Spectral distortions of the CMB dipole
S.A. Balashev, E.E. Kholupenko, J. Chluba, A.V. Ivanchik, D.A. Varshalovich
aa r X i v : . [ a s t r o - ph . C O ] S e p Spectral distortions of the CMB dipole
S.A. Balashev , , E.E. Kholupenko , J. Chluba , , A.V. Ivanchik , , and D.A. Varshalovich , Io ff e Institute, Polytekhnicheskaya 26, 194021 Saint-Petersburg, Russia; ∗ Peter The Great St.Petersburg Polytechnic University,Polytekhnicheskaya 29, 195251 Saint-Petersburg, Russia Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK and Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Steet, Baltimore, MD 21218, USA (Dated: March 5, 2018)We consider the distortions of the cosmic microwave background (CMB) dipole anisotropy related to theprimordial recombination radiation (PRR) and primordial y - and µ -distortions. The signals arise due to ourmotion relative to the CMB restframe and appear as a frequency-dependent distortion of the CMB temperaturedipole. To leading order, the expected relative distortion of the CMB dipole does not depend on the particularobservation directions and reaches the level of 10 − for the PRR- and µ -distortions and 10 − for the y -distortionin the frequency range 1 – 700 GHz. The temperature di ff erences arising from the dipole anisotropy of the relicCMB distortions depend on the observation directions. For mutually opposite directions, collinear to the CMBdipole axis, the temperature di ff erences because of the PRR- and µ -dipole anisotropy attain values ∆ T ≃
10 nKin the considered range. The temperature di ff erence arising from the y -dipole anisotropy may reach values ofup to 1 µ K. The key features of the considered e ff ect are as follow: (i) an observation of the e ff ect does notrequire absolute calibration; (ii) patches of sky with minimal foreground contamination can be chosen. Futuremeasurements of the CMB dipole distortion thus will provide an alternative method for direct detection of thePRR-, y -, and µ -distortions. The y -distortion dipole may be detectable with PIXIE at a few standard deviations. INTRODUCTION
Observations of the cosmic microwave background (CMB)temperature and polarization anisotropies constitute one of themajor pillars of the standard cosmological model. The CMBpower spectra provide an opportunity to determine the cosmo-logical parameters of the Universe, since these are mainly de-termined by cosmological recombination [1, 2], acoustic os-cillations [3, 4], di ff usion damping [5] and other well-studiedprocesses in the early Universe [6]. Measurements taken withthe Wilkinson Microwave Anisotropy Probe (WMAP) allowus to determine most of the cosmological parameters with aprecision better than 4% [7]. Planck improved these measure-ments, reaching an accuracy level of ≃ −
2% [8]. Whilecosmological constraints are driven by anisotropies at multi-poles ℓ >
1, the largest CMB anisotropy is caused by ourmotion relative to the CMB restframe, manifesting itself as atemperature dipole [e.g., 9, 10].It has been shown that the average CMB spectrum isPlanckian with a relative accuracy better than ≃ − [11–13].However, it is well known that tiny deviations of the CMBfrom the Planckian spectrum are expected [e.g., 1, 2, 14–21].One inevitable distortion was formed as a result of the un-compensated transitions of recombining electrons during thecosmological recombination era at redshifts z ≃
800 – 8000[22–30]. The other expected distortions are the primordial µ -and y -distortions [14, 15] caused by episodes of early energyrelease. Here, we argue that the spectrum of the CMB dipolecould be used to constrain and detect these signals.The physics of the primordial recombination process arevery clear and has been studied in detail [e.g., 31–43]. Wewill refer to the total spectrum from recombining electronsand ions as the primordial recombination radiation (PRR).The number of CMB photons is ≃ times larger than the number of baryons. Thus, the distortion of the CMB spec-trum is expected to be ≃ − – 10 − of the total CMB en-ergy density (it was shown that about 5 photons per hydro-gen atom were released as the PRR spectrum; Chluba andSunyaev 44). However, it was found that in some frequencyrange these distortions can exceed 10 − of the CMB spectrum[29, 44]. The PRR spectrum mainly consists of the Lyman,Balmer, Paschen, Brackett and other series [44, 45]. Also,there are contributions from various processes, such as 2s-1stwo-photon decay (Hirata 41, Chluba and Sunyaev 46; whichgives about 50% of the emission around 1 THz) and free-bound emission (Chluba and Sunyaev 44; about 20-30% atthe whole frequency range). In addition, it was found thatHe ii and He i recombination spectra contribute about 10%-15%, on average, to the PRR and up to 50% at specific fre-quencies [47, 48].Measuring of the PRR spectrum is very important for theseveral reasons: (i) it will allow us to measure the CMBmonopole temperature T [49]; (ii) it can be used to estimatecosmological parameters like the fraction of baryon matter, Ω b , and the helium to hydrogen abundance ratio, Y p [49]; (iii)additional energy release (e.g., annihilation or decay of darkmatter particles) can be constrained [e.g., 50–53], since thePRR formed before the epoch of last scattering (especially thePRR components arising from He ii and He i recombinations);and (iv) variations of fundamental constants could potentiallybe probed [52]. In addition, the non-detection of the PRRspectrum at the expected level and with the predicted shapewould be a serious challenge for the standard cosmologicalmodel.To create a µ -distortion [15] requires e ffi cient energy ex-change between matter and photons, so that this type of dis-tortion is only formed at redshifts z & × , while at lowerredshifts a y -type distortion, also known in connection withthe thermal Sunyaev-Zeldovich (SZ) e ff ect [14], is created[17, 18]. The amplitude of these signals is more uncertain, buteven within our standard cosmological paradigm, one expectsan average y -parameter of y ≃ − − − due to the large-scale structure and the reionization epoch [e.g., 54–57], withthe most recent computations giving y ≃ × − [58]. Theprimary contribution to the y -distortion is from galaxy groupsand clusters after reionization, while the contribution from thereionization era and the intergalactic medium is y ≃ × − [58]. For the µ -distortion, the dissipation of small-scale acous-tic modes [e.g., 15, 59, 60] in the standard slow-roll inflationscenario is expected to give rise to µ ≃ × − [61]. Bymeasuring the µ -distortion, we can learn about the small-scalepower spectrum at wavenumbers 1 Mpc − . k . Mpc − ,and thus constrain di ff erent early-universe models [62–65].To detect the average µ - and y -distortion, an absolute cali-bration and measurements in wide bands at high frequencies( ν ≃
30 GHz – 1 THz) are required. The spectral shapes ofthese signals are very broad and therefore measurements aremore challenging. Experimental concepts like PIXIE [66]and PRISM [67] may reach the required sensitivity and sta-bility to detect the aforementioned signals. In contrast, for thePRR, one can also make use of its unique spectral dependence[30], so that absolute calibration in frequency is not necessar-ily required, but one merely needs a su ffi cient inter-channelcalibration to extract the typical peak-to-peak amplitude of ≃
10 nK at low frequencies. In the future, this type of mea-surement may even be possible at low frequencies from theground [68], but an improved version of PIXIE could alsosucceed [69]. For these observations, the flux can be collectedfrom large regions of the sky which are minimally contami-nated by the Galaxy and other backgrounds. In addition, onecan make use of the fact that the distortion signals should beunpolarized [30].Dubrovich and Grachev [70] noted that the PRR signal be-comes much more prominent relative to the CMB if one ob-serves the di ff erential PRR spectrum and compares it with thedi ff erential spectrum of CMB. To measure derivatives, it isnecessary to subtract the spectrum at adjacent frequencies. Inprinciple, this also requires absolute calibration for di ff erentfrequency channels; however, the derivative of the averagespectrum can alternatively be obtained by measuring the CMBdipole spectrum. Recently, it was shown [71] that the SZ ef-fect, which introduces a shift and distortions of the CMB spec-trum through Comptonization, can also be used to calculatethe PRR spectrum derivative. However, it is di ffi cult to ob-serve this e ff ect with modern equipment because the expectedsignal is extremely weak due to the small collection area (itis determined by the angular size of the galaxy cluster chosenfor observation). Additionally, this e ff ect is model-dependent.It demands the determination of the galaxy cluster model forwhich the SZ e ff ect observation is performed.In this paper, we discuss the detailed spectrum of the CMBdipole anisotropy arising due to the motion of an observer rel-ative to the CMB restframe. Previously, this e ff ect was con-sidered by [72] and [73], but detailed calculations and numer- ical estimates considering modern technology and the moderncosmological model were not performed. The e ff ect can beused as a natural method to measure the derivatives of the PRRspectrum and the µ - and y -distortions, but also extragalactic(non-comoving) foreground signals. It should be emphasizedthat measurements of this e ff ect do not require absolute cali-bration. We show that the motion of the Solar System relativeto the CMB restframe is enough to consider the e ff ect for fu-ture experiments. SPECTRUM OF THE CMB DIPOLE
It is well known that relative to the CMB restframe, theSolar System is moving toward ( l , b ) = (263.99 ◦ ± ◦ ,48.26 ◦ ± ◦ ) [11, 74] in galactic coordinates. The speed ofthis motion is = . ± . / s and produces the CMBdipole anisotropy due to the Doppler e ff ect. Using the CMBdipole, the first derivative of the monopole spectrum can becalculated. To do this, one can subtract the spectra obtainedfrom the di ff erent directions using two identical instruments, aprocedure that directly cancels the dominant CMB monopole.Denoting the total occupation number of the monopolespectrum as η m ( ν ), at lowest order in β = / c , the associatedmotion-induced dipole spectrum is given by η d ( ν, Θ ) ≈ η d ( ν ) cos Θ , (1)where the angle Θ is measured with respect to the motion di-rection, and the occupation number amplitude is given by η d ( ν ) ≈ − ν∂ ν η m ( ν ) β. (2)Assuming that the CMB monopole spectrum is just a black-body with η CMB , m = / (e x − x = h ν/ ( k B T ) is the di-mensionless frequency, we obtain the well-known CMB tem-perature dipole η CMB , d ( ν ) = β G ( x ) , G ( x ) = x e x (e x − , (3)where the function G ( x ) exhibits the spectrum of thermal fluc-tuations. Note that ν is the measured frequency in the ob-server frame. The motion-induced temperature dipole causesthe dominant contribution to the dipole anisotropy.When calculating the contributions from the PRR, µ -, and y -distortions to the CMB dipole spectrum, we should specifythe spectral shapes of these distortions. For the PRR spectrum,we use the results given by Rubi˜no-Mart´ın et al. [47], withdigital data taken from personal web page of J.A. Rubi˜no-Mart´ın at Istituto de Astrof´ısica de Canarias site. [94] Sinceno simple analytic approximation for the PRR exists, we nu-merically evaluate the frequency derivative in Eq. (2). The as-sociated occupation number amplitude is denoted as η PRR , d ( ν ).For the y -distortion, the average occupation number wascalculated by the following formulae [75]: η y , m ( ν ) = yY ( x ) , Y ( x ) = G ( x ) [ x coth( x / − . (4)Therefore, for the motion-induced dipole signal related to theaverage y -distortion, we have η y , d ( ν ) ≈ − β y h G ( x ) + Y ( x ) − x G ( x ) − xG ( x ) i , (5)which was determined by simply evaluating ν∂ ν η y , m = x ∂ x Y ( x ). We will use a fiducial value of y = × − [58]as an estimate, neglecting the relativistic temperature correc-tions [76–79] caused by hotter group-size systems.For the µ -distortion, the average occupation number is η µ, m = x φ + µ − − x − ≈ µ M ( x ) , (6)where the function φ ( µ ) ≈ − . µ is the temperaturecorrection providing the total photon number density conser-vation and the function M ( x ) is M ( x ) = G ( x ) h . − x − i . (7)Thus, the related distortion to the dipole is given by η µ, d ( ν ) = βµ " + Y ( x ) G ( x ) ! M ( x ) − G ( x ) x . (8)We will use µ = × − as a fiducial value. In the fol-lowing, we neglect any corrections from free-free distortionat low frequencies due to reionization [e.g., 80, 81] and fore-grounds, which do not constitute primordial signals but couldalso be directly constrained using measurements of the dipolespectrum. The di ff erence between galactic (comoving) andextragalactic (non-comoving) signals may provide an addi-tional handle for component separation of the monopole sig-nals. We also omit corrections to the shapes of the µ - and y -distortion caused by the thermalization process at frequen-cies ν ≃ −
10 GHz [17–19]. Furthermore, we assume that the y -distortion of the dipole caused by the aberration and boost-ing [82, 83] of the CMB temperature quadrupole, an artifactof the map-making procedure [84–86], is separated. IMPORTANCE OF THE EFFECTS
To illustrate the e ff ects, let us define a measure of the rela-tive anisotropy by d ( ν, n , n ) = η ( ν, n ) η ( ν, n ) − , (9)where η ( ν, n ) is the total photon occupation number in the di-rection n . The dominant contribution to d ( ν, n , n ) is due tothe CMB temperature dipole itself and can be expressed as d CMB ( ν, n , n ) ≈ x e x e x − β [cos Θ − cos Θ ] , (10)where Θ and Θ are the angles between the direction of mo-tion and directions n and n , respectively. To estimate thetypical magnitude of the considered e ff ect, we will use direc-tions exactly along ( Θ =
0) and opposite ( Θ = π ) to the axisof the CMB dipole, which gives [cos Θ − cos Θ ] = -11 -10 -9 -8 -7 -6 -5 -3 -2 -1 y = - PRR y PRR dipole dipole y dipole d D / d C M B , D = - d C M B , GHz FIG. 1: Top panel: The relative distortions of the CMB monopole, δ D , and the relative distortions of the CMB dipole, ∆ d D / d CMB , asfunctions of frequency. Green, red, and blue lines correspond to thePRR-, µ -, and y -distortions, respectively. For each type of distortion,dashed lines indicate the positive part of the monopole distortion, δ D ,and dashed-dotted the negative part. Solid lines indicate the positivepart of the dipole distortion, ∆ d D / d CMB , and short dashed the neg-ative. We adopted y = × − and µ = × − as the fiducialvalues for the y - and µ -distortions, respectively. Bottom panel: therelative CMB dipole anisotropy, d CMB (calculated using mutually op-posite directions collinear to the CMB dipole axis), as a function offrequency.
To isolate the contributions arising only from the distortionsof the CMB monopole, it is useful to consider ∆ d D d CMB = d CMB + D d CMB − , (11)which defines the relative change of d ( ν, n , n ) caused by thedistortion of type ”D”, where ”D” takes the values ”PRR”,” y ”, and ‘ µ ’. It can be shown that the quantity ∆ d D / d CMB isdescribed by the following formula: ∆ d D d CMB ≈ α D γ CMB δ D , (12)where δ D ( ν ) = η D , m ( ν ) η CMB , m ( ν ) (13)is the relative distortion of the CMB monopole spectrumwith local spectral index α D = − ∂ ln δ D /∂ ln ν , and γ CMB = − ∂ ln η CMB , m /∂ ln ν = x e x / (e x −
1) is the local spectral in-dex of the occupation number η CMB , m . It should be noted that α D = ( γ D − γ CMB ), where γ D = − ∂ ln η D , m /∂ ln ν is the localspectral index of the occupation number η D , m . Note also thatthe spectral indices, γ , are defined corresponding to ∼ ν − γ ,such that positive values of γ correspond to a decreasing spec-trum.Both ∆ d D / d CMB and d CMB are presented in Fig. 1. Thedependence of ∆ d PRR / d CMB on observation frequency showsquasi-oscillations originating from line features in the PRRspectrum. The amplitude of these quasi-oscillations rangesfrom 5 . × − (at ν ≃ . . × − (at ν ≃ .
07 GHz), i.e. ∆ d PRR / d CMB increases at low frequen-cies. The value of ∆ d y / d CMB ranges from 3 . × − at 1 GHzto − . × − at 700 GHz with a local maximum at the levelof 9 . × − at 50.2 GHz, and null (change of sign) at 101.1GHz. Thus, in contrast to ∆ d PRR / d CMB , the relative distortionanisotropy due to the average y -distortion, ∆ d y / d CMB , tends tozero at lower frequencies. This implies that at ν . y -distortion. Thevalue of | ∆ d µ / d CMB | ranges from 9 . × − at 700 GHz to1 . × − at 1 GHz showing a monotonic increase at decreas-ing frequency.The following aspects should be emphasized.a. The relative distortion, ∆ d D / d CMB , is independent of theparticular observation direction to leading order.b. A comparison of ∆ d PRR / d CMB and δ PRR (the top panelof Fig. 1) shows that the relative distortion of the CMBdipole can be slightly larger than the relative distortionof the CMB monopole due to the PRR.c. At the same time, ∆ d y / d CMB is much smaller than δ y at ν .
100 GHz. Thus, the amplitude of | ∆ d PRR / d CMB | is larger than | ∆ d y / d CMB | at ν . | δ PRR | ≪ | δ y | in most part ofthe considered range (aside from around 217 GHz – thenull-point of δ y ).d. At low frequencies, the relation ∆ d µ / d CMB ≃ δ µ is valid.The statement (a) directly follows from Eq. (12). Similarly,(b) − (d) can be understood by using the analytical approxima-tion given above. To show this, let us introduce an amplifi-cation coe ffi cient (in the same sense as in Kholupenko et al.71) C D = ∆ d D / d CMB δ D ( ν ) ≈ α D γ CMB , (14)which characterizes how much the relative distortion of thedipole is larger than the corresponding relative contribution tothe monopole. In the Rayleigh-Jeans part of the CMB, γ CMB ≃ C D ≃ α D = γ D − ffi cient, C PRR , achievesvalues up to 6 (at the frequencies near 22, 26, and 107 GHz).For the y -distortion the amplification coe ffi cient tends to zero( C y →
0) at low frequencies ( x ≪
1) since the y -distortion ofthe CMB monopole has asymptotic δ y ≃ − y at x →
0, and,correspondingly, α y →
0. For the µ -distortion, the amplifi-cation coe ffi cient tends to unity ( C µ →
1) at low frequencies -4 -3 -2 -1 PRR distortions y distortions PRR dipole dipole y dipole T , n K , GHz P I X I E y = - = - FIG. 2: Monopole temperature deviations, ∆ T D , and the dipole tem-perature di ff erences, ∆ T dD (in nK), arising from di ff erent types of relicCMB distortions, as functions of frequency. Green, red, and bluelines correspond to the PRR-, µ -, and y - distortions, respectively. Foreach type of distortion, the dashed lines indicate the positive partof the monopole temperature deviations, ∆ T D , the dashed-dotted thenegative. The solid lines indicate the positive part of the dipole tem-perature di ff erences, ∆ T dD , and the short dashed the negative. Theblack dashed line correspond to the monopole sensitivity of PIXIE, ∆ I ν = − (the expected limit for dipole is √ ff erences were performed formutually opposite directions collinear to the CMB dipole axis. Weadopted y = × − and µ = × − as fiducial values for the y -and µ -distortions, respectively. ( µ ≪ x ≪ δ µ ≃ − µ/ x at x → x ≫ µ ) and,correspondingly, α µ →
1. Note that the change of sign of ∆ d D / d CMB (see Fig. 1) occurs at points of δ D -extrema, where α D = ff ective temperature di ff erencesfrom the individual distortion contributions to the dipole spec-trum, ∆ T dD ( ν, Θ , Θ ) ≈ T (cid:2) η D ( ν, Θ ) − η D ( ν, Θ ) (cid:3) / G ( x ),and compare it with the associated e ff ective temperature dif-ference of the monopole, ∆ T D ( ν ) ≈ T η D , m ( ν ) / G ( x ). Wenote that in contrast to ∆ d D / d CMB , the e ff ective temperaturedipole anisotropy, ∆ T dD , depends on direction, namely, it isdirectly proportional to (cos Θ − cos Θ ).The PRR monopole temperature deviation shows an in-crease at low frequencies with marked quasi-oscillations orig-inating from the wide overlapped lines in the PRR spectrum.This temperature deviation has a value in the range from0 . y -distortionmonopole temperature deviation ranges from − µ K at1 GHz to 45 µ K at 700 GHz, a signal that will be easily de-tected using PIXIE. The µ -distortion monopole temperaturedeviation ranges from − µ K at 1 GHz to 20 nK at 700 GHz.The PRR dipole anisotropy temperature di ff erence alsoshows marked quasi-oscillations due to the lines in the PRRspectrum. The amplitude of these quasi-oscillations has avalue in the range from 7 . × − nK at ν ≃
89 GHz to 5 . distortion monopole / dipole behavior frequency of min ∆ T , nK frequency of max ∆ T , nKminimum, GHz maximum , GHz y − monopole monotonic 1 -11 ×
700 45 × dipole monotonic 1 -27 700 1.1 × µ − monopole monotonic 1 -3 ×
700 20dipole monotonic 1 -15 700 0.6PRR monopole quasi-oscillating 270 0.3 1 755dipole quasi-oscillating 89 7.3 × − ff erences for the di ff erent types of distortions in frequency range 1-700GHz. Dipole temperature di ff erence we calculated using mutually opposite direction collinear to CMB dipole axis. We adopted y = × − and µ = × − as fiducial values for y − and µ − distortions, respectively. at ν ≃ .
07 GHz. The y -distortion dipole anisotropy temper-ature di ff erence ranges from −
27 nK at 1 GHz to 1 . µ K at700 GHz. The µ -distortion dipole anisotropy temperature dif-ference reaches −
15 nK at 1 GHz to 0 . . .
20 GHz, 0 . .
700 GHz (in whole considered range); (2) for y -distortion:4 µ K at frequencies .
165 GHz and &
265 GHz; (3) for µ -distortion: 1 µ K at frequencies . .
25 GHz,10 nK at .
90 GHz and &
200 GHz.On the other hand, from Fig. 2 one can also see that adetection of the dipole anisotropy due to the di ff erent dis-tortions requires measurements of the signal in several fre-quency channels with the following values of absolute temper-ature sensitivity: (1) for PRR: 1 nK at frequencies . . .
20 GHz, and 10 − nK at .
700 GHz; (2) for y -distortion: 4 nK at frequencies .
700 GHz (excluding the nar-row band 285 −
293 GHz around null of e ff ect at 289.6 GHz);(3) for µ -distortion: 1 nK at frequencies .
15 GHz, 0 . .
100 GHz, and &
270 GHz. Thus, observations of CMBdipole distortions require a sensitivity two to three orders ofmagnitude better than observations of corresponding distor-tions of CMB (monopole) spectrum in comparable ranges.The achievement of such an unprecedented level of sensitiv-ity is a challenging technological problem for a future exper-iment observing the distortion of the CMB dipole. Also, thelevel of foreground contamination to the dipole spectrum withassociated motion-induced e ff ects has to be considered morecarefully. CONCLUSION
The distortions of the dipole anisotropy due to several relicCMB distortions were studied. The temperature di ff erencesarising from the PRR-, y -, and µ -dipoles depend on the fre- quency and observation directions. The relative change of theCMB dipole due to the PRR-, y -, and µ -distortions is indepen-dent of the observation direction but depends on frequency.In the most promising range for observations of the PRR-and µ -dipoles, ν ≃ −
20 GHz, the absolute temperature dif-ferences arising from these signals have values in the range ≃ . −
10 nK for observations in mutually opposite directionsalong the CMB dipole axis. The relative changes of CMBdipole due to the PRR and µ -distortion in this range have val-ues in the range 10 − – 10 − . In the most appropriate rangefor observations of the y -dipole, ν ≃ −
700 GHz, the ab-solute value of the temperature di ff erence ranges from 28 nKto 1 . µ K, which could be detected at ≃ σ with PIXIE. Therelative change in the CMB dipole due to y -distortions in thisrange is − – 10 − .The considered e ff ect provides an additional method fordetecting the relic CMB distortions. The main idea of themethod is using an anisotropy technique that is already welldeveloped for the measurements of the CMB anisotropy, withdi ff erent systematics. One advantage of this method is that itdoes not require absolute, but only precise inter-channel cali-bration to observe spectral features, directly using the CMBsky as reference. This is especially important for the fre-quency range .
20 GHz, where absolute calibration is dif-ficult due to the size of cryogenic systems. The PRR dipoleanisotropy has a unique spectral shape that is related to the ini-tial PRR monopole spectrum. This can help experimentaliststo identify reliably the PRR dipole anisotropy. The method re-quires absolute temperature sensitivity of at least 0.1 − ffi cult to accesslarge scales because of atmospheric fluctuations. The generalconsensus is that these kind of distortion measurements willneed to be performed from space or possibly with the adventof the next generation of the bolometer telescopes (e.g. PIXIE,Kogut et al. 90; PRISM, PRISM Collaboration et al. 91; andCMB Stage-IV, Chang 92). At low frequencies ( .
20 GHz),the receivers become larger with increasing wavelength, so itis di ffi cult to observe with sensitive spectrometers in space.One possible resolution is the measurements using balloon-borne experiments, like ARCADE 2 [93], or finding a newpath to make observations from the ground.As mentioned above, the motion-induced dipole signal re-lated to the average y -distortion may be detected with PIXIE.Since the signal has a spectrum that di ff ers from the usual y -distortion, it cannot be mimicked by a dipolar modulation ofthe number of clusters (or generally scatters) across the sky.Thus, the distortion of the dipole can in principle be used toindependently confirm that the CMB temperature dipole iscaused primarily by our motion, thereby placing limits on aprimordial temperature dipole and large-scale perturbations. Acknowledgments J.C. thanks Yacine Ali-Ha¨ımoud for useful discussions.S.A.B., A.V.I. and D.A.V. are grateful to Russian Science Foundation grant14-12-00955. E.E.K. thanks RFBR grant 13-02-12017-ofi-m. J.C. is sup-ported by the Royal Society as a Royal Society University Research Fellowat the University of Cambridge, UK. ∗ Electronic address: [email protected][1] Y. B. Zeldovich, V. G. Kurt, and R. A. Syunyaev, ZhurnalEksperimentalnoi i Teoreticheskoi Fiziki , 278 (1968).[2] P. J. E. Peebles, ApJ , 1 (1968).[3] R. A. Sunyaev and Y. B. Zeldovich, Ap&SS , 3 (1970).[4] P. J. E. Peebles and J. T. Yu, ApJ , 815 (1970).[5] J. Silk, ApJ , 459 (1968).[6] W. Hu and S. Dodelson, ARA&A , 171 (2002), astro-ph / , 19 (2013), 1212.5226.[8] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Ar-naud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday,R. B. Barreiro, J. G. Bartlett, et al., ArXiv:1502.01589 (2015),1502.01589.[9] E. K. Conklin, Nature , 971 (1969).[10] D. J. Fixsen, E. S. Cheng, D. A. Cottingham, R. E. Eplee, Jr.,R. B. Isaacman, J. C. Mather, S. S. Meyer, P. D. Noerdlinger,R. A. Shafer, R. Weiss, et al., ApJ , 445 (1994).[11] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather,R. A. Shafer, and E. L. Wright, ApJ , 576 (1996), astro-ph / , 511 (1999), astro-ph / , 916 (2009), 0911.1955.[14] R. A. Sunyaev and Y. B. Zeldovich, Nature , 721 (1969).[15] R. A. Sunyaev and Y. B. Zeldovich, Ap&SS , 20 (1970).[16] A. F. Illarionov and R. A. Sunyaev, Soviet Astronomy , 413(1975).[17] C. Burigana, L. Danese, and G. de Zotti, A&A , 49 (1991).[18] W. Hu and J. Silk, Phys. Rev. D , 485 (1993).[19] J. Chluba and R. A. Sunyaev, MNRAS , 1294 (2012),1109.6552.[20] R. A. Sunyaev and R. Khatri, International Journal of ModernPhysics D , 1330014 (2013), 1302.6553.[21] H. Tashiro, Progress of Theoretical and Experimental Physics , 06B107 (2014). [22] V. K. Dubrovich, Pisma v Astronomicheskii Zhurnal , 3(1975).[23] I. N. Bernshtein, D. N. Bernshtein, and V. K. Dubrovich, Astro-nomicheskii Zhurnal , 727 (1977).[24] G. B. Rybicki and I. P. dell’Antonio, ApJ , 603 (1994),astro-ph / , 635 (1995).[26] V. K. Dubrovich and V. A. Stolyarov, Astronomy Letters ,565 (1997).[27] M. Spaans and C. A. Norman, ApJ , 27 (1997).[28] E. E. Kholupenko, A. V. Ivanchik, and D. A. Varshalovich,Gravitation and Cosmology , 161 (2005), astro-ph / , 1939 (2006), astro-ph / ,657 (2009), 0908.0435.[31] S. I. Grachev and V. K. Dubrovich, Astrophysics , 124(1991).[32] S. Seager, D. D. Sasselov, and D. Scott, ApJS , 407 (2000),astro-ph / , 359(2005).[34] J. Chluba and R. A. Sunyaev, A&A , 39 (2006), astro-ph / ,795 (2006), astro-ph / , L39 (2007), astro-ph / , 439(2008), 0801.3347.[38] E. R. Switzer and C. M. Hirata, Phys. Rev. D , 083006(2008), astro-ph / , 083008(2008), astro-ph / , 083007(2008), astro-ph / , 023001 (2008), 0803.0808.[42] D. Grin and C. M. Hirata, Phys. Rev. D , 083005 (2010),0911.1359.[43] Y. Ali-Ha¨ımoud and C. M. Hirata, Phys. Rev. D , 043513(2011), 1011.3758.[44] J. Chluba and R. A. Sunyaev, A&A , L29 (2006), astro-ph / , 1310 (2007), astro-ph / , 629 (2008),0705.3033.[47] J. A. Rubi˜no-Mart´ın, J. Chluba, and R. A. Sunyaev, A&A ,377 (2008), 0711.0594.[48] J. Chluba and R. A. Sunyaev, MNRAS , 1221 (2010),0909.2378.[49] J. Chluba and R. A. Sunyaev, A&A , L27 (2008),0707.0188.[50] J. Chluba and R. A. Sunyaev, A&A , 29 (2009), 0803.3584.[51] V. K. Dubrovich, S. I. Grachev, and V. G. Romanyuk, Astron-omy Letters , 723 (2009).[52] J. Chluba, MNRAS , 1195 (2010), 0910.3663.[53] S. I. Grachev and V. K. Dubrovich, Astronomy Letters , 293(2011), 1010.4455.[54] W. Hu, D. Scott, and J. Silk, Phys. Rev. D , 648 (1994),arXiv:astro-ph / , 1 (1999), arXiv:astro-ph / , 123001 (2000), arXiv:astro-ph / , L20(2003), arXiv:astro-ph / , 14 (1991).[60] W. Hu, D. Scott, and J. Silk, ApJ , L5 (1994), arXiv:astro-ph / , 1129(2012), 1202.0057.[62] J. Chluba, A. L. Erickcek, and I. Ben-Dayan, ApJ , 76(2012), 1203.2681.[63] R. Khatri and R. A. Sunyaev, J. Cosmology Astropart. Phys. ,026 (2013), 1303.7212.[64] J. Chluba and D. Jeong, MNRAS , 2065 (2014), 1306.5751.[65] S. Clesse, B. Garbrecht, and Y. Zhu, J. Cosmology Astropart.Phys. , 046 (2014), 1402.2257.[66] A. Kogut, D. J. Fixsen, D. T. Chuss, J. Dotson, E. Dwek,M. Halpern, G. F. Hinshaw, S. M. Meyer, S. H. Moseley, M. D.Sei ff ert, et al., J. Cosmology Astropart. Phys. , 25 (2011),1105.2044.[67] P. Andr´e, C. Baccigalupi, A. Banday, D. Barbosa, B. Barreiro,J. Bartlett, N. Bartolo, E. Battistelli, R. Battye, G. Bendo, et al.,J. Cosmology Astropart. Phys. , 006 (2014), 1310.1554.[68] M. Sathyanarayana Rao, R. Subrahmanyan, N. Udaya Shankar,and J. Chluba, ArXiv:1501.07191 (2015), 1501.07191.[69] V. Desjacques, J. Chluba, J. Silk, F. de Bernardis, and O. Dor´e,ArXiv:1503.05589 (2015), 1503.05589.[70] V. K. Dubrovich and S. I. Grachev, Astronomy Letters , 657(2004).[71] E. E. Kholupenko, S. A. Balashev, A. V. Ivanchik, and D. A.Varshalovich, MNRAS , 3593 (2015), 1402.3505.[72] L. Danese and G. de Zotti, A&A , L33 (1981).[73] G. Smoot, NATO Advanced Study Institute on the Cosmologi-cal Background Radiation p. 271 (1997).[74] G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson,C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N. Jarosik,et al., ApJS , 225 (2009), 0803.0732.[75] Y. B. Zeldovich and R. A. Sunyaev, Ap&SS , 301 (1969).[76] N. Itoh, Y. Kohyama, and S. Nozawa, ApJ , 7 (1998),arXiv:astro-ph / , 1 (1998).[78] A. Challinor and A. Lasenby, ApJ , 1 (1998), astro-ph / ,510 (2012), 1205.5778.[80] C. Burigana, G. De Zotti, and L. Feretti, New A Rev. , 1107(2004), astro-ph / , 2507 (2014),1310.6177.[82] A. Challinor and F. van Leeuwen, Phys. Rev. D , 103001(2002), astro-ph / , 389 (2004),arXiv:astro-ph / , 123504 (2014),1403.6117.[86] M. Quartin and A. Notari, ArXiv:1504.04897 (2015),1504.04897.[87] S. Naess, M. Hasselfield, J. McMahon, M. D. Niemack, G. E.Addison, P. A. R. Ade, R. Allison, M. Amiri, N. Battaglia, J. A.Beall, et al., J. Cosmology Astropart. Phys. , 007 (2014),1405.5524.[88] J. E. Austermann, K. A. Aird, J. A. Beall, D. Becker, A. Bender,B. A. Benson, L. E. Bleem, J. Britton, J. E. Carlstrom, C. L.Chang, et al., in SPIE Conference Series (2012), vol. 8452 ofSPIE Conference Series, p. 1, 1210.4970.[89] BICEP2 Collaboration, P. A. R. Ade, R. W. Aikin, M. Amiri,D. Barkats, S. J. Benton, C. A. Bischo ff , J. J. Bock, J. A. Brevik,I. Buder, et al., ArXiv e-prints (2014), 1403.4302.[90] A. Kogut, D. J. Fixsen, D. T. Chuss, J. Dotson, E. Dwek,M. Halpern, G. F. Hinshaw, S. M. Meyer, S. H. Moseley, M. D.Sei ff ert, et al., J. Cosmology Astropart. Phys. , 025 (2011),1105.2044.[91] PRISM Collaboration, P. Andr´e, C. Baccigalupi, A. Banday,D. Barbosa, B. Barreiro, J. Bartlett, N. Bartolo, E. Battistelli,R. Battye, et al., ArXiv e-prints (2013), 1310.1554.[92] C. Chang, conference report, CSS 2013, Showmass on the Mis-sissippi (2013).[93] J. Singal, D. J. Fixsen, A. Kogut, S. Levin, M. Limon, P. Lubin,P. Mirel, M. Sei ff ert, T. Villela, E. Wollack, et al., ApJ , 138(2011), 0901.0546.[94] http: // / galeria / jalberto / pages / research-interests //