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Draftversion February5,2008 PreprinttypesetusingLATEXstyleemulateapjv.6/22/04 ABERRATION OF THE COSMIC MICROWAVE BACKGROUND S. Burles1 and S. Rappaport 1 Draft versionFebruary 5, 2008 ABSTRACT Themotionofthesolarsystembarycenterwithrespecttothecosmicmicrowavebackground(CMB) induces a very large apparent dipole component into the CMB brightness map at the 3 mK level. In this Letter we discuss another kinematic effect of our motion through the CMB: the small shift in apparent angular positions due to the aberration of light. The aberration angles are only of order β ≃ 0.001, but this leads to a potentially measurable compression (expansion) of the spatial scale in 6 the hemisphere toward (away from) our motion through the CMB. In turn, this will shift the peaks 0 0 in the acoustic power spectrum of the CMB by a factor of order 1±β. For current CMB missions, 2 and even those in the foreseeable future, this effect is small, but should be taken into account. In principle,if the acousticpeak locationswerenotlimited by samplingnoise (i.e., the cosmic variance), n this effect could be used to determine the cosmic contribution to the dipole term. a J Subject headings: CMB 4 2 1. INTRODUCTION (Bradley 1729). The aberration angles are only of order β (where β =0.00123; Lineweaver et al. 1996), but this 1 Themotionofthesolarsystembarycenterwithrespect v to the cosmic microwave background (CMB) induces a leads to a potentially measurable compression (expan- 9 very large apparent dipole component into the CMB sion)ofthespatialscaleinthehemispheretoward(away 5 brightness map (3.353 mK, Bennett et al. 1996), and a from) our motion through the CMB. In turn, this will 5 shift the peaks in the angular temperature power spec- muchfainter,butstillreadilydetectable,quadrupolesig- 1 trum of the CMB by a factor of order 1±β. The effect natureatthe ∼1µKlevel(e.g.,Lineweaveretal. 1996). 0 amountstoabout1multipole(ℓ)binatthe5thharmonic The empirically fitted dipole term is subtracted out be- 6 of the CMB acoustic spectrum. In any event, the aber- fore computing the spherical harmonics to the residual 0 ration effect can and should be corrected even though it CMB fluctuations (e.g., Bennett et al. 2003). The con- / h comitant kinematically induced quadrupole term is con- leads to shifts in the position of the CMB features by ′ p siderably smaller than the cosmic quadrupole term and only up to ∼ 4. If it were not for inherent uncertain- - ties in determining the centroids of the acoustic peaks canbesubtractedoffaswell. Oneconsequenceofthisop- o due to sampling limitations (i.e., the cosmic variance), r erationisthattheresultantCMBpowerspectrumyields t no intrinsic dipole term, which would otherwise be of this effect could, in principle, yield an independent mea- s surementofthekinematiccontributiontothedipoleand a considerable interest for cosmologicalmodels. allow a correction to measure the intrinsic power in the : The intrinsic cosmic dipole is of particular interest as v cosmic dipole. it represents structure on the largest observable scale, i X and there may well be interesting phenomena to study 2. THEABERRATIONEFFECTONTHECMB r on the scales of the lowest multipoles. Recent analyses a of the first year all-sky WMAP datasets confirmthe low We can describe the aberration of the CMB radiation asfollows. Ifweusesphericalcoordinatesanddefineθ as quadrupoleamplitude (Gaztannãgaet al. 2003;Bennett the polar angle with respect to the direction of motion, et al. 2003;Tegmark et al. 2003)as was observedin the andφastheazimuthalangle,thenthetransformationof earlier COBE DMR maps (Smoot et. al. 1992; Hinshaw angles from the CMB to the barycenter frame is: et al. 1996). In addition, several papers have pointed out the unlikely alignment of the lowest multipoles for sinθ′ a randomGaussianfield, defining the so-called”Axis-of- sinθ=γ(1−βcosθ′) , (1) Evil” (Land & Magueijo 2005; de Oliveira-Costa et al. ′ 2004; Copi et al. 2004; Bielewicz et al. 2005). Possi- φ=φ , (2) ble explanations have been suggested, from selection of where the unprimed and primed frames are the CMB foreground-freesky (Slosar & Seljak 2004) to weak lens- and barycenter frames, respectively, β is the velocity of ing of the cosmic dipole by local structures (Vale 2005). the barycenter with respect to the CMB (∼0.001), γ is Future CMB datasets may settle the matter, but a mea- theusualLorentzfactor,andwhereθ =0correspondsto surementorconstraintof the amplitude anddirectionof the‘forward’direction. Forsmallβtheθtransformations theintrinsiccosmictemperaturedipolecouldbeessential can be expanded in a Taylor series to lowest order in β to reach a final resolution. as: In this Letter we discuss another kinematic effect of ′ ′ our motion throughthe CMB, i.e., the small shift in ap- sinθ =sinθ (1+βcosθ ) . (3) parent angular positions due to the aberration of light Finally, we can expand the arc sine function to find ′ θ(θ ): 1 Department of Physics and Kavli Institute for Astrophysics ′ ′ andSpaceResearch,MIT,Cambridge,MA02139; θ =θ +βsinθ . (4) 2 Burles and Rappaport At any location on the celestial sphere, this leads to a stretching or compression of the scales in the θ and φ directions by fractional amounts: dθ ′ =1+βcosθ (5) dθ′ sinθdφ ′ =1+βcosθ . (6) sinθ′dφ′ Fromthis we conclude that the two-dimensionalangular ′ scale size is compressed by a factor of 1−βcosθ in the forward hemisphere, and stretched by a factor of 1 + ′ β|cosθ | in the backward direction. 3. SHIFTINLOCATIONOFTHECMBACOUSTICPEAKS The full-sky map of the temperature anisotropies as measured by the WMAP satellite (Bennett et al. 2003; Page et al. 2003) is well described by adiabatic fluctuations in a ΛCDM cosmology (Spergel et al. 2003) after removal of foreground contamination to the cosmic signal (Hinshaw et al. 2003; Tegmark et al. 2003). Using thecurrentconcordancemodelofΛCDMwiththeBoltz- mann code CMBFAST (Seljak & Zaldarriaga1996), one can accuratelypredict the expected angular powerspec- Fig. 1.—Theoreticalplotofχaberr givenbyequation(9). Typi- trum of primary temperature anisotropies that will be calerrorbarsoneachofthe∼1500valuesofCℓ usedtoconstruct measuredby the Plancksatellite (Balbiet al. 2002;Piat suchaplotfromactual data wouldbegivenbyeq. (7)andareof etal. 2002;Sandri2004)outtomultipolesof∼1500. For order±10%intheratioofCf/Cb,orabout±0.1intheln. These ℓ ℓ the discussion below, we will assume this spectrum for uncertainties about 10 times larger than the aberration effect on individual values of χaberr, but the overall signature of the shifts the cosmic angular power spectrum in the CMB. shouldbedetectable withall1500points(seetext). IftheCMBpowerspectraarecomputedseparatelyfor portions of the forward and backward looking regions where the angular scale stretching factors are approxi- ′ ◦ ′ ◦ mately constant, e.g., for θ < 45 and θ > 135 , then since the expected shift in ℓ due to the aberration effect the ℓ values of the spherical harmonics at the acoustic peaks should shift (in opposite directions) for the seg- is of order δℓ ≃ βℓ, this implies that by ℓ ∼ 1000 we ments of the celestial sphere located in the forward and may expect to detect a significant shift in ℓn between backwarddirections. The shift in ℓ values is expected to the forward and backward hemispheres. If we utilize all the peaks up to the 5th harmonic, then we gain a small be δℓ = ±βℓ, where β represents the average value of f,b ′ additional factor in signal to noise. A more formal error β|cosθ | over the regions of interest (β ≃ 0.001). Thus, estimate is discussed below. in order to detect this effect – at all – requires an ac- Figure 1 illustrates how the aberration effect would curacy in determining the centroids of the peaks in the quantitatively affect the CMB power spectrum peaks. CMB harmonic structure to about one part in 1000. What is shown is a plot of the quantity The uncertainty in measuring the amplitude C due ℓ to the finite number of modes on the celestial sphere is Cfℓ (ℓ +1) given by: χ =ln ℓ f f , (9) aberr "Cℓbℓb(ℓb+1)# δC 2 1 ℓ = ≃ , (7) dlnCℓ Cℓ s(2ℓ+1)fsky sℓfsky ≃2β dlnℓ +2 , (10) (cid:20) (cid:21) (Scott et al. 1994) where fsky is the fraction of the ce- whichissimplyaconvenientwayofvisualizingtheeffects lestialsphereincludedin the analysis,andwherethe ex- of the aberration shifts. A ΛCDM model was assumed perimentalnoisecontributionisassumedtobenegligible in Figure 1, and was obtained through the package of compared to the cosmic variance. In the nth harmonic CMBEASY (Doran 2005), but the generic nature of the peak, centered at ℓ = ℓn there are ∼ ℓn/n values of Cℓ, derivativeoftheangularpowerspectruminthepresence allofwhich canbe determined with roughlycomparable of adiabatic fluctuations would be very similar for other fractional accuracy — given by eq. (7). This implies models. Wehavecarriedoutaformalstatisticalanalysis that the peak’s centroid can be determined in an ideal of the uncertainties in determining β using the function measurement with an uncertainty of approximately: givenbyeq. (9)(see Fig. 1)andthe assumeduncertain- ties in the individual C ’s given by eq. (7). We assumed ℓ 1 1 1 ℓ n δℓ≃ ≃ , (8) that Cf and Cb were both determined over 45◦ cones (cid:18)n (cid:19) ℓnfsky ℓn/n nfsky on theℓcelestialℓsphere in the forward and backward di- For observationalhpemispherpical caps ipn the forwardand rections, respectively. The result is that the formal 1σ backward directions of θ′ ∼ 45◦, the uncertainty in ℓn uncertaintyinβis3×10−4,andthustheaberrationeffect is about 1/(5f ) ≃ 1 for the 5th harmonic. And, should be detectable at a confidence level above 99.9%. sky p CMB Aberration 3 AdirectestimationofthesignificanceoftheratioinFig. between the locations of the peaks in the CMB power 1 gives a 2σ detection out to ℓ = 1000, and a 3.3σ de- spectrum for the forward and backward directions, then tection including all multipoles out to ℓ = 1500. This is this will serve to demonstrate the expected kinematic a slight overestimate as we have only included noise due effect due to motion of the barycenter. At present, it to cosmic variance, but an analysis of the full-sky would does not appear possible to measure this shift with bet- provide a measurement at slightly higher significance. terthanabout1partin4000accuracyduetothecosmic Therearetwocompetingeffects intryingto choosean variance. It may be worth noting, nonetheless, that if optimumfractionalhemisphericalcapsizeintheforward it were somehow possible to achieve a factor of ∼100 and backward directions in order to search for the kine- better in accuracy, this would be sufficient to allow the matic aberration effect. In the first, the larger the solid kinematic contribution to the dipole term to be deter- ′ of angle of the cap size (as characterized by θ ) the mined independently with ∼1% accuracy. This, in turn max greater is the available statistical precision. The frac- wouldenablea determinationofthe cosmic contribution tional solid angle in either of the hemispheres goes as to the dipole term. ′ ′ f =(1−cosθ )/2. Onthe otherhand, the largerθ Also, we would like to emphasize that a detection and sky max is, the smaller the average value of the shifts in the cen- measurement of the aberration of the CMB is comple- ′ ◦ troids of the acoustic peaks. Values of θ of 30 and mentarytoothereffectswhichariseduetothemotionof ◦ max 45 , for example, yield 6.7% and 14.6% of the celestial the solarsystem. Inparticular,the intensity quadrupole sphere, respectively, while preserving 93% and 85% of explored by Kamionkowski & Knox (2003) presents an the aberation effect, when averaged over the respective independent observable to constrain the contribution of hemispherical caps. the kinematic temperature dipole. Finally, we remarkthatin the same spiritof searching 4. SUMMARYANDCONCLUSIONS for direction dependent aberration effects, it might be We have shown that the aberration of the detected generally worthwhile to measure the CMB power spec- CMB radiation,due to the motion of the barycenterrel- trum over a number (∼10) of independent regions of ative to the CMB, can shift the centroids of the acous- the sky and intercompare the results. The isotropy of tic peaks in the forward vs. the backward directions the temperature power spectrum has been investigated by a measurable amount. This effect may already be (Eriksenet al. 2003,Hansen et al. 2004),and regions of marginallydetectablewiththeWMAP data,andshould excessasymmetryhavebeenreported. The next genera- be detectable with the data to be acquired with the tion of isotropic tests should account for the aberration PlanckMission. Toputthemagnitudeofthisaberration of the CMB to avoid systematic errors over the sky. effect into perspective, it is anticipated that the Planck ′ CMB maps will be constructed with ∼ 10 angular bins (Sandri et al. 2004). Locations in CMB features will be We thank Max Tegmark and Scott Hughes for helpful ′ ′ ′ ′ ◦ ◦ ◦ shifted by 2.1, 3.0, and 4.2 at θ = 30 , 45 , and 90 , discussions. SRacknowledgessupportfromNASAChan- respectively. draGrantNAG5-TM5-6003X.SBacknowledgessupport If there is, in fact, a significant measurable difference from NSF Grant AST-0307705. REFERENCES Balbi, A., de Gasperis, G., Natoli, P., & Vittorio, N. 2002, A&A, Hinshaw,G.,etal.2003,ApJS,148,135 395,417 Land, K., & Magueijo, J. 2005, Physical Review Letters, 95, Bennett, C.L.,etal.1996,ApJ,464,L1 071301. Bennett, C.L.,etal.2003,ApJS,148,1 Lineweaver,C.H.,Tenorio,L.,Smoot,G.F.,Keegstra,P.,Banday, Bielewicz,P.,Eriksen,H.K.,Banday,A.J.,Go´rski,K.M.,&Lilje, A.J.,&Lubin,P.1996,ApJ,470,38 P.B.2005,ApJ,635,750 Page.,L.,etal.2003, ApJS,148,233 Bond,J.R.,&Efstathiou,G.1987, MNRAS,226,655 Piat, M., Lagache, G., Bernard, J. P., Giard, M., & Puget, J. L. Bond,J.R.,Jaffe,A.H.,&Knox,L.1998,Phys.Rev.D,57,2117 2002,A&A,393,359 Bradley,J.1729,Phil.Trans.London, 135,637 Sandri,M.,etal.2004,MemoriedellaSocietaAstronomicaItaliana Copi, C. J., Huterer, D., & Starkman, G. D. 2004, Phys. Rev. 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