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Interpreting the Hi 21-cm cosmology
maps through Largest Cluster Statistics -
I: Impact of the synthetic SKA1-Low
observations
Saswata Dasgupta,𝑎Samit Kumar Pal,𝑎Satadru Bag,𝑏
Sohini Dutta,𝑎Suman Majumdar,𝑎,𝑐 Abhirup Datta,𝑎Aadarsh
Pathak,𝑑Mohd Kamran,𝑒Rajesh Mondal, 𝑓Prakash Sarkar𝑔
𝑎Department of Astronomy, Astrophysics & Space Engineering, Indian Institute of Technol-
ogy Indore, Indore 453552, India
𝑏Korea Astronomy and Space Science Institute, Daejeon, Republic of Korea
𝑐Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, U. K.
𝑑School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
𝑒Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala,
Sweden
𝑓School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, 69978, Israel
𝑔Department of Physics, Kashi Sahu College, Seraikella, Jharkhand - 833219, India
E-mail: saswata.iiti@gmail.com
Abstract. We analyse the evolution of the largest ionized region using the topological and
morphological evolution of the redshifted 21-cm signal coming from the neutral hydrogen
distribution during the different stages of reionization. For this analysis, we use the “Largest
Cluster Statistics" - LCS. We mainly study the impact of the array synthesized beam on the
LCS analysis of the 21-cm signal considering the upcoming low-frequency Square Kilometer
Array (SKA1-Low) observations using a realistic simulation for such observation based on
the 21cmE2E-pipeline using OSKAR. We find that bias in LCS estimation is introduced in
synthetic observations due to the array beam. This in turn shifts the apparent percolation
transition point towards the later stages of reionization. The biased estimates of LCS,
occurring due to the effect of the lower resolution (lack of longer baselines) and the telescope
synthesized beam will lead to a biased interpretation of the reionization history. This is
important to note while interpreting any future 21-cm signal images from upcoming or future
telescopes like the SKA, HERA, etc. We conclude that one may need denser 𝑢𝑣-coverage
at longer baselines for a better deconvolution of the array synthesized beam from the 21-cm
images and a relatively unbiased estimate of LCS from such images.
Keywords: Statistical sampling techniques, non-gaussianity, reionization, intergalactic
media, cosmological simulations
arXiv:2302.02727v1 [astro-ph.CO] 6 Feb 2023
Contents
1 Introduction 1
2 Simulating the SKA observations of the EoR 4
2.1 Semi-numerical simulation of reionization 4
2.2 Simulated 21-cm fields from the EoR 5
2.3 Adding observational effects to the simulated 21-cm maps 6
2.3.1 Impact of low-resolution 6
2.3.2 Impact of Noise and Beam 6
2.4 Simulating realistic SKA 21-cm maps 7
2.4.1 Telescope Model 10
2.4.2 OSKAR 10
2.4.3 Synthesis imaging using SKA 21cmE2E-pipeline 12
3 Methodology 14
3.1 Percolcation Transition and LCS 14
3.1.1 Binarization of the image cubes 15
4 Results 18
4.1 Effect of resolution on LCS 18
4.2 Effect of Noise and Model Array Synthesized Beam on LCS 19
4.3 LCS estimates through synthesis radio imaging with SKA 21
5 Summary and discussion 25
6 Acknowledgements 26
A Variation in the threshold to discern the ionized regions 27
1 Introduction
The two least understood eras in our universe’s history are the Cosmic Dawn (CD) and the
subsequent Epoch of Reionization (EoR). The Cosmic Dawn (CD) began when the Cosmic
Dark Ages came to an end and the first luminous sources began forming. The Intergalactic
Medium (IGM), which was primarily made up of neutral hydrogen ( Hi ), began to get ionized
as a result of the high-energy photons that emanated from these sources. This time period
is known as the Epoch of Reionization (EoR). Our comprehension of the “phase transition"
of baryon distribution in the universe from the epoch of the last scattering to the present
era is hampered by the absence of observations of this epoch [see e.g. 1–3]. In theory,
taking direct images of these epochs with various telescopes is the ideal method for studying
them. One of the main scientific objectives of the fairly young James Webb Space Telescope
(JWST) is to observe the first sources of light during the EoR. However, until a statistically
–1–
significant number of such observations are done, the inference from these observations will
remain skewed in favour of the brightest sources [4]. In order to explore this epoch, many
complementary probes, particularly those involving hydrogen, the most common baryonic
element present in the Universe, are used. By examining the Thompson scattering of CMBR
photons off free electrons [5,6], the luminosity function and clustering properties of the Ly𝛼
emitters [7–9] and the Lyman-emitting high redshift quasars [10–13], it has been determined
that the end of reionization occurred at a redshift of 𝑧∼6. However, these observations miss
out on a crucial part, that is the three-dimensional distribution of Hi in the IGM at different
stages of reionization. Thus, including topological analysis of the distribution of the regions
of ionized hydrogen ( Hii ) in the IGM will provide crucial inputs that can further add to the
inference on the ionizing sources (probed via e.g. JWST-like telescopes) and their impact on
the IGM.
The 21-cm line of neutral hydrogen caused by the spin-flip transition of an electron in
the ground state from a parallel to an anti-parallel state has the potential to answer a plethora
of unresolved questions related to the EoR as it directly probes neutral hydrogen [1,3]. It is
expected to be especially instrumental in tracking the evolving state of the IGM and via that
the reionization history. One of the major obstacles to the observation of this signal is the
foreground it is buried in. Since the galactic and extra-galactic foregrounds are ∼105times
brighter [14–20] than the feeble 21-cm signal, they easily obscure it and create hindrances
in its detection. The capacity to detect the signal is also further diminished by a variety of
additional factors, including instrument noise [21,22], telescope beam effect, and effect of the
ionosphere [e.g.[23,24], Pal et al., in prep ]. The upgraded Giant Metrewave Radio Telescope
(uGMRT)1[25], the Low Frequency Array (LOFAR) 2[26], the Murchison Widefield Array
(MWA)3[27], the Precision Array for Probing the Epoch of Reionization (PAPER) 4[28], and
the Hydrogen Epoch of Reionization Array (HERA) 5[29,30] are examples of modern radio
telescopes that are presently aiming to detect the 21-cm signal via the statistical fluctuations
in the signal in Fourier domain and constrain its spherically averaged power spectrum. The
Square Kilometre Array’s 6[31,32] low-frequency component (SKA1-Low), however, will
also be sensitive enough to produce images of the 21-cm signal. The EoR can be analysed as
a series of images, each from a different redshift, by observing over a range of frequencies.
21-cm tomography is the study of these images at various frequencies (or redshifts) [33].
We will be able to study the evolution of the signal and, consequently, the development of
reionization using these tomographic data sets.
It is only reasonable to employ Fourier statistics, such as the power spectrum [26,34–
46] and the multi-frequency angular power spectrum [47–50], as the target statistic for the
detection, since the basic observable in these interferometers is visibility, which is nothing but
the Fourier transform of sky brightness [51]. Due to their low sensitivity, the first-generation
telescopes have been able to produce only upper limits on the signal power spectra. However,
1http://www.gmrt.ncra.tifr.res.in
2http://www.lofar.org/
3http://www.mwatelescope.org/
4http://eor.berkeley.edu/
5https://reionization.org/
6http://www.skatelescope.org/
–2–
the power spectrum only offers a thorough statistical interpretation for a Gaussian random
field. The interaction between the underlying matter density and the changing distribution
and sizes of the ionized regions determines the fluctuations in the EoR 21-cm signal. This
complex origin makes the signal highly non-Gaussian [42,52–55]. This non-Gaussianity,
cannot be quantified through power spectrum and thus higher-order statistics are needed for
its proper quantification. Bispectrum [54,56–63] and Trispectrum [64] are two higher order
statistics which have been explored in this context. However, one should note that even
these higher-order Fourier statistics do not contain the phase information of the 21-cm signal
fluctuations.
As the future SKA1-low is expected to produce high-resolution tomographic images
of the EoR, a significant amount of work has already been done to find the optimal image
analysis method to extract a significant amount of information from such future observations.
These methods have been mostly tested using simulated tomographic images. These methods
mainly focus on tracking and analyzing the evolving topology and morphology of the 21-cm
field during the EoR. A good quantum of work related to analysis of 21-cm images has been
done via Minkowski Functionals (MFs) [65–70] and Minkowski Tensors [71]. There are
various other methods which are based on percolation theory that tracks the abrupt change
in the topology of the 21-cm field [69,70,72–74]. Along with these, granulometry [75] and
persistence theory [76] has also been used for analyzing the topological phases of ionized
hydrogen ( Hii ) regions during the EoR. Analyses involving the theory of Betti numbers
[77,78] and local variance [79] focus on the topology like the connectivity, genus and surface
area to quantify the IGM state.
However, it is commonly accepted that the conclusions obtained using these approaches
rely on the detection of numerous Hii regions at any stage with a wide range in their sizes.
The study done by Bag et al. [69] shows that the detection of only the largest ionized region is
sufficient to draw inferences on the percolation process. They use a novel statistic named the
“Largest Cluster Statistics (LCS)" along with Shapefinders to draw this conclusion. Pathak
et al. [80] have taken this tool a step further and have distinguished the different major
reionization scenarios (inside-out and outside-in) through the same analysis.
In this work, we test the ability of the LCS to extract optimal information from the
fluctuating 21-cm field under a realistic observation scenario. For this, we first reduce the
resolution of our simulated 21-cm maps to mimic the telescope resolution. Next, they are
corrupted with Gaussian random noise and then smoothed out with a Gaussian kernel as per
the specifications of a SKA1-low observation. To test the robustness of this analysis further
we simulate a mock SKA1-Low observation, starting from the distribution of its station
till the final visibility output, using the 21cmE2E-pipeline [81] which utilizes the OSKAR
software. Afterward, we create the final images using the CASA CLEAN algorithm from
these visibility data sets. Finally, after obtaining the clean images, we use the SURFGEN2
code [70,82] to calculate the LCS and draw inferences on the percolation process which tells
us about the sudden merger of the ionizing regions during the EoR.
This paper is structured as follows: In Section 2, we discuss the simulated EoR 21-cm
maps. Additionally, this section briefly discusses the 21cmE2E pipeline and the different
CLEAN algorithms employed using CASA to obtain simulated observational maps. Section
3comprises the method of binarization of the 21-cm maps. In Section 4, we discuss the
–3–
results obtained and what inferences can be drawn regarding the EoR. We summarize our
results and conclude in Section 5.
The cosmological parameters from the Wilkinson Microwave Anisotropy Probe (WMAP)
five-year data release have been used throughout the paper which details as follows: ℎ=0.7,
Ω𝑚=0.27,ΩΛ=0.73,Ω𝑏ℎ2=0.0226 [83].
2 Simulating the SKA observations of the EoR
2.1 Semi-numerical simulation of reionization
Simulation of the reionization era from the first principle is quite an expensive task in terms
of resources. To account for the large-scale density fluctuations of matter, the simulation
has to be done in a large cosmological volume (∼1 Gpc3). In order to correctly mimic the
properties of the reionizing sources, which are typically galaxies smaller than ∼10 kpc, it
is also required to identify these sources and mimic their internal physical processes. This
requires a high dynamic range in terms of mass and length scale in these simulations.
The complete intricacy of radiative transfer across the clumpy IGM can only be ac-
counted for by the numerical solution of the cosmic radiative transfer equation [73,84–87]
along the path of every ionizing and heating photon. On the other hand, radiative transfer
simulations are still computationally expensive, thus the equation is typically solved using
suitable approximations by calculating the ionized hydrogen’s temporal evolution [53,88].
One such method is employed by the radiative transfer algorithm “Conservative Causal Ray-
tracing methodology" (𝐶2-RAY), which functions by following rays from all sources and
iteratively solves for the time evolution of the ionized hydrogen fraction ( ¯𝑥HII ).
It is computationally very expensive to use these simulations to explore the vast multi-
dimensional reionization parameter space. Thus semi-numerical simulations [2,38,39,42,
89–91] can be used to account for these limits as a realistic trade-off. Based on the excursion
set formalism [92], the semi-numerical technique is computationally effective. One of these
approaches starts by creating the dark matter density field using perturbation theory, then
calculating the collapsed fraction in each grid cell (of size R and density fluctuation 𝛿)
using the analytical expression for the conditional mass function, followed by creating the
ionization field using the excursion set formalism [90]. An alternative tactic is to run a full
dark matter-only 𝑁-body simulation and apply a suitable group-finder technique to discover
the halos [2,89]. In this work, we have used the 21-cm maps that were generated using
semi-numerical simulation as per [40].
To summarize the methodology of our simulation as per [40], there are three key steps
involved:
•An 𝑁-Body dark matter gravity-only simulation is used to build the dark matter
distribution for any given redshift.
•The simulation then finds collapsing dark matter halo distribution, which can be
accomplished using means like Friends-of-Friends (FoF) algorithm.
–4–
•The ionization field is then generated using excursion set formalism and translated into
the 21-cm field as described in equation 2.1, with the consideration that the halos are
the most likely hosts of ionizing photon sources.
2.2 Simulated 21-cm fields from the EoR
The Hi 21-cm signal is observed in contrast with the Cosmic Microwave Background Radia-
tion (CMBR) quantified using the differential brightness temperature ( 𝛿𝑇𝑏) and is expressed
as:
𝛿𝑇𝑏≈27𝑥𝐻𝐼 (1+𝛿)1+𝑧
10 1
21−𝑇𝐶𝑀 𝐵 (𝑧)
𝑇𝑆 Ω𝑏
0.044
ℎ
0.7 Ω𝑚
0.27−1
2
𝑚𝐾 (2.1)
The neutral fraction of hydrogen is denoted by 𝑥𝐻 𝐼 , while the density fluctuation is
denoted by 𝛿. The CMB temperature at a redshift of 𝑧and the spin temperatures of the two
states of hydrogen are, respectively, denoted by 𝑇𝐶 𝑀 𝐵 (𝑧)and 𝑇𝑆. It can be clearly seen from
equation 2.1 that, when 𝑇𝐶 𝑀 𝐵 ≈𝑇𝑆, no signal can be detected. The 21-cm signal can be
detected against the background of the CMB because, according to the Wouthuysen Field
effect, the spin temperature will get coupled to the gas temperature during EoR. When the
gas temperature is higher than 𝑇𝐶𝑀 𝐵 , the signal can be detected in emission and when the gas
temperature is lower than 𝑇𝐶 𝑀 𝐵, it is seen in absorption. In our realization, we assume a high
spin-temperature limit ( 𝑇𝑆𝑇𝐶 𝑀 𝐵 ). Since even relatively small amounts of X-ray radiation
can cause the gas temperature to rise above the CMB temperature, it is generally accepted
that this is a reasonable assumption for the period when reionization is well underway. The
frequency axis is one of the three axes in this three-dimensional data set, while the other two
are assumed to indicate the position in the sky. However, in this work, we use simulated
coeval signal cubes rather than light cone cubes. It is also important to note that, initially in
our simulated signal cubes pixels having a value 𝛿𝑇𝑏=0are ionized pixels. The ionization
state for a particular redshift is quantified using the mass-averaged neutral fraction (¯𝑥HI) and
the Filling factor (or volume-averaged ionization fraction). Here, the mass averaged neutral
fraction (¯𝑥HI) is defined as:
¯𝑥HI(𝑧)=¯𝜌HI(𝑧)/ ¯𝜌H(𝑧),(2.2)
and, the Filling Factor (FF) is defined as follows:
FF =total volume of all the ionized regions
simulation volume (2.3)
In the previous work in [80] and in the initial portion of this paper, we consider the same
simulations as [40]. In [40], the 𝑁-body simulations were performed with the 𝐶𝑈 𝐵 𝐸 𝑃3𝑀
code [93], which is based on the earlier algorithm PMFAST, as part of the PRACE4LOFAR
project (PRACE projects 2012061089 and 2014102339) [?]. They used a simulation volume
of 500 ℎ−1Mpc =714 cMpc in length along each side. 69123particles of mass 4.0×107𝑀
on a 138243mesh is considered. This was then down-sampled to a 6003grid for reionization
modeling. For each redshift output of the 𝑁-body, halos were identified using a spherical
–5–
overdensity scheme. Minimum halo mass (𝑀ℎ𝑚𝑖𝑛 ) that was considered here is 2.02 ×109𝑀.
For this simulation, we use the Fiducial model of reionization as mentioned in [80].
In the later part of this paper, we simulate new Hi 21-cm cubes for our realization of
the EoR using 10243particles of mass 1.089 ×108𝑀on a 20483mesh grid. The final
reionization maps were generated by coarsening the initial mesh grid with a factor of 8
resulting in a 2563grid volume with 𝑀ℎ𝑚𝑖 𝑛 =1.089 ×109𝑀. The simulation volume for this
realization is 143.36 Mpc in comoving length along each side of the cube. In this simulation
as well, we use the Fiducial model as mentioned in [80].
2.3 Adding observational effects to the simulated 21-cm maps
2.3.1 Impact of low-resolution
The future SKA1-low telescope will typically have a lower resolution than the simulations
presented in [40], which has a resolution of ∼1.19 cMpc along each side of the simulation
volume. To investigate the impact of this lower resolution of observations on the derived
LCS, we coarse-grid the simulated signal maps and down-sample them to a resolution of
2.38 cMpc along each side of the box.
This down-sampling is done in the following manner, we first randomly choose one
of the 8 neighbouring cells in the simulated signal maps and only this one out of 8 cells
is kept in the down-sampled data. Through this method, our original simulated maps of
6003grids turn into 3003grids maps, while keeping the physical volume of the maps the
same (i.e 714 cMpc) for both cases. A pictorial representation of one of the slices from this
lower-resolution map can be seen in Figure 1.
Radio interferometric observations do not have zero-length baselines or visibilities
with 𝑘=0wave number. This results in the fact that the images obtained from radio
interferometric observations will not have absolute flux calibration. This in turn leads to an
image that has its mean subtracted. Hence, the ionized pixels in our simulations ( i.e 𝛿𝑇𝑏=0
) will be mapped to the minima of a mean subtracted field (i.e 𝛿𝑇𝑏=−¯𝜌𝐻). To mimic this
behaviour we subtract the mean from all of the simulated maps considered here. Figure 2
shows that after down-sampling and mean subtraction the reionization history for our maps
remains the same.
2.3.2 Impact of Noise and Beam
A thorough study on SKA1-low-like noise done by [94] and [95] showed that the noise can be
well estimated using a Gaussian random field. The RMS of such a field is expressed through
the radiometer equation [95]:
𝜎=√2𝐾𝐵𝑇𝑠𝑦𝑠
𝜖 𝐴𝐷√Δ𝜈𝑡𝑖𝑛𝑡
.(2.4)
The 𝑇𝑠𝑦𝑠 is the telescope system temperature, 𝐾𝐵is the Boltzmann constant, 𝜖is the
antenna efficiency and it is dependent on the frequency of observation. 𝐴𝐷is the physical
area of each dish, Δ𝜈and 𝑡𝑖𝑛𝑡 are the frequency resolution and the telescope integration time
respectively.
The authors of [94] have estimated the distribution of ionized bubbles in the 21-cm
maps, and have considered a noise of rms 2.82 mK for their analysis. As our work focuses
–6–
0 100 200 300 400 500 600 700
cMpc
0
100
200
300
400
500
600
700
cMpc
0
5
10
15
20
25
30
35
Tb
(
mK
)
Figure 1. A slice from the simulated and down-sampled Hi 21-cm image cube of 3003grid units
generated at neutral fraction ¯𝑥HI =0.2.
on the largest ionized region at any stage thus we start with a higher noise rms i.e 3.10 mK.
We simulate Gaussian random noise with the rms values ranging from 3.10 mK to 9.30 mK
and study its impact on the recovered LCS behaviour.
Further, the image generated by any radio interferometer will have the sky brightness
convolved with the beam of the telescope. This convolution process has a smoothing effect
on the image and it further degrades the image resolution. To imitate this effect we use a
3D Gaussian kernel with varying smoothing length-scales starting from 7.14 cMpc to 14.28
cMpc along each side of the simulation volume. In Figure 3, a visual representation of these
effects is shown in the three panels.
2.4 Simulating realistic SKA 21-cm maps
To analyse the robustness of LCS, we further simulate a synthetic radio observation as per
the characteristics of the upcoming SKA1-Low. The schematic diagram for the 21cmE2E-
–7–
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Neutral fraction (
XHI
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Filling Factor (FF)
Data of
6003
grid units
Data of
3003
grid units
Figure 2. FF vs ¯𝑥HI plots are shown for the data cubes of 6003grid units and 3003grid units. It can
be seen that the time evolution for Filling Factor (FF) remains unchanged even after the resolution of
the maps are lowered and the mean is subtracted.
0 100 200 300 400 500 600 700
0
100
200
300
400
500
600
700
cMpc
0 100 200 300 400 500 600 700
cMpc
0 100 200 300 400 500 600 700
5
0
5
10
15
Tb
(
mK
)
Figure 3. Pictorial representation of the different slices from an Hi 21-cm image cube of ¯𝑥HI =0.2.
Left: The simulated image cube after down-sampling and mean subtraction. It can be seen from
the colorbar on the right that the zero level has been shifted to a lower negative value. Centre:
Representation of the same image slice after the addition of a Gaussian random noise of 𝑁𝑟 𝑚𝑠 =
3.10mK. Random fluctuations in the 21-cm map is observed. Right: The observed 21-cm signal cube
slice after smoothing with a 3D Gaussian kernel on top of the noisy image map. It can be observed
that smoothing introduces some de-noising effect.
–8–
Input: Sky Model from simulations
Telescope Model
Observational Parameters
Simulation
Visibility
Image-based Analysis PS Estimation
21cmE2E
Thresholding and SURFGEN2 LCS
Figure 4. Schematic diagram of the 21cmE2E-pipeline [81] used for analysing the Hi 21-cm maps
from the future SKA observations.
pipeline is shown in Figure 4. These simulations are created using OSKAR7software for
SKA1-Low. Common Astronomy Software Application (CASA8) is used to create 21-cm
images from the visibility data generated from OSKAR. In this work, we have considered an
observation for half an hour (±0.5HA), with the phase centre at 𝛼=15h00m00s and 𝛿=−30◦.
The integration time for this observation is 120 seconds. One of the axes of the simulated
coeval 21-cm cubes was simply labeled to be as the Line-of-sight axis (LoS) or frequency
axis and the cubes were divided into slices according to their spatial resolution. Each of these
slices has a frequency label according to it’s comoving distance from the observer. These
slices of the simulated 21-cm maps are then converted from comoving coordinates to angular
coordinates in the sky plane before feeding into OSKAR. The 21cmE2E-pipeline outputs are
then stacked slice by slice according to their frequency label to generate the final visibility
cube.
7https://github.com/OxfordSKA/OSKAR/releases
8https://casaguides.nrao.edu/index.php?title=Main_Page
–9–
1500 1000 500 0 500 1000 1500
X[m]
1000
500
0
500
1000
1500
Y[m]
Figure 5. A simple layout of the telescope model used in this work. The stations residing at 2km
from the central station are operational.
2.4.1 Telescope Model
The Square Kilometer Array (SKA-1 Low) is a powerful radio telescope that is expected to
have the necessary sensitivity (Signal-to-Noise Ratio (SNR)) to create tomographic images
of the 21-cm signal from the EoR [32]. The SKA-1 Low will operate in the frequency range
50 to 350 MHz and will be located at the Murchison Radio-astronomy Observatory (MRO)
in Western Australia. The array is designed to have a dense core, spiral arms, and 512
stations with a maximum baseline length of 65 km. Each station will have 256 antennas and
a diameter of less than 40 m. The compact core of the array, which is composed of ∼85%
of all stations within a baseline of 2km, is designed to maximize the sensitivity of the target
signal. This is illustrated in Figure 5.
2.4.2 OSKAR
The OSKAR package contains a number of applications for simulating radio interferometric
observations. In order to produce simulated observations from aperture array-using tele-
scopes, such as those foreseen for the SKA, OSKAR was principally developed. OSKAR
reproduces digital beamforming for an aperture array of a SKA-sized station. The simulator
was developed as a versatile study platform to examine alternative computational beam-
former processing strategies and their impact on the output beam quality [96]. The radio
– 10 –
interferometer measurement equation as mentioned in [97] is used in OSKAR to generate
the visibility data from the sky model. The “end-to-end beamforming simulation" func-
tion of OSKAR generates antenna outputs for each element of the telescope array using a
point-source sky model. Because this mode simulates a full-size, authentic SKA station,
it is possible to replicate beam tracking and add several beams per level, noise, and other
time-variable effects.
Using the Rayleigh-Jeans equation as mentioned in equation 2.5, we convert the semi-
numerically simulated Hi 21-cm maps to maps of specific intensity. This enables us to use
these maps as an input to the SKA pipeline, namely, the 21cmE2E-pipeline.
𝑆=2𝑘𝐵𝑇
𝜆2Ω,(2.5)
here 𝑆denotes the specific intensity, 𝑘𝐵is the Boltzmann constant and 𝜆is the wavelength
of the corresponding central frequency of the observation. The calculation of the frequency
resolution requires a conversion from comoving coordinates to angular coordinates. This was
done using the formalism as mentioned by the authors of [94] and are described in equation
2.6 and equation 2.7.
Δ𝜃=
Δ𝑥
𝐷𝑐(𝑧),(2.6)
where Δ𝑥is the spatial resolution of the image slice in comoving units, 𝐷𝑐(𝑧)is the comoving
distance to redshift 𝑧, and
Δ𝜈=𝜈0𝐻(𝑧)Δ𝑥
𝑐(1+𝑧)2,(2.7)
where 𝑐is the speed of light, 𝐻(𝑧)is the Hubble parameter at redshift 𝑧, and 𝜈0is the rest
frequency of the 21-cm line. For instance, a slice from the simulated signal cube of 714
cMpc3at redshift 𝑧=7.221 has an angular resolution of 0.015 deg, and frequency resolution
of 125 KHz. The transformed images were used as the sky model for the OSKAR simulator
and then the visibility data corresponding to these images were obtained which incorporates
the telescope synthesized beam-effects using the van Cittert-Zernike equation as shown in
equation 2.8 [51]. The 𝑢𝑣-coverage of the simulated telescope for 30 minutes of observation
is shown in Figure 6.
𝐴(𝑙, 𝑚, 𝜈)𝐼(𝑙, 𝑚)=∫ ∫𝑠𝑘 𝑦
𝑉(𝑢, 𝑣, 𝜈)𝑒 𝑥 𝑝 [2𝜋𝑖(𝑢𝑙 +𝑣𝑚)]𝑑𝑢𝑑𝑣, (2.8)
here 𝐴denotes the primary beam pattern, 𝐼is the sky intensity and 𝑉is the observed visibility
from the interferometer. As this work does not deal with the primary beam correction, we
consider 𝐴(𝑙, 𝑚)=1. OSKAR employs a Fast Fourier Transform scheme to calculate the
visibility using FFTW [98]. Hence, it is reasonable to use a simulation volume of the order
of 23𝑛in grid units to avoid the effect of zero padding. The effects of this zero padding may
result in the creation of artefacts on the maps that is discussed in Section 4.
– 11 –
Figure 6. The 𝑢𝑣-coverage for the simulated SKA1-Low telescope using OSKAR for an observation
time of 30 mins.
2.4.3 Synthesis imaging using SKA 21cmE2E-pipeline
One of the major aims of this work is to study the impact of the array synthesized beam on the
recovered LCS information from the redshifted 21-cm maps from a simulated SKA1-Low
observation. One of the widely used deconvolution algorithms for imaging astronomical
sources is CLEAN, which has an implementation with the CASA software as well. The
CLEAN algorithm allowed for the synthesis imaging of complex structures even when the
Fourier plane had relatively weak coverage, as is the case with partial earth rotation synthesis
or arrays of few antennas. The sky image is considered to be a collection of point sources
in CLEAN. The algorithm works by iteratively calculating the brightness and location of
every point source in the sky from the image. The measured visibilities are weighted and
plotted on uniformly spaced grid points to determine the Fourier transform of the measured
visibilities from OSKAR. The terms uniform and natural weighting in CLEAN, relate to
two extreme weighting regimes. Natural weighting equalises the weights of all measured
values and adds them together. Natural weighting works by emphasising on the area of the
visibility plane with the most measurements. It is to be noted that the Fourier transform
of the sampling function multiplied by the weight kernel should be treated as the effective
dirty beam when implementing a weighting scheme. Contrarily, in uniform weighting, the
– 12 –
visibilities are weighed according to the spatial density of the observed visibilities prior to
screening. Although the Point Spread Function (PSF) is smaller, as a result of this weighting,
the sidelobe power is larger. The robust weighting technique [99] is an attempt to integrate
natural and uniform weighting. To support high-bandwidth, high-dynamic-range imaging, an
iterative weighting method that minimises PSF variation across frequencies while maximising
sensitivity is discussed in [100]. In [51], the specific gridding procedure, different weights,
and their effects are covered in detail.
Deconvolution is the process by which we construct the true sky brightness distribution
from the observed data. The main idea behind deconvolution is the subtraction of the
instrument’s point spread function (PSF) from the given dirty image. The CLEAN algorithm
is commonly used for this purpose and consists of two parts: the major cycle, which converts
the visibility data into image data, and the minor cycle, which performs the deconvolution
steps that separate the sky brightness distribution from the instrument’s PSF. There are many
algorithms for the deconvolution steps, such as Hogbom, Multiscale, Clark, and Clarkstokes.
We first applied the Hogbom method to reconstruct the sky emission which is represented
in Figure 7. However, this method is not extensive in reconstructing the images from the
original ones. To mitigate this problem, we applied the multiscale algorithm (see more
[101]), which assumes that the sky emission is a combination of Gaussian functions rather
than delta functions. This method is useful for images showing extended emission from the
true sky. A pictorial representation of the Multiscale cleaned image is shown in Figure 8. In
the context of radio interferometry, the CLEAN algorithm is considered greedy because it
aims to identify and remove the brightest point sources from the image as quickly as possible,
without paying much attention to the distribution of the sources in the image. The algorithm
starts by identifying the brightest point in the dirty image and then subtracting a scaled version
of the PSF at that location. This process is then repeated for the next brightest point, and
so on, until a stopping criterion is met. The algorithm is called greedy because it prioritizes
removing the brightest sources first, without considering the overall structure of the image.
This can lead to over-subtraction of the sources and loss of information, especially in the
case of extended sources. Our main goal is to recover the largest ionized region for further
statistical analysis. To improve the CLEAN process further, the Multiscale algorithm with
Briggs weighting (with robust parameter set to a value close to natural weighting) was used
to recover the images as shown in Figure 9. This method has some advantages, such as the
ability to reconstruct larger bubble sizes of an ionized region, resulting in higher sensitivity
for extended sources. However, the main disadvantage is that the resolution of the images
will be poorer. After deconvolution, the output sky model is restored by a Gaussian function
representing the instrumental resolution specified by the PSF main lobe but without the side
lobes. This deconvolution method creates another issue, the bimodal nature of the 21-cm
image histogram is lost. Therefore, the threshold finding for the LCS analysis becomes
challenging.
– 13 –
15h05m00m14h55m
-28°
-29°
-30°
-31°
-32°
Right Ascension (J2000)
Declination (J2000)
0.050
0.025
0.000
0.025
0.050
0.075
0.100
0.125
Jy/beam
Figure 7. Slice of a Hi 21-cm image cube at ¯𝑥HI ≈0.8obtained after performing Hogbom CLEAN
algorithm.
3 Methodology
3.1 Percolcation Transition and LCS
At the beginning of reionization (𝑧∼15 [102]), the newly formed luminous objects start to
emit copious amount of ionizing photons and gradually ionize the surrounding IGM which
was mostly abundant with neutral hydrogen. With reionization progressing, small pockets of
ionized regions start to appear and they grow in size and numbers. At some point in time, these
regions start to overlap and eventually merge to form a singly connected ionized region. This
“phase transition" when numerous small ionized regions abruptly merge together to form a
single large connected ionized region that extends through out the IGM is called “percolation
transition" [103,104]. In the context of this work, we identify the percolation transition
when the largest ionized region stretches from one end of the simulation volume to the other
end and it becomes formally infinitely extended due to periodic boundary conditions.
In this particular work, which is a follow-up of [69,80], we follow the largest ionized
region (LIR) with the progress of reionization. For our study we use a novel statistic called
the “Largest Cluster Statistic" (LCS) [103–105] to track the percolation process and it is
– 14 –
15h02m01m00m14h59m58m
-29°45'
-30°00'
15'
Right Ascension (J2000)
Declination (J2000)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Jy/beam
Figure 8. Slice of a simulated Hi 21-cm signal map at ¯𝑥HI =0.2cleaned using Multiscale deconvo-
lution with natural weighting.
defined as follows:
LCS =volume of the largest ionized region
total volume of all the ionized regions .(3.1)
From the above definition it can be stated that, LCS essentially denotes the ratio of
the largest ionized volume to the total ionized volume. Hence, at the onset of percolation
transition an abrupt increase in LCS is expected. The point at which this abrupt transition takes
place is defined as the percolation transition threshold. Results from [69] and [80] confirms
this fact and the authors of [80] also show that the two major types of reionization scenarios,
inside-out and outside-in can be distinguished from each other using LCS. This eventually
helped in distinguishing the different source models that the authors had considered.
3.1.1 Binarization of the image cubes
To determine LCS on the brightness temperature maps, we employ a sophisticated code called
SURFGEN2 [69,70] which is an advanced version of the original SURFGEN code developed
by the authors of [106–108] to study the large scale structures of the universe. SURFGEN2
– 15 –
15h02m01m00m14h59m58m
-29°45'
-30°00'
15'
Right Ascension (J2000)
Declination (J2000)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Jy/beam
Figure 9. Slice of a mock observational Hi 21-cm image slice at ¯𝑥HI =0.2obtained after deconvolving
with Briggs weighting scheme.
not only determines the LCS, but also helps to find the topological and morphological features
of the individual ionized regions using Shapefinders. The working principle of SURFGEN2
can be found in [69,70,80,109].
The working of SURFGEN2 necessitates a threshold to distinguish between the neutral
and the ionized pixels in the 21-cm maps. It is important to note that no partially ionized
regions are considered in the SURFGEN2 code. As mentioned earlier, for an ideal Hi 21-cm
field, 𝜌HI(x)=0can be considered to be the boundary between the ionized and the neutral
regions. But, due to computational constraints we choose 𝜌HI(x)to be a very small number
('0.1) as a threshold for the ideal case.
As we go to radio-interferometric images, we miss out on the zero-length baselines for
physical reasons. This results in the subtraction of the mean from the 21-cm images. Hence,
the threshold for these cases will be the minima of the fields (i.e 𝜌HI (x)=−¯𝜌). We use this
threshold on each of the 21-cm field and the results for the mean subtracted maps exactly
follow that of the ideal case, as expected.
As we add Gaussian random noise and run over a Gaussian smoothing kernel to create
realistic images, identification of the optimal threshold becomes a non-trivial task. Addition
– 16 –
0 5 10 15 20 25
0.0
0.5
1.0
1.5
Simulated image
1e7
xHI
0.2
0 10 20 30 40 50
0
1
2
3
1e6
xHI
0.71
0 10 20 30 40 50 60 70
0.0
0.5
1.0
1.5
2.0
1e6
xHI
0.9
20 10 0 10 20 30 40
0.0
0.5
1.0
1.5
2.0
2.5
Noisy image (
Nrms
= 3.10
mK
)
1e6
xHI
0.2
30 20 10 0 10 20 30 40 50
0.0
0.5
1.0
1.5
1e6
xHI
0.71
20 0 20 40
0.0
0.5
1.0
1.5
2.0
1e6
xHI
0.9
5 0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Smoothed image (length = 7.14 cMpc)
1e7
xHI
0.2
20 15 10 5 0 5 10 15
Brightness Temperature (
Tb
)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1e6
xHI
0.71
10 5 0 5 10 15
0.0
0.5
1.0
1.5
1e6
xHI
0.9
Pixel count
Figure 10. Evolution of the bimodality of the histogram of Hi 21-cm field with changing neutral
fraction ( ¯𝑥HI ). [Top panel: R-L] Each panel represents the histogram of the 21-cm field at a fixed
neutral fraction. As the universe gets ionized, the left peak, representing the ionized pixels, grows
in size. On the other hand, the peak on the right, representing the neutral pixels gets smaller as the
ionization fraction of the universe increases. The red vertical dashed line in each panel represents the
identified threshold by the gradient descent algorithm. [Middle panel: R-L] Adding a Gaussian random
noise introduces random fluctuation in the brightness temperature values in the 21-cm field which
shifts each of the pixel intensities by a random value defined by the noise RMS. This creates a bias
in the threshold identification by gradient descent. [Bottom panel: R-L] At a higher neutral fraction
(¯𝑥HI '0.9), the gradient descent algorithm fails to identify a proper threshold as the smoothing effect
of the beam washes out the bimodality of the 21-cm image histogram. The Gaussian smoothing kernel
creates an averaging effect on the 21-cm maps creating regions that appear as partially ionized regions.
This affects the choice of a threshold by the gradient descent, and therefore, a bias is introduced in the
selection of the threshold.
of noise generates random fluctuation in the pixel values that changes the brightness temper-
ature in each of the pixels with some amount defined by the rms of the noise. This, in turn
shifts the minima of the fields to a random point that is unknown to the user and some other
pixel gets mapped to the minima of the field. Hence, the minima of the fields could not be
used as a threshold to determine the LCS in such scenarios.
The histogram of an ideal Hi 21-cm field has a very sharp bimodal feature [110] that
evolves with time as seen in the top panels of Figure 10. The left delta function-like peak
in each panel represents the ionized pixels and the rest of the non-zero histogram represents
neutral pixels. It can be seen that as reionization progresses, the delta-like peak, representing
– 17 –
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xHI
0.0
0.2
0.4
0.6
0.8
1.0
LCS
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Filling factor (FF)
Original image
Image of
3003
grid volume
Image of
2563
grid volume
Figure 11. The Largest Cluster Statistic (LCS) is plotted with changing neutral fraction (¯𝑥HI ) and
Filling Factor (FF) for 21-cm image cubes of different resolutions. In both panels, the solid blue line
denotes the development of LCS for the original simulated data cubes of 6003grid volume. The red
dashed line represents the coarsened maps of 3003grid volume. The green dotted line represents the
data cubes of 2563grid volume. It is observed that the percolation transition threshold is consistent
for all three resolutions.
the ionized pixels grows. Alongside, the peak on the right-hand side loses its height as the
neutral regions get diminished by the incoming ionizing photons. We utilize this feature and
set our threshold as the local minima between these two peaks. To find the local minima, we
employ the method of Gradient Descent.
To utilize the Gradient Descent algorithm, we smooth the histograms after choosing an
optimum bin size for the histograms. We verify our method by running Gradient Descent
on the mean-subtracted image cubes and find that the chosen threshold is in-line with the
minima of the ideal fields. We apply the method of Gradient Descent to find the threshold
for the Noisy and Gaussian smoothed image-cubes to calculate the LCS. We apply the same
technique on the maps produced by the 21cmE2E-pipeline and calculate LCS for each of the
image cubes.
As we add a higher noise rms, we lose the bimodal feature of the 21-cm image histogram
at higher ¯𝑥HI and the gradient descent algorithm identifies a threshold value that is biased
(shown in the middle panel of Figure 10). Similar cases happen for the Gaussian smoothing
case with larger smoothing length scales as an averaging effect is employed on the maps.
This effect is shown in the bottom panel of Figure 10. This necessitates the search for a better
threshold-finding algorithm that is independent of the histogram of the 21-cm image cubes.
We discuss these possibilities in Section 5.
4 Results
4.1 Effect of resolution on LCS
The original simulated Hi 21-cm image cubes having a resolution of 1.19 cMpc3were
coarsened down to a resolution half the original resolution (i.e. 2.38 cMpc along each side).
It is expected that the future SKA1-Low observations will have a similar resolution [80].
At the later part of this work, we observe edge/spill-over effects to appear on the
observational maps produced with synthetic observations of the SKA1-Low. We suspect that
– 18 –
these effects appear due to zero padding in the FFTW scheme as our sky-model maps did not
have a grid resolution of the order of 23𝑛. Hence, we use our newly simulated maps of 2563
grids. It is important to note that the data cubes of 6003grids and 3003grids had a volume
of ∼714 cMpc3, whereas the maps of 2563grid has a volume of 143.36 cMpc3.
We explore the percolation transition process by following the development of the LCS
with changing neutral fraction ¯𝑥HI in Figure 11. Alongside, we also explore the evolution
of LCS with Filling Factor (FF) which has been defined in equation (2.3). Analysing LCS
with ¯𝑥HI and FF will enable us to understand the development of the Largest Ionized Region
(LIR) with the changing ionization state of the universe both in mass averaged and volumetric
terms.
In Figure 11 we plot the LCS at different ¯𝑥HI and FF in the two panels. Initially, the size
of the LIR is very small, hence a small value of LCS is seen. As reionization progresses,
small ionized regions start to overlap and that causes the LIR to grow in size, leading to an
increase in the LCS. It is observed that the LCS of the cubes of both 3003and 2563grid units
follow the original 6003grid units with changing ¯𝑥HI and FF. As the percolation transition
approaches, an abrupt change in the LCS is observed as the LIR suddenly grows in size
and volume. It is also observed that the percolation transition threshold point for all three
coincides at the same point. The percolation transition threshold for each of these cases is
observed at ¯𝑥HI ≈0.75 and FF ≈0.096. This result is consistent with the previous findings
of the authors of [69,72,74,80,111]. After percolation occurs, the individual LIRs also
start to merge with one another and then form a singly connected LIR. At this time as the
entire simulation volume consists of the LIR, we see that LCS saturates to unity.
4.2 Effect of Noise and Model Array Synthesized Beam on LCS
In Section 2.3.2, we discuss the prescription we follow for the addition of a Gaussian noise
of varying rms values. In Figure 12, we plot the LCS with changing ¯𝑥HI and FF for different
noise rms values. For low noise level (i.e ∼3.10 mK), we see that the LCS is almost
unaffected and it retains the same development as seen for the case of the original 3003
grid volume image. It is also observed that the percolation transition threshold point is also
consistent with the original image cube.
As we go to higher noise levels, we see that the LCS could no longer be computed after a
certain point as the random fluctuations introduced starts dominating the image pixels. This,
as a result, masks the small ionized regions that are created in the early stages of reionization
(∼¯𝑥HI ≥0.7). LCS could only be computed at a time when the LIR is sufficiently large
enough to not get masked by the noise.
For even higher noise levels it is observed that the LCS could not be computed even
for higher neutral fractions as noise starts dominating even more. Adding higher levels of
Gaussian noise pushes the histogram of the 21-cm maps to a more Gaussian type and the
bimodal feature is lost. Hence, the method of gradient descent fails to identify the boundary
between the ionized and neutral pixels. This leads to a biased computation of LCS at very
high noise levels and they can be observed in the case of 𝑁𝑟𝑚𝑠 =6.20 mK and 𝑁𝑟 𝑚𝑠 =9.30
mK in Figure 12.
To create mock observational effects of the array synthesized beam, we simulate a
Gaussian smoothing kernel of varying FWHM as discussed in Section 2.3.2. We plot the
– 19 –
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xHI
0.0
0.2
0.4
0.6
0.8
1.0
LCS
0.0 0.2 0.4 0.6 0.8 1.0
Filling factor (FF)
Original image
Noise rms = 3.10 mK
Noise rms = 4.50 mK
Noise rms = 6.20 mK
Noise rms = 9.30 mK
Figure 12. The LCS of the ionized hydrogen is displayed in the left and right panels as a function
of ¯𝑥HI and the filling factor (FF), respectively. The LCS for a noise rms of 3.10 mK follows the
LCS of the original image cube. But as the noise rms goes higher, the LCS could not be computed.
For extremely high noise rms cases, the gradient descent algorithm fails to binarize the 21-cm field.
Hence, biased LCS results are achieved.
LCS by changing ¯𝑥HI and FF to see the impact of the beam on the detection and development
of the LIR in Figure 13. We start with a Gaussian smoothing kernel having a smoothing
length-scale of 7.14 cMpc and go all the way up to a smoothing length-scale of 21.42 cMpc.
For a relatively smaller smoothing length scale of 7.14 cMpc, it is observed that as the
small individual ionized regions are smoothed out, numerous partially ionized regions start
to appear due to the effect of averaging. These newly introduced partially ionized regions fall
either into the category of ionized or neutral regions as the binarization scheme is imposed
on the maps. This leads to a bias in the LCS results and that can be observed in the crimson
red dotted lines in both the panels of Figure 13.
We tend to lose out on the features of LCS for a higher smoothing length scale at earlier
stages of reionization. This occurs due to the fact that the averaging effect creates even more
partially ionized regions in this case and the bimodality of the Hi maps is lost. The method
of gradient descent identifies these partially ionized regions as neutral regions and hence the
volume of the LIR is zero, leading to no estimation of LCS.
For even higher smoothing length scales, we see that the behaviour of the LCS goes
erratic. This is due to two major reasons.
•The partially ionized regions appearing due to smoothing shifts the histogram of the
21-cm field to a Gaussian-like nature.
•Due to the above-mentioned reason, the binarization scheme does not work properly
with gradient descent algorithm and bias is introduced in the estimation of LCS.
In Figure 14, we choose a constant noise rms of 3.10 mK and a constant smoothing
radius of 7.14 cMpc, in-line with the future SKA observations [94] and plot LCS with varying
¯𝑥HI . We see that although the features of LCS could be computed up to a certain ¯𝑥HI, the
percolation transition threshold point shifts heavily to a lower ¯𝑥HI value. This is due to the
combined effect of random fluctuation and the introduction of partially ionized regions in the
Hi 21-cm field. As the percolation transition threshold moves to a lower ¯𝑥HI value, it leads
to a biased interpretation of the reionization history.
– 20 –
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xHI
0.0
0.2
0.4
0.6
0.8
1.0
LCS
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Filling factor (FF)
Original image
Smoothing scale = 7.14 cMpc
Smoothing scale = 9.52 cMpc
Smoothing scale = 14.28 cMpc
Smoothing scale = 21.42 cMpc
Figure 13. The two panels from left to right represents the variation of LCS with changing neutral
fraction (¯𝑥HI) and the filling factor (FF) respectively. Bias is introduced as the radius of the Gaussian
smoothing kernel is increased. This occurs majorly due to the fact that the averaging effect caused by
smoothing introduces numerous partially ionized regions which is not identifiable by SURFGEN2.
0.2 0.3 0.4 0.5 0.6 0.7 0.8
xHI
0.0
0.2
0.4
0.6
0.8
1.0
LCS
Noise rms = 3.10 mK
Smoothing Lengthscale = 7.14 cMpc
Original Data
Noise + Gaussian smoothing
Figure 14. LCS is computed with varying ¯𝑥HI for a noise level of rms 3.10 mK and a smoothing
length-scale of 7.14 cMpc. It is seen that the percolation process is identified at a later stage due to
these effects and a biased interpretation of the reionization history is obtained.
4.3 LCS estimates through synthesis radio imaging with SKA
In section 2.4 we discuss the synthesis radio imaging procedure that we have used in this work
in detail. The 21cmE2E-pipeline creates visibility data from the input sky models (which are
– 21 –
the simulated signal maps in this case) and finally, we use CASA to make the image cubes
from the visibility. We run SURFGEN2 on these coeval image cubes and estimate LCS
for each of them. As seen in Figure 7, edge effects occur in the synthetic maps due to the
zero-padding operation in OSKAR, when the input data grid is not equal to 23𝑛. Alongside,
other artefacts as seen in Figure 7also occur in the images. Among these artefacts, a notable
one is the central dark patch, which is identified as a neutral region by SURFGEN2 and thus
affects the estimation of LCS severely. The smoothing of the sky model due to the array
synthesized beam, as implemented in the 21cmE2E-pipeline, mimics features of partially
ionized regions in the output maps, which also contribute to the biased estimation of LCS. For
this reason, LCS could not be estimated for higher neutral fractions as observed in Figure 15.
As SURFGEN2 identifies the dark patches to be islands of neutral hydrogen, we see an erratic
behaviour in the LCS plot with varying ¯𝑥HI values which are counter-intuitive in nature. The
CLEAN method that has been used to deconvolve the dirty beam from the image, in this case,
is the Hogbom CLEAN [101]. We see that the Hogbom CLEAN method could not properly
deconvolve the beam from the image and it leaves residues of the dirty beam. These residues
appear to be as ionized regions in the binarization scheme and hence, the counter-intuitive
behaviour of LCS is observed. Additionally, we suspect that several edge effects appear in
the image when the synthesis radio imaging is done using the 21cmE2E-pipeline. This is due
to the fact that as OSKAR employs an FFTW operation, it is optimal to use images having
a size of 23𝑛. But, the simulated images that we used, in this case, have a grid size of 3003,
leading to a zero padding effect. This further adds a bias in the LCS estimation.
To mitigate the effect of zero padding, we rerun our simulations of 21-cm signal from
EoR and generate signal maps of size 2563grids (more details are discussed in Section 2).
Additionally, for this set of simulated sky models, we also employ the Multiscale deconvolu-
tion CLEAN process [101] instead of Hogbom CLEAN to achieve a better deconvolution of
the dirty beam.
Figure 7shows that the Hogbom CLEAN algorithm does not effectively deconvolve the
dirty beam from the image as it assumes the sky emission to be a bunch of delta functions.
This is mitigated using Multiscale CLEAN algorithm (discussed in section 2.4.3). Figure
16 shows the evolution of LCS with ¯𝑥HI for two different types of Multiscale CLEAN
algorithms employed on the simulated SKA1-Low observational maps. In the case of natural
weighted Multiscale cleaned maps, LCS could not be estimated for ¯𝑥HI values larger than
0.6. Natural weighting equalises the weights on all measured pixel values and adds them
together (discussed in detail in section 2.4.3). Hence, the contrast between the neutral and
ionized pixels gets reduced. This, in turn, forces a significant number of ionized regions
to appear partially neutral. Effectively it diminishes the number of small ionized pixels at
higher ¯𝑥HI and subsequently, SURFGEN2 could not identify ionized clusters which leads
to an estimation of zero LCS. Furthermore, as we consider a closed-packed interferometric
array distribution within a 2km radius within the central core of SKA1-Low for our mock
observations, the obtained images suffer from poor resolution due to the lack of longer
baselines. This, in turn, adds to the bias in the estimation of LCS at the earlier stages of
reionization as the small ionized bubbles can no longer be resolved. Besides, the natural
weighted deconvolution is not the optimal weighting scheme for extended sources as it
leaves residues of the dirty beam on the cleaned images. The gradient descent algorithm for
– 22 –
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
xHI
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
LCS
Original image
Image obtained from SKA pipeline
Figure 15. Effect of the artefacts in LCS introduced in the 21-cm maps as they are run through the
21cmE2E-pipeline. A combined effect of the dirty beam and zero padding leads to a counter-intuitive
nature in the LCS plot.
identification of the optimal threshold for LCS estimation is further affected as the bimodal
feature of the 21-cm histograms gradually disappears (due to the presence of residues of the
dirty beam) and becomes more like a Gaussian structure. This further adds to the bias in
the estimation of LCS and the LCS curve thus shifts towards a lower ¯𝑥HI for percolation
transition (compared to the scenario when the impact of the beam has not been considered
i.e. the blue curve in Figure 16) as observed by the red dashed line in Figure 16.
We next use an extensive weighting scheme named Briggs weighting by setting the
robust parameter in our CLEAN algorithm to be 0.5to further push the contrast of the
sky maps closer to the scenario when the impact of the beam has not been considered.
This CLEAN scheme does this by enhancing the contrast between the ionized and neutral
pixels. As the contrast between the neutral and ionized pixels is enhanced, the non-zero LCS
estimation now could be pushed to even higher neutral fractions, ¯𝑥HI ≈0.7, than in the case of
natural weighting. However, the effect of telescope resolution still persists for ¯𝑥HI >0.7and
hence LCS could not be estimated for the same reasons as mentioned for the case of natural
weighted CLEAN. Even though the estimation of LCS at a lower neutral fraction ¯𝑥HI ≤0.7
could be done (as shown by the green dotted lines in Figure 16), it is still significantly biased
due to the assumed Gaussian PSF and the telescope resolution. As the functional form of the
PSF is unknown, the Briggs weighting scheme also leaves the residue of the dirty beam on
– 23 –
0.2 0.3 0.4 0.5 0.6 0.7 0.8
xHI
0.0
0.2
0.4
0.6
0.8
1.0
LCS
Simulated image of
2563
grid volume
Image after Multiscale CLEAN (Robust parameter = 2)
Image after Multiscale CLEAN (Robust parameter = 0.5)
Figure 16. Comparison shown for the LCS obtained with changing neutral fraction after natural
weighted Multiscale CLEAN and Multiscale weighted CLEAN with Briggs weighting by setting the
robust parameter = 0.5. It can be seen that Briggs weighted CLEAN performs superiorly in two ways.
Firstly, LCS values upto almost 0.7could be calculated and secondly, the percolation transition is not
as heavily shifted as of the natural weighted CLEAN case.
the images and that further adds to the bias in the LCS estimation due to the sidelobes of the
synthesized beam. This effectively shifts the critical ¯𝑥HI value (percolation threshold point)
to later stages of reionization. Therefore, it ultimately leads to a biased interpretation of the
reionization history.
It is important to note that the results shown in Figure 16 incorporate a more accurate
model of the array synthesized beam for SKA1-Low as compared to the one assumed in the
case of Figure 14. We see that even without incorporating noise in the 21cmE2E-pipeline,
the percolation transition threshold shifts almost as much as in the case when the effect of
telescope noise and beam was implemented directly on the image plane. However, both cases
will make the observer draw a biased inference that the percolation process happens at a later
stage of EoR than its true value. This suggests that for the SKA1-Low to be able to create
images with the required resolution to perform credible image analysis with techniques e.g.
LCS, denser 𝑢𝑣-sampling is needed at longer baselines. Additionally, a longer observation
period is also needed to fill the 𝑢𝑣-coverage in order to reduce the impact of sidelobes and
provide a more accurate estimation of the PSF.
– 24 –
5 Summary and discussion
In this work we investigate if it is possible to probe the percolation transition period of the
EoR, i.e. the period when a group of small ionized regions merges together to form a large
singly-connected ionized region, using a synthetic SKA1-Low observation (via 21cmE2E-
pipeline) of the redshifted 21-cm signal from EoR. We use LCS to evaluate the evolution of the
Largest Ionized Region (LIR) during the percolation process. To estimate the LCS, we use the
SURFGEN2 algorithm that uses a Marching Cube 33 scheme to find the boundaries between
the neutral and ionized regions in the Hi 21-cm maps. The findings of this investigation can
be summarized as follows:
•The determination of the threshold brightness (to differentiate between ionized and
neutral regions) for a Hi 21-cm map obtained from a realistic radio interferometric
observation of the EoR is a non-trivial task. This is due to the fact that a combined
effect of noise and beam on the interferometric (mean-subtracted) images shifts the
brightness temperature in each pixel by a random value. The histogram of an ideal
EoR 21-cm field has a very sharp bimodal feature where the two modes represent the
ionized and the neutral pixels respectively. Hence, we employ a Gradient Descent
scheme on the radio interferometric EoR 21-cm image histograms to binarize these
fields by identifying the local minima between the two peaks of the histogram.
•We assume the presence of a Gaussian random system noise in these observations and
add it to the EoR 21-cm field in the image domain with varying rms values to study
its impact on the LCS analysis. For a relatively low noise rms level of 3.10 mK,
we find that the LCS is unaffected and the percolation transition threshold remains
unchanged. However, as we go to higher noise levels, due to higher noise fluctuations,
the bimodal nature of the 21-cm histogram gets lost. This, in turn, creates a bias in
the selection of the threshold by the Gradient Descent scheme. Alongside, as noise
starts dominating the 21-cm image field, LCS could not be computed for higher neutral
fractions (¯𝑥HI) as the contrast between the tiny ionized bubbles and their surroundings
become comparable with the amplitude of the noise and it becomes difficult to identify
them via SURFGEN2.
•We next mimic the impact of the array synthesized beam, by smoothing the EoR
21-cm images with a Gaussian smoothing kernel of varying length. We observe that
for smaller smoothing scales, the features of the LCS could be fully recovered and
the percolation threshold remains unaltered. However, as we go to higher smoothing
length scales, due to the averaging effect by the smoothing kernel, a lot of patches start
to appear in the smoothed image which mimics the characteristics of partially ionized
regions. For images from the earlier stages of the EoR i.e. at higher neutral fractions,
this effect smooths out the small ionized regions and makes them appear as neutral
regions. Hence, for larger smoothing scales LCS could not be estimated at these early
stages. Further, at later stages of the EoR, this averaging effect results in a bias in the
determination of the threshold between the ionized and neutral regions. Therefore, one
observes erratic behaviour in the LCS evolution in such scenarios.
– 25 –
•To investigate the impact of the interferometric array synthesized beam on the LCS
analysis under a more realistic condition, we simulate observations of the EoR 21-cm
signal with the upcoming SKA1-Low using the 21cmE2E-pipeline. We observe that
how the dirty beam is deconvolved from the observed data has a significant impact
on the resulting 21-cm images and thus on the estimated values of the LCS. We find
that due to the absence of knowledge of the PSF, it is not possible to fully recover the
important features of the 21-cm maps which are crucial to identify the largest ionized
region, i.e. its volume and shape. We show that Hogbom CLEAN algorithm for
deconvolution produces a very biased image of the sky as it assumes the sky emission
to be a bunch of delta functions. We next demonstrate that this issue can be mitigated
to some extent by using a Multiscale CLEAN algorithm. However, even a Multiscale
CLEAN algorithm needs to adopt a non-trivial weighting scheme such as Briggs
weighting to be able to make the SURFGEN2 algorithm work effectively on 21-cm
images obtained from the early stages of the reionization. Even after adopting such
an advanced CLEAN algorithm for deconvolution, still, the LCS cannot be computed
for significantly early stages of the EoR i.e. ¯𝑥HI ≥0.7. This is due to the lack of the
presence of long baselines in our observations, which does not allow us to resolve the
smaller ionized regions in the early stages of the reionization and thus leads to a biased
estimate of the critical neutral fraction when the percolation transition takes place. We
conclude that we would require a much better sampling of the longer baselines in the
𝑢−𝑣plane to be able to have a less biased estimate of the LCS at the early stages of
the reionization.
In this particular work, we have focused mostly on the impact of the array synthesized
beam of SKA1-Low and its optimal deconvolution techniques on the LCS image analysis
of the EoR 21-cm observations. Here we use the gradient descent algorithm to find the
optimal threshold for LCS analysis. However, it was observed that gradient descent may not
be the optimal algorithm to identify an optimum threshold. For this reason, we would like to
investigate other thresholding methods in our follow-up work. Moreover, here we have not
considered several other observational issues, e.g., the presence of strong foregrounds in the
observed data, incompleteness of the foreground modelling, ionospheric distortions to the
signal and presence of more realistic noise, all of which will make the LCS analysis even
more complicated. We plan to explore these issues in our follow-up work.
6 Acknowledgements
SB thanks Varun Sahni, and Santanu Das for their contributions to developing SURFGEN2
in its initial phase. SM and AD acknowledge financial support through the project titled
“Observing the Cosmic Dawn in Multicolour using Next Generation Telescopes” funded by
the Science and Engineering Research Board (SERB), Department of Science and Tech-
nology, Government of India through the Core Research Grant No. CRG/2021/004025.
RM is supported by the Israel Academy of Sciences and Humanities & Council for Higher
Education Excellence Fellowship Program for International Postdoctoral Researchers. The
entire analysis of the simulated 21-cm maps presented here were done using the computing
– 26 –
0246810
Threshold Tb
300-sim
Noise (3mK)
Smoothed (7.14 cMpc)
Noise+Smoothing
Hogbom - 21cmE2E
256-sim
Natural - 21cmE2E
Robust - 0.5 - 21cmE2E
Figure 17.Bottom-Up: Threshold identified by gradient descent algorithm for different cases starting
from the mean subtracted 21-cm field at the bottom-most position are shown. Subsequently, the effect
of noise and smoothing directly on the image plane can also be seen. Additionally, the effect of the
array synthesized beam was simulated using the 21cmE2E-pipeline and the corresponding change in
the histogram is reflected by the choice of threshold for these cases. These thresholds are shown for a
21-cm signal map obtained from the late stages of reionization i.e. ¯𝑥HI =0.2.
facilities available with the Cosmology with Statistical Inference (CSI) research group at IIT
Indore.
A Variation in the threshold to discern the ionized regions
A crucial aspect of the EoR 21-cm image analysis lies within the threshold selection to
identify neutral and ionized pixels. In our work, we use the gradient descent algorithm
to choose an appropriate threshold for our image analysis. Figure 17 shows the estimated
threshold by this algorithm for different aspects of our analyses done in this paper. We
plot all of the thresholds for a 21-cm map corresponding to a late stage of reionization,
specifically when mass average neutral fraction ¯𝑥HI =0.2. We plot the difference between
the chosen threshold identified by gradient descent and the mean of the simulated brightness
temperature map for each of these cases. In the case of a simulated 21-cm map of 3003
grids, as shown in Figure 17, the gradient descent identifies the threshold very accurately;
ergo, the difference is almost zero. After the map was corrupted using a Gaussian noise,
the histogram of the 21-cm image changed significantly (as shown in the centre left panel
of Figure 10) as random fluctuations arose in the brightness temperature value of the image
pixels. Due to this change, the identified threshold differs from that of the simulated 21-cm
– 27 –
map. A similar case is observed for the case of the 21-cm map smoothed with a Gaussian
kernel of FWHM of 7.14 cMpc as shown in the bottom left panel of Figure 10. Due to the
averaging effect introduced by the Gaussian smoothing kernel, many pixels in the 21-cm map
appear as partially ionized pixels, resulting in a distortion in the histogram. This, in turn,
creates difficulty in identifying the threshold using a gradient descent algorithm as we go to
higher neutral fractions i.e. earlier stages of reionization. Due to the aforementioned reason,
a combined impact of the noise and smoothing affects the histogram of the 21-cm image and
the chosen threshold thereafter.
The edge effects and other artefacts introduced on the image created via the 21cmE2E-
pipeline using Hogbom CLEAN severely corrupt the histogram of the 21-cm map. Naturally,
this, in turn, affects the choice of threshold using gradient descent. The bias introduced by the
pipeline imposes a bias on the threshold as well, shown in Figure 17. To mitigate the effects
of zero padding and other artefacts, Hi 21-cm image cubes of 2563grids were simulated.
The threshold chosen for one such cube of ¯𝑥HI =0.2is shown in Figure 17. These maps
were then cleaned using Multiscale deconvolution CLEAN with robust parameters set to 2
and 0.5 to deconvolve the dirty beam from the maps. However, even though these CLEAN
algorithms are advanced, residues of the dirty beam remain on the cleaned maps and impose
a bias on these images. Subsequently, when the gradient descent algorithm is applied to these
images, the chosen threshold contains this bias. This, in turn, leads to a biased estimation of
LCS and the inferred reionization history obtained, therefore, is heavily biased.
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