Exploring universe with neutral hydrogen + machine learning

1. Exploring early universe with neutral hydrogen + machine learning @東京⼤学宇宙理論（2022/5/16） ©Aman Chokshi Hayato Shimabukuro(島袋隼⼠) （Yunnan university, Nagoya university） Aman Chokshi 1

2. •Born in Okinawa •Ph.D from Nagoya university(2016) •Postdoc at Paris observatory(2016-2018) About me •Postdoc at Tsinghua University(2018-2019) 2

3. •Born in Okinawa •Ph.D from Nagoya university(2016) •Postdoc at Paris observatory(2016-2018) About me •Postdoc at Tsinghua University(2018-2019) •Yunnan university（2019ー） 2

4. How is Yunnan province? 3

5. How is Yunnan province? High altitude (~2000m) 3

6. How is Yunnan province? High altitude (~2000m) Some old cities 3

7. How is Yunnan province? High altitude (~2000m) Some old cities Mushroom ! 3

8. Outline • Introduction (4 pages) • Basics of 21cm line (7 pages) • Current 21cm cosmology status and future ( 4 pages) • 21cm signal analysis with machine learning (17 pages) • Summary 4

9. Introduction 5

10. The history of the universe ©NAOJ Dark Ages・・・No luminous object exists. Epoch of Reionization(EoR)・・・UV photons by luminous objects ionize neutral hydrogen in the IGM (z~6-15). Cosmic Dawn・・・First stars and galaxies form (z~20-30). 6

11. The history of the universe ©NAOJ Dark Ages・・・No luminous object exists. Epoch of Reionization(EoR)・・・UV photons by luminous objects ionize neutral hydrogen in the IGM (z~6-15). Cosmic Dawn・・・First stars and galaxies form (z~20-30). 6

12. （C)Kenji Hasegawa（Nagoya University） Credit: M. Alvarez, R. Kaehler and T.Abel

13. （C)Kenji Hasegawa（Nagoya University） Credit: M. Alvarez, R. Kaehler and T.Abel

14. (Naidu et al 2020) Current observations tell us Current observations such as Quasar absorption lines and Lyman-alpha emitter galaxies constraint on average neutral(ionized) hydrogen fraction (Global history). 8

15. (Naidu et al 2020) Current observations tell us Current observations such as Quasar absorption lines and Lyman-alpha emitter galaxies constraint on average neutral(ionized) hydrogen fraction (Global history). Current observations tell us only global history. 8

16. We want to know EoR much more (ex) •EoR theory •Morphology and topology of ionized bubble •Ionizing sources •Relation to galaxy formation and evolution etc… 9

17. We want to know EoR much more We should observe IGM at the EoR directly ! (ex) •EoR theory •Morphology and topology of ionized bubble •Ionizing sources •Relation to galaxy formation and evolution etc… 9

18. We want to know EoR much more We should observe IGM at the EoR directly ! (ex) •EoR theory •Morphology and topology of ionized bubble •Ionizing sources •Relation to galaxy formation and evolution etc… ✕ Synergy with galaxy observation by ALMA, JWST, Subaru 9

21. Basics of 21cm line 10

22. 21cm line •21cm line radiation : Neutral hydrogen atom in IGM emits the radiation due to the hyperfine structure. z=6 → 1.5m or 202 MHz z=20 → 4.4m or 68MHz Radio wavelength. Proton Electron 21cm line emission(1.4GHz) (Neutral) hydrogen atom is good tracer for IGM. 11 signlet Triplet

23. Spin temperature n"" n"# = 3 exp ✓ h⌫21cm kTS ◆ Key quantity in 21cm line physics T 1 S = T 1 CMB + xcT 1 K + x↵T 1 c 1 + xc + x↵ Spin temperature is determined by •interaction with CMB photons •collision with hydrogen atoms •interaction with Ly-alpha photons (TCMB) (TK, xc) (Tc ⇠ TK, x↵) Properties of X-ray sources (e.g. spectral energy distribution (SED)) Relevant astrophysics Properties of first stars (e.g. Initial mass function) 12 Spin temperature

24. Mesinger et al 2010 heating WF e ff ect Wouthuysen-Field(WF) effect Spin temperature couples to IGM kinetic temperature via Ly-alpha photons from first stars. Thermal history X-ray heating X-ray photons drastically heat kinetic temperature of the IGM Spin temperature Kinetic temperature CMB temperature 13

25. 21cm line signal Red : cosmology Blue : astrophysics Tb = TS T 1 + z (1 exp(⌧⌫)) ⇠ 27xH(1 + m) ✓ H dvr/dr + H ◆ ✓ 1 T TS ◆ ✓ 1 + z 10 0.15 ⌦mh2 ◆1/2 ✓ ⌦bh2 0.023 ◆ [mK] Brightness temperature *We actually observe brightness temperature We can map the distribution of HI in the IGM with 21cm line 14

26. 21cm line signal Red : cosmology Blue : astrophysics Tb = TS T 1 + z (1 exp(⌧⌫)) ⇠ 27xH(1 + m) ✓ H dvr/dr + H ◆ ✓ 1 T TS ◆ ✓ 1 + z 10 0.15 ⌦mh2 ◆1/2 ✓ ⌦bh2 0.023 ◆ [mK] Brightness temperature *We actually observe brightness temperature We can map the distribution of HI in the IGM with 21cm line 14

27. 21cm power spectrum (PS) : Scale dependence Pober et al (2014) EoR X-ray heating WF effect z Redshift dependence 21cm power spectrum h Tb(k) Tb(k 0 )i = (2⇡)3 (k + k 0 )P21 We first try to detect the 21cm line signal statistically with ongoing telescopes. 15

28. Current upper limits on 21cm PS Current 21cm experiments put upper limit of the 21cm line power spectrum 2-3 order of magnitude higher than theoretical expectation. Challenges: ionosphere, RFI, foreground, etc Shimabukuro et al 2022b 16

29. 21cm line global signal Global signal has characteristic peaks and troughs according to key epochs Global signal (sky averaged brightness temperature) So far, we introduced 21cm line power spectrum to measure 21cm line signal, global signal is another observable. These behavior inherits the behavior of the spin temperature (and ionization history) 17

30. Current 21cm cosmology status and future project 18

31. EDGES (Bouman et al 2018) Too deep trough Too flat We detected the 21cm line signal? 19

32. EDGES (Bouman et al 2018) Too deep trough Too flat We detected the 21cm line signal? SARAS3 did not detect signal （Singh + 2022, Nature astronomy） 19

33. EDGES (Bouman et al 2018) Too deep trough Too flat We detected the 21cm line signal? SARAS3 did not detect signal （Singh + 2022, Nature astronomy） Very strange result ! Need exotic physics? mis-calibration? unknown systematics? 19

34. Current radio interferometers MWA LOFAR HERA GMRT Radio interferometer •Array of radio telescope antennas •Measure time delay between antennas •Work together as a single telescope 20

35. 21

36. SKA-Low •Frequency 50-350MHz(z=3~27) & High sensitivity Wide FoV & 22

39. 21cm signal analysis with machine learning 24

40. Artificial Neural Network (ANN) An ANN is a mathematical model of human brain network. ex.) Rumelhart et. al (1986) LeCun et. al (1989) Recently, it has been applied to field of astronomy. 25

41. Artificial Neural Network (ANN) •Training network with training dataset, ANN can approximate any function which associates input and output values. y = f(x) • Applying trained network to unknown data(test data) for prediction. yANN = f(xtest) • ANN consists of input layer, hidden layer and output layer. Each layer has neurons. non linear regression Problem 26

42. •Emulator •parameter estimate •Distinguish EoR sources (e.g) Hassan +2019 •Others (e.g.) Li + 2019, Chardin + 2019, Yoshiura + 2020, Shimabukuro + 2022 (e.g.) Kern + 2017, Schmit + 2018, Aviad + 2020, Bevins + 2021, Bevins+ 2021 (e.g.) Shimabukuro + 2017, Gilet+ 2018, Nicolas +2019, Doussot +2019, Choudhury+ 2020,2021a,b, Zhao+ 2022a,b 21cm study+machine learning 27

43. 1.EoR parameter estimation with ANN Based on Shimabukuro and Semelin 2017 28

44. Statistical challenge in 21cm cosmology (Mesinger 2018) Cosmology CMB map (angular) power spectrum cosmological parameter 21cm 21cm 3D map 21cm power spectrum astrophysical parameter Based on Bayesian inference 29

45. Statistical challenge in 21cm cosmology (Mesinger 2018) Cosmology CMB map (angular) power spectrum cosmological parameter 21cm 21cm 3D map 21cm power spectrum astrophysical parameter Based on Bayesian inference We proposed alternative method. 29

46. Dataset ⇣ : the ionizing efficiency. : the minimum viral temperature of halos producing ionizing photons : the mean free path of ionizing photons through the IGM (Maximum HII bubble size) Tvir Rmfp ~ d = [P(k), ~ ✓] 21cm power spectrum (input) EoR parameter (output) EoR Parameter ✓EoR = f(P21) 30

47. z=11, PS without any noise Reconstructed by 21cmPS at z=11 10 20 30 40 50 60 10 20 30 40 50 60 R mfp,ANN [Mpc] Rmfp,true[Mpc] z=12 10 20 30 40 50 60 10 20 30 40 50 60 ANN true z=12 1 10 100 1 10 100 T vir,ANN [K/10 3 ] Tvir,true[K/103 ] z=12 14 neurons, 100’000 iterations • True value .vs. Reconstructed value •The scatter of is large. Rmfp ⇣ Tvir •Other reconstructed parameters match true one relatively well. Shimabukuro & Semelin (2017) Rmfp 31

48. z=11, PS without any noise Reconstructed by 21cmPS at z=11 10 20 30 40 50 60 10 20 30 40 50 60 R mfp,ANN [Mpc] Rmfp,true[Mpc] z=12 10 20 30 40 50 60 10 20 30 40 50 60 ANN true z=12 1 10 100 1 10 100 T vir,ANN [K/10 3 ] Tvir,true[K/103 ] z=12 14 neurons, 100’000 iterations • True value .vs. Reconstructed value •The scatter of is large. Rmfp ⇣ Tvir •Other reconstructed parameters match true one relatively well. Shimabukuro & Semelin (2017) Rmfp 31

49. z=9, 10, 11. PS with thermal noise and cosmic variance Reconstructed by 21cm PS at z=9,10,11 Rmfp ⇣ Tvir 10 20 30 40 50 60 10 20 30 40 50 60 R mfp,ANN [Mpc] Rmfp,true[Mpc] 10 20 30 40 50 60 10 20 30 40 50 60 ANN true 1 10 100 1 10 100 T vir,ANN [K/10 3 ] Tvir,true[K/10 3 ] Red : z=9,10,11 Blue : z=9 The parameters obtained by the ANN match true values. ANN work well ! 32

50. Emulator EoR parameters 21cmPS ANN MCMC Before : 2.5days on 6 cores After: 4minutes speed up by 3 orders of magnitude (Schmit et al 2018) (input) (output) 33

51. parameters 21cm map ANN (input) (output) Parameter estimate Gillet +2018 34

52. 2.Recovering HII bubble size distribution with ANN Based on Shimabukuro et al 2022 35

53. Bubble size distribution (BSD) ''How large bubbles are distributed ?’' Giri 2019 What can we learn from BSD? Giri et al 2017 •EoR source (galaxy or AGN?) •ionizing efficiency, recombination, radiative feedback. (ex.) 36

54. BSD from 21cm observation Kakiichi et al 2017 IFT 21cm Image BSD Incomplete IFT due to limited number of antenna in interferometer. visibility We do not observe 21cm image directly by radio interferometer! We first observe visibility and perform Inverse Fourier Transformation (IFT) to obtain 21cm image. Then, compute BSD. 37

55. BSD from 21cm PS Kakiichi et al 2017 21cm power spectrum BSD visibility We can directly compute 21cm power spectrum from visibility without Inverse Fourier Transformation. Avoid information loss by incomplete IFT. 38

56. BSD from 21cm PS Kakiichi et al 2017 21cm power spectrum BSD visibility We can directly compute 21cm power spectrum from visibility without Inverse Fourier Transformation. Can we recover BSD from 21cm PS ? Avoid information loss by incomplete IFT. 38

57. 21cm power spectrum Input Output ionised bubble size distribution Our datasets consist of 21cm power spectrum as input data and bubble size distribution as output data. Our strategy We try to recover ionised bubble size distribution from 21cm PS 39

58. Recovered BSD Black: Distribution obtained by 21cm 3D image directly. Red: Distribution obtained by ANN. 40

59. Different stage of reionization 41

60. Effect of thermal noise 42 21cm PS with thermal noises (SKA level) Errors are estimated by 10 realizations thermal noises

61. Accuracy for all test data Relative error between two size distributions at fixed bubble radius for all test data. Good recovery for all test data. 43

62. Reconstruction of HI distribution from LAE map is marked in angles (degrees) and the projected distances (comoving megaparsecs). Fig. 5. Same as Figure 4, but for the LAEs z = 6.6. The large red open squares indicate the LAEs with spatially extended Lyα emission including Himiko (Ouchi et al. 2009a) and CR7 (Sobral et al. 2015). See Shibuya et al. (2017b) for more details. Input : Lyman-alpha emitter galaxies Output : HI distribution Yoshiura,HS +2021 44 cGAN

63. Take home messages of my talk are… •The epoch from the Dark Ages to cosmic reionization is the frontier in the history of the universe. •21cm signal is a promising tool to study this epoch. •SKA will bring us fruitful information on the epoch through Dark Ages to EoR •We proposed a method based on machine learning to analyze the 21cm line signal. 45

64. •"21cm cosmology" (Prithcard & Loeb, astro-ph/1109.6012) Textbook Review paper •"Cosmology at low frequencies" (Furlanetto et al, astro-ph/0608032) •''In the beginning : the first sources of light and the deionization of the universe” R,Bakana & A,Loeb (astro-ph/0010468) References 46

65. Coming soon 47 Shimabukuro et al 2022b, accepted in PASJ

66. bakcup

67. Challenging issue

68. Foreground problem Jelic et al 2008 The 21cm signal is buried under strong foreground ! Remove foreground ? or Avoid (strong)foreground? Santos 2005 ~8 order Dillon et al 2013

69. HII bubble

70. Red: 21cm power spectrum > ANN > bubble size distribution Blue: 21cm power spectrum > MCMC > parameter > bubble size distribution 21cm PS > parameter > BSD

71. Algorithm for calculating bubble size distribution 2. Generating density field 3. Generating ionization field from density field with excursion-set formalism for modeling Reionization 1.Input EoR & cosmological parameters 21cmFAST Roughly speaking, it evaluates whether isolated region is ionized or not (Furlanetto+2004, Zahn+ 2010). 4. Evaluating ionized bubble size distribution (Zahn+2007, Mesinger & Furlanetto 2007, See also Giri+ 2018) •Randomly choose a pixel of ionized region. •Record the distance from that pixel to neutral region along randomly chosen direction. •Repeat Monte Carlo procedure times. 107 Ionized region R Neutral region

72. Introduction & 21cm basic

73. Current observations for EoR •Lyman alpha emitter galaxies（LAE） •Lyman alpha forest Konno et al (2014) (http://pages.astronomy.ua.edu/keel/agn/forest.html) QSO HI cloud >The number of ionizing photons Sensitive to neutral hydrogen fraction

76. n"" n"# = 3 exp ✓ h⌫21cm kTS ◆ The spin temperature is determined by following equilibrium T 1 S = T 1 CMB + xcT 1 K + x↵T 1 c 1 + xc + x↵ de-excitation rate by collision de-excitation rate by UV photons excitation rate by UV photons excitation rate by collision Stimulated by CMB photons Spontaneous de- excitation with Einstein coefficient

77. Wouthuysen Field (WF)effect •The mechanism that couples the spin temperature of neutral hydrogen atom to Lyman-alpha photons(Wouthuysen 1952,Field 1959) •The hyperfine state is changed via 2P state Solid lines : allowed path Dashed lines : not allowed path

78. The impact of IMF on 21cm signal Jones+ 2022, Magg+ 2021 IMF Lower mass dominated IMF causes delayed 21cm signal.

79. 21cm PS with SKA SKA covers wide epoch and range of the 21cm PS !! Redshift evolution Scale dependence z=8.95 z=15.98 Koopmans et al. (2014) Pritchard et al. (2014)

80. EoR parameter with ANN

81. • 1000 EoR models • 48000 training datasets (20% of which is used for validation) • 2000 test datasets • 21cm PS is ranged from k=0.11/Mpc to 1.1/Mpc with 14 bins • 5 hidden layers • 212 neurons at each hidden layer • 2000 iterations Setup

82. Evaluate accuracy: noise We evaluate accuracy of obtained parameters by chi-square. Smaller chi-square means better accuracy. single z As expected, accuracy becomes worse if we add noise to 21cm power spectrum. without noise with noise

83. Evaluate accuracy: redshift We evaluate accuracy of obtained parameters by chi-square. Smaller chi-square means better accuracy. multiple z The accuracy of parameter estimation is improved when we consider redshift evolution of 21cm power spectrum. Single z Both include noise

84. HERA results “HERA Phase I Limits on the cosmic 21cm signal” (astro-ph/2108.07282)

85. First HERA results HERA（8/4） Constraints on EoR and cosmology（8/16）

86. First HERA results HERA（8/4） Constraints on EoR and cosmology（8/16）

87. Observation •HERA phase I instrument (HERA/PAPER hybrid) •HERA Phase I observation (2017/10- 2018/4) •2017/12/10-2017/12/28. 18 nights. 10 hours/night. 18*10=180 hours •52 antennas at the time of observation. (350 antennas in the end). However, we can only use 39 antennas

88. Calibration procedure No treatment for foreground such as foreground removal. Just use foreground avoidance (EoR window)

89. Observational results band 1 (z=10.4) band 2 (z=7.9) Consistent with thermal noise level

90. Data Likelihood :21cm power spectrum data z=7.9,10.3 :Likelihood function :Posterior via Bayesʼ theorem u: extant systematics W: window function m: cosmic 21cm signal for given parameter θ Γ: inverse of covariance matrix of the data

91. Marginalizing over systematics We have no explicit way of modeling u We marginalize directly over the binned values u Assuming Γ is diagonal and writing t=d-Wm

92. Inverse Likelihood Given upper limits presented by HERA are still roughly two orders of magnitude above fi ducial 21cm models We introduce inverse likelihood *LOFAR and MWA also adopted same manner “With the inverse likelihood, the resulting marginalized distributions identify the parameter combinations that can be ruled out by the HERA limits alone”

93. Disfavored parameter space !

94. Lower limits on spin temperature Assumptions •Full WF coupling (i.e. T_s=T_k) •x_H=1 •Performing a spherical average of RSD : Density driven approach At z=7.9, 95% con fi dence level is above adiabatic cooling !

95. Galaxy-driven models of the cosmic 21cm signal “standard” galaxy formation models used by 21cmFAST (Park et al 2019) •Stellar to Halo mass relation empirical galaxy relations, which reproduces the observed UV luminosity function of galaxies during the EoR •ionizing escape fraction •X-ray SED of high-z galaxies

96. Bayesian inference Performing Bayesian inference by 21cmMC (Greig et al 2015,2017,2018) Including… •Observed faint galaxy UV luminosity functions at z=6-10 •High redshift QSO spectra at z~5.9 •CMB optical depth •Length of 250cMpc •128^3 grids Note that this should not be interpreted as a Bayesian posterior of disfavored models. The models that exceed HERA reside in the parameter space.

97. Spin temperature and neutral fraction disfavored by HERA •At z=7.9, T_s <3 K is disfavored for T_s/T_{radio} <0.1 for 0.1 <x_H < 0.9. •These constraints are somewhat tighter than LOFAR and MWA.

98. With HERA Without HERA(galaxy UV luminosity function, Lyman alpha forest, CMB) Posterior •HERA limits do not have notable impact over most of the astrophysical parameters except X-ray parameters. First galaxies were more X- ray luminous than their local counterparts

99. <latexit sha1_base64="SuHINk+leEIw/jPRex7X06bbW+s=">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</latexit> 27K < TS < 630K(2.3K < TS < 640K) 8.9K < TK < 1.3 ⇥ 103 K(1.5K < TK < 3.3 ⇥ 103 K) HERA observation disfavors low spin temperature peaks seen at observations without HERA.

100. Motivated from EDGESʼs results •milli-charged dark matter (mQDM) •Extra radio background •HERA Band2 (z=7.9) result indicates some heating by z=7.9 (above adiabatic cooling) •An EDGES detection of mQDM is only compatible withe HERA if heating takes place between z=17-10.4

101. If there are extra-radio back ground, 21cm fl uctuations are enhanced •A fl uctuating, time-variable radio background generated by galaxies •A smooth synchrotron background that decays with time Models with an additional radio background can easily exceed HERA upper limits

102. •A fl uctuating, time-variable radio background generated by galaxies <latexit sha1_base64="9rI0RlrwheQ4zRZXMwp/4fe8jzk=">AAACAXicbVDLSsNAFJ3UV42vqBvBzWARXIWkFR8gUnTjsoKthSaEyXTSDp1MwsxEKKVu/BU3LhRx61+482+ctFlo64ELh3Pu5d57wpRRqRzn2ygtLC4tr5RXzbX1jc0ta3unJZNMYNLECUtEO0SSMMpJU1HFSDsVBMUhI/fh4Dr37x+IkDThd2qYEj9GPU4jipHSUmDtRUEbXkLHrtWg55lRIOAFrJ27ZmBVHNuZAM4TtyAVUKARWF9eN8FZTLjCDEnZcZ1U+SMkFMWMjE0vkyRFeIB6pKMpRzGR/mjywRgeaqULo0To4gpO1N8TIxRLOYxD3Rkj1ZezXi7+53UyFZ35I8rTTBGOp4uijEGVwDwO2KWCYMWGmiAsqL4V4j4SCCsdWh6CO/vyPGlVbffErt4eV+pXRRxlsA8OwBFwwSmogxvQAE2AwSN4Bq/gzXgyXox342PaWjKKmV3wB8bnD8Lekz0=</latexit> fX > 0.33 fr < 391 We constrain with 1 sigma con fi dence level high T_r is excluded when T_k is less than 1000 K

103. •A smooth synchrotron background that decays with time

104. PAPER calibration

105. 観測

106. データ較正：Antenna metrics 相関器からのデータは最初、antenna metricsステージに送られ、誤ったアンテナはフラッグされる。 antenna metricsは１０分ごとに計算される。 Antenna metrics antenna metricsはアンテナゲインが他のアンテナゲインと⽐べて値が低いかを計算する。（ロストパワー、接続不良などが原因） 5σより⼤きければアンテナをフラッグする。

107. データ較正：Redundant baseline calibration g: antennae gain 21cm cosmologyで重要なチャレンジの⼀つが装置由来のゲインを精密に⾏うことこの式を解いてantenna gainを得たい。 ”(HERA’s) Redundant-base line calibration uses the principle that every redundant baseline should measure the same visibility” 理想的なχ⼆乗分布からのズレ＞熱雑⾳以上の系統誤差が⼊っていることを⽰唆。

108. データ較正：Absolute calibration sky modelを使ったデータ較正。CASAを使⽤。また、bright point sourceに関してはMWA GLEAM surveyを使⽤。

109. データ較正：RFI fl agging 時間と周波数のデータポイント周りの分布でoutlierを検出する。 Raw HERA data, the gain solutions from redundant-baseline calibration and redundant visibility solutions, its chi-square distribution, the absolute calibration gain and their chi-square distribution Input Z-scoreを計算してFlagging

110. データ較正：Gain smoothing redundant calibrationではgain phaseにギャップが⾒られるが、absolute calibration ではギャップが解消されている。また、RFIを同定後、 fl ag maskをFourier fi lter algorithmに⽤いてdeconvolution。さらにmissing pixelをsmoothly varying componentモデルを⽤いて埋めた（Gain smoothing）

111. データ較正：LST Binning 夜な夜な観測していると、LSTグリッドからわずかにずれが⽣じる。 0−24 hourを21.4秒間隔でbinning. bin内のデータはbinの中⼼とする。毎晩のデータでconsistentになっているのを確認。

112. データ較正：Data inpainting Inpaiting(画像の修復) •強いサイドローブのせいで、フーリエドメインでのパワースペクトル解析は困難。 •強いサイドローブをマスクしたり、取り除いたデータから画像を修復するためによく使われるのがCLEAN（Hogbom 1974） •CLEANと似た⼿法を使ってinpainting (Parsons & Backer 2009, Ali et al 2015, Kerrigan et al 2018)

113. データ較正：Systematic modeling •HERA phase Iシステムではケーブル反射やchain cross couplingが重⼤な系統誤差として存在。 •re fl ection calibrationとcross-coupling fi lteringで系統誤差を軽減したが、もしこの較正が適切でなければ、cosmological signal lossが起きるかもしれない。 Kern et al (2020b)

114. データ較正：Faraday rotated foreground emission instrumental linear polarization pseudo-Stokes polarization Faraday depth instrumental systematicsを除去した後もpower spectrumにlow level excess Faraday rotation。ストークスラパメータを調べた Stokes Q > stokes I leakageはstokes I power spectrum に影響を与える。

115. パワースペクトル解析 band 1 (z=10.4) band 2 (z=7.9)

116. パワースペクトル解析ビジビリティーをdelay domainにフーリエ変換 Quadratic estimator (alpha band) x: ビジビリティー R: 重み⾏列 Q:共分散⾏列のバンドパワー微分。

117. 結果 z＝10.4

118. z＝7.9 結果

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