) − ν (cid:48) ddν (cid:48) (cid:19)(cid:21) a (cid:96)m ( ν (cid:48) ) − β s ξ ( (cid:96) +1) m (cid:20) ( (cid:96) −
1) + ν (cid:48) ddν (cid:48) (cid:21) a ( (cid:96) +1) m ( ν (cid:48) ) − β s ξ (cid:96)m (cid:20) − ( (cid:96) + 2) + ν (cid:48) ddν (cid:48) (cid:21) a ( (cid:96) − m ( ν (cid:48) ) (6)where s ξ (cid:96)m ≡ (cid:115) ( (cid:96) − m )( (cid:96) − s ) (cid:96) (2 (cid:96) + 1)(2 (cid:96) − . Here, the multipole moments are the frequency dependent intensity coefficients. The more familiartemperature coefficients can be deduced from the coefficients above through the Stefan-Boltzmannlaw, which for fluctuations and a series expansion to O ( β ) reads: δT (ˆ n ) T = 14 δI (ˆ n ) I (7)where I (ˆ n ) = (cid:90) ∞ I ( ν, ˆ n ) dν (8)is the incident CMB power per unit area per solid angle and T and I are sky averages (monopoles).Therefore, in order to get the temperature multipole moments we integrate Eq.(6) over all frequen-cies (making use of the fact that the a (cid:48) (cid:96)m ( ν (cid:48) ) are Planck distributed) and divide the result by fourtimes the average intensity per solid angle [15]. To first order in β we have for the temperature a (cid:48) (cid:96)m ( s = 0): a (cid:48) (cid:96)m = a (cid:96)m + βξ + (cid:96)m a ( (cid:96) +1) m + βξ − (cid:96)m a ( (cid:96) − m (9)where the conversion factor from temperature to intensity was absorbed into the multipolar coef-ficients and ξ + (cid:96)m ≡ − ( (cid:96) − ξ ( (cid:96) +1) m , ξ − (cid:96)m ≡ ( (cid:96) + 3) ξ (cid:96)m . (10)We would now like to estimate the bias induced by the boost on the temperature power spec-trum. Before getting into the details of the calculation, it is first necessary to define two quantities.Throughout this paper we will make the distinction between the theoretical power spectrum, givenby (cid:104) a (cid:96)m a ∗ (cid:96) (cid:48) m (cid:48) (cid:105) = δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) C (cid:96) , (11)and the measured power spectrum (also referred to as the power spectrum estimator), given by C (cid:96) ≡ (cid:96) + 1 (cid:88) m | a (cid:96)m | . (12)To estimate this bias we need to go second order in the expansion (9) since the C (cid:96) s are quadraticin a (cid:96)m (see the appendix for details). The key point here is that, on the assumption of statisticalisotropy of the unboosted a (cid:96)m , the smallest β correction to the boosted power spectrum is givenby [4]: (cid:104)C (cid:48) (cid:96) (cid:105) ≈ C (cid:96) (1 + 4 β + O ( β )) . (13)Note that the main effect is a rescaling of the spectrum by an overall amplitude 1 + 4 β . Moreover,since β ∼ − , the boosted power spectrum is essentially unbiased. We will now show that thisconclusion may not be true in the case where the boosted spectrum is reconstructed from a cut-sky. III. MASKING EFFECTS ON THE BOOSTED ANGULAR POWER SPECTRUM
When analyzing CMB temperature maps, it is common practice to mask regions of the sky thatare believed to be contaminated. The region masked most often is the galaxy, where the observedtemperature signal is known not to come from the surface of last scattering. In this case, we havea new expression for the measured temperature fluctuations:∆ ˜ T ( n ) = (cid:88) (cid:96)> (cid:96) (cid:88) m = − (cid:96) a (cid:96)m W ( n ) Y (cid:96)m ( n ) , (14)where W ( n ) is a window function described by the mask. One can then decompose the left handside of Eq.(14) and solve for ˜ a (cid:96)m in terms of a (cid:96)m . We then get the linear relation˜ a (cid:96) m = (cid:88) (cid:96) ,m a (cid:96) m K (cid:96) m (cid:96) m . (15)Here K (cid:96) m (cid:96) m is a general kernel that contains all of the information about the window function, W ( n ) = (cid:80) (cid:96) m w (cid:96) m Y (cid:96) m ( n ). It is given explicitly by K (cid:96) m (cid:96) m ≡ (cid:88) (cid:96) ,m w (cid:96) m ( − m (cid:20) (2 (cid:96) + 1)(2 (cid:96) + 1)(2 (cid:96) + 1)4 π (cid:21) × (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) m − m m , (16)where the 3 × C (cid:96) = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) | ˜ a (cid:96)m | . (17)These ˜ C (cid:96) s are not unbiased estimators for the true C (cid:96) s, but assuming statistical isotropy theirexpectation values are related by a real and symmetric mode coupling matrix, (cid:104) ˜ C (cid:96) (cid:105) = (cid:88) (cid:96) M (cid:96) (cid:96) (cid:104)C (cid:96) (cid:105) = (cid:88) (cid:96) M (cid:96) (cid:96) C (cid:96) , (18)where M (cid:96) (cid:96) = 2 (cid:96) + 14 π (cid:88) (cid:96) (2 (cid:96) + 1) W (cid:96) (cid:96) (cid:96) (cid:96) (19)and W (cid:96) = 1 / (2 (cid:96) + 1) (cid:80) m | w (cid:96)m | is the power spectrum of the window function [10].We can therefore obtain an estimator for C (cid:96) by approximately re-writing the relation with (cid:104) ˜ C (cid:96) (cid:105) replaced by its observed value on the sky ˜ C obs(cid:96) (cid:39) (cid:80) (cid:96) M (cid:96) (cid:96) C (cid:96) . We approximately invert Eq.(18)to obtain the true C (cid:96) s from the pseudo ones. However, this prescription is complicated by Lorentzboosts. Since the masking procedure is unavoidably performed in the boosted frame, we no longerget a simple mode-coupling relation as in Eq.(18). Instead, we will now have a more generalexpression: (cid:104) ˜ C (cid:48) (cid:96) (cid:105) = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) | ˜ a (cid:48) (cid:96) m | = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) (cid:88) (cid:96) m (cid:88) (cid:96) m (cid:104) a (cid:48) (cid:96) m a ∗(cid:48) (cid:96) m (cid:105) K (cid:96) m (cid:96) m K ∗ (cid:96) m (cid:96) m , (20)where the multipolar coefficients are given by Eq.(9). Note that the kernels in this expressioncouple all the elements of the non-diagonal covariance matrix (cid:104) a (cid:48) (cid:96) m a ∗(cid:48) (cid:96) m (cid:105) . As a consequence,linear β terms which were absent in Eq.(9) will now contribute to the window function multipolarcoefficients.Going back to expression Eq.(20), note that to first order in β there will only be couplingbetween (cid:96) and (cid:96) satisfying | (cid:96) − | ≤ (cid:96) ≤ (cid:96) + 1. Therefore this expression can be re-written as (cid:104) ˜ C (cid:48) (cid:96) (cid:105) = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) (cid:88) (cid:96) ,m (cid:96) +1 (cid:88) m ,(cid:96) = (cid:96) − (cid:104) a (cid:48) (cid:96) m a ∗(cid:48) (cid:96) m (cid:105) K (cid:96) m (cid:96) m K ∗ (cid:96) m (cid:96) m . (21)When writing out each term in the sum over (cid:96) and plugging in the expression for a (cid:48) (cid:96) m (seeEq.(9)), we arrive at the following relation between the boosted pseudo- C (cid:96) ’s and the expectationvalues of the CMB rest frame, C (cid:96) ’s: (cid:104) ˜ C (cid:48) (cid:96) (cid:105) = (cid:88) (cid:96) ( M (cid:96) (cid:96) C (cid:96) + N (cid:96) (cid:96) C (cid:96) + P (cid:96) (cid:96) C (cid:96) +1 + Q (cid:96) (cid:96) C (cid:96) − ) (22)where N (cid:96) (cid:96) = β (cid:96) + 1 (cid:88) m ,m K (cid:96) m (cid:96) m (cid:16) K ∗ (cid:96) m ( (cid:96) +1) m ξ − ( (cid:96) +1) m + K ∗ (cid:96) m ( (cid:96) − m ξ +( (cid:96) − m (cid:17) , (23) P (cid:96) (cid:96) = β (cid:96) + 1 (cid:88) m ,m K (cid:96) m (cid:96) m K ∗ (cid:96) m ( (cid:96) +1) m ξ + (cid:96) m , (24) Q (cid:96) (cid:96) = β (cid:96) + 1 (cid:88) m ,m K (cid:96) m (cid:96) m K ∗ (cid:96) m ( (cid:96) − m ξ − (cid:96) m . (25)Ignoring ”fence-post” terms and re-indexing the sum, we can re-write Eq.(26) as (cid:104) ˜ C (cid:48) (cid:96) (cid:105) ≈ (cid:88) (cid:96) ( M (cid:96) (cid:96) + N (cid:96) (cid:96) + P (cid:96) (cid:96) − + Q (cid:96) (cid:96) +1 ) C (cid:96) (26)Clearly, the boosted pseudo power spectra in Eq.(26) are far from being unbiased estimators ofthe C (cid:96) s. If we neglect the boost effects on the C (cid:96) s and only multiply by M − to solve for thevalues of the C (cid:96) s, then we will obtain an unbiased estimate since N , P , and Q are β -dependent.Furthermore, the bias in Eq.(26) produces more than just an overall multiplicative constant, sincethe matrices M , N , P and Q are not in general peaked at (cid:96) ≈ (cid:96) . The correct reconstructionof the true C (cid:96) s depends on the careful inversion of the sum of the four matrices above. Beforecarrying a complete numerical analysis, we must analyze the type of couplings that the matrices N , P and Q induce on the pseudo-power spectrum. This is the subject of the next section. A. Example: Equatorial Strip
Before we calculate these couplings explicitly, we should mention two important points. First, inthis section we will examine only the special case where both the boost and the normal to the maskare in the z -direction. This is of course unlikely to be correct. It is also likely to underestimate themagnitude of the effects we are examining. A thorough analysis that also includes the angle betweenthe boost and the galactic plane as a free parameter is being carried out and will be reported on ina future work. Our point here is to note that β corrections exist regardless of this direction, andmight in turn affect the reconstruction of the true C (cid:96) s if the boost is not correctly accounted for.Second, given that linear corrections will induce couplings of the form ( (cid:96) (1 , , (cid:96) (2 , ±
1) and becauseof the restriction that (cid:96) + (cid:96) + (cid:96) must be even in Eq.(16), the effect uncovered is a mixed-parityone. This means that β corrections are identically zero for masks which are even or odd functions.Real CMB masks do not possess well-defined parity, a fact which further points to the importanceof the analysis being carried out here.To illustrate the type of coupling induced by the matrices (23-25), we have carried out a nu-merical analysis using a north-south asymmetric equatorial strip as a mask with different widthsand degrees of asymmetry. Figure 1 show some plots of these matrices for (cid:96) = 5 as a function of (cid:96) . Noticeably, the main difference between (23-25) and the matrix (19) is that the former assumeboth positive and negative values, whereas the latter is a positive-definite matrix. Regarding theiramplitude, we have checked that matrices N , P and Q can be as large as 15% of matrix M for therange of (cid:96) shown in Fig. 1. Including the β factor, this should amount to a correction of order10 − to Eq.(18). This may look smaller than the known 10 − signal present in the off-diagonalterms of full-sky correlation matrices [4, 7]. However, we emphasize that here we are proposing acorrection to the reconstructed true C (cid:96) s, and not cross-terms of the correlation matrix as discussed FIG. 1: The top figure shows the effect of moving the mask north by 5 degrees with respect to the galacticequator. Note that there are significant changes in contribution to the correction matrix elements as themask is moved to 76-105 degrees (blue dots), 86-115 degrees (green triangles), and 96-125 degrees (redcircles). The bottom figure shows the effect of varying mask size while keeping it only 1 degree north-southasymmetric for 76-105 (green triangles), 79-100 (blue dots), and 84-95 (red circles) degree masks. Here wenotice slight changes in the matrix elements when the masks decrease in size, however the effect is very smallwhen compared to the effect of moving the mask off-center. All of these values were generated with (cid:96) = 5. in [7]; the former are not only less noise contaminated but also have a smaller cosmic variance. IV. DISCUSSION AND FUTURE WORK
Determining our direction and velocity with respect to the CMB rest frame is a fundamentalquest for the standard cosmological model. While the main observable effects, noticeably thedipole, was already detected in the late 70’s [13], higher multipolar distortions in the temperaturefield may yet be uncovered by future high-resolution data from the Planck satellite. In this paperwe have shown that the CMB power spectrum may be systematically contaminated by this effectalready at first order in the boost parameter if the boost effect is not taken into account whenreconstructing full-sky data from cut-skies. We estimated this effect as a 10 − signal in the pseudo C (cid:96) s. Although this may seem smaller than the signal in off-diagonal full-sky CMB observables,only a careful reconstruction of full-sky data will be able to set this issue. We have also shown,using a simple strip as a mask, that the boost couples different multipoles in a way which is insharp contrast to the couplings induced by the mask alone. Since the masking procedure is knownto induce large-angle correlations, reconstructing the C (cid:96) s correctly may have a sizable impact intracing systematics and/or accounting for large-angle anomalies.In a companion paper we will carry a more complete analysis of these effects, including the anglebetween the boost and the galactic plane as a free parameter. This angle dependence will lead tofurther couplings between the boost and the mask which might be used as a further estimator ofthe boost direction.Since not all of the available information is contained in the C (cid:96) ’s (due to the coupling betweennearby modes of the a (cid:48) (cid:96)m ), we will define off-diagonal estimators of the covariance matrix. Addi-tionally, we will determine the likelihood of finding the direction of β from our boosted covariancematrix. Both of these issues have been discussed in part by [7]. We will also be looking into theeffect of cut skies on reconstructing polarization estimators, which can be carried out starting fromexpression 6 and setting s = 2. Appendix A: Second order results1. Multipolar coefficients
For completeness, we present here the expansion in the brightness multipolar coefficients tosecond order in the boost parameter. Similar expansions can also be found in [4, 7] We now want0to expand Eq. 5 to O ( β ). We begin by re-writing γ (1 + β cos θ ) (cid:39) β cos θ + 12 β (A1)and a (cid:96) m ( ν ) using a (cid:96)m ( ν ) (cid:39) a (cid:96)m ( ν (cid:48) ) + (cid:18) β cos θ − β − β cos θ (cid:19) ν (cid:48) ddν (cid:48) a (cid:96)m ( ν (cid:48) ) + (cid:0) β cos θ (cid:1) ν (cid:48) d dν (cid:48) a (cid:96)m ( ν (cid:48) ) (A2)We will also expand s Y ∗ (cid:96) m (ˆ n (cid:48) ) about ˆ n to get s Y (cid:96)m (ˆ n (cid:48) ) = s Y ∗ (cid:96)m (ˆ n ) − β sin θ (1 + β cos θ ) ∂ s Y ∗ (cid:96)m (ˆ n ) ∂ cos θ + β sin θ ∂ s Y ∗ (cid:96)m (ˆ n ) ∂ cos θ . (A3)There is a general expression for writing the derivatives of a spherical harmonic in terms of otherspherical harmonics (valid for (cid:96) ≥ θ ∂ s Y (cid:96)m ∂ cos θ = (cid:96) (cid:115) ( (cid:96) + 1) − m (2 (cid:96) + 1)(2 (cid:96) + 3) s Y (cid:96) +1 ,m + sm(cid:96) ( (cid:96) + 1) s Y (cid:96),m − ( (cid:96) + 1) (cid:115) (cid:96) − m (2 (cid:96) + 1)(2 (cid:96) − s Y (cid:96) − ,m (A4)We also see that there will be some β cos θ cross terms to deal with, which can be done using theformulacos θ s Y (cid:96)m = (cid:115) ( (cid:96) + 1) − m (2 (cid:96) + 1)(2 (cid:96) + 3) s Y (cid:96) +1 ,m − sm(cid:96) ( (cid:96) + 1) s Y (cid:96),m + (cid:115) (cid:96) − m (2 (cid:96) + 1)(2 (cid:96) − s Y (cid:96) − ,m . (A5)Putting all this together and working to order β , we arrive at an expression for the multipolemoments measured in the moving frame as a function of those in the CMB rest frame, noting thatthe usage of A4 and A5 limits this expression to (cid:96) ≥ a (cid:48) (cid:96)m ( ν (cid:48) ) = (cid:26) − βsm(cid:96) ( (cid:96) + 1) (cid:18) − ν (cid:48) dd ν (cid:48) (cid:19) + β (cid:20) s ξ (cid:96) +1) m (cid:18) − (cid:96) ( (cid:96) + 4) + 2 (cid:96)ν (cid:48) dd ν (cid:48) + ν (cid:48) d d ν (cid:48) (cid:19) + 12 s ξ (cid:96)m (cid:18) − ( (cid:96) + 1)( (cid:96) − − (cid:96) + 1) ν (cid:48) dd ν (cid:48) + ν (cid:48) d d ν (cid:48) (cid:19) + 12 (cid:18) − ν (cid:48) dd ν (cid:48) (cid:19) + s m (cid:96) ( (cid:96) + 1) (cid:18) − − ν (cid:48) dd ν (cid:48) + ν (cid:48) d ν (cid:48) (cid:19)(cid:21)(cid:27) a (cid:96)m ( ν (cid:48) ) − β s ξ ( (cid:96) +1) m (cid:20) ( (cid:96) −
1) + ν (cid:48) dd ν (cid:48) − βsm(cid:96) ( (cid:96) + 2) (cid:18) (cid:96) − − ( (cid:96) − ν (cid:48) dd ν (cid:48) − ν (cid:48) d d ν (cid:48) (cid:19)(cid:21) a ( (cid:96) +1) m ( ν (cid:48) ) − β s ξ (cid:96)m (cid:20) − ( (cid:96) + 2) + ν (cid:48) dd ν (cid:48) − βsm ( (cid:96) + 1)( (cid:96) − (cid:18) − (cid:96) + 2) + ( (cid:96) + 3) ν (cid:48) dd ν (cid:48) − ν (cid:48) d d ν (cid:48) (cid:19)(cid:21) a ( (cid:96) − m ( ν (cid:48) )+ 12 β s ξ ( (cid:96) +2) ms ξ ( (cid:96) +1) m (cid:18) (cid:96) ( (cid:96) −
1) + 2 (cid:96)ν (cid:48) dd ν (cid:48) + ν (cid:48) d d ν (cid:48) (cid:19) a ( (cid:96) +2) m ( ν (cid:48) )+ 12 β s ξ (cid:96)ms ξ ( (cid:96) − m (cid:18) ( (cid:96) + 1)( (cid:96) + 2) − (cid:96) + 1) ν (cid:48) dd ν (cid:48) + ν (cid:48) d d ν (cid:48) (cid:19) a ( (cid:96) − m ( ν (cid:48) ) (A6)1where s ξ (cid:96)m = (cid:115) ( (cid:96) − m ) ( (cid:96) − s ) (cid:96) (2 (cid:96) + 1)(2 (cid:96) − . (A7)Note that this result differs slightly from the one shown in [4]. We have checked that they are thesame up to some re-arranging.
2. Expansion of the Pseudo Angular Power Spectrum
Using expression (A6) and (20) we can show after a straightforward algebra that: (cid:104) ˜ C (cid:48) (cid:96) (cid:105) = (cid:88) (cid:96) C (cid:96) M (cid:96) (cid:96) + 1(2 (cid:96) + 1) (cid:88) (cid:96) (cid:88) m m K (cid:96) m (cid:96) m (cid:110) β (cid:2) ξ (cid:96) +1 m ( (cid:96) + 4) K (cid:96) m (cid:96) +1 m − ξ (cid:96) m ( (cid:96) − K (cid:96) m (cid:96) − m (cid:3) C (cid:96) + (cid:2) − βξ (cid:96) +1 m ( (cid:96) − K (cid:96) m (cid:96) +1 (cid:3) C (cid:96) +1 + (cid:2) βξ (cid:96) m ( (cid:96) + 3) K (cid:96) m (cid:96) − m (cid:3) C (cid:96) − (cid:111) . (A8) Acknowledgments
We would like to thank Anthony Challinor, Craig Copi, Arthur Kosowsky, Dominik J. Schwarz,and Pascal Vaudrevange for useful conversations during the preparation of this work. TSP thanksBrazilian agency FAPESP for partial support and the physics department of Case Western Re-serve University for its hospitality during the initial stages of this work. GDS and AY are sup-ported by a grant from the US Department of Energy and by NASA under cooperative agreementNNX07AG89G. MS is supported by the Friedrich- Ebert-Foundation and thanks the physics de-partment of Case Western Reserve University for its hospitality. [1] D. Larson et al. , (2010), 1001.4635.[2] B. Gold et al. , (2010), 1001.4555.[3] WMAP, C. Bennett et al. , Astrophys. J. Suppl. , 97 (2003), astro-ph/0302208.[4] A. Challinor and F. van Leeuwen, Phys. Rev.
D65 , 103001 (2002), astro-ph/0112457.[5] M. Kamionkowski and L. Knox, Phys. Rev.
D67 , 063001 (2003), astro-ph/0210165.[6] A. Kosowsky and T. Kahniashvili, (2010), 1007.4539.[7] L. Amendola et al. , (2010), 1008.1183.[8] WMAP, G. Hinshaw et al. , Astrophys. J. Suppl. , 135 (2003), astro-ph/0302217. [9] D. J. Schwarz, G. D. Starkman, D. Huterer, and C. J. Copi, Phys. Rev. Lett. , 221301 (2004),astro-ph/0403353.[10] E. Hivon et al. , (2001), astro-ph/0105302.[11] C. Misner, K. Thorne, and J. Wheeler, Gravitation (WH Freeman & co, 1973).[12] J. McKinley, American Journal of Physics , 612 (1980).[13] G. F. Smoot, M. V. Gorenstein, and R. A. Muller, Phys. Rev. Lett. , 898 (1977).[14] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables [15] The monopole I0