Motion induced second order temperature and y-type anisotropies after the subtraction of linear dipole in the CMB maps
aa r X i v : . [ a s t r o - ph . C O ] F e b P repared for submission to JCAP
Motion induced second ordertemperature and y -type anisotropiesafter the subtraction of linear dipole inthe CMB maps Rashid A. Sunyaev, a , b Rishi Khatri a a Max Planck Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1 85741, Garching, Germany b Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84 /
32, 117997 Moscow,RussiaE-mail: [email protected]
Abstract. y -type spectral distortions of the cosmic microwave background allow us to detect clus-ters and groups of galaxies, filaments of hot gas and the non-uniformities in the warm hot intergalacticmedium. Several CMB experiments (on small areas of sky) and theoretical groups (for full sky) haverecently published y -type distortion maps. We propose to search for two artificial hot spots in such y -type maps resulting from the incomplete subtraction of the e ff ect of the motion induced dipole on thecosmic microwave background sky. This dipole introduces, at second order, additional temperatureand y -distortion anisotropy on the sky of amplitude few µ K which could potentially be measured byPlanck HFI and Pixie experiments and can be used as a source of cross channel calibration by CMBexperiments. This y -type distortion is present in every pixel and is not the result of averaging thewhole sky. This distortion, calculated exactly from the known linear dipole, can be subtracted fromthe final y -type maps, if desired. Keywords: cosmic background radiation, cosmology:theory, Sunyaev-Zeldovich e ff ect, CMBRexperiments ontents Recently South Pole Telescope (SPT) [1] and Atacama Cosmology Telescope (ACT) [2] teams pub-lished their power spectra for the thermal y -type distortions [3] for high ℓ multipoles. There arealso many theoretical computations of y -type distortion maps going down to y ∼ − − − [4–7],where y is the amplitude of the y -type distortion, with the expectation that Planck Surveyor’s HighFrequency Instrument (Planck-HFI) [8] will be able to create maps of y -type distortion for the fullsky. Future proposed experiments Cosmic Origins Explorer (CoRE) [9], Lite satellite for the stud-ies of B-mode polarization and Inflation from cosmic background Radiation Detection (LiteBIRD)[10] and Primordial Inflation Explorer (Pixie) [11] will be able to improve the sensitivity by morethan an order of magnitude over Planck-HFI and several orders of magnitude compared to CosmicBackground Explorer’s Far Infrared Absolute Spectrophotometer (COBE-FIRAS) [12]. Pixie willhave an angular resolution of approximately a degree. The purpose of this short note is to remind theobservers about an additional artificial component having a y -type spectrum which arises in the usualprocedure of subtracting the dipole component from cosmic microwave background (CMB) maps.It is well known that sum of blackbody spectra is not a blackbody [13] and that the averagingof dipole component of CMB over the whole sky should lead to y -type distortion with amplitude y ≈ . × − [14, 15]. In this note we give the exact formulae for the angular dependence andamplitude of the y -type distortions in CMB sky maps after the usual subtraction of the linear dipolecomponent described in the Planck pre-launch and early results papers [16, 17]. It is important thatthe experimentalists will take special search of two maxima on the y -type distortion maps peaked,one towards and the other exactly opposite, in the direction of our motion with respect to CMBwith the characteristic dimensions of 38 deg for y >
80% of maximum and 105 deg for y > y maps, it would provide a source of calibration fordi ff erent channels of instruments such as Planck-HFI, in addition to the solar dipole and the orbitaldipole of the spacecraft. Relative small angular dimensions of the hot spots open a possibility oftheir detection by experiments like SPT and ACT or even experiments with much poorer angularresolution. Planck-HFI, CoRE, LiteBIRD and Pixie are unique because they are able to produce fullsky y -type distortion maps.In addition to the space missions Relikt [18], COBE [19] and Wilkinson Microwave AnisotropyProbe (WMAP) [20], there have been numerous ground based and airborne measurements of theCMB dipole, see [21] for a complete list. Since we know the dipole amplitude (3 . ± .
008 mK)and direction (galactic longitude 263 . ± .
14 deg, latitude 48 . ± .
03 deg) at high precisionfrom COBE Di ff erential Microwave Radiometers (DMR) and (WMAP) measurements [19, 20], theseartificial second order temperature and y -type quadrupoles can be easily subtracted from the CMBmaps.It is important to mention that the evaluation of the terms of second order in β = v / c , where v isour motion w.r.t. CMB and c is the speed of light, gives us an additional term with blackbody spec-trum but with characteristic quadrupolar angular dependence over the sky in the large scale CMB– 1 –aps. This term also has two positive maxima coinciding with the axes of dipole and the y -typedistortion is comparable in magnitude to this well known second order quadrupole e ff ect. The black-body part of the motion induced quadrupole will add to the primary CMB quadrupole, but both thesecomponents would be absent in the y -type distortion maps.This frequency dependence of the second order terms in the Taylor series expansion of motioninduced anisotropies in the CMB was studied by several authors [14, 22–26]. However, the connec-tion with the y -type distortion, in particular the possibility of the residual y -type anisotropy due to ourmotion in the full sky y -type maps produced by highly sensitive experiments like Planck-HFI, was notmade in the previous studies. See also [26] on additional aspects of the motion induced quadrupolenot discussed here. In each direction ˆn in the sky, CMB consists of a blackbody with temperature T + T ( ˆn ), where T ≈ .
725 K is the average temperature and T + T ( ˆn ) ≡ T + T ( θ ) = T (1 + β cos( θ )) − (1 − β ) / [27, 28] is the velocity induced anisotropy, where β = . × − ± .
2% is the velocity of sun w.r.t.CMB in units of speed of light in the direction ˆn = ( ℓ = . , b = .
26) in galactic coordinates[19, 20] and cos( θ ) = ˆn . ˆn . Taylor expanding to second order in β , we get for the velocity inducedmonopole, dipole and quadrupole, T ( θ ) ≈ T h β cos( θ ) + β (cid:16) cos ( θ ) − / (cid:17)i . Therefore the intensityor equivalently the occupation number ( I ν = h ν / c n ( ν )) in each direction in the sky is given by,following [22], n Pl ( T + T ( ˆn )) ≡ e h ν k B ( T + T ˆn ) ) − ≈ n Pl ( T ) + ln " + T ( ˆn ) T ∂ n Pl ( T ) ∂ ln[ T ] + ln " + T ( ˆn ) T ∂ n Pl ( T ) ∂ (ln[ T ]) = n Pl ( T ) + T ( ˆn ) T + T ( ˆn ) T ! T ∂ n Pl ( T ) ∂ T + T ( ˆn ) T ! T ∂∂ T T ∂ n Pl ( T ) ∂ T = n Pl ( T ) + T ( ˆn ) T + T ( ˆn ) T ! T ∂ n Pl ( T ) ∂ T + Y ( x ) T ( ˆn ) T ! , (2.1)where h is the Planck’s constant and k B is the Boltzmann’s constant. Planck-HFI is insensitive to theaverage intensity and only measures the fluctuating part, n Pl ( T + T ( ˆn )) − n Pl ( T ). We are thereforesubtracting, and mixing, two blackbodies of di ff erent temperature which should give us a y -typedistortion + change in the temperature [13–15]. c h ν ∆ I ν = T ( ˆn ) T + T ( ˆn ) T ! G ( x ) + Y ( x ) T ( ˆn ) T ! = " β cos( θ ) + β ( θ ) − ! G ( x ) + Y ( x ) ( β cos( θ )) , (2.2)– 2 –here x = h ν/ k B T is the dimension less frequency, G ( x ) = xe x ( e x − and Y ( x ) = xe x ( e x − x e x + e x − − ! (2.3)is the y -type distortion.We can also subtract, and absorb in T , an average term, ( β / G ( x ) to make the β G ( x ) termproportional to the Legendre polynomial P (cos( θ )). Similarly, we should also subtract average y -type distortion. Subtracting the linear dipole thus leaves the following residual in the map, c h ν I residual ν = β ( θ ) − ! G ( x ) + Y ( x ) β cos ( θ ) − ! (2.4)The first term is just change in the blackbody temperature, the second term is the y -distortion ampli-tude and might become visible in the future y-distortion maps of Pixie [11] and CoRE [9], and possi-bly in the Planck-HFI [8] maps. Pixie will also be sensitive to the monopole, y = β / = . × − .This average distortion is larger than what is expected from reionization [11, 14]. Note that aftersubtracting the monopole, the quadrupolar y -type distortion parameter is negative in part of the skywith minimum around θ = π/ , π/ θ = , π and is given by, in CMB temperature units, c h ν I residual ν T G ( x ) = T " β + β Y ( x ) / G ( x ) . (2.5)The maximum change in the blackbody temperature is T t max = T β = . µ K. The maximum y -distortion amplitude is y max = β = × − ± . µ K [20]. Pixie will haveabsolute sensitivity of few nK [11] and the y -distortion quadrupole, also known with a similar preci-sion, might be an important external source of calibration for it. Table 1 gives the maximum signalfor di ff erent Planck-HFI and Low Frequency Instrument (LFI) channels. Figure 1 shows the lin-ear dipole, additional blackbody temperature anisotropy t = β (cid:16) ( θ ) − (cid:17) and y -type anisotropy y = β (cid:16) cos ( θ ) − (cid:17) and their relative amplitude with respect their maxima as a function of an-gle θ of the line of sight direction ˆn with the dipole ˆn . Figure 2 shows the 3d visualization of thedipole, and second order temperature and y -distortion (353 GHz) anisotropies with Mollweide equalarea projection on the x-y plane and amplitude in color scale as well as the z axes. The temperaturequadrupole ( β G ( x ) term) in particular has a di ff erent orientation than the primary CMB quadrupole[29] but much smaller amplitude, Q t rms = (5 / (4 π ) C ℓ = ) / = β / (3 √ T = . µ K. The area on thesky covered by a spot with y > . y max is 38 deg and the region with y > . y max covers 105 deg .Thus even though this signal may be not be detectable in individual pixels, the peaks, with morethan a 1000 pixels, might be detected. This y -type distortion is artificial, resulting from incompletesubtraction of the e ff ect of our motion from the CMB maps, but it still opens a way to calibrate CMBexperiments, as was already noted by Refs. [14, 26] but Ref. [26] did not separate out y -type part. References [1] K. T. Story et al.,
A Measurement of the Cosmic Microwave Background Damping Tail from the2500-square-degree SPT-SZ survey , ArXiv e-prints (2012) [ arXiv:1210.7231 ]. – 3 –hannel ∆ T = T β ( µ K) y-distortion ( µ K) total ( µ K)30 -2.7 2.844 -2.6 2.970 -2.4 3.1100 5.5 -2.1 3.4143 -1.4 4.1217 0.0 5.5353 3.1 8.6
Table 1 . Maximum residuals in CMB temperature units in Planck-HFI and LFI channels after removingthe linear dipole. The blackbody temperature part is absent from the y -type distortion maps produced bySPT,ACT,Planck-HFI.[2] J. Sievers et al., The Atacama Cosmology Telescope: Cosmological parameters from three seasons ofdata , ArXiv e-prints (2013) [ arXiv:1301.0824 ].[3] R. A. Sunyaev and I. B. Zeldovich,
Microwave background radiation as a probe of the contemporarystructure and history of the universe , ARA & A (1980) 537–560.[4] H. Trac, P. Bode, and J. P. Ostriker, Templates for the Sunyaev-Zel’dovich Angular Power Spectrum , ApJ (2011) 94, [ arXiv:1006.2828 ].[5] N. Battaglia, J. R. Bond, C. Pfrommer, and J. L. Sievers,
On the Cluster Physics of Sunyaev-Zel’dovichand X-Ray Surveys. II. Deconstructing the Thermal SZ Power Spectrum , ApJ (2012) 75,[ arXiv:1109.3711 ].[6] L. D. Shaw, D. Nagai, S. Bhattacharya, and E. T. Lau,
Impact of Cluster Physics on theSunyaev-Zel’dovich Power Spectrum , ApJ (2010) 1452–1465, [ arXiv:1006.1945 ].[7] K. Dolag and R. Sunyaev,
Relative velocity of dark matter and barions in clusters of galaxies andmeasurements of their peculiar velocities , ArXiv e-prints (2013) [ arXiv:1301.0024 ].[8] J. A. Tauber et al.,
Planck pre-launch status: The Planck mission , A & A (2010) A1.[9] The COrE Collaboration, COrE (Cosmic Origins Explorer) A White Paper , ArXiv e-prints (2011)[ arXiv:1102.2181 ].[10] M. Hazumi et al.,
LiteBIRD: a small satellite for the study of B-mode polarization and inflation fromcosmic background radiation detection , in
Society of Photo-Optical Instrumentation Engineers (SPIE)Conference Series , vol. 8442 of
Society of Photo-Optical Instrumentation Engineers (SPIE) ConferenceSeries , Sept., 2012.[11] A. Kogut et al.,
The Primordial Inflation Explorer (PIXIE): a nulling polarimeter for cosmic microwavebackground observations , JCAP (2011) 25, [ arXiv:1105.2044 ].[12] D. Fixsen et al., The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set , ApJ (1996) 576.[13] Y. B. Zeldovich, A. F. Illarionov, and R. A. Sunyaev,
The E ff ect of Energy Release on the EmissionSpectrum in a Hot Universe , Soviet Journal of Experimental and Theoretical Physics (1972) 643– + .[14] J. Chluba and R. A. Sunyaev, Superposition of blackbodies and the dipole anisotropy: A possibility tocalibrate CMB experiments , A & A (2004) 389–408, [ astro-ph/0404067 ].[15] R. Khatri, R. A. Sunyaev, and J. Chluba, Mixing of blackbodies: entropy production and dissipation ofsound waves in the early Universe , A & A (2012) A136.[16] S. M. Leach et al., Component separation methods for the PLANCK mission , A & A (2008) 597–615,[ arXiv:0805.0269 ]. – 4 – π /2 0 π /2 π π /2dipolet/t max y/y max -1.5x10 -6 -1.0x10 -6 -5.0x10 -7 -7 -6 -6 -6 -6 - π /2 0 π /2 π π /2 θ =n.n -3 dipolety Figure 1 . Temperature and y -distortion anisotropy as a function of angle from the CMB dipole axes, θ ≡ ˆn . ˆn .Upper panel shows the angular distribution of dipole, temperature and y -type anisotropies relative to theirrespective maxima while the bottom panel shows their actual values.[17] J. Delabrouille et al., The pre-launch Planck Sky Model: a model of sky emission at submillimetre tocentimetre wavelengths , ArXiv e-prints (2012) [ arXiv:1207.3675 ].[18] I. A. Strukov, D. P. Skulachev, M. N. Boyarskii, and A. N. Tkachev, a Spacecraft Determination of theDipole Anisotropy in the Microwave Background , Soviet Astronomy Letters (Apr., 1987) 65.[19] C. L. Bennett et al., Four-Year COBE DMR Cosmic Microwave Background Observations: Maps andBasic Results , ApJL (1996) L1, [ astro-ph/9601067 ].[20] N. Jarosik et al.,
Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps,Systematic Errors, and Basic Results , ApJS (2011) 14, [ arXiv:1001.4744 ].[21] C. H. Lineweaver,
The CMB dipole: the most recent measurement and some history. , in
MicrowaveBackground Anisotropies (F. R. Bouchet, R. Gispert, B. Guiderdoni, and J. Trˆan Thanh Vˆan, eds.),pp. 69–75, 1997. astro-ph/9609034 . – 5 –
22] R. A. Sunyaev and I. B. Zeldovich,
The velocity of clusters of galaxies relative to the microwavebackground - The possibility of its measurement , MNRAS (1980) 413–420.[23] P. de Bernardis, M. Epifani, G. Guarini, S. Masi, F. Melchiorri, and B. Melchiorri,
On the dipole andquadrupole kinematic anisotropy in the brightness of the cosmic background radiation , ApJ (1990)145–148.[24] S. Bottani, P. de Bernardis, and F. Melchiorri,
On the origin of the dipole anisotropy as determined byquadrupole measurements , ApJL (1992) L1–L3.[25] S. Y. Sazonov and R. A. Sunyaev,
Microwave polarization in the direction of galaxy clusters induced bythe CMB quadrupole anisotropy , MNRAS (1999) 765–772, [ astro-ph/9903287 ].[26] M. Kamionkowski and L. Knox,
Aspects of the cosmic microwave background dipole , Phys.Rev.D (2003), no. 6 063001, [ astro-ph/0210165 ].[27] L. D. Landau and E. M. Lifshitz, The classical theory of fields . Pergamon Press, Oxford, 1962.[28] P. J. Peebles and D. T. Wilkinson,
Comment on the Anisotropy of the Primeval Fireball , Physical Review (1968) 2168–2168.[29] C. L. Bennett et al.,
Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: AreThere Cosmic Microwave Background Anomalies? , ApJS (Feb., 2011) 17, [ arXiv:1001.4758 ]. – 6 – igure 2 . Full sky maps in equal-area Mollweide projection on the x-y plane for the dipole, second order mo-tion induced blackbody temperature quadrupolar anisotropy and motion induced y -type quadrupolar anisotropyis shown. Only the y -type quadrupolar component would be present in the y -type distortion maps produced byCMB experiments such as Planck-HFI.-type distortion maps produced byCMB experiments such as Planck-HFI.