A forecast of the sensitivity on the measurement of the optical depth to reionization with the GroundBIRD experiment
Kyungmin Lee, Ricardo T. Génova-Santos, Masashi Hazumi, Shunsuke Honda, Hiroki Kutsuma, Shugo Oguri, Chiko Otani, Mike W. Peel, Yoshinori Sueno, Junya Suzuki, Osamu Tajima, Eunil Won
DDraft version February 8, 2021
Typeset using L A TEX twocolumn style in AASTeX63
A FORECAST OF THE SENSITIVITY ON THE MEASUREMENT OF THE OPTICAL DEPTHTO REIONIZATION WITH THE GROUNDBIRD EXPERIMENT
K. Lee, R. T. G´enova-Santos,
2, 3
M. Hazumi,
4, 5, 6, 7
S. Honda, H. Kutsuma,
9, 10
S. Oguri, C. Otani,
9, 10
M. W. Peel,
11, 3
J. Suzuki, O. Tajima, and E. Won Department of Physics, Korea University, Seoul, 02841, Republic of Korea Instituto de Astrof´ısica de Canarias, E38205 La Laguna, Tenerife, Canary Islands, Spain Departamento de Astrof´ısica, Universidad de La Laguna, E38206 La Laguna, Tenerife, Canary Islands, Spain High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, 305-0801, Japan Japan Aerospace Exploration Agency (JAXA), Institute of Space and Astronautical Science (ISAS), Sagamihara, Kanagawa 252-5210,Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba277-8583, Japan The Graduate University for Advanced Studies (SOKENDAI), Miura District, Kanagawa 240-0115, Hayama, Japan Physics Department, Kyoto University, Kyoto, 606-8502, Japan Department of Physics, Tohoku University, 6-3 Aramaki-Aoba, Aoba-ku, Sendai, Miyagi 980-8578, Japan The Institute of Physical and Chemical Research (RIKEN), 519-1399 Aramaki-Aoba, Aoba-ku, Sendai, Miyagi 980-0845, Japan Instituto de Astrof´ısica de Canarias, E38205 - La Laguna, Tenerife, Canary Islands, Spain (Received; Revised February 8, 2021; Accepted)
Submitted to ApJABSTRACTWe compute the expected sensitivity on measurements of optical depth to reionization for a ground-based experiment at Teide Observatory. We simulate polarized partial sky maps for the GroundBIRDexperiment at the frequencies 145 and 220 GHz. We perform fits for the simulated maps with ourpixel-based likelihood to extract the optical depth to reionization. The noise levels of polarizationmaps are estimated as 131 µ K arcmin and 826 µ K arcmin for 145 and 220 GHz, respectively, byassuming a three-year observing campaign and sky coverages of 52% for 145 GHz and 45% at 220GHz. Our sensitivities for the optical depth to reionization are found to be σ τ = 0 .
031 with thesimulated GroundBIRD maps, and σ τ = 0 .
011 by combining with the simulated QUIJOTE maps at11, 13, 17, 19, 30, and 40 GHz.
Keywords: early universe, cosmic microwave background, cosmological parameters, photon decoupling INTRODUCTIONA model of rapid accelerated expansion of the earlyuniverse (Starobinsky 1980; Guth 1981; Sato 1981; Al-brecht & Steinhardt 1982; Linde 1982), required in or-der to resolve the horizon and flatness problems of thestandard model of cosmology, predicts the production oftwo distinct polarized patterns in the cosmic microwavebackground (CMB) radiation due to density fluctuation
Corresponding author: Eunil [email protected] and the production of primordial gravitational waves(Kamionkowski et al. 1997a; Seljak & Zaldarriaga 1997;Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997b).Along with this possible inflationary expansion scenarioof the early universe, the standard model of cosmology,or a spatially-flat six-parameter model of the universe(ΛCDM), having a cosmological constant and dark mat-ter as dominant forms of the energy density in the uni-verse at present, agrees remarkably well with observa-tional data to date. In both cases, the CMB polarizationplays a key role in discovering the physics of the earlyuniverse. Among six parameters in the ΛCDM model ofthe standard cosmology, the optical depth to reioniza- a r X i v : . [ a s t r o - ph . C O ] F e b tion ( τ ), a dimensionless quantity that provides a mea-sure of the line-of-sight free-electron opacity to CMBradiation, has important cosmological implications.Of the six parameters in the ΛCDM model mentioned, τ is the only one that is measured with sensitivity worsethan 1% at present, τ = 0 . ± .
007 (Planck Collab-oration 2020a). This measurement also constrains thesum of the neutrino masses, one of the critical parame-ters in describing the evolution of the early universe andunderstanding the physics beyond the standard modelof particle physics (Hannestad 2010). Note that, as ofnow, the non-zero neutrino mass is the only experimen-tal observation beyond the reach of the standard modelof particle physics. Therefore, it is of prime importanceto improve the measurement uncertainty of the opticaldepth to reionization in order to understand the natureof the neutrino.At large angular scales in the polarization power spec-trum of the CMB ( (cid:96) (cid:46)
10 where (cid:96) is the multipole indexin the angular power spectrum), anisotropies are createdby the rescattering of the local temperature quadrupole.These lead to a “bump” today in the large-scale po-larization power spectrum at the Hubble scale duringreionization (Hu & White 1997).The amplitude of the bump scales like a power lawof τ (WMAP Collaboration 2007) and τ is thereforemostly constrained by the large-scale polarization mea-surements from the Planck satellite (Planck Collabora-tion 2020b) (at present to 0 . ± . τ have a clear systematictendency for the central value to decrease, the large an-gular scale polarization measurements become crucial.In the near future, we expect recently-started ground-based experiments such as ACTpol/SPTpol (Swetzet al. 2011; McMahon et al. 2009), BICEP/Keck Array(Ogburn IV et al. 2012), CLASS (Watts et al. 2018),GroundBIRD (Tajima et al. 2012; Won 2018), POLAR-BEAR (Lee et al. 2008; Kermish et al. 2012), SimonsArray (Stebor et al. 2016), Simons Observatory (Adeet al. 2019), STRIP (Franceschet et al. 2018), and QUI-JOTE (G´enova-Santos et al. 2015) to provide such mea-surements. Among these, CLASS, GroundBIRD, andQUIJOTE are unique owing to their large sky coverage(greater than 40%).In this paper, we calculate the expected sensitivityof the GroundBIRD experiment to τ , based on simu-lated maps with the GroundBIRD configuration. A jointmeasurement with QUIJOTE, which has six frequencybands between 11 and 40 GHz, will provide better sys-tematic error control for the foregrounds. We also esti-mate the sensitivity of this combined data analysis. In section 2, we describe the GroundBIRD and QUI-JOTE experiments; in section 3, we describe our pro-cess to prepare the data; in section 4, we describe theforeground cleaning; in section 5, we describe our pixel-likelihood method; in section 6, we discuss the result ofthe analysis; and in section 7, we present our conclu-sions. THE GROUNDBIRD AND QUIJOTETELESCOPESGroundBIRD (Ground-based Background Image Ra-diation Detector) is a ground-based experiment thatmeasures the polarization of the CMB radiation. It willcover about 45% of the sky from the Teide Observatory(28 ◦ (cid:48) N and 16 ◦ (cid:48) W in the northern hemisphere,2400 m above mean sea level) in Tenerife, at two fre-quencies (145 and 220 GHz). This large sky coverageis obtained by an off-axis rotation of the telescope, asfast as 20 revolutions per minute (rpm), with the aimof minimizing impact from the instrument and atmo-spheric 1 /f noise. Here, we refer to f as the frequencyin the power spectral density of the time-ordered data(TOD). The GroundBIRD telescope also has cryogenicmirrors in a crossed-Dragone optical set-up at a tem-perature of 4 K to minimize unwanted radiation fromthe optics reaching the photon sensors (Tajima et al.2012). GroundBIRD’s photon sensors are kinetic in-ductance detectors (KIDs) (Day et al. 2003), and weplan to use six modules with 138 and one module with23 polarization-sensitive detectors for 145 and 220 GHz,respectively. The GroundBIRD telescope will also be atestbed for the newly adopted technologies: high speedrotation scanning, cryogenic mirror, and KIDs to deter-mine how well they will work in future experiments.The QUIJOTE (Q-U-I JOint Tenerife) is a polariza-tion experiment (Rubi˜no-Mart´ın et al. 2012), also lo-cated at Teide Observatory, consisting of two 2.3-mCross-Dragone telescopes which are equipped with threeradiometer-based instruments: the MFI (multifrequencyinstrument, 11, 13, 17 and 19 GHz), the TGI (thirty-gigahertz instrument, 30 GHz), and the FGI (forty-gigahertz instrument, 40 GHz). By observing at eleva-tions down to 30 deg, QUIJOTE achieves a sky coverageof up to f sky ∼ GENERATION AND RECONSTRUCTION OFMAPSIn the experiment, the TOD correlates with the in-tensities of the sky signal given the direction of the tele-scope and the polarization antenna directions. With theTOD, we construct the sky maps for different frequen-cies through a map-making process. For our simulationstudy, however, we first simulate the template sky mapsof different frequencies. Our TOD is then obtained bysimulating observations with these template sky maps.We obtain our final maps through map-making from fur-ther procedures described below.3.1.
CMB
We use the
CAMB (Lewis et al. 2000) software for gen-erating the theoretical CMB angular power spectra. Allthe cosmological parameters are taken from the latestPlanck result (Planck Collaboration 2020b) unless oth-erwise stated. The maps are synthesized using SYNFAST in HEALPix (G´orski et al. 2005) software with the par-tition parameter N side = 1024 in HEALPix , which cor-responds to an angular resolution of 3.4 arcmin. Fig-ure 1 shows the synthesized CMB polarization maps.The Stokes parameters of linear polarization are shownas Q and U . The input spectra for the map synthe-sis are generated with τ = 0 .
05. The synthesized mapsare convolved with a Gaussian beam with 0 . ◦ FWHM(full width at half maximum), which is the designedbeamwidth of the 145 GHz detectors of the Ground-BIRD telescope. The beam width of the 145 GHz de-tectors is wider than that of the 220 GHz detectors, sowe choose to smooth both maps with the beam width of0 . ◦ FWHM to have a common resolution.3.2.
Foregrounds
The foregrounds are generated with the public versionof the Python Sky Model (
PySM ) (Thorne et al. 2017). The polarization maps include only the synchrotron anddust foregrounds. Other foreground components such asfree–free or anomalous microwave emission componentsare mostly unpolarized (G´enova-Santos et al. 2015). TheWMAP 23 GHz and the Planck 353 GHz polarizationmaps, shown in Fig. 2 (a), (b), (c), and (d), are usedas the synchrotron and dust templates (Thorne et al.2017). The foreground maps for a given frequency are https://camb.info/ http://healpix.sf.net https://github.com/bthorne93/PySM public obtained by extrapolating these template maps. Forboth of the foreground components, spatially-varyingspectral indices, shown in Fig. 2 (e) and (f), are used toscale the templates of the components. The foregroundmaps obtained for GroundBIRD frequencies are shownin Fig. 3 for each component for each frequency band.Here N side = 1024, and the Galactic coordinate systemis used. Note that the temperature units are the CMBtemperatures, µ K CMB , for all the maps. As shown inFig. 3, the dust component is dominant for Ground-BIRD frequencies.In order to improve the foreground removal further, wealso simulate other frequency maps, for the QUIJOTEfrequencies 11, 13, 17, 19, 30, and 40 GHz. These willbe described in the foreground cleaning stage.3.3.
TOD generation
We simulate the TOD based on the observationmethod of the GroundBIRD telescope, which scans thesky with a continuous azimuthal rotation at a fixed ele-vation. The CMB and foreground maps that are gener-ated as in Fig. 1 and 3 are used for the input to the TODsimulation. The lines of sight and the polarimeter direc-tions on the sky map are computed using rotation andthe lines of sight of the detectors. We obtain the linesof sight and parallactic angles of the detectors with asimulation of the optics of the telescope. We use a com-mercial tool, LightTools (Synopsis 2017), for ray opticssimulation.We construct two rotational matrices to compute theobservation points in celestial coordinates. One ma-trix accounts for the boresight angle, elevation, and az-imuthal rotation of the telescope. The other one ac-counts for the location of the observing site and therotation of the Earth. As a net result, we obtain theobservation points in the equatorial coordinate system.For each step of the computation, we use the routinesin healpy (Zonca et al. 2019) and astropy (AstropyCollaboration et al. 2013, 2018).We set the elevation of the telescope at 70 ◦ , and therotation speed at 20 rpm with a data sampling rate of1000 samples per second. The angular distance betweentwo consecutive samples is 2.4 arcmin on the sky, whichis smaller than the pixel size of the template maps. Itdoes not cause any sizable effect because our beamwidthis much larger than the pixel size. The TOD is expressedin µ K CMB . 3.4.
Noise
The noise in TOD contains white and 1 /f noise. Thewhite noise is a frequency-independent term due to ran-dom fluctuation. The 1 /f noise, however, is a frequency-dependent term that is higher at low frequencies. It Q -1 1 µ K CMB (a) Simulated CMB Q U -1 1 µ K CMB (b) Simulated CMB U Figure 1.
Simulated CMB polarization maps: (a) and (b) show the Q and U maps, respectively. The temperature unit, µ K CMB , is the CMB temperature or thermodynamic temperature. These maps are used as the template for CMB TOD. Here N side = 1024 and the Galactic coordinate system are used. All the maps are smoothed with a beamwidth of 0 . ◦ . contains any low-frequency drifts due to detector condi-tion changes or fluctuation of atmospheric radiation. Forthis study, we assume the knee frequency, the frequencywhere the 1 /f noise is equal to the white noise, to be0.1 Hz. Note that we also assume that the exponentof the 1 /f noise component is equal to unity. We esti-mate our noise level by considering the KID response,the optical efficiency, and the atmospheric emission, thenoise equivalent power (NEP) to be given by (Zmuidzi-nas 2012; Flanigan et al. 2016)NEP = (cid:115) hνP rad (1 + η opt η em ¯ n bb ) + 4∆ P rad /η pb η opt , (1)where h is the Planck constant, P rad is the radiationpower reaching the telescope window at a given elevationangle, η opt is the optical efficiency of our optical system, η em is the sky emissivity, ¯ n bb is the mean photon num-ber emitted from blackbody, ∆ is the gap energy, and η pb is the Cooper pair breaking efficiency. Here ν is thefrequency of the incoming radiation. This calculationassumes the atmosphere to be a blackbody source. Thenoise equivalent temperature (NET) in antenna temper-ature is then given by NET = √ NEP k ∆ ν , where k is theBoltzmann constant and ∆ ν the frequency band width.A conservative estimation for NET values of the de-tector and atmospheric noise for the GroundBIRD is1000 µ K √ s for 145 GHz and 2782 µ K √ s for 220 GHz in µ K CMB . If we assume the three-year observation time,the numbers of detectors to be 138 for 145 GHz and23 for 220 GHz, 52% and 45% sky coverage, respec-tively, and 70% observation efficiency, then the meanobservation times are 118 s / arcmin for 145 GHz and22 . / arcmin for 220 GHz. These assumptions lead toexpected pixel noise levels for Q and U (Wu et al. 2014)of 131 µ K arcmin for 145 GHz and 826 µ K arcmin for220 GHz. For the polarization maps, we need to multi- ply these values by √
2. We use these pixel noise levelsto generate uniform white noise maps with
SYNFAST . Wecan also make noise maps by map-making from noiseTOD. For this study we choose the former method togenerate multiple noise realizations at low cost.3.5.
Map-making
The sky maps are constructed from the TOD by themap-making process. The 1 /f component of the TODneeds to be removed before map-making otherwise ringpatterns from the combination of the instrumental vari-ations, change in the atmospheric condition, and thescan strategy will remain in the maps. The process ofremoving 1 /f noise is referred to as destriping. For de-striping and map-making, we use MADAM (Keih¨anenet al. 2005, 2010).Figure 4 shows the result of map-making. Since theGroundBIRD telescope observes the same sky area everyday, we use one-day TOD for map-making to economizecomputing resources and time. Accordingly, the map-making residuals and the various types of noise are sup-pressed by assuming three-year observations. We per-form the map-making for the six modules for 145 GHzand one module for 220 GHz individually. The mapsfor 145 GHz are combined by averaging the individualmaps, weighted with the observation time on each pixel.The resulting maps are also pixelized with N side = 1024.There are ill-defined pixels at the edges of the observedregions due to a lack of polarization angle informationat such pixels. To compute the T , Q , and U from TODwe need at least two observations at different parallac-tic angles to get the correct polarization angle. Theill-defined pixels have antenna directions that are notdistributed over a wide range of angles and have toogreat an uncertainty to obtain correct polarization com-ponents. We exclude such pixels from our analysis, andthis affects pixels within 0 . ◦ of the map edge. Q -100 100 µ K CMB (a) Synchrotron template at 23 GHz, Q U -100 100 µ K CMB (b) Synchrotron template at 23 GHz, U Q -100 100 µ K CMB (c) Dust template at 353 GHz, Q U -100 100 µ K CMB (d) Dust template at 353 GHz, U -3.3 -2.7 (e) Spectral index map for synchrotron (f) Spectral index map for dust Figure 2.
Template maps for the foreground simulations. The top two-row plots show polarization Q and U maps for thesynchrotron and dust templates. The bottom-row plots show the spectral index maps for synchrotron and dust components.Here N side = 1024, and the Galactic coordinate system is used. The units for the template maps are µ K CMB . After the destriping, there are tiny residuals from the1 /f noise. These residuals leave unwanted non-zero ringpattern originating from the scanning pattern on themaps. The residuals from the CMB and foregrounds areobtained by making the differences between the compo-nent maps from the map-making and the TOD templatemaps. For the 1 /f noise, the residual is made fromthe TOD with only 1 /f noise included. The root meansquare (RMS) values of polarization intensity maps ofthe CMB, foregrounds, and 1 /f residuals are 0.03, 0.07,and 3.4 µ K arcmin, respectively, in the 145 GHz map.We also check potential bias due to the residuals fromthe map-making. 3.6.
Sky coverage and mask for the analysis
The observing strategy sets the sky coverage ofGroundBIRD. Our simulation gives the sky coverages of52% for 145 GHz and 45% for 220 GHz. Note that thesky coverage for a given module depends on the locationof the module on the focal plane because our telescoperotates at a fixed elevation. We use the 220 GHz skycoverage mask for the analysis, which masks the outsideof a 50 ◦ wide ring area, as shown in Fig. 5 (a). In Table1, we summarize the number of detectors, sky coverage,and the pixel noise level of the GroundBIRD telescope. Q -2.5 2.5 µ K CMB (a) Synchrotron at 145 GHz, Q U -2.5 2.5 µ K CMB (b) Synchrotron at 145 GHz, U Q -2.5 2.5 µ K CMB (c) Dust at 145 GHz, Q U -2.5 2.5 µ K CMB (d) Dust at 145 GHz, U Q -2.5 2.5 µ K CMB (e) Synchrotron at 220 GHz, Q U -2.5 2.5 µ K CMB (f) Synchrotron at 220 GHz, U Q -2.5 2.5 µ K CMB (g) Dust at 220 GHz, Q U -2.5 2.5 µ K CMB (h) Dust at 220 GHz, U Figure 3.
A set of simulated foreground maps. The Q and U polarization maps for synchrotron at 145 GHz, dust at 145GHz, synchrotron at 220 GHz, and dust at 220 GHz are shown. Here N side = 1024, and the Galactic coordinate system is used.The units are µ K CMB . All the maps are smoothed with a beamwidth of 0 . ◦ , as for CMB maps. Q -2.5 2.5 µ K CMB (a) Frequency map at 145 GHz, Q U -2.5 2.5 µ K CMB (b) Frequency map at 145 GHz, U Q -2.5 2.5 µ K CMB (c) Frequency map at 220 GHz, Q U -2.5 2.5 µ K CMB (d) Frequency map at 220 GHz, U Figure 4.
Results of map-making for the one-day TOD are displayed. The frequency maps at 145 GHz for the Q and U polarizations are shown in (a) and (b); and the maps at 220 GHz are shown in (c) and (d). The equatorial coordinate systemis used. The gray region is not observable by the GroundBIRD telescope. (a) GroundBIRD sky coverage (b) Galactic plane mask Figure 5.
The GroundBIRD sky coverage and the Galactic plane mask are shown in (a) and (b), respectively. The equatorialcoordinate system is used for visualization. N side = 1024 is assumed. The gray regions are masked. Table 1.
A summary of the number of detectors, sky cov-erage, mean observation time, and the pixel noise level forthe polarization maps of the GroundBIRD telescope. Anobservation efficiency of 0.7 is assumed.145 GHz 220 GHzNumber of detectors 138 23Sky coverage 0.54 0.45Mean observation time (s / arcmin ) 117 23Pixel noise level ( µ K arcmin) 131 826
To mask the Galactic plane, we apply the Galacticmask for polarization with a coverage of 52% usedin the Planck analysis (Planck Collaboration 2016), asshown in Fig. 5 (b). The combination of the Ground-BIRD sky coverage and the Galactic mask results in acoverage of 22.8% of the full sky. FOREGROUND CLEANINGWe use the internal linear combination (ILC) methodto remove the foreground components (Eriksen et al.2004). A frequency map m ν from observationsat frequency ν contains CMB ( m CMB ), foreground( m ν { sync , dust } ), and noise ( m ν noise ) components. The mapmay be expressed in the form m ν = m CMB + m ν sync + m ν dust + m ν noise , (2)where the units of all the maps are µ K CMB . The fore-ground components have different frequency dependen-cies, whereas the CMB component is constant over fre-quencies.We find the ILC coefficients for the linear combinationof observed frequency maps that minimizes the varianceof the combined map ( m combined ), as m combined = (cid:88) ν c ν m ν , (3)where c ν is the ILC coefficient given the frequency ν .The linear combination of frequency maps minimizesthe foreground components, assuming the sum of thecoefficients is 1, or (cid:80) ν c ν = 1. LIKELIHOOD FITSWe use the exact likelihood function in real space (oralso called the pixel-based likelihood) defined as L ( C (cid:96) ) = 1 | π M | / exp (cid:18) − m T M − m (cid:19) , (4) COM Mask Likelihood-polarization-143 2048 R2.00.fits fromthe Planck Legacy Archive https://pla.esac.esa.int/pla/ where m is a vector of the temperature map T andand two Stokes parameters Q and U . The pixel covari-ance matrix M is defined by Tegmark & Oliveria-Costa(2001) M = (cid:104) T i T j (cid:105) (cid:104) T i Q j (cid:105) (cid:104) T i U j (cid:105)(cid:104) T i Q j (cid:105) (cid:104) Q i Q j (cid:105) (cid:104) Q i U j (cid:105)(cid:104) T i U j (cid:105) (cid:104) Q i U j (cid:105) (cid:104) U i U j (cid:105) , (5)where indices i and j run over pixels. Each T i is a linearcombination of normal spherical harmonics while Q i and U i are linear combinations of spin-2 spherical harmonics(Tegmark & Oliveria-Costa 2001). The operation of (cid:104) (cid:105) is the average over our ensemble. The ensemble is a setof possible realizations of T , Q , and U maps when theCMB maps have a multivariate Gaussian distribution.Since only the polarization maps are used as input datain our analysis, the covariances for polarizations QQ , QU , U Q , and
U U are used. We compute the covari-ance matrix analytically using the formulae in Zaldar-riaga (1998). The theoretical angular spectra obtainedby
CAMB are used in this calculation.Note that use of the real-space likelihood approach en-sures mathematical rigor but is extremely costly from acomputational point of view when N side becomes largefor the inversion of a large covariance matrix. Forthat reason, the real-space likelihood is more feasiblefor studying large angular scales, which is the case forGroundBIRD. In an alternative approach, the likelihoodin the harmonic space produces unwanted E → B mix-ing for partial sky maps. This mixing can be removedby modifying the likelihood (Chon et al. 2004; Smith &Zaldarriaga 2007). However, we choose to use the exactreal-space likelihood to study directly the sensitivity ofthe optical depth to reionization with GroundBIRD inorder to simplify our study. For that reason, we degradeall the input maps to N side = 8, which corresponds themaximum multipole moment, (cid:96) max = 23 and an angularresolution of 7 . ◦ . We therefore need to compute thecovariance matrix for (cid:96) ≤ CAMB creates a time-consuming bottleneck at each it-eration of the pixel-based likelihood fit. We use a lin-ear interpolation algorithm to improve the computationspeed, which is motivated by Watts et al. (2018). A setof the pre-computed angular power spectra is evaluatedby sweeping the parameter space with
CAMB . The angu-lar power spectra for specific parameters are obtainedby linear interpolation between the pre-computed spec-tra. This method is proper for less than four parametersbecause the time consumption and the storage space forthe pre-computed data are exponentially proportional tothe number of parameters. We make the pre-computeddata set for τ by sampling 100 points. This method im-proves the calculation speed by a factor of 100 with onlya negligible difference from the result obtained by CAMB .We perform an ensemble test for 10 000 realizationsof the CMB and noise to examine possible fit bias andestimate the expected uncertainty on τ . For each re-alization, we generate a set of input maps. If we fol-low the whole simulation procedure for every realization,the TOD simulation and the map-making will take sev-eral years and require inaccessible computing resources.We therefore simplify the data simulations by skippingthe time-consuming steps. For the ensemble test inputmaps, we make CMB and noise maps for every realiza-tion. We make the CMB maps directly by SYNFAST aswe did for the TOD template. We also use
SYNFAST tomake the white noise maps for each frequency with theassumed pixel noise levels. The foregrounds are con-stant over the realizations and need to be made onlyonce. The input map combines the CMB map and thewhite noise map for the realization, and the fixed fore-ground map. Additionally, we include the map-makingresiduals for the 145 and 220 GHz maps to evaluate thesystematic effect of the 1 /f noise. With the simulatedmaps, we remove the foreground components with theILC method and then perform the maximum likelihoodfit. The fit parameters are τ and the total noise level σ p . The total noise level is the pixel noise level of thecombined polarization map after foreground cleaning.We verify the above technique using a simulated fre-quency maps for the Planck frequencies. We simulatethe maps with our simulation pipeline for all of thePlanck frequency bands (30, 44, 70, 100, 145, 217, and353 GHz). The white noises are generated with the noiselevels obtained from the published Planck frequencymaps. The uncertainty of τ for the simulated Planckdata is found as σ τ = 0 . τ = 0 . +0 . − . .
04 0 .
08 0 .
12 0 . τ σ p
420 450 480 510 540 σ p σ p = +18 − Figure 6.
The result of the ensemble test with the two-frequency ILC. The simulated GroundBIRD frequency mapsare used. The contours for 1 σ and 2 σ are drawn, and thevertical solid lines in the τ histogram and contours displaythe input value of τ . The dashed curve in the τ histogramis the best-fit Gaussian curve of the histogram without thepeak at the lower boundary. The estimated τ is 0 . ± . ± µ K arcmin. small discrepancy between our value and the publishedvalue arises because the analysis methods are differentand our simulated maps are based on the aforementionedassumptions. RESULTSFirst, we show the result with the GroundBIRD sim-ulations only and then the result with three frequencymaps including a synchrotron template. We also showthe result with the GroundBIRD and QUIJOTE simu-lations here. The cases without foregrounds or withoutthe map-making residuals are summarized in AppendixA. 6.1.
GroundBIRD only
We first fit the GroundBIRD maps. The ensemble testis performed with 10 000 samples. For each realization,we remove the foreground first and then estimate τ froma pixel-based likelihood fit.The ILC with the two GroundBIRD maps leaves alarge foreground residual mainly from the synchrotroncomponent. This residual prevents the correct estima-tion of τ . To solve this problem, we add a mask toexclude the sky region dominated by large synchrotronresiduals from the input map for the likelihood fit. This0mask is obtained by applying a threshold to the fore-ground cleaned map. This extra mask excludes an ad-ditional 5% of the sky and enables estimation of theunbiased τ . After applying this mask, the ILC coeffi-cients for 145 and 220 GHz are c = 1 . ± .
004 and c = − . ± . τ versus σ p and two his-tograms of them are shown. The estimated τ is foundto be 0 . ± .
031 when the input τ is 0 .
05. Thenoise level of the foreground cleaned map is found tobe 482 ± µ K arcmin, to which the white noise andmap-making residuals are contributing.A small peak near the lower boundary of the τ his-togram in Fig. 6 originates from our high noise level.Fluctuation in an ensemble, which simulates the statis-tical uncertainty of τ , sometimes pushes τ down to itslower limit that is 0, making such a peak. This may in-troduce small bias in the τ estimate. One way to avoidit is to allow the negative value of τ in the likelihood.However, since it is not only unclear how to correctlyconstruct the pixel likelihood with the unphysical nega-tive τ values from the CAMB , but is also not of the mainissue of this paper, we limit τ to non-negative values.Even after applying the extra mask for the syn-chrotron, the dominant foreground residual at this stageoriginates from the synchrotron foreground. We findthat the non-zero foreground residuals introduce a biasin τ , which tends to be overestimated by around 10%but is still within 1 σ of the statistical uncertainty.To remove the synchrotron foreground further, we addthe simulated QUIJOTE 30 GHz map as a synchrotrontemplate. The ILC coefficients for the three-frequencycase are c = 1 . ± . c = − . ± . c = − . ± . c ) is suppressed because this map is scaled with thespectral index of the synchrotron foreground. There-fore, the synchrotron template removes the synchrotronforeground without significantly changing the map. Theestimated τ is 0 . ± .
030 and the noise level is foundto be 483 ± µ K arcmin.Note that the map-making residuals become prac-tically an additional white noise source because thering patterns are diluted as the maps are degraded to N side = 8. We find that the existence of the map-makingresiduals increases the noise level effectively by 13%, andthat the uncertainty of the τ is increased by around 5%accordingly. We discuss the results with and withoutmap-making residuals in Appendix A.6.2. GroundBIRD and QUIJOTE τ = 0 . +0 . − . .
025 0 .
050 0 .
075 0 . τ σ p
120 130 140 150 160 σ p σ p = . +5 . − . Figure 7.
The result of the ensemble test with the eight-frequency ILC. The simulated GroundBIRD maps for { } GHz and the simulated QUIJOTE maps for {
11, 13, 17,19, 30, 40 } GHz are used. The contours for 1 σ and 2 σ aredrawn, and the vertical solid lines in τ histogram and con-tours display the input value of τ . The dashed curve in the τ histogram is the best-fit Gaussian curve of the histogramwithout the peak at the lower boundary. The estimated τ is0 . ± . We estimate τ again by including the simulated QUI-JOTE frequency maps at 11, 13, 17, 19, 30, and 40GHz in our analysis. We generate the foreground andnoise maps for QUIJOTE in the same way as for theGroundBIRD maps. The pixel noise levels of the po-larization maps and the resulting ILC coefficients aresummarized in Table 2. In this case, the map-makingresiduals are included only in the GroundBIRD maps,and the ILC coefficients of the GroundBIRD maps arerelatively smaller than in previous cases. The effect ofthe map-making residuals therefore becomes small.The result of the eight-frequency case is shown in Fig.7. The estimated τ is found to be 0 . ± .
011 withthe total noise level of 141 . ± . µ K arcmin. The noiselevel is reduced by 60% compared to the three-frequencyresult because the contribution of the 220 GHz map witha high noise level is suppressed dominantly by adding 30and 40 GHz maps. Accordingly, the sensitivity on the τ is improved by 50% compared to the three-frequencycase. 6.3. Limitations of current analysis
We include several sources of systematic errors for re-alistic simulations: atmospheric noise, detector calibra-1
Table 2.
Noise levels of polarization maps and ILC coeffi-cients for the eight-frequency result are summarized.Frequency Noise level ILC coefficients(GHz) ( µ K arcmin)11 3600 − ± − ± − ± − ± − ± ± ± − ± tion, and map-making residuals. However, there are po-tential sources of systematic error that are not includedin this study.We assume that our fast-scanning technique, in com-bination with the sparse wire-grid calibration removesmost of the atmospheric noise. Therefore, we do notconsider the effect of frequency-dependent atmosphericfluctuations (Lay & Halverson 2000) in this work. Thesystematic effects from the optical system are incom-plete. For example, the uncertainty of the polarizationangle and imperfect focusing due to the aberration ofthe reflectors are not included in our analysis. We willaddress these in a future publication. CONCLUSIONSWe have performed estimates of the optical depth toreionization ( τ ) with simulated data for the Ground-BIRD experiment. We started the simulation from TODwith the inclusion of 1 /f noise in our simulation. Theinput maps cover 45% of the sky (22% after maskingthe Galactic plane). These maps included the CMB re-alization, foregrounds, expected noise, and map-makingresiduals. We assumed a three-year observation with161 detectors in total. We cleaned the polarization fore-grounds using the ILC method, which requires at leastthree frequency maps to remove two foreground compo- nents; namely, synchrotron and dust foregrounds. Wealso added the simulated QUIJOTE maps for betterforeground cleaning. To estimate the τ values, we usedour pixel-based likelihood.The estimated τ is 0 . ± .
031 with GroundBIRDmaps only and 0 . ± .
029 with a synchrotron tem-plate included, where the input value is 0.05. AddingQUIJOTE frequency maps to the GroundBIRD mapsfurther removed the foregrounds and reduced the noiselevel. The estimated τ with eight-frequency (11, 13,17, 19, 30, 40, 145, and 220 GHz) ILC is found to be0 . ± . τ estimate. The uncertainty inthe τ estimate is constrained mainly by the pixel noiselevel of the foreground cleaned map. The map-makingresiduals increase the noise level and degrade the sen-sitivity to τ . In our case, τ sensitivity is degraded byabout 5%, which depends on the observing strategy anddetector set-up.ACKNOWLEDGMENTSThe authors would like to thank the Instituto de As-trof´ısica de Canarias for their support and hospital-ity during the installation of the GroundBIRD tele-scope at Teide Observatory. This work was par-tially supported by the National Research Foundationof Korea (NRF) grant funded by the Korea govern-ment (MSIT) (No. NRF-2017R1A2B3001968). Thiswork was also supported by the JSPS Grant NumberJP15H05743, JP18H05539, and Heiwa Nakajima Foun-dation. The computing network was partially supportedfrom KISTI/KREONET. Some of the results in this pa-per have been derived using the HEALPix package.
Software: astropy (Astropy Collaboration et al.2013, 2018),
CAMB (Lewis et al. 2000), corner (Foreman-Mackey 2016)
HEALPix (G´orski et al. 2005), healpy (Zonca et al. 2019), iminuit (iminuit team 2018), python (Python Core Team 2015),
PySM (Thorne et al.2017)APPENDIX A. LIKELIHOOD FIT RESULTS FOR OTHER SET-UPSIn this section, we summarize the results of ensemble tests for several foreground set-ups. The estimates of τ andtotal noise level σ p are summarized in Table 3. The simplest case is the “No foregrounds” case, where it is assumedthat the foreground is cleaned completely. The input map contains CMB and noise only, and the same noise level asthe “Two-frequency ILC” case is assumed. The estimated τ is 0 . ± .
032 when the input value is 0 .
05, with an2
Table 3.
Comparison of estimates of τ and the pixel noise level of the combined map σ p for different foreground cleaningmethods. For the “Fisher matrix” case, the same noise level as the two-frequency map case is assumed and the τ uncertainty isestimated from the Fisher matrix approach. Other estimates are from the pixel-based likelihood approach. The “No foregrounds”is the case with an input map combining only the CMB and noise. The same noise with “Two-frequency ILC” case is assumed.The “Two-frequency ILC” is the case performing the foreground cleaning with only two frequency maps of GroundBIRD. For the“Three-frequency ILC” case, a 30 GHz map is added to the two-frequency case, as a synchrotron template. The“Eight-frequencyILC” is the case with eight frequency maps, including all the QUIJOTE frequency maps. The cases with foregrounds are testedwith and without the map-making residuals.Foreground cleaning method τ σ p ( µ K arcmin)Fisher matrix 0.050 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± estimate of the white noise level of 426 ± µ K arcmin. This uncertainty is 15% greater than the estimate by Fishermatrix, σ τ = 0 . τ is 0 . ± . σ p is 428 ± µ K arcmin without the map-making residuals. With the map-makingresiduals, the estimated τ is 0 . ± . σ p is 482 ± µ K arcmin. The map-making residuals increase the noiselevel by 13% and also increase the τ uncertainty by 6%.In the three-frequency cases, we expect the synchrotron to be removed effectively by adding a 30 GHz map. Withoutthe map-making residuals, the estimated τ is 0 . ± . ± µ K arcmin.The τ uncertainty and the white noise level are at the same level as the two-frequency map case, but the τ valuehas a smaller bias because of the smaller foreground residuals. With the map-making residuals, the estimated τ is0 . ± .
030 which has slightly larger uncertainty compared to the value without map-making residuals. The noiselevel is almost same with the two-frequency case.The eight-frequency map case uses all six QUIJOTE frequency maps with the GroundBIRD frequency maps. Themap-making residuals are included only for the GroundBIRD frequency maps because we are not doing map-makingfor the QUIJOTE maps. The estimated value of τ is 0 . ± .
011 and noise level is 139 . ± . µ K arcmin without themap-making residuals and τ is 0 . ± .
011 and noise level is 141 . ± . µ K arcmin with the map-making residuals.In this case, the map-making residuals slightly increase the noise level, but have a negligible effect on the estimation of τ because the contribution of the GroundBIRD 220 GHz map, which has a large white noise level and the map-makingresiduals, becomes small by adding the QUIJOTE maps.REFERENCES Ade, P., Aguirre, J., Ahmed, Z., et al. 2019, Journal ofCosmology and Astroparticle Physics, 2019, 056Albrecht, A., & Steinhardt, P. J. 1982, Phys. Rev. Lett., 48,1220Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,et al. 2013, A&A, 558, A33Astropy Collaboration, Price-Whelan, A. M., SipHocz,B. M., et al. 2018, aj, 156, 123Chon, G., Challinor, A., Prunet, S., Hivon, E., & Szapudi,I. 2004, MNRAS, 350, 914 Day, P. K., LeDuc, H. G., Mazin, B. A., Vayonakis, A., &Zmuidzinas, J. 2003, Nature, 425, 817Eriksen, H. K., Banday, A. J., Gorski, K. M., & Lilje, P. B.2004, The Astrophysical Journal, 612, 633Flanigan, D., McCarrick, H., Jones, G., et al. 2016, AppliedPhysics Letters, 108, 083504Foreman-Mackey, D. 2016, The Journal of Open SourceSoftware, 1, 243