Exploring universe with neutral hydrogen + machine learning

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

•Born in Okinawa
•Ph.D from

Nagoya university(2016)
•Postdoc at

Paris observatory(2016-2018)
About me
•Postdoc at

Tsinghua University(2018-2019)
2

•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

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

Some old cities
3

Some old cities
Mushroom !
3

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

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

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

(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

(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

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

We should observe IGM at the EoR directly !
(ex)

•EoR theory


•Ionizing sources


etc…
9

(ex)

•EoR theory


•Ionizing sources


etc…
✕
Synergy with galaxy observation by ALMA, JWST, Subaru
9

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

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

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

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

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

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

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

Current 21cm cosmology
status and future project
18

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

Too deep trough
Too flat
SARAS3 did not detect signal

（Singh + 2022, Nature astronomy）
19

Too deep trough
Too flat
SARAS3 did not detect signal

（Singh + 2022, Nature astronomy）
Very strange result ! Need exotic physics?
mis-calibration? unknown systematics?
19

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

SKA-Low
•Frequency 50-350MHz(z=3~27)

&

High sensitivity

Wide FoV
&
22

(ex)

•EoR theory


•Ionizing sources


etc…
✕
Synergy with galaxy observation by ALMA, JWST, Subaru
23

21cm signal analysis with
machine learning
24

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

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

•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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Different stage of reionization
41

Effect of thermal noise
42
21cm PS with thermal noises
(SKA level)

Errors are estimated by 10
realizations thermal noises

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

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

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

•"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

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

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

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

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

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

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

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

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

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)

• 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

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

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

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

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

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

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

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

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

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

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”

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 !

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

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.

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.

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

<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.

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

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

•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

•A smooth synchrotron background that decays with time

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。

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。

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。

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

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）

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。

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)

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)

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。

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

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。

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Exploring universe with neutral hydrogen + machine learning