FSD: Frequency Space Differential measurement of CMB spectral distortions
MMNRAS , 1–9 (2018) Preprint 17 January 2018 Compiled using MNRAS L A TEX style file v3.0
FSD: Frequency Space Differential measurement of CMBspectral distortions
Suvodip Mukherjee, , , (cid:63) Joseph Silk, , , , † Benjamin D. Wandelt , , , ‡ Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA Institut d’Astrophysique de Paris98bis Boulevard Arago, 75014 Paris, France Sorbonne Universit´es, Institut Lagrange de Paris98 bis Boulevard Arago, 75014 Paris, France The Johns Hopkins University, Department of Physics & Astronomy,Bloomberg Center for Physics and Astronomy, Room 366, 3400 N. Charles Street, Baltimore, MD 21218, USA Beecroft Institute for Cosmology and Particle Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK Departments of Physics and Astronomy, University of Illinois at Urbana-Champaign, 1002 W Green St, Urbana, IL 61801, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Although the Cosmic Microwave Background agrees with a perfect blackbody spectrumwithin the current experimental limits, it is expected to exhibit certain spectraldistortions with known spectral properties. We propose a new method, FrequencySpace Differential (FSD) to measure the spectral distortions in the CMB spectrumby using the inter-frequency differences of the brightness temperature. The differencebetween the observed CMB temperature at different frequencies must agree withthe frequency derivative of the blackbody spectrum, in the absence of any distortion.However, in the presence of spectral distortions, the measured inter-frequency differenceswould also exhibit deviations from blackbody which can be modeled for known sourcesof spectral distortions like y & µ . Our technique uses FSD information for the CMBblackbody, y , µ or any other sources of spectral distortions to model the observedsignal. Successful application of this method in future CMB missions can provide analternative method to extract spectral distortion signals and can potentially make itfeasible to measure spectral distortions without an internal blackbody calibrator. Key words:
CMB spectral distortion, measurement technique
Imprints of spectral distortions in the Cosmic MicrowaveBackground (CMB) are a prediction of the Standard Cosmo-logical Model (Zeldovich & Sunyaev 1969; Chluba & Sunyaev2012a; Chluba et al. 2012; Khatri & Sunyaev 2012a; Khatriet al. 2012; Khatri & Sunyaev 2012b; Chluba 2016; Hill et al.2015; Emami et al. 2015). Measurement of signals such as y and µ distortions will help to validate our standard cos-mological model. Indeed the essential ansatz of structureformation by gravitational instability predicts weak but po-tentially measurable µ distortions (Hu et al. 1994; Pajer &Zaldarriaga 2012). Discoveries of any other kinds of spectraldistortions can open up a window to new physics. One of themain goals of several next generation cosmology missions is (cid:63) E-mail: smukherjee@flatironinstitute.org † E-mail: [email protected] ‡ E-mail: [email protected] to measure the spectral distortions in the CMB blackbodyspectrum. The first observational bound on the spectral dis-tortion was given by FIRAS (Mather et al. 1994; Fixsen et al.1997; Smoot et al. 1991) with µ < × − and y < × − at C.L. FIRAS used an absolute blackbody internalcalibrator to measure the monopole of the CMB temperaturefield and constrained its temperature T = . ± . K(Mather et al. 1999; Fixsen 2009). With the recent CMBanisotropy data, measurements of Sunyaev-Zel’dovich (SZ)clusters (Hasselfield et al. 2013; Bleem et al. 2015; Ade et al.2016; Staniszewski et al. 2009) and bounds on the fluctuating y and µ have also been obtained (Khatri & Sunyaev 2015a,b).Several concepts are under discussion for a post-PlanckCMB polarization mission in space, including spectrometry(PIXIE (Kogut et al. 2011) or PRISM (Andr´e et al. 2014)),high-resolution imaging (CMBPOL) (Dunkley et al. 2009),or a mission with modest resolution focusing on the large-angle primordial anisotropy (LiteBIRD) (Matsumura et al.2016). While imaging and spectroscopy are often presented as © a r X i v : . [ a s t r o - ph . C O ] J a n Mukherjee, Silk & Wandelt mutually exclusive concepts we propose a hybrid approach toimage spectral distortions which we term the Frequency SpaceDifferential (FSD) method. This technique uses a differentialmeasurement of the CMB between different frequencies andtherefore does not require an absolute calibrator.In this paper, we propose a new technique for measuringthe spectral distortions in the CMB which can avoid using anabsolute blackbody calibrator. We describe the possibility ofmeasuring the spectral distortion in CMB by measuring theinter-frequency differences of the sky intensity and matchingit with the theoretical prediction of the signal frequencyspectrum. The blackbody spectrum predicts a well-knownintensity or brightness temperature at every frequency. Asa result, the difference of the blackbody intensity betweentwo different frequencies can predict a unique spectral shape,and we can obtain an all-sky frequency space derivative mapof the blackbody spectrum. In the presence of any spectraldistortions in the blackbody intensity, the derivative of theobserved intensity is a composite signature of frequencyderivatives of the blackbody spectrum and other sources ofspectral distortions.We will see that the key idea is to design the measure-ment such that any gain fluctuations couple only to thefrequency derivative of the blackbody spectrum rather thanthe blackbody spectrum itself. A similar approach was dis-cussed recently by Sironi (2017) who also proposed a detectordesign.Even in the absence of an absolute calibrator, an overallcalibration can be obtained from the time-dependent velocitydipole due to the orbital motion around the sun. This effectcan be extracted from a multi-year campaign due to its annualmodulation. It’s frequency spectrum only depends on well-known relativistic effects and is directly proportional to thederivative of the blackbody spectrum. It can therefore serveas an absolute and robust calibrator for the FSD technique.This effect was used in Fixsen (2009) to recalibrate the FIRASdata using the WMAP time ordered data.In this paper we discuss the main idea of using the fre-quency derivative of the spectral distortion signal to measurethe µ and y distortions without an absolute calibrator (in ananalogous way to how WMAP created a map from a purelydifferential measurement of the anisotropies, in contrast toPlanck which used an internal reference). We prescribe pos-sible measurement strategies and statistical techniques toimplement this method for future CMB missions. The im-plementation of this method to a particular mission putsrequirements on measurement technique, scan strategy, de-tector properties and calibration techniques.This paper is organized as follows: Sec. 2 sets out theform of the expected signals in the sky when observed dif-ferentially in frequency space. In Sec. 3 we discuss how toform differential combinations of nearby frequency channelssuch that inter-channel calibration errors do not couple tothe CMB monopole but only to the derivative of the Planckspectrum. Once this major source of noise is removed, theremaining signal needs to be cleaned from foreground con-tamination. A method for removal of those contaminants andrecovery of the spectral distortion signal is given in Sec. 4. InSec. 5 we discuss the main requirements our approach placeson instrument design. We conclude in Sec. 6. The all-sky average temperature field of the CMB ex-hibits a blackbody spectrum ( S brightness ≡ c B ν / k B ν = h ν / k B ( e h ν / k B T CMB -1) with a brightness temperature S brightness = . K in the RJ limit ( h ν / k B T CMB < ). Anydeviation from the blackbody spectrum can be parametrisedas (Mather et al. 1994; Fixsen et al. 1997) I o ν = B ν ( T CMB ) + ∆ T CMB ∂ B ∂ T + ∆ I gal ν + u ∂ B ∂ u , (1)where I o ν is the observed intensity in the sky and the first andsecond terms are the blackbody spectrum and fluctuationsin the blackbody due to CMB temperature fluctuations. Thethird term indicates the galactic contamination and the lastterm is the spectral distortion due to cosmological processes(like u ≡ µ, y ). The observed intensity of the sky at everyfrequency should be compared with an internal blackbodycalibrator fixed at a particular temperature to deduce thetemperature of the CMB field and also any departure fromblackbody. The FIRAS (Mather et al. 1994; Fixsen et al.1997) experiment used an internal blackbody calibrator tomeasure the CMB temperature field and also provided thefirst observational constraints on µ, y distortions as × − and × − at C.L. respectively. Measurement ofany well-motivated CMB spectral distortions to values ofcosmological interest (Zeldovich & Sunyaev 1969; Hu et al.1994; Chluba 2010; Chluba & Sunyaev 2012b; Chluba &Sunyaev 2012a; Chluba et al. 2012; Khatri & Sunyaev 2012a;Khatri et al. 2012; Khatri & Sunyaev 2012b; Chluba 2016;Hill et al. 2015; Emami et al. 2015) requires a much betterabsolute blackbody calibrator than FIRAS.We will show how to estimate cosmological spectraldistortions with any given spectrum using the FSD tech-nique. Astrophysical sources add contaminations with ap-proximately known spectra. We will find that these have asimilar effect on the FSD technique as on absolutely cali-brated spectral distortion measurements.
The observed CMB blackbody intensity, along with spec-tral distortions like y & µ , also gets contaminated by severalgalactic astrophysical emissions in the CMB frequency rangeby processes like synchrotron, free-free, spinning dust, ther-mal dust, etc. The total emission can be written in termsof a brightness temperature at a particular frequency ν ina particular pixel ( ˆ p ) as a superimposition of various effectswhich can be written in the form S ν ( ˆ p ) = K pl ν + A CMB ( ˆ p ) K T ν + A µ ( ˆ p ) K µν + A y ( ˆ p ) K y ν + A dust ( ˆ p ) K dust ν + A syn ( ˆ p ) K syn ν + A free ( ˆ p ) K free ν + A spin-dust ( ˆ p ) K spin-dust ν , (2) MNRAS , 1–9 (2018)
SD: Frequency Space Differential measurement of CMB spectral distortions where with x = h ν / k B T CMB = ν / ν CMB , we can write (Adamet al. 2016b) K pl ν = xT CMB ( e x − ) , K T ν = x e x ( e x − ) , K µν = − xe x T CMB ( e x − ) , K y ν = x e x T CMB ( e x − ) (cid:18) x (cid:0) e x + e x − (cid:1) − (cid:19) , K Dust ν = (cid:18) νν (cid:19) β d + (cid:18) e ν / T d − e ν / T d − (cid:19) , T d = K , ν = GHz , β d = . , K syn ν = (cid:18) ν ν (cid:19) f s ( ν / α ) f s ( ν / α ) ν = MHz , f s = templates , K free ν = T e ( − e τ ) ,τ = . T − / e ν − log ( e [ . −√ π log ( ν T − / )] + e ) ,ν = ν GHz , T = T e / , K spin-dust ν = (cid:18) ν ν (cid:19) f sd ( ν.ν p / ν p ) f sd ( ν .ν p / ν p ) ν p = GHz , f sd = templates . (3)The all-sky average measurement of Eq. (2) obtainsthe contribution only from the monopole term, whereas thedifferential measurement of Eq. (2) between different pix-els gets the contribution only from the fluctuation parts( ∆ T CMB ( ˆ p ) , ∆ µ ( ˆ p ) and ∆ y ( ˆ p ) ), and captures no contributionsfrom the monopole of the CMB or any other spectral dis-tortion signals. However, the differential measurement infrequency space S ν ji = S ν j − S ν i has non-zero contributionsboth from the monopole and from the fluctuation part, whichcan be written as S ν ji ( ˆ p ) = K pl ν ji + A CMB ( ˆ p )K T ν ji + A µ ( ˆ p )K µν ji + A y ( ˆ p )K y ν ji + A dust ( ˆ p )K dust ν ji + A syn ( ˆ p )K syn ν ji + A free ( ˆ p )K free ν ji + A spin-dust ( ˆ p )K spin-dust ν ji . (4)For closely spaced frequency channels, i.e. small ∆ ν ji = ν j − ν i this can be related to the derivative of the theoreticalfrequency spectrum K x ν ji ( ˆ p ) = ∂ K x / ∂ν | ν ji ∆ ν ji evaluated atthe midpoint ν ji = ( ν j + ν i )/ . The theoretical FrequencySpace Derivative (FSD) spectrum of the various sources can Figure 1.
We plot the kernel of the signal for different sources ofspectral distortions for y -distortions (black) which is multiplied bya factor of two in the amplitude, µ -distortions (red) and blackbodyspectrum (blue) at T = . Kelvin. be expressed in terms of D x ν ( ˆ p ) ≡ ( ∂ K x / ∂ν )| ν as D pl ν = A CMB ν CMB ( e x − ) (cid:20) − xe x ( e x − ) (cid:21) , D T ν = ν CMB x e x ( e x − ) (cid:20) x + − e x ( e x − ) (cid:21) , D µν = T CMB ν CMB e x ( e x − ) (cid:20) − − x + xe x ( e x − ) (cid:21) , D y ν = T CMB ν CMB (cid:20) ∆ n y ν (cid:18) + x − xe x ( e x − ) (cid:19) + x e x ( e x − ) (cid:18)(cid:18) e x + e x − (cid:19) + (cid:18) xe x e x − (cid:19) − (cid:18) xe x ( e x + )( e x − ) (cid:19)(cid:19)(cid:21) , D Dust ν = (cid:18) νν (cid:19) β d + (cid:18) e ν / T d − e ν / T d − (cid:19) (cid:20) β d + ν − γ e γν ( e γν − ) (cid:21) , D syn ν = (cid:18) ν ν (cid:19) (cid:18) − ν f s ( ν / α ) f s ( ν / α ) + ∂ f s ( ν / α ) ∂ν f s ( ν / α ) (cid:19) , D free ν = − T e e τ ∂τ∂ν , D spin-dust ν = (cid:18) ν ν (cid:19) (cid:18) − ν f sd ( ν.ν p / ν p ) f sd ( ν .ν p / ν p ) + ∂ f sd ( ν.ν p / ν p ) ∂ν f sd ( ν .ν p / ν p ) (cid:19) , (5)where ∆ n y ν = K y ν / xT CMB . Using Eq. (5), we can writeEq. (4) as S ν ( ˆ p ) = D pl ν ∆ ν + A CMB ( ˆ p )D T ν ∆ ν + A µ ( ˆ p )D µν ∆ ν + A y ( ˆ p )D y ν ∆ ν + A dust ( ˆ p )D dust ν ∆ ν + A syn ( ˆ p )D syn ν ∆ ν + A free ( ˆ p )D free ν ∆ ν + A spin-dust ( ˆ p )D spin-dust ν ∆ ν, (6)or in matrix notation S( ˆ p ) = DA ( ˆ p ) . (7)Here D is the matrix of the FSD spectrum with components MNRAS000
We plot the kernel of the signal for different sources ofspectral distortions for y -distortions (black) which is multiplied bya factor of two in the amplitude, µ -distortions (red) and blackbodyspectrum (blue) at T = . Kelvin. be expressed in terms of D x ν ( ˆ p ) ≡ ( ∂ K x / ∂ν )| ν as D pl ν = A CMB ν CMB ( e x − ) (cid:20) − xe x ( e x − ) (cid:21) , D T ν = ν CMB x e x ( e x − ) (cid:20) x + − e x ( e x − ) (cid:21) , D µν = T CMB ν CMB e x ( e x − ) (cid:20) − − x + xe x ( e x − ) (cid:21) , D y ν = T CMB ν CMB (cid:20) ∆ n y ν (cid:18) + x − xe x ( e x − ) (cid:19) + x e x ( e x − ) (cid:18)(cid:18) e x + e x − (cid:19) + (cid:18) xe x e x − (cid:19) − (cid:18) xe x ( e x + )( e x − ) (cid:19)(cid:19)(cid:21) , D Dust ν = (cid:18) νν (cid:19) β d + (cid:18) e ν / T d − e ν / T d − (cid:19) (cid:20) β d + ν − γ e γν ( e γν − ) (cid:21) , D syn ν = (cid:18) ν ν (cid:19) (cid:18) − ν f s ( ν / α ) f s ( ν / α ) + ∂ f s ( ν / α ) ∂ν f s ( ν / α ) (cid:19) , D free ν = − T e e τ ∂τ∂ν , D spin-dust ν = (cid:18) ν ν (cid:19) (cid:18) − ν f sd ( ν.ν p / ν p ) f sd ( ν .ν p / ν p ) + ∂ f sd ( ν.ν p / ν p ) ∂ν f sd ( ν .ν p / ν p ) (cid:19) , (5)where ∆ n y ν = K y ν / xT CMB . Using Eq. (5), we can writeEq. (4) as S ν ( ˆ p ) = D pl ν ∆ ν + A CMB ( ˆ p )D T ν ∆ ν + A µ ( ˆ p )D µν ∆ ν + A y ( ˆ p )D y ν ∆ ν + A dust ( ˆ p )D dust ν ∆ ν + A syn ( ˆ p )D syn ν ∆ ν + A free ( ˆ p )D free ν ∆ ν + A spin-dust ( ˆ p )D spin-dust ν ∆ ν, (6)or in matrix notation S( ˆ p ) = DA ( ˆ p ) . (7)Here D is the matrix of the FSD spectrum with components MNRAS000 , 1–9 (2018)
Mukherjee, Silk & Wandelt
Figure 2.
The Frequency Space Derivative (FSD) spectrum of the y -distortions (black), µ -distortions (red) and blackbody spectrum(blue) are depicted over a wide frequency range which is usuallyaccessible by CMB missions. D ji = D j ν i ∆ ν i and A is the column matrix composed of thesignals. This equation relates the Frequency Space Differen-tial (FSD) with the known theoretical spectrum of severalsources. The FSD spectrum for different sources are plottedin Fig. 2. As is clear from Fig. 2, the spectrum for each ofthe sources is distinct. For µ distortions, the FSD signal ismainly strong at low frequencies and decays rapidly. The y distortions peak at higher frequencies with a much widerFSD spectrum ( ν ∈ − GHz) than µ . A mission to con-strain both y and µ therefore requires a combination of lowand high frequency channels. We will leave a detailed designstudy of an optimal distribution of channel frequencies andbandwidths to future work. An estimator such as ModifiedInternal Linear Combination (Hurier et al. 2013) combinesall frequency channels to reject foreground contaminationand improve Signal to Noise (SNR). We will develop theformalism of such an estimator in the context of the FSDtechnique in Sec. 4.In the next section, we describe the dominant source ofsystematic error in this method and how to mitigate it. Usage of the FSD technique to measure CMB spectral distor-tions is only possible if the temperature differences betweenfrequency channels can be determined with sufficient system-atic error control. Though the measurement techniques anddetector properties depend on specific missions, we discussthe basic requirements which should be addressed in orderto use the FSD method in this section.
To illustrate the problem we will first discuss why non-differential methods will not be able to provide useful con-straints on spectral distortions. The radiation impinging on apixel ( ˆ p ) at a frequency ν i of the detector produces a voltage V ν i ( ˆ p ) which is related to the observed temperature field bythe gain factor G ν i as T ν i ( ˆ p ) = G ν i V ν i ( ˆ p ) + T off ν i , (8)where T off ν i is the instrumental off-set temperature. In theabsence of mean detector noise, the measured temperatureat a pixel ˆ p in frequency ν i is related to the theoretical FSDsignal by the relation, T ν i ( ˆ p ) = S ν i ( ˆ p ) T off ν i . (9)However, detectors even with a known and stable gain factor G ν i and offset temperature T off ν i = , exhibit variations δ G ν i and δ T off ν i which are the sources of systematic errors thatpropagate through the measurements. As a result, the sys-tematic error associated with the temperature field (Eq. (8))are due to the gain and off-set error, which can be writtenas ( σ ν i ) sys ≡ ( δ T ν i ( ˆ p )) = (cid:18) δ G ν i G ν i (cid:19) ( G ν i V ν i ( ˆ p )) + ( δ T off ν i ) . (10)After using Eq. (9), above equation can be expressed as ( σ ν i ) sys = (cid:18) δ G ν i G ν i (cid:19) ( S ν i ( ˆ p )) + ( δ T off ν i ) . (11)The dominant contribution to S ν i ( ˆ p ) is the blackbody tem-perature field of CMB ( K pl ν i in Eq. (2)). As a result, thedominant source of systematic error in Eq. (11) is inducedby the coupling between gain error δ G ν i and K pl ν i . For thetypical values of gain error (of order . − . ) , thecontribution of the systematic error is greater than the usualsignal strength of µ and y distortions. So to measure thespectral distortion signals, an absolute internal blackbodycalibrator with a precisely known reference temperature isrequired under this method of measurement.In the following subsection, we will introduce a new dif-ferential method which can reduce the systematic error with-out an absolute calibrator and also use the cross-calibrationbetween the frequency channels to minimize the budget ofthe systematic error. The incoming electromagnetic waves (composed of multiplecomponents) from the sky at a particular frequency channel ν i falls on the detector (operating at this frequency) will havean induced voltage, which we define as V ν i . In the FSD tech-nique, we propose the measurement of the difference in theamplitude of electromagnetic field at two different frequen-cies by taking the difference between the induced voltages( V ν j − V ν i ). This differential measurement of the signal carriesthe information of the change in the electromagnetic fieldof CMB (and also in other contaminations) with variationin the frequency. The measured differential voltage can beconverted into a temperature difference by a known gainfactor G ν i and G ν j by the relation T ν ji ( ˆ p ) = G ν j δ V ν ji ( ˆ p ) + G ν i δ V ν ji ( ˆ p ) , (12) A detailed description of the systematic error is given in Sec.3.4 MNRAS , 1–9 (2018)
SD: Frequency Space Differential measurement of CMB spectral distortions where, δ V ν ji ( ˆ p ) = ( V ν j − V ν i )/ , ν ji = ( ν j + ν i )/ and the off-settemperature difference between the two channels is assumedto be zero. If G ν i = G ν j , then the above equation is directlyrelated to the theoretical FSD signal S ν ji (Eq. (6)) as T ν ji ( ˆ p ) ≡ G ν j ( δ V ν ji ( ˆ p ) + δ V ν ji ( ˆ p )) = S ν ji ( ˆ p ) . (13)However, if G ν i = G ν j + ∆ G ν ji , then T ν ji ( ˆ p ) = S ν ji ( ˆ p ) + ∆ GG ν j G ν j δ V ν ji ( ˆ p ) , (14)where, the second term is an extra bias originating from thedifference of the gain factors between two frequency channels.This indicates that any variation in gain factor will affect themeasurement by coupling it with the difference in the voltagesand not with the absolute value of the voltages. As thevoltage difference between two channels have the dominantcontribution from the FSD spectrum of the blackbody S pl ν ji ,the above equation can be approximated as T ν ji ( ˆ p ) ≈ S ν ji ( ˆ p ) + ∆ GG ν j S pl ν ji ( ˆ p ) . (15)The variance of Eq. 12, only due to the uncorrelatedsystematic errors in the gain factor and off-set temperaturecan be written as ( σ ν ji ) sys ≡ ( δ T ν ji ) = (cid:18) δ G ν i G ν i (cid:19) G ν i δ V ν ji + (cid:18) δ G ν j G ν j (cid:19) G ν j δ V ν ji + ( δ T off ν j ) + ( δ T off ν i ) . (16)Here, ( δ T off ν i ) denotes the variance in the offset measurement.The systematic error is related to the voltage difference whichaccording to Eq. (14), have the major contribution from theFSD spectrum of CMB blackbody.The comparison of Eq. (10) and Eq. (16) exhibits the keydifference between the non-differential technique and the FSDtechnique. The systematic error is related to the absoluteblackbody signal in the former case and to the derivative ofthe blackbody in the latter case. As depicted in Fig. 2, for ∆ ν = GHz, the amplitude of FSD spectrum of blackbody istwo orders of magnitude below the blackbody signal. Hence,the systematic error between these two methods will alsodiffer by two orders of magnitude.The total error due to both the systematic and thestatistical error can be written as ( σ ν ji ) tot = ( σ ν ji ) sys + ( σ ν ji ) stat , (17)where, we define the statistical error in terms of the uncorre-lated instrumental noise as ( σ ν ji ) stat ( ˆ p ) = ( δ T N ν i ( ˆ p )) + ( δ T N ν j ( ˆ p )) . (18) The above-mentioned FSD technique is a differential measure-ment of the imaging signal obtained from the high resolutionfrequency bands to construct the deviations from blackbody.Implementation of this method along with the standardimaging method (by using low resolution frequency bands)is required to achieve the science goals from the spectral dis-tortions as well as the anisotropic part of CMB. So we need a hybrid composition of frequency resolution to implementboth FSD technique and imaging technique, such that wecan obtain the spectral distortion signal & anisotropic signalfrom the same conceptual framework and also with minimumcost and minimum error.To minimize the sources of systematic error, it is re-quired to reduce the contribution of CMB blackbody in thedifferential measurement between two different frequencychannels. So we need high resolution frequency channels tosubtract the blackbody part substantially so that the totalsystematic error is smaller than the spectral distortion signal.The high resolution FSD technique needs to be implementedon the frequency range which have the large values of FSDkernel for µ distortion (approximately − GHz) and y distortion (approximately − GHz), which can be iden-tified from Fig. 2. The remaining frequency ranges can havelarge bandwidth to perform the scientific studies related toimaging. A detailed case study of the FSD technique can bedone for a specific mission with the knowledge of the detectorproperties, calibration error, data read-out frequency, size ofthe focal plane etc.For multiple frequency channels, we require to esti-mate the covariance matrix consisting of contributions fromsystematic errors ( C sys ), statistical error ( C N ) and er-ror due to each cosmological and astrophysical component( C A ≡ (cid:104) AA † (cid:105) ). So the total covariance matrix becomes (cid:104)T T † (cid:105) ≡ C T = DC A D T + C sys + C N . (19)The covariance matrix is not diagonal and needs to be evalu-ated for every mission with the particular instrumental noise,systematic errors and frequency coverage. The essential re-quirement to implement the FSD technique is to reduce thecontribution of the total error on the signals of spectral dis-tortion. The total contribution from the systematic and theinstrumental noise matrix can be written as C ˜ N = C sys + C N , (20)which can be decomposed as C ˜ N = E Γ E T , (21)where, Γ is a diagonal matrix of eigenvalues γ , γ , . . . , γ n such that γ > γ > . . . > γ n and the matrix E contains thecorresponding eigenvectors. An experimental design whichcan achieve the condition that the eigenvectors with largesteigenvalues have a minimum projection on the FSD kernel(like D µν and D y ν for µ and y respectively) can significantlyimprove the SNR of the measurement. In Sec. 4, we elaboratemore on this and also explain the procedures to extract thesignal. For CMB experiments, there are several standard calibrationsources (Adam et al. 2016a) like the CMB solar dipole, theorbital dipole and planets. These are used for calibration bythe Planck mission (Adam et al. 2016a). The orbital dipoleis a very good calibrator due to the well-known value ofthe satellite velocity and gives a very small calibration error Bold fonts denotes matricesMNRAS000
SD: Frequency Space Differential measurement of CMB spectral distortions where, δ V ν ji ( ˆ p ) = ( V ν j − V ν i )/ , ν ji = ( ν j + ν i )/ and the off-settemperature difference between the two channels is assumedto be zero. If G ν i = G ν j , then the above equation is directlyrelated to the theoretical FSD signal S ν ji (Eq. (6)) as T ν ji ( ˆ p ) ≡ G ν j ( δ V ν ji ( ˆ p ) + δ V ν ji ( ˆ p )) = S ν ji ( ˆ p ) . (13)However, if G ν i = G ν j + ∆ G ν ji , then T ν ji ( ˆ p ) = S ν ji ( ˆ p ) + ∆ GG ν j G ν j δ V ν ji ( ˆ p ) , (14)where, the second term is an extra bias originating from thedifference of the gain factors between two frequency channels.This indicates that any variation in gain factor will affect themeasurement by coupling it with the difference in the voltagesand not with the absolute value of the voltages. As thevoltage difference between two channels have the dominantcontribution from the FSD spectrum of the blackbody S pl ν ji ,the above equation can be approximated as T ν ji ( ˆ p ) ≈ S ν ji ( ˆ p ) + ∆ GG ν j S pl ν ji ( ˆ p ) . (15)The variance of Eq. 12, only due to the uncorrelatedsystematic errors in the gain factor and off-set temperaturecan be written as ( σ ν ji ) sys ≡ ( δ T ν ji ) = (cid:18) δ G ν i G ν i (cid:19) G ν i δ V ν ji + (cid:18) δ G ν j G ν j (cid:19) G ν j δ V ν ji + ( δ T off ν j ) + ( δ T off ν i ) . (16)Here, ( δ T off ν i ) denotes the variance in the offset measurement.The systematic error is related to the voltage difference whichaccording to Eq. (14), have the major contribution from theFSD spectrum of CMB blackbody.The comparison of Eq. (10) and Eq. (16) exhibits the keydifference between the non-differential technique and the FSDtechnique. The systematic error is related to the absoluteblackbody signal in the former case and to the derivative ofthe blackbody in the latter case. As depicted in Fig. 2, for ∆ ν = GHz, the amplitude of FSD spectrum of blackbody istwo orders of magnitude below the blackbody signal. Hence,the systematic error between these two methods will alsodiffer by two orders of magnitude.The total error due to both the systematic and thestatistical error can be written as ( σ ν ji ) tot = ( σ ν ji ) sys + ( σ ν ji ) stat , (17)where, we define the statistical error in terms of the uncorre-lated instrumental noise as ( σ ν ji ) stat ( ˆ p ) = ( δ T N ν i ( ˆ p )) + ( δ T N ν j ( ˆ p )) . (18) The above-mentioned FSD technique is a differential measure-ment of the imaging signal obtained from the high resolutionfrequency bands to construct the deviations from blackbody.Implementation of this method along with the standardimaging method (by using low resolution frequency bands)is required to achieve the science goals from the spectral dis-tortions as well as the anisotropic part of CMB. So we need a hybrid composition of frequency resolution to implementboth FSD technique and imaging technique, such that wecan obtain the spectral distortion signal & anisotropic signalfrom the same conceptual framework and also with minimumcost and minimum error.To minimize the sources of systematic error, it is re-quired to reduce the contribution of CMB blackbody in thedifferential measurement between two different frequencychannels. So we need high resolution frequency channels tosubtract the blackbody part substantially so that the totalsystematic error is smaller than the spectral distortion signal.The high resolution FSD technique needs to be implementedon the frequency range which have the large values of FSDkernel for µ distortion (approximately − GHz) and y distortion (approximately − GHz), which can be iden-tified from Fig. 2. The remaining frequency ranges can havelarge bandwidth to perform the scientific studies related toimaging. A detailed case study of the FSD technique can bedone for a specific mission with the knowledge of the detectorproperties, calibration error, data read-out frequency, size ofthe focal plane etc.For multiple frequency channels, we require to esti-mate the covariance matrix consisting of contributions fromsystematic errors ( C sys ), statistical error ( C N ) and er-ror due to each cosmological and astrophysical component( C A ≡ (cid:104) AA † (cid:105) ). So the total covariance matrix becomes (cid:104)T T † (cid:105) ≡ C T = DC A D T + C sys + C N . (19)The covariance matrix is not diagonal and needs to be evalu-ated for every mission with the particular instrumental noise,systematic errors and frequency coverage. The essential re-quirement to implement the FSD technique is to reduce thecontribution of the total error on the signals of spectral dis-tortion. The total contribution from the systematic and theinstrumental noise matrix can be written as C ˜ N = C sys + C N , (20)which can be decomposed as C ˜ N = E Γ E T , (21)where, Γ is a diagonal matrix of eigenvalues γ , γ , . . . , γ n such that γ > γ > . . . > γ n and the matrix E contains thecorresponding eigenvectors. An experimental design whichcan achieve the condition that the eigenvectors with largesteigenvalues have a minimum projection on the FSD kernel(like D µν and D y ν for µ and y respectively) can significantlyimprove the SNR of the measurement. In Sec. 4, we elaboratemore on this and also explain the procedures to extract thesignal. For CMB experiments, there are several standard calibrationsources (Adam et al. 2016a) like the CMB solar dipole, theorbital dipole and planets. These are used for calibration bythe Planck mission (Adam et al. 2016a). The orbital dipoleis a very good calibrator due to the well-known value ofthe satellite velocity and gives a very small calibration error Bold fonts denotes matricesMNRAS000 , 1–9 (2018)
Mukherjee, Silk & Wandelt of typically . − . (Adam et al. 2016a). The motionof the solar barycentre (dipole) or the orbital motion (ofthe satellite) also exhibits a known spectrum which can bewritten as ∆ T dip K T ν i , where ∆ T dip is the magnitude of theinduced temperature due to the solar or the orbital motionand K T ν i is the derivative of the blackbody spectrum withrespect to the temperature. So using the known value ofthe brightness temperature of the CMB dipole and (time-dependent) orbital motion, we can calibrate the detectorsfor each frequency channel. For the remaining discussion inthe paper we will focus only on the orbital dipole becauseof several factors like (i) ease of modelling accurately, (ii)measurement with very high SNR by current detectors and(iii) clean demodulation from multi-year data due to itsannual variation. Schematically, the CMB dipole can bemeasured within each frequency channel through its pixel-to-pixel variation δ S dip ν i ≡ S ν i ( ˆ p ) − S ν i ( ˆ p ) = G ν i ( V ν i ( ˆ p ) − V ν i ( ˆ p )) ≡ G ν i ∆ V ν i , (22)Due to the known frequency spectrum of δ S dip ν i , one can writethis as G ν i = K T ν i ∆ T dip ∆ V ν i . (23)As a result, the gain error in terms of the error associated withthe dipole measurement ( δ ( ∆ T dip ) ) and voltage measurement( δ ( ∆ V ) ) can be written as (cid:18) δ G ν i G ν i (cid:19) (cid:39) (cid:18) δ ( ∆ T dip ) ∆ T dip (cid:19) + (cid:18) δ ( ∆ V ) ∆ V (cid:19) . (24)So the gain error of each channel is related to the errorassociated with the measurement of dipole amplitude, evenif the error in the measurement of voltage is negligible.We can accurately obtain the relative gain coefficients atdifferent frequencies by cross-calibrating between frequencychannels. By equating the dipole amplitude fluctuation ∆ T dip between any two frequency channels, we can write G ν j ∆ V ν j K T ν j = G ν i ∆ V ν i K T ν i , (25)which implies G ji ≡ G ν j G ν i = ∆ V ν i ∆ V ν j K T ν j K T ν i . (26)This indicates that the relative calibration depends only onthe measured voltage difference. Therefore, the correspondingerror in the ratio of the gain is affected only by the errorassociated with the measurement of voltage difference andnot that associated with the orbital dipole measurement. As aresult, the error on the relative gain ratio can be reduced. Thisalso indicates that the accurate calibration of the gain factorat any one frequency channel translates into an accuratecalibration at all channels. The FSD measurement of the all-sky intensity (or equiva-lently brightness temperature) at different frequency channels is an addition of several signals due to cosmological and astro-physical sources and also instrumental noise. With the knownspectrum of the FSD and a high spectral resolution measure-ment over a wide frequency band, we can estimate the best-fitparameter ˆ A x (where x ∈ [ y , µ, . . . ] ), which minimizes thechi-square defined as χ y ,µ = (cid:213) ν,ν (cid:48) (cid:18) ¯ T ν − ˆ A y ,µ D y ,µν (cid:19) ( C − T ) νν (cid:48) (cid:18) ¯ T ν (cid:48) − ˆ A y ,µ D y ,µν (cid:48) (cid:19) . (27)Addition over a wide range of frequencies increases the over-all SNR of the signal. The corresponding error bar on ˆ A y ,µ is a standard result given by σ y ,µ = (cid:20) (cid:213) ν,ν (cid:48) D y ,µν ( C − T ) νν (cid:48) D y ,µν (cid:48) (cid:21) − . (28)As mentioned before, the covariance matrix C T is non-diagonal and is a quantity which depends upon instrumentalnoise, scanning strategy, systematic errors, etc. For a par-ticular mission, these quantities need to be evaluated forsuccessfully implementing the FSD technique. After the removal of the coupling between gain errors andthe CMB monopole the main remaining hurdle to measuring µ & y distortions is foreground contamination. At low fre-quencies, the main sources of contamination are synchrotronemission and spinning dust emission from our galaxy. At highfrequency, foreground contamination is mainly due to dust.Since the FSD spectrum of µ and y are not degenerate withthese foregrounds we will now discuss how to use combina-tions of frequency channels over a wide range of frequenciesto project out foreground contamination.We at first address the extraction of the monopole part ofthe spectral distortion signal by an all-sky average of the FSDspectrum. The all-sky average value of the distortion signalcan be extracted using the known FSD spectrum (Eq. (5))by the Internal Linear Combination (ILC) (Remazeilles et al.2011) and Modified Linear Combination (MILC) (Hurieret al. 2013). With M ν frequency channels over which T i isestimated, we can write T ν i ≡ T ν i ( ˆ p ) = D j ν i ∆ ν i A j ( ˆ p ) + N ν i ( ˆ p ) , (29)where, i ∈ [ , M ν ] and A j ≡ [ A CMB , A µ , A y , . . . , A N s − ] . A j contains both cosmological signal and also foreground con-taminations. In terms of the M ν × column vector T , N s × column vector A and M ν × N s mixing kernel D , we can write T = DA + N . (30)In the presence of a non-zero value of ∆ G (introduced inthe previous section), there is also an additional componentgiven by T ν i = D j ν i ∆ ν i A j ( ˆ p ) + N ν i ( ˆ p ) + ∆ G ν i G ν i S pl ν i , (31) T = DA + N + J G , (32)where, S pl ν i is the FSD blackbody spectrum at frequency ν i and J G is the residual column matrix which can arisedue to difference in the gain of the frequency channels. Theextraction of the signal is achievable with the requirement MNRAS , 1–9 (2018)
SD: Frequency Space Differential measurement of CMB spectral distortions that we recover only one component and follow the constraintthat all other components do not contribute to the signal.For N s rejected components, we can define weights w suchthat u = w T f = , u = w T f = ,... ... u j = w T f j = ,... ... u N s = w T f N s = , u N s + = w T J G = , (33)where f j are the frequency dependence of the j th signaldefined as f j = D x j . f j is a column vector with M ν × elementsand x j = [ , , . . . , , . . . , ] T with only j th element equal toone. The last condition of Eq. (33) also put constraints onthe nature of relative gain difference ∆ G i / G i . For the FSDtechnique to work, J G should not behave like any of thespectral signatures like µ , y , etc. and hence needs to satisfythe condition (cid:2) J G f j (cid:3) i = ∆ G ν i G ν i S pl ν i f j ν i = . (34)As S pl ν i is the known FSD spectrum of blackbody at fre-quency ν i , so the required frequency dependence of ∆ G ν i / G ν i to minimize the residual contaminations in the signal is man-ifested by Eq. (34). A special case with ∆ G ν i = is a trivialsolution of this equation and is sufficient but not necessary tobe satisfied by the detectors. In the remaining of the paper,we will assume that the correction from ∆ G ν i / G ν i can bemade and we restrict only to N s values of u .With the requirement that the variance in the extracted signal map C ˆ A = (cid:104) ˆ A ˆ A T (cid:105) is minimum, the weight matrix canbe obtained by solving the equation (cid:20) C T − DD T (cid:21) (cid:20) w λ (cid:21) = (cid:20) x (cid:21) , (35)where λ is the Lagrange multiplier and the covariance matrix C T is a M ν × M ν dimension matrix which can be expressedas C T = D T C A D + C ¯N , (36)where C ¯N is defined in Eq.(20) and have the contributionsfrom instrumental noise, systematic errors and covariancematrix of the cosmological and astrophysical sources.The weight matrix which satisfies Eq. (35) can be ex-pressed as W = C − T D ( D T C − T D ) − , (37)and the corresponding j th component of the map can beobtained as ˆ A j = x Tj W T T . (38)The error estimate of the signal map ˆ R j can be written as C ˆ A j = x Tj W T C T Wx j . (39)Using the above formalism for every component of the signal, we can obtain the weight matrix W which minimizesthe variance of the signal. To further reduce the error of thesignal, we can satisfy the condition similar to Eq. (21) forthe covariance matrix C T such that weight matrix projectsminimally with the eigenvector corresponding to the largesteigenvalue of the covariance matrix. So the error estimate onthe j th component in terms of the eigenvector decomposition( C ˜ T = E ˜ T Γ ˜ T E ˜ T− ) can be written as C ˆ A j = x Tj W T E ˜ T Γ ˜ T E ˜ T− Wx j , (40)which satisfies the condition x Tj W T E i ˜ T ≈ ∀ γ i > γ min . (41)where, γ min is the smallest eigenvalue of the covariancematrix C T . The methods described previously have a particular appli-cation for approaching the monopole part of the spectraldistortion signal. However this approach can be readily ex-tendable to measure the fluctuations in the spectral distortionsignal. Measurement of the FSD signal at every frequencychannel gives a pixel space map of the signal, which in generalcan be written as T ν i ( ˆ p ) = (cid:213) j D j ν i ∆ ν i A j ( ˆ p ) , (42) (T ν i ) lm = (cid:213) lm (cid:213) j D j ν i ∆ ν i ( A j ) lm , (43)where, T ν i ) lm and ( A j ) lm are the spherical harmonics trans-formation of T ν i ( ˆ p ) and A j ( ˆ p ) respectively. The fluctua-tions in the signal can be captured by the power spectrum ν C T T l = (cid:205) m (T ν ) lm (T ∗ ν ) lm ( l + ) which is a composite effect of allthe mechanisms. The dominant source of fluctuations in thespectral distortion is due to the y-distortion (Hill et al. 2015).With this technique, we can access the spatial fluctuationsin the spectral distortions which are expected to be strongerthan µ distortions. The intrinsic temperature fluctuationsexhibit a very different FSD spectrum from y and henceare easily separable. The ILC method for FSD signal dis-cussed previously is also directly applicable to reconstructingthe signal at every pixel and to generating a map of thefluctuations. While not using an absolute internal calibrator to measureCMB spectral distortions has clear practical advantages, theboth approaches have their unique features and challenges.For an absolute internal calibrator, it is essential that thecalibrator is stable in temperature and is a perfect blackbodyso that it matches the blackbody distribution of CMB. Evena tiny departure from the blackbody spectrum of the absoluteinternal calibrator can act as a source of systematic errorand obscure any cosmological spectral distortion. In theabsence of an absolute internal calibrator we have to achievegood control of the systematic errors in the temperaturemeasurement and excellent relative calibration of different
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Mukherjee, Silk & Wandelt frequency channels, but for a potentially significant reductionin mission complexity and hence cost.We now discuss the necessary requirements to use FSDfor detecting CMB spectral distortions.(i) Instrument and measurement technique should be de-signed such that the final output is calibrated only with therelative voltage difference between two frequency channelsas discussed in Sec. 3.2.(ii) The FSD signal due to y and µ distortions peak atdifferent frequency ranges as shown in Fig. 2. So, multiplehigh spectral resolution channels in those frequency rangeshould be implemented with minimum instrumental noise.Use of high spectral resolution channels can help in reducingthe contaminations from other sources and also improve thesystematic errors in the measurement.(iii) Measurement techniques should be devised such thatthe coupling of the FSD spectrum of the signal with theeigenvectors corresponding to the largest eigenvalues of thenoise covariance matrix is minimized. This can improve themeasurability of the spectral distortion signal and reduce thecontamination from systematic error and instrumental noise.(iv) The relative difference in the gain factor G betweenfrequency channels should satisfy the condition given in Eq.(34).(v) A stable gain factor G for the complete frequencyrange is required with a very small relative calibration errorof δ G / G between different frequencies. The requirement fora controlled gain error is provided in Sec. 3.4.(vi) The systematic errors due to off-set temperature ofthe detectors must be controlled below the desired signal S µν and S y ν at every frequency channel. The rich domain of cosmological information embedded inthe spectral distortion of the CMB spectrum is going tobe unveiled by the next generation of CMB missions. TheCMB absolute intensity is usually compared with an inter-nal blackbody calibrator to search for any deviations fromblackbody. In the presence of an internal blackbody calibra-tor, the observed intensity of the sky is compared at everyfrequency with the intensity from the internal blackbody cal-ibrator and any departure of the observed sky intensity fromthe blackbody can be modeled with the known spectrum ofspectral distortions. As a result, a successful measurementof spectral distortion signals with a high SNR requires theinternal blackbody calibrator to be extremely stable at afixed temperature and also should obey a perfect blackbodyspectrum over the complete frequency range of a mission(typically − GHz). The departure of the internal cali-brator from blackbody can induce a systematic error and canalso be misunderstood with the spectral distortion signal.We propose an alternative strategy called the FrequencySpace Differential (FSD) to measure spectral distortions inCMB. This technique measures the difference in the observedbrightness temperature at different frequencies and modelsthe observed difference with the theoretically predictableFSD kernel for different components in Eq. (5). The FSDspectrum for expected sources of spectral distortions like µ & y are different and not degenerate, which makes iteasily distinguishable and extractable. The µ spectrum is stronger at low frequencies and decreases rapidly at higherfrequencies, whereas y distortion FSD spectrum is dominantat high frequency range as depicted in Fig. 2.Our proposed method uses the CMB itself between theneighboring channels as a calibrator to measure the devia-tions from blackbody. This method does not directly measurethe absolute blackbody spectrum, but only measures the fre-quency space derivative of a blackbody signal. In the presenceof spectral distortions, the FSD signal exhibits a combinationof effects from blackbody along with other sources and canbe fitted uniquely for a known FSD spectrum. Successfulimplementation of the FSD method needs several instrumen-tal controls in order to reduce contaminations by systematicerrors and instrumental noise, which we listed in Sec. 5. Mea-surement of the spectral distortion signal without an internalabsolute blackbody calibrator can be possible in implement-ing this formalism via suitable instrumental engineering forfuture missions.The main insight of this paper is to explore signaturesof spectral distortion and measuring any deviations fromblackbody through the FSD spectrum. This process enablesone to measure the spectral distortion signal in the samespirit as WMAP measured the CMB anisotropies through adifferential measurement without an internal reference. Themain advantage of our method is that it does not require aninternal blackbody calibrator to measure the signal. Secondly,this approach opens up an alternative way of measuring thespectral distortion signal which can be useful for comparingresults from other missions which use an internal blackbodycalibration method. Next-generation CMB missions with up-graded detector technologies can implement this method tomeasure spectral distortions without using an absolute cali-brator. Estimation of the noise properties and experimentalrequirements in order to implement this method for a futureCMB mission like LiteBIRD (Matsumura et al. 2016) will beaddressed in a follow-up paper. Acknowledgements
This work has been done withinthe Labex ILP (reference ANR-10-LABX-63) part of the IdexSUPER, and received financial state aid managed by theAgence Nationale de la Recherche, as part of the programmeInvestissements d’avenir under the reference ANR-11-IDEX-0004-02. The work of JS has been supported in part by ERCProject No. 267117 (DARK) hosted by the Pierre and MarieCurie University-Paris VI, Sorbonne Universities. The workof SM and BDW is supported by the Simons Foundation.The authors acknowledge valuable comments from MasashiHazumi on this draft.
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